From d5b678620138c27204d5d2977db72028ee01a06a Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 15:11:30 -0400 Subject: [PATCH 01/34] rename, remove _output.yml --- .gitignore | 3 +- 01_warmup.Rmd => 01_warmup.qmd | 0 02_functions.Rmd => 02_functions.qmd | 0 03_limits.Rmd => 03_limits.qmd | 0 04_calculus.Rmd => 04_calculus.qmd | 0 05_optimization.Rmd => 05_optimization.qmd | 0 06_probability.Rmd => 06_probability.qmd | 0 ...inear-algebra.Rmd => 07_linear-algebra.qmd | 0 ...nting.Rmd => 11_data-handling_counting.qmd | 0 ...ation.Rmd => 12_matricies-manipulation.qmd | 0 ...bj_loops.Rmd => 13_functions_obj_loops.qmd | 0 14_visualization.Rmd => 14_visualization.qmd | 0 ...ct-dempeace.Rmd => 15_project-dempeace.qmd | 0 16_simulation.Rmd => 16_simulation.qmd | 0 17_non-wysiwyg.Rmd => 17_non-wysiwyg.qmd | 0 17_non-wysiwyg.quarto_ipynb | 333 ++++++++++++++++++ 18_text.Rmd => 18_text.qmd | 0 ...nd-line_git.Rmd => 19_command-line_git.qmd | 21 +- ...ions-warmup.Rmd => 21_solutions-warmup.qmd | 0 ...ramming.Rmd => 23_solution_programming.qmd | 0 _bookdown.yml => _quarto.yml | 0 index.Rmd => index.qmd | 2 +- _output.yml => old-output.yml | 0 23 files changed, 348 insertions(+), 11 deletions(-) rename 01_warmup.Rmd => 01_warmup.qmd (100%) rename 02_functions.Rmd => 02_functions.qmd (100%) rename 03_limits.Rmd => 03_limits.qmd (100%) rename 04_calculus.Rmd => 04_calculus.qmd (100%) rename 05_optimization.Rmd => 05_optimization.qmd (100%) rename 06_probability.Rmd => 06_probability.qmd (100%) rename 07_linear-algebra.Rmd => 07_linear-algebra.qmd (100%) rename 11_data-handling_counting.Rmd => 11_data-handling_counting.qmd (100%) rename 12_matricies-manipulation.Rmd => 12_matricies-manipulation.qmd (100%) rename 13_functions_obj_loops.Rmd => 13_functions_obj_loops.qmd (100%) rename 14_visualization.Rmd => 14_visualization.qmd (100%) rename 15_project-dempeace.Rmd => 15_project-dempeace.qmd (100%) rename 16_simulation.Rmd => 16_simulation.qmd (100%) rename 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written by Shiro Kuriwaki] {#nonwysiwyg}\n", + "\n", + "\n", + "### Where are we? Where are we headed? {-}\n", + "\n", + "Up till now, you should have covered:\n", + "\n", + "* Statistical Programming in `R`\n", + "\n", + "This is only the beginning of `R` -- programming is like learning a language, so learn more as we use it. And yet `R` is of likely not the only programming language you will want to use. While we cannot introduce everything, we'll pick out a few that we think are particularly helpful.\n", + "\n", + "Here will cover\n", + "\n", + "* Markdown\n", + "* LaTeX (and BibTeX)\n", + "\n", + "as examples of a non-WYSIWYG editor\n", + "\n", + "and the next chapter (you can read it without reading this LaTeX chapter) covers\n", + "\n", + "* command-line\n", + "* git\n", + "\n", + "command-line are a basic set of tools that you may have to use from time to time. It also clarifies what more complicated programs are doing. Markdown is an example of compiling a plain text file. LaTeX is a typesetting program and git is a version control program -- both are useful for non-quantitative work as well.\n", + "\n", + "\n", + "\n", + "### Check your understanding {-}\n", + "\n", + "Check if you have an idea of how you might code the following tasks:\n", + "\n", + "* What does \"WYSIWYG\" stand for? How would a non-WYSIWYG format text?\n", + "* How do you start a header in markdown?\n", + "* What are some \"plain text\" editors?\n", + "* How do you start a document in `.tex`?\n", + "* How do you start a environment in `.tex`?\n", + "* How do you insert a figure in `.tex`?\n", + "* How do you reference a figure in `.tex`?\n", + "* What is a `.bib` file? \n", + "* Say you came across a interesting journal article. How would you want to maintain this reference so that you can refer to its citation in all your subsequent papers?\n", + "\n", + "\n", + "\n", + "\n", + "## Motivation\n", + "\n", + "Statistical programming is a fast-moving field. The beta version of `R` was released in 2000, `ggplot2` was released on 2005, and `RStudio` started around 2010. Of course, some programming technologies are quite \"old\": (`C` in 1969, `C++` around 1989, `TeX` in 1978, `Linux` in 1991, Mac OS in 1984). But it is easy to feel you are falling behind in the recent developments of programming. Today we will do a **brief** and rough overview of some fundamental and new tools other than `R`, with the general aim of having you break out of your comfort zone so you won't be shut out from learning these tools in the future.\n", + "\n", + "\n", + "## Markdown\n", + "\n", + "Markdown is the text we have been using throughout this course! At its core markdown is just plain text. Plain text does not have any formatting embedded in it. Instead, the formatting is coded up as text. Markdown is _not_ a WYSIWYG (What you see is what you get) text editor like Microsoft Word or Google Docs. This will mean that you need to explicitly code for `bold{text}` rather than hitting Command+B and making your text look __bold__ on your own computer. \n", + "\n", + "Markdown is known as a \"light-weight\" editor, which means that it is relatively easy to write code that will compile. It is quick and easy and satisfies most presentation purposes; you might want to try `LaTeX` for more involved papers.\n", + "\n", + "### markdown commands\n", + "For italic and bold, use either the asterisks or the underlines, \n", + "\n", + "```\n", + "*italic* **bold**\n", + "_italic_ __bold__\n", + "```\n", + "\n", + "And for headers use the hash symbols, \n", + "```\n", + "# Main Header\n", + "## Sub-headers\n", + "```\n", + "\n", + "### your own markdown \n", + "\n", + "RStudio makes it easy to compile your very first markdown file by giving you templates. Got to `New > R Markdown`, pick a document and click Ok. This will give you a skeleton of a document you can compile -- or \"knit\".\n", + "\n", + "Rmd is actually a slight modification of real markdown. It is a type of file that R reads and turns into a proper `md` file. Then, it uses a document-conversion called pandoc to compile your `md` into documents like PDF or HTML.\n", + "\n", + "![How Rmds become PDFs or HTMLs](images/RMarkdownFlow.png)\n", + "\n", + "### Quarto\n", + "\n", + "R Markdown (`.Rmd`) files have long been the go-to for reproducible writing workflows for R users.\n", + "In 2022, [Posit, PBC](https://posit.co/), who created R Markdown announced a new generation of markdown extensions, with Quarto.\n", + "Quarto (`.qmd`) files are a variation on R Markdown which allows for including R, python, Observable, Julia, and more within a document.\n", + "Quarto is largely compatible with older `.Rmd` files, just by changing the extension.\n", + "As such, you can integrate LaTeX and markdown seamlessly.\n", + "\n", + "Some benefits of using Quarto include:\n", + "\n", + "* [ease of customization with template partials](https://quarto.org/docs/journals/templates.html#template-partials)\n", + "* [journal submission templates for many journals](https://quarto.org/docs/extensions/listing-journals.html)\n", + "* [dozen of output types](https://quarto.org/docs/reference/)\n", + "* [the ability to make websites interacting only with Quarto](https://quarto.org/docs/websites/)\n", + "\n", + "\n", + "### A note on plain-text editors\n", + "\n", + "Multiple software exist where you can edit plain-text (roughly speaking, text that is not WYSIWYG). \n", + "\n", + "* [RStudio](https://posit.co/products/open-source/rstudio/) (especially for R-related links)\n", + "* TeXMaker, TeXShop (especially for TeX)\n", + "* [emacs](https://www.gnu.org/software/emacs/), aquamacs (general)\n", + "* [vim](http://www.vim.org/download.php) (general)\n", + "* [Sublime Text](https://www.sublimetext.com) (general)\n", + "\n", + "Each has their own keyboard shortcuts and special features. You can browse a couple and see which one(s) you like.\n", + "\n", + "Since June 2021, RStudio has offered a visual editor which tries to bridge the gap between plain-text and WYSIWYG.\n", + "While writing, it transforms plain markdown, RMarkdown, or Quarto documents into a \"WYSISWYM\" version, What You See Is What You Mean.\n", + "Formatting choices, like bold or italicized text are shown as **bold** or *italicized* text, rather than as intermediate markdown.\n", + "Lists, enumerations, and images are shown inline, rather than the code that includes them.\n", + "This is not a final form though, as styling still occurs when rendering the final document.\n", + "\n", + "## LaTeX\n", + "\n", + "LaTeX is a typesetting program. You'd engage with LaTeX much like you engage with your `R` code. You will interact with LaTeX in a text editor, and will writing code which will be interpreted by the LaTeX compiler and which will finally be parsed to form your final PDF.\n", + "\n", + "### compile online\n", + "\n", + "1. Go to \n", + "2. Scroll down and go to \"CREATE A NEW PAPER\" if you don't have an account.\n", + "3. Let's discuss the default template.\n", + "4. Make a new document, and set it as your main document. Then type in the Minimal Working Example (MWE):\n", + "\n", + "\n", + "\n", + "\n", + "```{bash, eval = FALSE}\n", + "\\documentclass{article}\n", + "\\begin{document}\n", + "Hello World\n", + "\\end{document}\n", + "```\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "### compile your first LaTeX document locally\n", + "LaTeX is a very stable system, and few changes to it have been made since the 1990s. The main benefit: better control over how your papers will look; better methods for writing equations or making tables; overall pleasing aesthetic. \n", + "\n", + "1. Open a plain text editor. Then type in the MWE\n", + "\n", + "\n", + "\n", + "\n", + "```{bash, eval = FALSE}\n", + "\\documentclass{article}\n", + "\\begin{document}\n", + "Hello World\n", + "\\end{document}\n", + "```\n", + "\n", + "\n", + "\n", + "\n", + "2. Save this as `hello_world.tex`. Make sure you get the file extension right. \n", + "3. Open this in your \"LaTeX\" editor. This can be `TeXMaker`, `Aqumacs`, etc..\n", + "4. Go through the click/dropdown interface and click compile.\n", + "\n", + "\n", + "### main LaTeX commands\n", + "LaTeX can cover most of your typesetting needs, to clean equations and intricate diagrams. \n", + "\n", + "Some main commands you'll be using are below, and a very concise cheat sheet here: \n", + "\n", + "Most involved features require that you begin a specific \"environment\" for that feature, clearly demarcating them by the notation `\\begin{figure}` and then `\\end{figure}`, e.g. in the case of figures.\n", + "\n", + "```\n", + "\\begin{figure}\n", + "\\includegraphics{histogram.pdf}\n", + "\\end{figure}\n", + "```\n", + "where `histogram.pdf` is a path to one of your files. \n", + "\n", + "Notice that each line starts with a backslash `\\` -- in LaTeX this is the symbol to run a command.\n", + "\n", + "The following syntax at the endpoints are shorthand for math equations.\n", + "```\n", + "\\[\\int x^2 dx\\]\n", + "```\n", + "these compile math symbols: $\\displaystyle \\int x^2 dx.$^[Enclosing with `$$` instead of `\\[` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.]\n", + "\n", + "\n", + "The `align` environment is useful to align your multi-line math, for example. \n", + "```\n", + "\\begin{align}\n", + "P(A \\mid B) &= \\frac{P(A \\cap B)}{P(B)}\\\\\n", + "&= \\frac{P(B \\mid A)P(A)}{P(B)}\n", + "\\end{align}\n", + "```\n", + "\\begin{align}\n", + "P(A \\mid B) &= \\frac{P(A \\cap B)}{P(B)}\\\\\n", + "&= \\frac{P(B \\mid A)P(A)}{P(B)}\n", + "\\end{align}\n", + "\n", + "Regression tables should be outputted as `.tex` files with packages like `xtable` and `stargazer`, and then called into LaTeX by `\\input{regression_table.tex}` where `regression_table.tex` is the path to your regression output.\n", + "\n", + "Figures and equations should be labelled with the tag (e.g. `label{tab:regression}` so that you can refer to them later with their tag `Table \\ref{tab:regression}`, instead of hard-coding `Table 2`).\n", + "\n", + "For some LaTeX commands you might need to load a separate package that someone else has written. Do this in your preamble (i.e. before `\\begin{document}`):\n", + "```\n", + "\\usepackage[options]{package}\n", + "```\n", + "where `package` is the name of the package and `options` are options specific to the package. \n", + "\n", + "\n", + "### Further Guides {-}\n", + "\n", + "For a more comprehensive listing of LaTeX commands, Mayya Komisarchik has a great tutorial set of folders: \n", + "\n", + "There is a version of LaTeX called Beamer, which is a popular way of making a slideshow. Slides in markdown is also a competitor. The language of Beamer is the same as LaTeX but has some special functions for slides.\n", + "\n", + "\n", + "## BibTeX\n", + "\n", + "BibTeX is a reference system for bibliographical tests. We have a `.bib` file separately on our computer. This is also a plain text file, but it encodes bibliographical resources with special syntax so that a program can rearrange parts accordingly for different citation systems. \n", + "\n", + "### what is a `.bib` file?\n", + "For example, here is the Nunn and Wantchekon article entry in `.bib` form.\n", + "\n", + "```{}\n", + "@article{nunn2011slave,\n", + " title={The Slave Trade and the Origins of Mistrust in Africa},\n", + " author={Nunn, Nathan and Wantchekon, Leonard},\n", + " journal={American Economic Review},\n", + " volume={101},\n", + " number={7},\n", + " pages={3221--3252},\n", + " year={2011}\n", + "}\n", + "```\n", + "\n", + "The first entry, `nunn2011slave`, is \"pick your favorite\" -- pick your own name for your reference system. The other slots in this `@article` entry are entries that refer to specific bibliographical text.\n", + "\n", + "### what does LaTeX do with .bib files?\n", + "Now, in LaTeX, if you type \n", + "\n", + " \\textcite{nunn2011slave} argue that current variation in the trust among citizens of African countries has historical roots in the European slave trade in the 1600s.\n", + " \n", + "as part of your text, then when the `.tex` file is compiled the PDF shows something like \n", + "\n", + "![](images/biblatex_inline.png)\n", + "\n", + "in whatever citation style (APSA, APA, Chicago) you pre-specified! \n", + "\n", + "\n", + "Also at the end of your paper you will have a bibliography with entries ordered and formatted in the appropriate citation.\n", + "\n", + "![](images/biblatex_bibliography.png)\n", + "\n", + "This is a much less frustrating way of keeping track of your references -- no need to hand-edit formatting the bibliography to conform to citation rules (which biblatex already knows) and no need to update your bibliography as you add and drop references (biblatex will only show entries that are used in the main text).\n", + "\n", + "\n", + "### stocking up on your .bib files\n", + "You should keep your own `.bib` file that has all your bibliographical resources. Storing entries is cheap (does not take much memory), so it is fine to keep all your references in one place (but you'll want to make a new one for collaborative projects where multiple people will compile a `.tex` file).\n", + "\n", + "For example, Gary's BibTeX file is here: \n", + "\n", + "Citation management software (Mendeley or Zotero) automatically generates .bib entries from your library of PDFs for you, provided you have the bibliography attributes right. \n", + "\n", + "## Exercise {-}\n", + "\n", + "Create a LaTeX document for a hypothetical research paper on your laptop and, once you've verified it compiles into a PDF, come show it to either one of the instructors. \n", + "\n", + "You can also use overleaf if you have preference for a cloud-based system. But don't swallow the built-in templates without understanding or testing them.\n", + "\n", + "Each student will have slightly different substantive interests, so we won't impose much of a standard. But at a minimum, the LaTeX document should have:\n", + "\n", + "* A title, author, date, and abstract\n", + "* Sections\n", + "* Italics and boldface\n", + "* A figure with a caption and in-text reference to it. \n", + "\n", + "\n", + "Depending on your subfield or interests, try to implement some of the following:\n", + "\n", + "* A bibliographical reference drawing from a separate `.bib` file\n", + "* A table\n", + "* A math expression\n", + "* A different font\n", + "* Different page margins\n", + "* Different line spacing\n", + "\n", + "\n", + "\n", + "\n", + "## Concluding the Prefresher {-}\n", + "\n", + "Math may not be the perfect tool for every aspiring political scientist, but hopefully it was useful background to have at the least:\n", + "\n", + "

Historians think this totally meaningless and nonsensical statistic is the product of an early-modern epistemological shift in which numbers and quantifiable data became revered above other kinds of knowledge as the most useful and credible form of truth https://t.co/wVFyAQGxEv

— Gina Anne Tam 譚吉娜 (@DGTam86) May 29, 2018
\n", + "\n", + "\n", + "\n", + "\n", + "But we should be aware that too much slant towards math and programming can miss the point:\n", + "\n", + "

To be clear, PhD training in Econ (first year) is often a disaster-- like how to prove the Central Limit Theorem (the LeBron James of Statistics) with polar-cooardinates. This is mostly a way to demoralize actual economists and select a bunch of unimaginative math jocks.

— Amitabh Chandra (@amitabhchandra2) August 14, 2018
\n", + "\n", + "\n", + "\n", + "\n", + "Keep on learning, trying new techniques to improve your work, and learn from others! \n", + "\n", + "

What #rstats tricks did it take you way too long to learn? One of mine is using readRDS and saveRDS instead of repeatedly loading from CSV

— Emily Riederer (@EmilyRiederer) August 19, 2017
\n", + "\n", + "\n", + "\n", + "### Your Feedback Matters {-}\n", + "\n", + "_Please tell us how we can improve the Prefresher_: The Prefresher is a work in progress, with material mainly driven by graduate students. Please tell us how we should change (or not change) each of its elements:\n", + "\n", + "https://harvard.az1.qualtrics.com/jfe/form/SV_esbzN8ZFAOPTqiV" + ], + "id": "86efe863" + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} \ No newline at end of file diff --git a/18_text.Rmd b/18_text.qmd similarity index 100% rename from 18_text.Rmd rename to 18_text.qmd diff --git a/19_command-line_git.Rmd b/19_command-line_git.qmd similarity index 98% rename from 19_command-line_git.Rmd rename to 19_command-line_git.qmd index ca3f84b..02cacb2 100644 --- a/19_command-line_git.Rmd +++ b/19_command-line_git.qmd @@ -1,6 +1,9 @@ -# Command-line, git^[Module originally written by Shiro Kuriwaki] {#commandline-git} - +--- +execute: + eval: false +--- +# Command-line, git^[Module originally written by Shiro Kuriwaki] {#commandline-git} ## Where are we? Where are we headed? @@ -31,12 +34,12 @@ Elementary programming operations are done on the command-line, or by entering c Open up `Terminal` in a Mac. (`Command Prompt` in Windows) Running this command in a Mac (`dir` in Windows) should show you a list of all files in the directory that you are currently in. -```{bash} +```bash ls ``` `pwd` stands for present working directory (`cd` in Windows) -```{bash} +```bash pwd ``` @@ -44,14 +47,14 @@ pwd Or you can go up to your parent directory. The syntax for that are two periods, `..` . One period `.` refers to the current directory. -```{bash} +```bash cd .. pwd ``` `~/` stands for your home directory defined by your computer. -```{bash} +```bash cd ~/ ls ``` @@ -71,7 +74,7 @@ cat("Hello World") Then in command-line, go to the directory that contains `hello_world.R` and enter -```{bash, eval = FALSE} +```bash Rscript hello_world.R ``` @@ -80,7 +83,7 @@ This should give you the output `Hello World`, which verifies that you "executed ### why do command-line? If you know exactly what you want to do your files and the changes are local, then command-line might be faster and be more sensible than navigating yourself through a GUI. For example, what if you wanted a single command that will run 10 R scripts successively at once (as Gentzkow and Shapiro suggest you should do in your research)? It is tedious to run each of your scripts on Rstudio, especially if running some take more than a few minutes. Instead you could write a "batch" script that you can run on the terminal, -```{bash, eval = FALSE} +```bash Rscript 01_read_data.R Rscript 02_merge_data.R Rscript 03_run_regressions.R @@ -89,7 +92,7 @@ Rscript 05_maketable.R ``` Then run this single file, call it `run_all_Rscripts.sh`, on your terminal as -```{bash, eval = FALSE} +```bash sh run_all_Rscripts.sh ``` diff --git a/21_solutions-warmup.Rmd b/21_solutions-warmup.qmd similarity index 100% rename from 21_solutions-warmup.Rmd rename to 21_solutions-warmup.qmd diff --git a/23_solution_programming.Rmd b/23_solution_programming.qmd similarity index 100% rename from 23_solution_programming.Rmd rename to 23_solution_programming.qmd diff --git a/_bookdown.yml b/_quarto.yml similarity index 100% rename from _bookdown.yml rename to _quarto.yml diff --git a/index.Rmd b/index.qmd similarity index 99% rename from index.Rmd rename to index.qmd index 0cda87e..6f9af30 100644 --- a/index.Rmd +++ b/index.qmd @@ -8,7 +8,7 @@ site: "bookdown::bookdown_site" documentclass: book geometry: "margin=1.5in" biblio-style: apalike -link-citations: yes +link-citations: true cover-image: "./images/logo.png" --- # About this Booklet {-} diff --git a/_output.yml b/old-output.yml similarity index 100% rename from _output.yml rename to old-output.yml From eb8ed21be675f53dde137cf76b1f257d9dbb3979 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 15:34:44 -0400 Subject: [PATCH 02/34] use quarto unnumbered, move chunks with knitr --- 01_warmup.qmd | 24 ++-- 02_functions.qmd | 36 ++++-- 03_limits.qmd | 79 +++++++++---- 04_calculus.qmd | 120 ++++++++++++++------ 05_optimization.qmd | 67 +++++++---- 06_probability.qmd | 177 +++++++++++++++++++++--------- 07_linear-algebra.qmd | 69 ++++++++---- 11_data-handling_counting.qmd | 58 ++++++---- 12_matricies-manipulation.qmd | 61 ++++++---- 13_functions_obj_loops.qmd | 40 ++++--- 14_visualization.qmd | 45 +++++--- 15_project-dempeace.qmd | 18 +-- 16_simulation.qmd | 26 +++-- 17_non-wysiwyg.qmd | 18 +-- 18_text.qmd | 27 +++-- 21_solutions-warmup.qmd | 26 ++--- 23_solution_programming.qmd | 45 +++++--- R_exercises/04_visualization.R | 4 +- R_exercises/05_project-dempeace.R | 2 +- _quarto.yml | 14 ++- index.qmd | 8 +- 21 files changed, 631 insertions(+), 333 deletions(-) diff --git a/01_warmup.qmd b/01_warmup.qmd index a1eb85b..a0d53e4 100644 --- a/01_warmup.qmd +++ b/01_warmup.qmd @@ -1,4 +1,4 @@ -# Pre-Prefresher Exercises {-} +# Pre-Prefresher Exercises {.unnumbered} Before our first meeting, please try solving these questions. They are a sample of the very beginning of each math section. We have provided links to the parts of the book you can read if the concepts are new to you. @@ -7,9 +7,9 @@ The goal of this "pre"-prefresher assignment is not to intimidate you but to set -## Linear Algebra {-} +## Linear Algebra {.unnumbered} -### Vectors {-} +### Vectors {.unnumbered} Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pmatrix} 4\\5\\6 \end{pmatrix}$, and the scalar $c = 2$. Calculate the following: @@ -34,7 +34,7 @@ Are the following sets of vectors linearly independent? If you are having trouble with these problems, please review Section \@ref(linearindependence). -### Matrices {-} +### Matrices {.unnumbered} \[{\bf A}=\begin{pmatrix} 7 & 5 & 1 \\ @@ -82,9 +82,9 @@ If you are having trouble with these problems, please review Section \@ref(matri -## Operations {-} +## Operations {.unnumbered} -### Summation {-} +### Summation {.unnumbered} Simplify the following @@ -95,7 +95,7 @@ Simplify the following 3. \(\sum\limits_{i= 1}^4 (3k + i + 2)\) -### Products {-} +### Products {.unnumbered} 1. \(\prod\limits_{i= 1}^3 i\) @@ -105,7 +105,7 @@ Simplify the following To review this material, please see Section \@ref(sum-notation). -### Logs and exponents {-} +### Logs and exponents {.unnumbered} Simplify the following @@ -124,7 +124,7 @@ To review this material, please see Section \@ref(logexponents) -## Limits {-} +## Limits {.unnumbered} Find the limit of the following. @@ -135,7 +135,7 @@ Find the limit of the following. To review this material please see Section \@ref(limitsfun) -## Calculus {-} +## Calculus {.unnumbered} For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\frac{d}{dx}f(x)$ @@ -148,7 +148,7 @@ For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\fra For a review, please see Section \@ref(derivintro) - \@ref(derivpoly) -## Optimization {-} +## Optimization {.unnumbered} For each of the followng functions $f(x)$, does a maximum and minimum exist in the domain $x \in \mathbf{R}$? If so, for what are those values and for which values of $x$? @@ -159,7 +159,7 @@ For each of the followng functions $f(x)$, does a maximum and minimum exist in t If you are stuck, please try sketching out a picture of each of the functions. -## Probability {-} +## Probability {.unnumbered} 1. If there are 12 cards, numbered 1 to 12, and 4 cards are chosen, how many distinct possible choices are there? (unordered, without replacement) 2. Let $A = \{1,3,5,7,8\}$ and $B = \{2,4,7,8,12,13\}$. What is $A \cup B$? What is $A \cap B$? If $A$ is a subset of the Sample Space $S = \{1,2,3,4,5,6,7,8,9,10\}$, what is the complement $A^C$? diff --git a/02_functions.qmd b/02_functions.qmd index afdf896..5faebbe 100644 --- a/02_functions.qmd +++ b/02_functions.qmd @@ -1,4 +1,4 @@ -# (PART) Math {-} +# (PART) Math {.unnumbered} # Functions and Operations @@ -60,7 +60,9 @@ __Modulo:__ Tells you the remainder when you divide the first number by the seco -```{example, name = "Operators", operators} +```{example} +#| name: Operators +#| label: operators 1. $\sum\limits_{i=1}^{5} i =$ @@ -75,7 +77,9 @@ __Modulo:__ Tells you the remainder when you divide the first number by the seco ``` -```{exercise, name = "Operators", operators1} +```{exercise} +#| name: Operators +#| label: operators1 Let $x_1 = 4, x_2 = 3, x_3 = 7, x_4 = 11, x_5 = 2$ 1. $\sum\limits_{i=1}^{3} (7)x_i$ @@ -123,7 +127,9 @@ Function of two variables: $f: {\bf R}^2\to{\bf R}^1$. We often use variable $x$ as input and another $y$ as output, e.g. $y=x+1$ -```{example, name = "Functions", functions} +```{example} +#| name: Functions +#| label: functions For each of the following, state whether they are one-to-one or many-to-one functions. @@ -136,7 +142,9 @@ For each of the following, state whether they are one-to-one or many-to-one func ``` -```{exercise, name = "Functions", functions1} +```{exercise} +#| name: Functions +#| label: functions1 For each of the following, state whether they are one-to-one or many-to-one functions. @@ -247,7 +255,9 @@ Therefore, you can see that the log of a product is equal to the sum of the logs -```{example, name = "Logarithmic Functions", log} +```{example} +#| name: Logarithmic Functions +#| label: log Evaluate each of the following logarithms @@ -265,7 +275,9 @@ Simplify the following logarithm. By "simplify", we actually really mean - use a ``` -```{exercise, name = "Logarithmic Functions", log1} +```{exercise} +#| name: Logarithmic Functions +#| label: log1 Evaluate each of the following logarithms @@ -302,7 +314,9 @@ What can a graph tell you about a function? Sometimes we're given a function $y=f(x)$ and we want to find how $x$ varies as a function of $y$. Use algebra to move $x$ to the left hand side (LHS) of the equation and so that the right hand side (RHS) is only a function of $y$. -```{example, name = "Solving for Variables", solvevar} +```{example} +#| name: Solving for Variables +#| label: solvevar Solve for x: @@ -328,7 +342,9 @@ __Quadratic Formula:__ For quadratic equations $ax^2+bx+c=0$, use the quadratic -```{exercise, name = "Finding Roots", solvevar1} +```{exercise} +#| name: Finding Roots +#| label: solvevar1 Solve for x: @@ -380,7 +396,7 @@ __Empty__: The empty (or null) set is a unique set that has no elements, denoted * Examples: The set of squares with 5 sides; the set of countries south of the South Pole. -## Answers to Examples and Exercises {-} +## Answers to Examples and Exercises {.unnumbered} Answer to Example \@ref(exm:operators): diff --git a/03_limits.qmd b/03_limits.qmd index 62f547d..2e7de2b 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -1,6 +1,7 @@ # Limits {#limits-precalc} -```{r, include = FALSE} +```{r} +#| include: false library(ggplot2) library(tibble) library(scales) @@ -11,11 +12,13 @@ library(patchwork) Solving limits, i.e. finding out the value of functions as its input moves closer to some value, is important for the social scientist's mathematical toolkit for two related tasks. The first is for the study of calculus, which will be in turn useful to show where certain functions are maximized or minimized. The second is for the study of statistical inference, which is the study of inferring things about things you cannot see by using things you can see. -## Example: The Central Limit Theorem {-} +## Example: The Central Limit Theorem {.unnumbered} Perhaps the most important theorem in statistics is the Central Limit Theorem, -```{theorem, clt-lim, name = "Central Limit Theorem (i.i.d. case)"} +```{theorem} +#| label: clt-lim +#| name: Central Limit Theorem (i.i.d. case) For any series of independent and identically distributed random variables $X_1, X_2, \cdots$, we know the distribution of its sum even if we do not know the distribution of $X$. The distribution of the sum is a Normal distribution. \[\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),\] @@ -28,11 +31,13 @@ That is, the limit of the distribution of the lefthand side is the distribution The sign of a limit is the arrow "$\rightarrow$". Although we have not yet covered probability (in Section \@ref(probability-theory)) so we have not described what distributions and random variables are, it is worth foreshadowing the Central Limit Theorem. The Central Limit Theorem is powerful because it gives us a _guarantee_ of what would happen if $n \rightarrow \infty$, which in this case means we collected more data. -## Example: The Law of Large Numbers {-} +## Example: The Law of Large Numbers {.unnumbered} A finding that perhaps rivals the Central Limit Theorem is the Law of Large Numbers: -```{theorem, lln-lim, name = "(Weak) Law of Large Numbers"} +```{theorem} +#| label: lln-lim +#| name: (Weak) Law of Large Numbers For any draw of identically distributed independent variables with mean $\mu$, the sample average after $n$ draws, $\bar{X}_n$, converges in probability to the true mean as $n \rightarrow \infty$: \[\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0\] @@ -44,7 +49,11 @@ A shorthand of which is \(\bar{X}_n \xrightarrow{p} \mu\), where the arrow is r Intuitively, the more data, the more accurate is your guess. For example, the Figure \@ref(fig:llnsim) shows how the sample average from many coin tosses converges to the true value : 0.5. -```{r, llnsim, fig.cap="As the number of coin tosses goes to infinity, the average probabiity of heads converges to 0.5", echo= FALSE} +```{r} +#| label: llnsim +#| echo: false +#| fig-cap: As the number of coin tosses goes to infinity, the average probabiity of +#| heads converges to 0.5 set.seed(02138) n <- 1e3 Xs <- rbinom(n = n, size = 1, prob = 0.5) @@ -71,7 +80,9 @@ A **sequence** \[\{x_n\}=\{x_1, x_2, x_3, \ldots, x_n\}\] is an ordered set of r where the subscript and superscript are read together as "from 1 to infinity." -```{example, seqbehav, name = "Sequences"} +```{example} +#| label: seqbehav +#| name: Sequences How do these sequences behave? @@ -85,7 +96,10 @@ How do these sequences behave? We find the sequence by simply "plugging in" the integers into each $n$. The important thing is to get a sense of how these numbers are going to change. Example 1's numbers seem to come closer and closer to 2, but will it ever surpass 2? Example 2's numbers are also increasing each time, but will it hit a limit? What is the pattern in Example 3? Graphing helps you make this point more clearly. See the sequence of $n = 1, ...20$ for each of the three examples in Figure \@ref(fig:seqabc). -```{r, seqabc, fig.cap = "Behavior of Some Sequences", echo = FALSE} +```{r} +#| label: seqabc +#| echo: false +#| fig-cap: Behavior of Some Sequences seq <- 1:20 df <- tibble(n = seq, A = 2 - 1/(seq^2), B = (seq^2 + 1)/(seq), C = (-1)^seq * (1 - 1/seq)) @@ -135,7 +149,8 @@ This looks reasonable enough. The harder question, obviously is when the parts o It is nice for a sequence to converge in limit. We want to know if complex-looking sequences converge or not. The name of the game here is to break that complex sequence up into sums of simple fractions where $n$ only appears in the denominator: $\frac{1}{n}, \frac{1}{n^2}$, and so on. Each of these will converge to 0, because the denominator gets larger and larger. Then, because of the properties above, we can then find the final sequence. -```{example, name = "Simplifying a Fraction into Sums"} +```{example} +#| name: Simplifying a Fraction into Sums Find the limit of \[\lim_{n\to \infty} \frac{n + 3}{n},\] ``` @@ -150,7 +165,8 @@ so, the limit is actually 1. After some practice, the key to intuition is whether one part of the fraction grows "faster" than another. If the denominator grows faster to infinity than the numerator, then the fraction will converge to 0, even if the numerator will also increase to infinity. In a sense, limits show how not all infinities are the same. -```{exercise, limseq2} +```{exercise} +#| label: limseq2 Find the following limits of sequences, then explain in English the intuition for why that is the case. 1. \(\lim\limits_{n\to\infty} \frac{2n}{n^2 + 1}\) @@ -169,7 +185,8 @@ A function $f$ is a compact representation of some behavior we care about. Like For a limit $L$ to exist, the function $f(x)$ must approach $L$ from both the left (increasing) and the right (decreasing). -```{definition, name = "Limit of a function"} +```{definition} +#| name: Limit of a function Let $f(x)$ be defined at each point in some open interval containing the point $c$. Then $L$ equals $\lim\limits_{x \to c} f(x)$ if for any (small positive) number $\epsilon$, there exists a corresponding number $\delta>0$ such that if $0<|x-c|<\delta$, then $|f(x)-L|<\epsilon$. ``` @@ -185,7 +202,9 @@ Properties: Let $f$ and $g$ be functions with $\lim\limits_{x \to c} f(x)=k$ and Simple limits of functions can be solved as we did limits of sequences. Just be careful which part of the function is changing. -```{example, limfun1, name ="Limits of Functions"} +```{example} +#| label: limfun1 +#| name: Limits of Functions Find the limit of the following functions. 1. \(\lim_{x \to c} k\) @@ -199,7 +218,9 @@ Limits can get more complex in roughly two ways. First, the functions may become The function can be thought of as a more general or "smooth" version of sequences. For example, -```{exercise, limfunmax, name = "Limits of a Fraction of Functions"} +```{exercise} +#| label: limfunmax +#| name: Limits of a Fraction of Functions Find the limit of @@ -211,7 +232,8 @@ Find the limit of Now, the functions will become a bit more complex: -```{exercise, discontlim} +```{exercise} +#| label: discontlim Solve the following limits of functions 1. $\lim\limits_{x\to 0} |x|$ @@ -231,7 +253,10 @@ So there are a few more alternatives about what a limit of a function could be: The distinction between left and right becomes important when the function is not determined for some values of $x$. What are those cases in the examples below? -```{r, echo=FALSE,fig.cap="Functions which are not defined in some areas", warning = FALSE} +```{r} +#| echo: false +#| warning: false +#| fig-cap: Functions which are not defined in some areas range1 <- tibble::tibble(x = c(-2, 16)) range2 <- tibble::tibble(x = c(-2, 2)) @@ -255,13 +280,16 @@ fx1 + fx2 + plot_layout(nrow = 1) To repeat a finding from the limits of functions: $f(x)$ does not necessarily have to be defined at $c$ for $\lim\limits_{x \to c}$ to exist. Functions that have breaks in their lines are called discontinuous. Functions that have no breaks are called continuous. Continuity is a concept that is more fundamental to, but related to that of "differentiability", which we will cover next in calculus. -```{definition, name = "Continuity"} +```{definition} +#| name: Continuity Suppose that the domain of the function $f$ includes an open interval containing the point $c$. Then $f$ is continuous at $c$ if $\lim\limits_{x \to c} f(x)$ exists and if $\lim\limits_{x \to c} f(x)=f(c)$. Further, $f$ is continuous on an open interval $(a,b)$ if it is continuous at each point in the interval. ``` To prove that a function is continuous for all points is beyond this practical introduction to math, but the general intuition can be grasped by graphing. -```{example, contdiscont, name = "Continuous and Discontinuous Functions"} +```{example} +#| label: contdiscont +#| name: Continuous and Discontinuous Functions For each function, determine if it is continuous or discontinuous. @@ -279,7 +307,11 @@ In Figure \@ref(fig:fig-contdiscont), we can see that the first two functions ar ``` -```{r, fig-contdiscont, echo=FALSE,fig.cap="Continuous and Discontinuous Functions", warning = FALSE} +```{r} +#| label: fig-contdiscont +#| echo: false +#| warning: false +#| fig-cap: Continuous and Discontinuous Functions range1 <- tibble(x = c(0, 10)) range2 <- tibble(x = c(-2, 2)) range3 <- tibble(x = c(-4, 4)) @@ -320,7 +352,9 @@ Some properties of continuous functions: -```{exercise, discontdraw, name = "Limit when Denominator converges to 0"} +```{exercise} +#| label: discontdraw +#| name: Limit when Denominator converges to 0 Let \[f(x) = \frac{x^2 + 2x}{x}.\] @@ -332,7 +366,7 @@ Let \[f(x) = \frac{x^2 + 2x}{x}.\] -## Answers to Examples {-} +## Answers to Examples {.unnumbered} Example \@ref(exm:seqbehav) @@ -397,7 +431,10 @@ Divide each part by $x$, and we get $x + \frac{2}{x}$ on the numerator, $1$ on t ``` -```{r, fig-hole-0, fig.cap = "A function undedefined at x = 0", echo=FALSE} +```{r} +#| label: fig-hole-0 +#| echo: false +#| fig-cap: A function undedefined at x = 0 range0 <- tibble(x = c(-4, 2)) ggplot(range0, aes(x = x)) + diff --git a/04_calculus.qmd b/04_calculus.qmd index a7adaf6..2a69850 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -1,6 +1,7 @@ # Calculus {#derivatives} -```{r, include = FALSE} +```{r} +#| include: false library(ggplot2) library(tibble) library(dplyr) @@ -11,7 +12,7 @@ library(patchwork) Calculus is a fundamental part of any type of statistics exercise. Although you may not be taking derivatives and integral in your daily work as an analyst, calculus undergirds many concepts we use: maximization, expectation, and cumulative probability. -## Example: The Mean is a Type of Integral {-} +## Example: The Mean is a Type of Integral {.unnumbered} The average of a quantity is a type of weighted mean, where the potential values are weighted by their likelihood, loosely speaking. The integral is actually a general way to describe this weighted average when there are conceptually an infinite number of potential values. @@ -35,7 +36,8 @@ even more concretely, if the potential values of $X$ are finite, then we can wri The derivative of $f$ at $x$ is its rate of change at $x$: how much $f(x)$ changes with a change in $x$. The rate of change is a fraction --- rise over run --- but because not all lines are straight and the rise over run formula will give us different values depending on the range we examine, we need to take a limit (Section \@ref(limits-precalc)). -```{definition, name = "Derivative"} +```{definition} +#| name: Derivative Let $f$ be a function whose domain includes an open interval containing the point $x$. The derivative of $f$ at $x$ is given by \[\frac{d}{dx}f(x) =\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x} = \lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h} @@ -52,7 +54,10 @@ There are a two main ways to denote a derivate: If $f(x)$ is a straight line, the derivative is the slope. For a curve, the slope changes by the values of $x$, so the derivative is the slope of the line tangent to the curve at $x$. See, For example, Figure \@ref(fig:derivsimple). -```{r, derivsimple, echo = FALSE, fig.cap="The Derivative as a Slope"} +```{r} +#| label: derivsimple +#| echo: false +#| fig-cap: The Derivative as a Slope range <- tibble::tibble(x = c(-3, 3)) fx0 <- ggplot(range, aes(x = x)) + @@ -83,7 +88,7 @@ If $f'(x)$ exists at a point $x_0$, then $f$ is said to be __differentiable__ at -### Properties of derivatives {-} +### Properties of derivatives {.unnumbered} Suppose that $f$ and $g$ are differentiable at $x$ and that $\alpha$ is a constant. Then the functions $f\pm g$, $\alpha f$, $f g$, and $f/g$ (provided $g(x)\ne 0$) are also differentiable at $x$. Additionally, __Constant rule:__ \[\left[k f(x)\right]' = k f'(x)\] @@ -103,7 +108,9 @@ __Power rule:__ \[\left[x^k\right]^\prime = k x^{k-1}\] For any real number $k$ (that is, both whole numbers and fractions). The power rule is proved __by induction__, a neat method of proof used in many fundamental applications to prove that a general statement holds for every possible case, even if there are countably infinite cases. We'll show a simple case where $k$ is an integer here. -```{proof, pwrinduct, name = "Proof of Power Rule by Induction"} +```{proof} +#| label: pwrinduct +#| name: Proof of Power Rule by Induction We would like to prove that @@ -143,7 +150,9 @@ To show that it holds for real fractions as well, we can prove expressing that e These "rules" become apparent by applying the definition of the derivative above to each of the things to be "derived", but these come up so frequently that it is best to repeat until it is muscle memory. -```{exercise, introderivatives, name="Derivative of Polynomials"} +```{exercise} +#| label: introderivatives +#| name: Derivative of Polynomials For each of the following functions, find the first-order derivative $f^\prime(x)$. @@ -179,7 +188,8 @@ Similarly, the derivative of $f''(x)$ would be called the third derivative and i -```{example, name="Succession of Derivatives"} +```{example} +#| name: Succession of Derivatives \begin{align*} f(x) &=x^3\\ f^{\prime}(x) &=3x^2\\ @@ -224,7 +234,9 @@ The chain rule can be thought of as the derivative of the "outside" times the de * The chain rule can also be written as \[\frac{dy}{dx}=\frac{dy}{dg(x)} \frac{dg(x)}{dx}\] This expression does not imply that the $dg(x)$'s cancel out, as in fractions. They are part of the derivative notation and you can't separate them out or cancel them.) -```{example,tothesix, name = "Composite Exponent"} +```{example} +#| label: tothesix +#| name: Composite Exponent Find $f^\prime(x)$ for $f(x) = (3x^2+5x-7)^6$. ``` @@ -245,7 +257,8 @@ If $f(x)=[g(x)]^p$ for any rational number $p$, \[f^\prime(x) =p[g(x)]^{p-1}g^\p Natural logs and exponents (they are inverses of each other; see Section \@ref(logexponents)) crop up everywhere in statistics. Their derivative is a special case from the above, but quite elegant. -```{theorem, derivexplog} +```{theorem} +#| label: derivexplog The functions $e^x$ and the natural logarithm $\log(x)$ are continuous and differentiable in their domains, and their first derivate is \[(e^x)^\prime = e^x\] @@ -267,7 +280,7 @@ We will relegate the proofs to small excerpts. -### Derivatives of natural exponential function ($e$) {-} +### Derivatives of natural exponential function ($e$) {.unnumbered} To repeat the main rule in Theorem \@ref(thm:derivexplog), the intuition is that @@ -277,7 +290,10 @@ To repeat the main rule in Theorem \@ref(thm:derivexplog), the intuition is that 4. Chain Rule: When the exponent is a function of $x$, remember to take derivative of that function and add to product. $\frac{d}{dx}e^{g(x)}= e^{g(x)} g^\prime(x)$ -```{r fig-derivexponent, echo = FALSE, fig.cap = "Derivative of the Exponential Function"} +```{r} +#| label: fig-derivexponent +#| echo: false +#| fig-cap: Derivative of the Exponential Function range <- tibble::tibble(x = c(-3, 3)) fx0 <- ggplot(range, aes(x = x)) + @@ -294,7 +310,9 @@ fprimex <- fx0 + stat_function(fun = function(x) exp(x), size = 0.5) + fx + fprimex + plot_layout(nrow = 1) ``` -```{example, exmderivexp, name="Derivative of exponents"} +```{example} +#| label: exmderivexp +#| name: Derivative of exponents Find the derivative for the following. 1. $f(x)=e^{-3x}$ @@ -305,7 +323,7 @@ Find the derivative for the following. -### Derivatives of $\log$ {-} +### Derivatives of $\log$ {.unnumbered} The natural log is the mirror image of the natural exponent and has mirroring properties, again, to repeat the theorem, @@ -316,7 +334,11 @@ The natural log is the mirror image of the natural exponent and has mirroring p -```{r fig-derivlog, echo = FALSE, fig.cap = "Derivative of the Natural Log", warning=FALSE} +```{r} +#| label: fig-derivlog +#| echo: false +#| warning: false +#| fig-cap: Derivative of the Natural Log range <- tibble::tibble(x = c(-0.1, 3)) fx0 <- ggplot(range, aes(x = x)) + @@ -333,7 +355,9 @@ fprimex <- fx0 + stat_function(fun = function(x) ifelse(x <= 0, NA, 1/x), size = fx + fprimex + plot_layout(nrow = 1) ``` -```{example, exmderivlog, name="Derivative of logs"} +```{example} +#| label: exmderivlog +#| name: Derivative of logs Find $dy/dx$ for the following. 1. $f(x)=\log(x^2+9)$ @@ -346,7 +370,7 @@ Find $dy/dx$ for the following. -### Outline of Proof {-} +### Outline of Proof {.unnumbered} We actually show the derivative of the log first, and then the derivative of the exponential naturally follows. @@ -397,7 +421,8 @@ Only the $i$th variable changes --- the others are treated as constants. We can take higher-order partial derivatives, like we did with functions of a single variable, except now the higher-order partials can be with respect to multiple variables. -```{example, name = "More than one type of partial"} +```{example} +#| name: More than one type of partial Notice that you can take partials with regard to different variables. Suppose $f(x,y)=x^2+y^2$. Then @@ -464,7 +489,8 @@ The more derivatives that are added, the smaller the remainder $R$ and the more So far, we've been interested in finding the derivative $f=F'$ of a function $F$. However, sometimes we're interested in exactly the reverse: finding the function $F$ for which $f$ is its derivative. We refer to $F$ as the antiderivative of $f$. -```{definition, name ="Antiderivative"} +```{definition} +#| name: Antiderivative The antiverivative of a function $f(x)$ is a differentiable function $F$ whose derivative is $f$. \[F^\prime = f.\] @@ -477,7 +503,8 @@ Another way to describe is through the inverse formula. Let $DF$ be the derivati This definition bolsters the main takeaway about integrals and derivatives: They are inverses of each other. -```{exercise, name = "Antiderivative"} +```{exercise} +#| name: Antiderivative Find the antiderivative of the following: 1. $f(x) = \frac{1}{x^2}$ @@ -490,7 +517,8 @@ Find the antiderivative of the following: We know from derivatives how to manipulate $F$ to get $f$. But how do you express the procedure to manipulate $f$ to get $F$? For that, we need a new symbol, which we will call indefinite integration. -```{definition, name = "Indefinite Integral"} +```{definition} +#| name: Indefinite Integral The indefinite integral of $f(x)$ is written \[\int f(x) dx \] @@ -517,7 +545,10 @@ The Indefinite Integral of the function $f(x) = (x^2-4)$ can, for example, be $F Some of these functions are plotted in the bottom panel of Figure \@ref(fig:integralc) as dotted lines. -```{r, integralc, fig.cap="The Many Indefinite Integrals of a Function", echo = FALSE} +```{r} +#| label: integralc +#| echo: false +#| fig-cap: The Many Indefinite Integrals of a Function range1 <- tibble(x = c(-4, 4)) @@ -537,7 +568,7 @@ fx + Fx + plot_layout(ncol = 1) Notice from these examples that while there is only a single derivative for any function, there are multiple antiderivatives: one for any arbitrary constant $c$. $c$ just shifts the curve up or down on the $y$-axis. If more information is present about the antiderivative --- e.g., that it passes through a particular point --- then we can solve for a specific value of $c$. -### Common Rules of Integration {-} +### Common Rules of Integration {.unnumbered} Some common rules of integrals follow by virtue of being the inverse of a derivative. @@ -552,7 +583,8 @@ Some common rules of integrals follow by virtue of being the inverse of a deriva 1. Remember the derivative of a log of a function: $\int \frac{f^\prime(x)}{f(x)}dx=\log f(x) + c$ -```{example, name="Common Integration"} +```{example} +#| name: Common Integration Simplify the following indefinite integrals: * \(\int 3x^2 dx\) @@ -570,7 +602,10 @@ If there is a indefinite integral, there _must_ be a definite integral. Indeed t Suppose we want to determine the area $A(R)$ of a region $R$ defined by a curve $f(x)$ and some interval $a\le x \le b$. -```{r, defintfig, echo = FALSE, fig.cap="The Riemann Integral as a Sum of Evaluations"} +```{r} +#| label: defintfig +#| echo: false +#| fig-cap: The Riemann Integral as a Sum of Evaluations f3 <- function(x) -15*(x - 5) + (x - 5)^3 + 50 d1 <- tibble(x = seq(0, 10, 1)) %>% mutate(f = f3(x)) @@ -602,7 +637,8 @@ Figure \@ref(fig:defintfig) shows that illustration. The curve depicted is $f(x) This is how we define the "Definite" Integral: -```{definition, name = "The Definite Integral (Riemann)"} +```{definition} +#| name: The Definite Integral (Riemann) If for a given function $f$ the Riemann sum approaches a limit as $\Delta x \to 0$, then that limit is called the Riemann integral of $f$ from $a$ to $b$. We express this with the $\int$, symbol, and write $$\int\limits_a^b f(x) dx= \lim\limits_{\Delta x\to 0} \sum\limits_{i=1}^n f(x_i)\Delta x$$ The most straightforward of a definite integral is the definite integral. That is, we read @@ -611,7 +647,8 @@ The most straightforward of a definite integral is the definite integral. That i The fundamental theorem of calculus shows us that this sum is, in fact, the antiderivative. -```{theorem, name = "First Fundamental Theorem of Calculus"} +```{theorem} +#| name: First Fundamental Theorem of Calculus Let the function $f$ be bounded on $[a,b]$ and continuous on $(a,b)$. Then, suggestively, use the symbol $F(x)$ to denote the definite integral from $a$ to $x$: \[F(x)=\int\limits_a^x f(t)dt, \quad a\le x\le b\] @@ -624,7 +661,8 @@ This is again a long way of saying that that differentiation is the inverse of The second theorem gives us a simple way of computing a definite integral as a function of indefinite integrals. -```{theorem, name = "Second Fundamental Theorem of Calculus"} +```{theorem} +#| name: Second Fundamental Theorem of Calculus Let the function $f$ be bounded on $[a,b]$ and continuous on $(a,b)$. Let $F$ be any function that is continuous on $[a,b]$ such that $F^\prime(x)=f(x)$ on $(a,b)$. Then $$\int\limits_a^bf(x)dx = F(b)-F(a)$$ ``` @@ -633,7 +671,9 @@ So the procedure to calculate a simple definite integral $\int\limits_a^b f(x)dx 1. Find the indefinite integral $F(x)$. 2. Evaluate $F(b)-F(a)$. -```{example, defintmon, name="Definite Integral of a monomial"} +```{example} +#| label: defintmon +#| name: Definite Integral of a monomial Solve $\int\limits_1^3 3x^2 dx.$ Let $f(x) = 3x^2$. @@ -649,7 +689,7 @@ What is the value of $\int\limits_{-2}^2 e^x e^{e^x} dx$? -### Common Rules for Definite Integrals {-} +### Common Rules for Definite Integrals {.unnumbered} The area-interpretation of the definite integral provides some rules for simplification. @@ -660,7 +700,8 @@ The area-interpretation of the definite integral provides some rules for simplif 4. Areas can be combined as long as limits are linked: \[\int\limits_a^b f(x) dx +\int\limits_b^c f(x)dx = \int\limits_a^c f(x)dx\] -```{exercise, name="Definite integral shortcuts"} +```{exercise} +#| name: Definite integral shortcuts Simplify the following definite intergrals. 1. $\int\limits_1^1 3x^2 dx =$ @@ -697,7 +738,9 @@ where $c=u(a)$ and $d=u(b)$. -```{example, intsub1, name ="Integration by Substitution I"} +```{example} +#| label: intsub1 +#| name: Integration by Substitution I Solve the indefinite integral \[\int x^2 \sqrt{x+1}dx.\] @@ -712,7 +755,9 @@ For the above problem, we could have also used the substitution $u=\sqrt{x+1}$. Another case in which integration by substitution is is useful is with a fraction. -```{example, intsub2, name="Integration by Substitutiton II"} +```{example} +#| label: intsub2 +#| name: Integration by Substitutiton II Simplify \[\int\limits_0^1 \frac{5e^{2x}}{(1+e^{2x})^{1/3}}dx.\] ``` @@ -735,7 +780,8 @@ Our goal here is to find expressions for $u$ and $dv$ that, when substituted int -```{example,name="Integration by Parts"} +```{example} +#| name: Integration by Parts Simplify the following integrals. These seemingly obscure forms of integrals come up often when integrating distributions. \[\int x e^{ax} dx\] @@ -753,7 +799,9 @@ Let $u=x$ and $\frac{dv}{dx} = e^{ax}$. Then $du=dx$ and $v=(1/a)e^{ax}$. Subst ``` -```{exercise, intparts-adv, name = "Integration by Parts II"} +```{exercise} +#| label: intparts-adv +#| name: Integration by Parts II 1. Integrate \[\int x^n e^{ax} dx\] @@ -766,7 +814,7 @@ Let $u=x$ and $\frac{dv}{dx} = e^{ax}$. Then $du=dx$ and $v=(1/a)e^{ax}$. Subst ``` -## Answers to Examples and Exercises {-} +## Answers to Examples and Exercises {.unnumbered} Exercise \@ref(exr:introderivatives) diff --git a/05_optimization.qmd b/05_optimization.qmd index 1a97903..4b8b17f 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -1,7 +1,8 @@ # Optimization {#optim} -```{r, include = FALSE} +```{r} +#| include: false library(ggplot2) library(tibble) library(patchwork) @@ -12,7 +13,7 @@ To optimize, we use derivatives and calculus. Optimization is to find the maximu Optimization also comes up in Economics, Formal Theory, and Political Economy all the time. A go-to model of human behavior is that they optimize a certain utility function. Humans are not pure utility maximizers, of course, but nuanced models of optimization -- for example, adding constraints and adding uncertainty -- will prove to be quite useful. -## Example: Meltzer-Richard {-} +## Example: Meltzer-Richard {.unnumbered} A standard backdrop in comparative political economy, the Meltzer-Richard (1981) model states that redistribution of wealth should be higher in societies where the median income is much smaller than the average income. More to the point, typically income distributions wher ethe median is very different from the average is one of high inequality. In other words, the Meltzer-Richard model says that highly unequal economies will have more re-distribution of wealth. Why is that the case? Here is a simplified example that is not the exact model by Meltzer and Richard^[Allan H. Meltzer and Scott F. Richard. ["A Rational Theory of the Size of Government"](https://www.jstor.org/stable/1830813). _Journal of Political Economy_ 89:5 (1981), p. 914-927], but adapted from Persson and Tabellini^[Adapted from Torsten Persson and Guido Tabellini, _Political Economics: Explaining Economic Policy_. MIT Press. ] @@ -100,7 +101,9 @@ The first derivative, $f'(x)$, quantifies the slope of a function. Therefore, it So for example, $f(x) = x^2 + 2$ and $f^\prime(x) = 2x$ -```{r, echo = FALSE, fig.cap="Maxima and Minima"} +```{r} +#| echo: false +#| fig-cap: Maxima and Minima range <- tibble::tibble(x = c(-3, 3)) fx0 <- ggplot(range, aes(x = x)) + labs(x = expression(x), y = expression(f(x)), @@ -116,13 +119,16 @@ fx + fprimex + plot_layout(nrow = 1) ``` -```{exercise, name = "Plotting a mazimum and minimum"} +```{exercise} +#| name: Plotting a mazimum and minimum Plot $f(x)=x^3+ x^2 + 2$, plot its derivative, and identifiy where the derivative is zero. Is there a maximum or minimum? ``` -```{r, echo = FALSE, eval = FALSE} +```{r} +#| echo: false +#| eval: false range <- tibble::tibble(x = c(-3, 3)) fx0 <- ggplot(range, aes(x = x)) + labs(x = expression(x), y = expression(f(x))) + @@ -169,7 +175,8 @@ __Global Maxima and Minima__ Sometimes no global max or min exists --- e.g., $f( -```{example, name = "Maxima and Minima by drawing"} +```{example} +#| name: Maxima and Minima by drawing Find any critical points and identify whether they are a max, min, or saddle point: @@ -187,13 +194,15 @@ Find any critical points and identify whether they are a max, min, or saddle poi Concavity helps identify the curvature of a function, $f(x)$, in 2 dimensional space. -```{definition, name = "Concave Function"} +```{definition} +#| name: Concave Function A function $f$ is strictly concave over the set S \underline{if} $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) > af(x_1) + (1-a)f(x_2)$$ \textit{Any} line connecting two points on a concave function will lie \textit{below} the function. ``` -```{r, echo = FALSE} +```{r} +#| echo: false range1 <- tibble(x = c(-4, 4)) fx1 <- ggplot(range1, aes(x = x)) + stat_function(fun = function(x) -x^2, size = 0.5) + @@ -206,7 +215,8 @@ fx1 + fx2 + plot_layout(nrow = 1) ``` -```{definition, name = "Convex Function"} +```{definition} +#| name: Convex Function Convex: A function f is strictly convex over the set S \underline{if} $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) < af(x_1) + (1-a)f(x_2)$$ Any line connecting two points on a convex function will lie above the function. @@ -219,7 +229,8 @@ Convex: A function f is strictly convex over the set S \underline{if} $\forall x Sometimes, concavity and convexity are strict of a requirement. For most purposes of getting solutions, what we call quasi-concavity is enough. -```{definition, name = "Quasiconcave Function"} +```{definition} +#| name: Quasiconcave Function A function f is quasiconcave over the set S if $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) \ge \min(f(x_1),f(x_2))$$ No matter what two points you select, the \textit{lowest} valued point will always be an end point. @@ -230,7 +241,8 @@ A function f is quasiconcave over the set S if $\forall x_1,x_2 \in S$ and $\for \parbox{2in}{\includegraphics[scale=.4]{Quasiconcave.pdf}} \end{comment} -```{definition, name = "Quasiconvex"} +```{definition} +#| name: Quasiconvex A function f is quasiconvex over the set $S$ if $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) \le \max(f(x_1),f(x_2))$$ No matter what two points you select, the \textit{highest} valued point will always be an end point. ``` @@ -249,7 +261,7 @@ f''(x) > 0 & \Rightarrow & \text{Convex} \end{array}\] -### Quadratic Forms {-} +### Quadratic Forms {.unnumbered} Quadratic forms is shorthand for a way to summarize a function. This is important for finding concavity because @@ -295,7 +307,7 @@ For example, the Quadratic on $\mathbf{R}^2$: -### Definiteness of Quadratic Forms {-} +### Definiteness of Quadratic Forms {.unnumbered} When the function $f(\mathbf{x})$ has more than two inputs, determining whether it has a maxima and minima (remember, functions may have many inputs but they have only one output) is a bit more tedious. Definiteness helps identify the curvature of a function, $Q(\textbf{x})$, in n dimensional space. @@ -310,13 +322,14 @@ zero, or sometimes both over all $\mathbf{x}\ne 0$. We can see from a graphical representation that if a point is a local maxima or minima, it must meet certain conditions regarding its derivative. These are so commonly used that we refer these to "First Order Conditions" (FOCs) and "Second Order Conditions" (SOCs) in the economic tradition. -### First Order Conditions {-} +### First Order Conditions {.unnumbered} When we examined functions of one variable $x$, we found critical points by taking the first derivative, setting it to zero, and solving for $x$. For functions of $n$ variables, the critical points are found in much the same way, except now we set the partial derivatives equal to zero. Note: We will only consider critical points on the interior of a function's domain. In a derivative, we only took the derivative with respect to one variable at a time. When we take the derivative separately with respect to all variables in the elements of $\mathbf{x}$ and then express the result as a vector, we use the term Gradient and Hessian. -```{definition, name = "Gradient"} +```{definition} +#| name: Gradient Given a function $f(\textbf{x})$ in $n$ variables, the gradient $\nabla f(\mathbf{x})$ (the greek letter nabla ) is a column vector, where the $i$th element is the partial derivative of $f(\textbf{x})$ with respect to $x_i$: @@ -328,7 +341,8 @@ Given a function $f(\textbf{x})$ in $n$ variables, the gradient $\nabla f(\mathb Before we know whether a point is a maxima or minima, if it meets the FOC it is a "Critical Point". -```{definition, name = "Critical Point"} +```{definition} +#| name: Critical Point $\mathbf{x}^*$ is a critical point if and only if $\nabla f(\mathbf{x}^*)=0$. If the partial derivative of f(x) with respect to $x^*$ is 0, then $\mathbf{x}^*$ is a critical point. To solve for $\mathbf{x}^*$, find the gradient, set each element equal to 0, and solve the system of equations. $$\mathbf{x}^* = \begin{pmatrix} x_1^*\\x_2^*\\ \vdots \\ x_n^*\end{pmatrix}$$ @@ -371,10 +385,11 @@ So \[\mathbf{x}^* = (1,0)\] -### Second Order Conditions {-} +### Second Order Conditions {.unnumbered} When we found a critical point for a function of one variable, we used the second derivative as a indicator of the curvature at the point in order to determine whether the point was a min, max, or saddle (second derivative test of concavity). For functions of $n$ variables, we use _second order partial derivatives_ as an indicator of curvature. -```{definition, name = "Hessian"} +```{definition} +#| name: Hessian Given a function $f(\mathbf{x})$ in $n$ variables, the hessian $\mathbf{H(x)}$ is an $n\times n$ matrix, where the $(i,j)$th element is the second order @@ -411,7 +426,8 @@ Given a function $f(\mathbf{x})$ and a point $\mathbf{x}^*$ such that $\nabla f( -```{example, name = "Max and min with two dimensions"} +```{example} +#| name: Max and min with two dimensions We found that the only critical point of $f(\mathbf{x})=(x_1-1)^2+x_2^2+1$ is at $\mathbf{x}^*=(1,0)$. Is it a min, max, or @@ -446,7 +462,7 @@ For any $\mathbf{x}\ne 0$, $2(x_1^2+x_2^2)>0$, so the Hessian is positive defin -### Definiteness and Concavity {-} +### Definiteness and Concavity {.unnumbered} Although definiteness helps us to understand the curvature of an n-dimensional function, it does not necessarily tell us whether the function is globally concave or convex. @@ -585,7 +601,7 @@ It is easy to see that the \textit{unconstrained} maximum occurs at $(x_1, x_2) -### Equality Constraints {-} +### Equality Constraints {.unnumbered} Equality constraints are the easiest to deal with because we know that the maximum or minimum has to lie on the (intersection of the) constraint(s). @@ -624,7 +640,8 @@ We can then solve this system of equations, because there are $n+k$ equations an **Second-order Conditions and Unconstrained Optimization:** There may be more than one critical point, i.e. we need to verify that the critical point we find is a maximum/minimum. Similar to unconstrained optimization, we can do this by checking the second-order conditions. -```{example, name = "Constrained optimization with two goods and a budget constraint"} +```{example} +#| name: Constrained optimization with two goods and a budget constraint Find the constrained optimization of $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 = 4$$ @@ -860,7 +877,8 @@ In a two-dimensional set-up, this means we must check the following cases: -```{example, name = "Kuhn-Tucker with two variables"} +```{example} +#| name: Kuhn-Tucker with two variables Solve the following optimization problem with inequality constraints \[\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)\] @@ -921,7 +939,8 @@ Three of the critical points violate the requirement that $\lambda \geq 0$, so t -```{exercise, name = "Kuhn-Tucker with logs"} +```{exercise} +#| name: Kuhn-Tucker with logs Solve the constrained optimization problem, diff --git a/06_probability.qmd b/06_probability.qmd index d1b0779..c11197c 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -3,7 +3,8 @@ Probability and Inferences are mirror images of each other, and both are integral to social science. Probability quantifies uncertainty, which is important because many things in the social world are at first uncertain. Inference is then the study of how to learn about facts you don't observe from facts you do observe. -```{r, include = FALSE} +```{r} +#| include: false library(ggplot2) options(knitr.graphics.auto_pdf = TRUE) # use pdf for images ``` @@ -16,7 +17,9 @@ Probability in high school is usually really about combinatorics: the probabilit __Fundamental Theorem of Counting__: If an object has $j$ different characteristics that are independent of each other, and each characteristic $i$ has $n_i$ ways of being expressed, then there are $\prod_{i = 1}^j n_i$ possible unique objects. -```{example, name = "Counting Possibilities", countingrules} +```{example} +#| name: Counting Possibilities +#| label: countingrules Suppose we are given a stack of cards. Cards can be either red or black and can take on any of 13 values. There is only one of each color-number combination. In this case, @@ -52,7 +55,9 @@ If replacement is allowed, there are always the same $n$ objects to select from. Expression $\binom{n}{k}$ is read as "n choose k" and denotes $\frac{n!}{(n-k)!k!}$. Also, note that $0! = 1$. -```{example, name = "Counting", counting} +```{example} +#| name: Counting +#| label: counting There are five balls numbered from 1 through 5 in a jar. Three balls are chosen. How many possible choices are there? @@ -66,7 +71,9 @@ There are five balls numbered from 1 through 5 in a jar. Three balls are chosen. ``` -```{exercise, name = "Counting", counting1} +```{exercise} +#| name: Counting +#| label: counting1 Four cards are selected from a deck of 52 cards. Once a card has been drawn, it is not reshuffled back into the deck. Moreover, we care only about the complete hand that we get (i.e. we care about the set of selected cards, not the sequence in which it was drawn). How many possible outcomes are there? @@ -111,7 +118,9 @@ Properties of set operations: * __Disjointness__: Sets are disjoint when they do not intersect, such that $A \cap B = \emptyset$. A collection of sets is pairwise disjoint (**mutually exclusive**) if, for all $i \neq j$, $A_i \cap A_j = \emptyset$. A collection of sets form a partition of set $S$ if they are pairwise disjoint and they cover set $S$, such that $\bigcup_{i = 1}^k A_i = S$. -```{example, name = "Sets", sets} +```{example} +#| name: Sets +#| label: sets Let set $A$ be {1, 2, 3, 4}, $B$ be {3, 4, 5, 6}, and $C$ be {5, 6, 7, 8}. Sets $A$, $B$, and $C$ are all subsets of the sample space $S$ which is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} @@ -128,7 +137,9 @@ Write out the following sets: ``` -```{exercise, name = "Sets", sets1} +```{exercise} +#| name: Sets +#| label: sets1 @@ -152,11 +163,15 @@ Consider subsets A {2, 8} and B {2,3,7} of the sample space you found. What is ## Probability {#probdef} -```{r prob-image, fig.cap='Probablity as a Measure^[Images of Probability and Random Variables drawn by Shiro Kuriwaki and inspired by Blitzstein and Morris]', echo = FALSE} +```{r} +#| label: prob-image +#| echo: false +#| fig-cap: Probablity as a Measure^[Images of Probability and Random Variables drawn +#| by Shiro Kuriwaki and inspired by Blitzstein and Morris] knitr::include_graphics('images/probability.png') ``` -### Probability Definitions: Formal and Informal {-} +### Probability Definitions: Formal and Informal {.unnumbered} Many things in the world are uncertain. In everyday speech, we say that we are _uncertain_ about the outcome of random events. Probability is a formal model of uncertainty which provides a measure of uncertainty governed by a particular set of rules (Figure \@ref(fig:prob-image)). A different model of uncertainty would, of course, have a set of rules different from anything we discuss here. Our focus on probability is justified because it has proven to be a particularly useful model of uncertainty. @@ -165,7 +180,8 @@ __Probability Distribution Function__: a mapping of each event in the sample spa Formally, -```{definition, name = "Probability"} +```{definition} +#| name: Probability Probability is a function that maps events to a real number, obeying the axioms of probability. @@ -173,7 +189,8 @@ Probability is a function that maps events to a real number, obeying the axioms The axioms of probability make sure that the separate events add up in terms of probability, and -- for standardization purposes -- that they add up to 1. -```{definition, name = "Axioms of Probability"} +```{definition} +#| name: Axioms of Probability 1. For any event $A$, $P(A)\ge 0$. @@ -194,7 +211,7 @@ P(A_1 \cup A_2) = P(A_1) + P(A_2) \quad\text{for disjoint } A_1, A_2 -### Probability Operations {-} +### Probability Operations {.unnumbered} Using these three axioms, we can define all of the common rules of probability. @@ -207,7 +224,9 @@ B)=P(A)+P(B)-P(A\cap B)$ 6. Boole's Inequality: For any sequence of $n$ events (which need not be disjoint) $A_1,A_2,\ldots,A_n$, then $P\left( \bigcup\limits_{i=1}^n A_i\right) \leq \sum\limits_{i=1}^n P(A_i)$. -```{example, name = "Probability", prob} +```{example} +#| name: Probability +#| label: prob Assume we have an evenly-balanced, six-sided die. @@ -225,7 +244,9 @@ Then, ``` -```{exercise, name = "Probability", prob1} +```{exercise} +#| name: Probability +#| label: prob1 Suppose you had a pair of four-sided dice. You sum the results from a single toss. Let us call this sum, or the outcome, X. @@ -249,7 +270,9 @@ __Conditional Probability__: The conditional probability $P(A|B)$ of an event $ Note that conditional probabilities are probabilities and must also follow the Kolmagorov axioms of probability. -```{example, name = "Conditional Probability 1", condprobexm1} +```{example} +#| name: Conditional Probability 1 +#| label: condprobexm1 Assume $A$ and $B$ occur with the following frequencies: $\quad$ @@ -272,7 +295,9 @@ and let $n_{ab}+n_{a^Cb}+n_{ab^C}+n_{(ab)^C}=N$. Then -```{example, name = "Conditional Probability 2", condprobexm2} +```{example} +#| name: Conditional Probability 2 +#| label: condprobexm2 A six-sided die is rolled. What is the probability of a 1, given the outcome is an odd number? @@ -281,7 +306,8 @@ and let $n_{ab}+n_{a^Cb}+n_{ab^C}+n_{(ab)^C}=N$. Then You could rearrange the fraction to highlight how a joint probability could be expressed as the product of a conditional probability. -```{definition, name = "Multiplicative Law of Probability"} +```{definition} +#| name: Multiplicative Law of Probability The probability of the intersection of two events $A$ and $B$ is $P(A\cap B)=P(A)P(B|A)=P(B)P(A|B)$ which follows directly from the definition of conditional probability. More generally, @@ -293,7 +319,8 @@ Sometimes it is easier to calculate these conditional probabilities and sum them -```{definition, name = "Law of Total Probability"} +```{definition} +#| name: Law of Total Probability Let $S$ be the sample space of some experiment and let the disjoint $k$ events $B_1,\ldots,B_k$ partition $S$, such that $P(B_1\cup ... \cup B_k) = P(S) = 1$. If $A$ is some other event in $S$, then the events $A\cap B_1, A\cap B_2, \ldots, A\cap B_k$ will form a partition of $A$ and we can write $A$ as \[A=(A\cap B_1)\cup\cdots\cup (A\cap B_k)\]. Since the $k$ events are disjoint, @@ -318,13 +345,17 @@ Bayes' rule determines the posterior probability of a state $P(B_j|A)$ by calc __Prior and Posterior Probabilities__: Above, $P(B_1)$ is often called the prior probability, since it's the probability of $B_1$ before anything else is known. $P(B_1|A)$ is called the posterior probability, since it's the probability after other information is taken into account. -```{example, name = "Bayes' Rule", bayesrule} +```{example} +#| name: Bayes' Rule +#| label: bayesrule In a given town, 40% of the voters are Democrat and 60% are Republican. The president's budget is supported by 50% of the Democrats and 90% of the Republicans. If a randomly (equally likely) selected voter is found to support the president's budget, what is the probability that they are a Democrat? ``` -```{exercise, name = "Conditional Probability", condprobexr} +```{exercise} +#| name: Conditional Probability +#| label: condprobexr Assume that 2% of the population of the U.S. are members of some extremist militia group. We develop a survey that positively classifies someone as being a member of a militia group given that they are a member 95% of the time and negatively classifies someone as not being a member of a militia group given that they are not a member 97% of the time. What is the probability that someone positively classified as being a member of a militia group is actually a militia member? @@ -337,7 +368,8 @@ Assume that 2% of the population of the U.S. are members of some extremist milit ## Independence -```{definition, name = "Independence"} +```{definition} +#| name: Independence If the occurrence or nonoccurrence of either events $A$ and $B$ have no effect on the occurrence or nonoccurrence of the other, then $A$ and $B$ are independent. ``` @@ -375,12 +407,16 @@ Perhaps more counter-intuitively: If two events are already independent, then it Most questions in the social sciences involve events, rather than numbers per se. To analyze and reason about events quantitatively, we need a way of mapping events to numbers. A random variable does exactly that. -```{r rv-image, fig.cap='The Random Variable as a Real-Valued Function', echo = FALSE} +```{r} +#| label: rv-image +#| echo: false +#| fig-cap: The Random Variable as a Real-Valued Function knitr::include_graphics('images/rv.png') ``` -```{definition, name = "Random Variable"} +```{definition} +#| name: Random Variable A random variable is a measurable function $X$ that maps from the sample space $S$ to the set of real numbers $R.$ It assigns a real number to every outcome $s \in S$. @@ -411,14 +447,16 @@ We now have two main concepts in this section -- probability and random variable The concept of distributions is the natural bridge between these two concepts. -```{definition, name = "Distribution of a random variable"} +```{definition} +#| name: Distribution of a random variable A distribution of a random variable is a function that specifies the probabilities of all events associated with that random variable. There are several types of distributions: A probability mass function for a discrete random variable and probability density function for a continuous random variable. ``` Notice how the definition of distributions combines two ideas of random variables and probabilities of events. First, the distribution considers a random variable, call it $X$. $X$ can take a number of possible numeric values. -```{example, name = "Total Number of Occurrences"} +```{example} +#| name: Total Number of Occurrences Consider three binary outcomes, one for each patient recovering from a disease: $R_i$ denotes the event in which patient $i$ ($i = 1, 2, 3$) recovers from a disease. $R_1$, $R_2$, and $R_3$. How would we represent the total number of people who end up recovering from the disease? @@ -434,11 +472,12 @@ Recall that with each of these numerical values there is a class of *events*. In In other words, a random variable $X$ *induces a probability distribution* $P$ (sometimes written $P_X$ to emphasize that the probability density is about the r.v. $X$) -### Discrete Random Variables {-} +### Discrete Random Variables {.unnumbered} The formal definition of a random variable is easier to given by separating out two cases: discrete random variables when the numeric summaries of the events are discrete, and continuous random variables when they are continuous. -```{definition, name ="Discrete Random Variable"} +```{definition} +#| name: Discrete Random Variable $X$ is a discrete random variable if it can assume only a finite or countably infinite number of distinct values. Examples: number of wars per year, heads or tails. ``` @@ -446,7 +485,8 @@ $X$ is a discrete random variable if it can assume only a finite or countably in The distribution of a discrete r.v. is a PMF: -```{definition, name = "Probability Mass Function"} +```{definition} +#| name: Probability Mass Function For a discrete random variable $X$, the probability mass function (Also referred to simply as the "probability distribution.") (PMF), $p(x)=P(X=x)$, assigns probabilities to a countable number of distinct $x$ values such that @@ -489,19 +529,21 @@ For a fair die with its value as $Y$, What are the following? -### Continuous Random Variables {-} +### Continuous Random Variables {.unnumbered} We also have a similar definition for _continuous_ random variables. -```{definition, name = "Continuous Random Variable"} +```{definition} +#| name: Continuous Random Variable $X$ is a continuous random variable if there exists a nonnegative function $f(x)$ defined for all real $x\in (-\infty,\infty)$, such that for any interval $A$, $P(X\in A)=\int\limits_A f(x)dx$. Examples: age, income, GNP, temperature. ``` -```{definition, name = "Probability Density Function"} +```{definition} +#| name: Probability Density Function The function $f$ above is called the probability density function (pdf) of $X$ and must satisfy \[f(x)\ge 0\] @@ -519,7 +561,8 @@ The function $f$ above is called the probability density function (pdf) of $X$ a For both discrete and continuous random variables, we have a unifying concept of another measure: the cumulative distribution: -```{definition, name = "Cumulative Density Function"} +```{definition} +#| name: Cumulative Density Function Because the probability that a continuous random variable will assume any particular value is zero, we can only make statements about the probability of a continuous random variable being within an interval. The cumulative distribution gives the probability that $Y$ lies on the interval $(-\infty,y)$ and is defined as $$F(x)=P(X\le x)=\int\limits_{-\infty}^x f(s)ds$$ Note that $F(x)$ has similar properties with continuous distributions as it does with discrete - non-decreasing, continuous (not just right-continuous), and $\lim\limits_{x \to -\infty} F(x) = 0$ and $\lim\limits_{x \to \infty} F(x) = 1$. ``` @@ -574,7 +617,8 @@ __Conditional Probability Distribution__: probability distribution for one varia * Continuous: $f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)},\quad f_X(x) > 0$ -```{exercise, name= "Discrete Outcomes"} +```{exercise} +#| name: Discrete Outcomes Suppose we are interested in the outcomes of flipping a coin and rolling a 6-sided die at the same time. The sample space for this process contains 12 elements: $$\{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}$$ We can define two random variables $X$ and $Y$ such that $X = 1$ if heads and $X = 0$ if tails, while $Y$ equals the number on the die. We can then make statements about the joint distribution of $X$ and $Y$. What are the following? @@ -599,14 +643,17 @@ We can then make statements about the joint distribution of $X$ and $Y$. What ar We often want to summarize some characteristics of the distribution of a random variable. The most important summary is the expectation (or expected value, or mean), in which the possible values of a random variable are weighted by their probabilities. -```{definition, name = "Expectation of a Discrete Random Variable"} +```{definition} +#| name: Expectation of a Discrete Random Variable The expected value of a discrete random variable $Y$ is \[E(Y)=\sum\limits_{y} y P(Y=y)= \sum\limits_{y} y p(y)\] In words, it is the weighted average of all possible values of $Y$, weighted by the probability that $y$ occurs. It is not necessarily the number we would expect $Y$ to take on, but the average value of $Y$ after a large number of repetitions of an experiment. ``` -```{example, name = "Expectation of a Discrete Random Variable", expectdiscrete} +```{example} +#| name: Expectation of a Discrete Random Variable +#| label: expectdiscrete What is the expectation of a fair, six-sided die? @@ -624,7 +671,9 @@ Hence, the expected value of the continuous variable $Y$ is defined by \[E(Y)=\int\limits_{y} y f(y) dy\] -```{example, name = "Expectation of a Continuous Random Variable", expectconti} +```{example} +#| name: Expectation of a Continuous Random Variable +#| label: expectconti Find $E(Y)$ for $f(y)=\frac{1}{1.5}, \quad 00\] @@ -856,7 +918,8 @@ Border disputes occur between two countries through a Poisson Distribution, at a Two _continuous_ distributions used often are: -```{definition, name = "Uniform Distribution"} +```{definition} +#| name: Uniform Distribution A random variable $Y$ has a continuous uniform distribution on the interval $(\alpha,\beta)$ if its density is given by $$f(y)=\frac{1}{\beta-\alpha}, \quad \alpha\le y\le \beta$$ The mean and variance of $Y$ are $E(Y)=\frac{\alpha+\beta}{2}$ and $\text{Var}(Y)=\frac{(\beta-\alpha)^2}{12}$. ``` @@ -877,7 +940,8 @@ For $Y$ uniformly distributed over $(1,3)$, what are the following probabilities \parbox{1.5in}{\hfill \epsffile{unifpdf.eps}} \end{comment} -```{definition, name = "Normal Distribution"} +```{definition} +#| name: Normal Distribution A random variable $Y$ is normally distributed with mean $E(Y)=\mu$ and variance $\text{Var}(Y)=\sigma^2$ if its density is @@ -889,7 +953,10 @@ A random variable $Y$ is normally distributed with mean $E(Y)=\mu$ and variance See Figure \@ref(fig:normaldens) are various Normal Distributions with the same $\mu = 1$ and two versions of the variance. -```{r normaldens, echo = FALSE, fig.cap = "Normal Distribution Density"} +```{r} +#| label: normaldens +#| echo: false +#| fig-cap: Normal Distribution Density range <- tibble::tibble(x = c(-5, 5)) fx0 <- ggplot(range, aes(x = x)) + @@ -990,7 +1057,9 @@ Why is this important? We rarely know the true process governing the events we We are now finally ready to revisit, with a bit more precise terms, the two pillars of statistical theory we motivated Section \@ref(limitsfun) with. -```{theorem, clt, name = "Central Limit Theorem (i.i.d. case)"} +```{theorem} +#| label: clt +#| name: Central Limit Theorem (i.i.d. case) Let $\{X_n\} = \{X_1, X_2, \ldots\}$ be a sequence of i.i.d. random variables with finite mean ($\mu$) and variance ($\sigma^2$). Then, the sample mean $\bar{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n}$ increasingly converges into a Normal distribution. \[\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),\] @@ -1006,7 +1075,9 @@ This result means that, as $n$ grows, the distribution of the sample mean $\bar Intuitively, this result also justifies that whenever a lot of small, independent processes somehow combine together to form the realized observations, practitioners often feel comfortable assuming Normality. -```{theorem, lln, name = "Law of Large Numbers (LLN)"} +```{theorem} +#| label: lln +#| name: Law of Large Numbers (LLN) For any draw of independent random variables with the same mean $\mu$, the sample average after $n$ draws, $\bar{X}_n = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)$, converges in probability to the expected value of $X$, $\mu$ as $n \rightarrow \infty$: \[\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0\] @@ -1043,7 +1114,7 @@ denominator <- exp(n_seq) numerator / denominator <= (1 * denominator) / denominator \end{comment} -## Answers to Examples and Exercises {-} +## Answers to Examples and Exercises {.unnumbered} Answer to Example \@ref(exm:counting): @@ -1194,4 +1265,4 @@ Answer to Exercise \@ref(exr:expvar3): 1. mean = 2, standard deviation = $\sqrt(\frac{2}{3})$ -2. $\frac{1}{8}(2 - \sqrt(\frac{2}{3}))^2$ \ No newline at end of file +2. $\frac{1}{8}(2 - \sqrt(\frac{2}{3}))^2$ diff --git a/07_linear-algebra.qmd b/07_linear-algebra.qmd index 0a15dae..daf551f 100644 --- a/07_linear-algebra.qmd +++ b/07_linear-algebra.qmd @@ -36,7 +36,9 @@ __Vector Inner Product__: The inner product (also called the dot product or scal __Vector Norm__: The norm of a vector is a measure of its length. There are many different ways to calculate the norm, but the most common is the Euclidean norm (which corresponds to our usual conception of distance in three-dimensional space): $$ ||{\bf v}|| = \sqrt{{\bf v}\cdot{\bf v}} = \sqrt{ v_1v_1 + v_2v_2 + \cdots + v_nv_n}$$ -```{example, name = "Vector Algebra", vectors} +```{example} +#| name: Vector Algebra +#| label: vectors Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{pmatrix}$. Calculate the following: @@ -49,7 +51,9 @@ Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{p ``` -```{exercise, name = "Vector Algebra", vectors1} +```{exercise} +#| name: Vector Algebra +#| label: vectors1 Let $u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}$, $v = \begin{pmatrix} 9&-3&2&8 \end{pmatrix}$, $w = \begin{pmatrix} 1&13& -7&2 &15 \end{pmatrix}$, and $c = 2$. Calculate the following: @@ -86,7 +90,9 @@ A set $S$ of vectors is linearly dependent if and only if at least one of the ve Since $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$, these 4 vectors constitute a linearly dependent set. -```{example, name = "Linear Independence", linearindep} +```{example} +#| name: Linear Independence +#| label: linearindep Are the following sets of vectors linearly independent? @@ -99,7 +105,9 @@ Are the following sets of vectors linearly independent? -```{exercise, name = "Linear Independence", linearindep1} +```{exercise} +#| name: Linear Independence +#| label: linearindep1 Are the following sets of vectors linearly independent? @@ -140,7 +148,8 @@ __Matrix Addition__: Let $\bf A$ and $\bf B$ be two $m\times n$ matrices. Note that matrices ${\bf A}$ and ${\bf B}$ must have the same dimensionality, in which case they are __conformable for addition__. -```{example, matrixaddition} +```{example} +#| label: matrixaddition $${\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & 2 \end{pmatrix}$$ ${\bf A+B}=$ @@ -164,7 +173,8 @@ __Scalar Multiplication__: Given the scalar $s$, the scalar multiplication of $ s a_{m1} & s a_{m2} & \cdots & s a_{mn} \end{pmatrix}$$ -```{example, scalarmulti} +```{example} +#| label: scalarmulti $s=2, \qquad {\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ @@ -179,7 +189,8 @@ $s {\bf A} =$ __Matrix Multiplication__: If ${\bf A}$ is an $m\times k$ matrix and $\bf B$ is a $k\times n$ matrix, then their product $\bf C = A B$ is the $m\times n$ matrix where $$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ik}b_{kj}$$ -```{example, matrixmulti} +```{example} +#| label: matrixmulti 1. $\begin{pmatrix} a&b\\c&d\\e&f \end{pmatrix} \begin{pmatrix} A&B\\C&D \end{pmatrix} @@ -242,7 +253,9 @@ Example of $({\bf AB})^T = {\bf B}^T{\bf A}^T$: -```{exercise, name = "Matrix Multiplication", matrixmulti1} +```{exercise} +#| name: Matrix Multiplication +#| label: matrixmulti1 Let $$A = \begin{pmatrix} 2&0&-1&1\\1&2&0&1 \end{pmatrix}$$ @@ -324,7 +337,9 @@ Methods to solve linear systems: 2. Elimination of variables 3. Matrix methods -```{exercise, name = "Linear Equations", lineareq} +```{exercise} +#| name: Linear Equations +#| label: lineareq Provide a system of 2 equations with 2 unknowns that has @@ -375,7 +390,9 @@ __Augmented Matrix__: When we append $\bf b$ to the coefficient matrix $\bf A$, \end{pmatrix}$$ -```{exercise, name="Augmented Matrix", augmatrix} +```{exercise} +#| name: Augmented Matrix +#| label: augmatrix Create an augmented matrix that represent the following system of equations: @@ -446,7 +463,8 @@ __Adding (subtracting) Rows__: If we add (subtract) the first row of matrix $\w \end{pmatrix}$$ which represents a linear system equivalent to that represented by matrix $\widehat{\bf A}$. -```{example, solvesys} +```{example} +#| label: solvesys Solve the following system of equations by using elementary row operations: @@ -463,7 +481,9 @@ $\begin{matrix} -```{exercise, name = "Solving Systems of Equations", solvesys1} +```{exercise} +#| name: Solving Systems of Equations +#| label: solvesys1 Put the following system of equations into augmented matrix form. Then, using Gaussian or Gauss-Jordan elimination, solve the system of equations by putting the matrix into row echelon or reduced row echelon form. @@ -514,7 +534,9 @@ $\begin{pmatrix} 1 & 2 & 3 \\ Rank = 2 -```{exercise, name = "Rank of Matrices", rank} +```{exercise} +#| name: Rank of Matrices +#| label: rank Find the rank of each matrix below: @@ -589,7 +611,8 @@ To summarize: To calculate the inverse of ${\bf A}$ b. If ${\bf C}\ne{\bf I}_n$, then $\bf C$ has a row of zeros. This means ${\bf A}$ is singular and ${\bf A}^{-1}$ does not exist. -```{example, inverse} +```{example} +#| label: inverse Find the inverse of the following matricies: @@ -603,7 +626,9 @@ Find the inverse of the following matricies: -```{exercise, name = "Finding the inverse of matrices", inverse1} +```{exercise} +#| name: Finding the inverse of matrices +#| label: inverse1 Find the inverse of the following matrix: @@ -636,7 +661,9 @@ Hence, given $\bf{A}$ and $\bf{b}$ and given that $\bf{A}$ is nonsingular, then -```{exercise, invlinsys, name = "Solve linear system using inverses"} +```{exercise} +#| label: invlinsys +#| name: Solve linear system using inverses Use the inverse matrix to solve the following linear system: @@ -712,7 +739,9 @@ For example, in figuring out whether the following matrix has an inverse? -```{exercise, name = "Determinants and Inverses", determinants} +```{exercise} +#| name: Determinants and Inverses +#| label: determinants Determine whether the following matrices are nonsingular: @@ -787,7 +816,9 @@ Recall, \end{pmatrix}\] -```{exercise, name = "Calculate Inverse using Determinant Formula", calcinverse} +```{exercise} +#| name: Calculate Inverse using Determinant Formula +#| label: calcinverse Caculate the inverse of A @@ -804,7 +835,7 @@ $$A = \begin{pmatrix} -## Answers to Examples and Exercises {-} +## Answers to Examples and Exercises {.unnumbered} diff --git a/11_data-handling_counting.qmd b/11_data-handling_counting.qmd index c86a72c..05cc610 100644 --- a/11_data-handling_counting.qmd +++ b/11_data-handling_counting.qmd @@ -1,4 +1,4 @@ -# (PART) Programming {-} +# (PART) Programming {.unnumbered} # Orientation and Reading in Data^[Module originally written by Shiro Kuriwaki] {#dataimport} @@ -12,14 +12,14 @@ Welcome to the first in-class session for programming. Up till this point, you s * Successfully signed up for the University wi-fi: (Access Harvard Secure with your HarvardKey. Try to get a HarvardKey as soon as possible.) -## Motivation: Data and You {-} +## Motivation: Data and You {.unnumbered} The modal social science project starts by importing existing datasets. Datasets come in all shapes and sizes. As you search for new data you may encounter dozens of file extensions -- csv, xlsx, dta, sav, por, Rdata, Rds, txt, xml, json, shp ... the list continues. Although these files can often be cumbersome, its a good to be able to find a way to encounter any file that your research may call for. Reviewing data import will allow us to get on the same page on how computer systems work. -### Where are we? Where are we headed? {-} +### Where are we? Where are we headed? {.unnumbered} Today we'll cover: @@ -28,7 +28,7 @@ Today we'll cover: * How to read in data * Comment on coding style on the way -### Check your understanding {-} +### Check your understanding {.unnumbered} * What is the difference between a file and a folder? * In the RStudio windows, what is the difference between the "Source" Pane and the "Console"? What is a "code chunk"? @@ -39,7 +39,9 @@ Today we'll cover: * How would you read in a `csv` file, a `dta` file, a `sav` file? -```{r, include = FALSE, message = FALSE} +```{r} +#| include: false +#| message: false library(ggplot2) library(dplyr) library(fs) @@ -160,7 +162,7 @@ Although you do not need to choose one over the other, for beginners it is confu The following side-by-side comparison of commands for a particular function compares some tidyverse and non-tidyverse functions (which we refer to loosely as base-R). This list is not meant to be comprehensive and more to give you a quick rule of thumb. -### Dataframe subsetting {-} +### Dataframe subsetting {.unnumbered} | In order to ... | in tidyverse: | in base-R: | |----------------------|:------------------------|:----------------------| @@ -177,7 +179,7 @@ The following side-by-side comparison of commands for a particular function comp Remember that tidyverse applies to _dataframes_ only, not vectors. For subsetting vectors, use the base-R functions with the square brackets. -### Read data {-} +### Read data {.unnumbered} Some non-tidyverse functions are not quite "base-R" but have similar relationships to tidyverse. For these, we recommend using the _tidyverse_ functions as a general rule due to their common format, simplicity, and scalability. @@ -191,7 +193,7 @@ Some non-tidyverse functions are not quite "base-R" but have similar relationshi | Merge `data1` and `data2` on variables `x1` and `x2` | `left_join(data1, data2, by = c("x1", "x2"))` | `merge(data1, data2, by.x = "x1", by.y = "x2", all.x = TRUE)` | -### Visualization {-} +### Visualization {.unnumbered} Plotting by ggplot2 (from your tutorials) is also a tidyverse family. @@ -214,7 +216,8 @@ This POLIS dataset was generously provided by Professor Josiah Ober of Stanford ### Locating the Data What files do we have in the `data/input` folder? -```{r, echo = FALSE} +```{r} +#| echo: false dir_ls("data/input") ``` @@ -228,19 +231,22 @@ A typical file format is Microsoft Excel. Although this is not usually the best In Rstudio, a good way to start is to use the GUI and the Import tool. Once you click a file, an option to "Import Dataset" comes up. RStudio picks the right function for you, and you can copy that code, but it's important to eventually be able to write that code yourself. For the first time using an outside package, you first need to install it. -```{r, eval = FALSE} +```{r} +#| eval: false install.packages("readxl") ``` After that, you don't need to install it again. But you __do__ need to load it each time. -```{r, eval = FALSE} +```{r} +#| eval: false library(readxl) ``` The package `readxl` has a website: https://readxl.tidyverse.org/. Other packages are not as user-friendly, but they have a help page with a table of contents of all their functions. -```{r, eval = FALSE} +```{r} +#| eval: false help(package = readxl) ``` @@ -300,12 +306,18 @@ These `tidyverse` commands from the `dplyr` package are newer and not built-in, Although this is a bit beyond our current stage, it's hard to resist the temptation to see what you can do with data like this. For example, you can map it.^[In mid-2018, changes in Google's services made it no longer possible to render maps on the fly. Therefore, the map is not currently rendered automatically (but can be rendered once the user registers their API). Instead, you now need to register with Google. See the [change](https://github.com/dkahle/ggmap/blob/e55c0b22b0d16a010b4b45dd2fce800ff0ef19b8/NEWS#L6-L12) to the pacakge ggmap.] Using the `ggmap` package -```{r message=FALSE, warning=FALSE, eval = FALSE} +```{r} +#| message: false +#| warning: false +#| eval: false library(ggmap) ``` First get a map of the Greek world. -```{r, message=FALSE, warning=FALSE, eval = FALSE} +```{r} +#| message: false +#| warning: false +#| eval: false greece <- get_map(location = c(lon = 22.6382849, lat = 39.543287), zoom = 5, source = "stamen", @@ -319,7 +331,9 @@ I chose the specifications for arguments `zoom` and `maptype` by looking at the Ober's data has the latitude and longitude of each polis. Because the map of Greece has the same coordinates, we can add the polei on the same map. -```{r, warning=FALSE, eval = FALSE} +```{r} +#| warning: false +#| eval: false gg_ober <- ggmap(greece) + geom_point(data = ober, aes(y = Latitude, x = Longitude), @@ -335,9 +349,9 @@ gg_ober + -## Exercises {-} +## Exercises {.unnumbered} -### 1 {-} +### 1 {.unnumbered} What is the Fame value of Delphoi? @@ -346,7 +360,7 @@ What is the Fame value of Delphoi? ``` -### 2 {-} +### 2 {.unnumbered} Find the polis with the top 10 Fame values. @@ -356,7 +370,7 @@ Find the polis with the top 10 Fame values. -### 3 {-} +### 3 {.unnumbered} Make a scatterplot with the number of colonies on the x-axis and Fame on the y-axis. @@ -365,7 +379,7 @@ Make a scatterplot with the number of colonies on the x-axis and Fame on the y-a ``` -### 4 {-} +### 4 {.unnumbered} Find the correct function to read the following datasets (available in your rstudio.cloud session) into your R window. @@ -383,7 +397,7 @@ Our Recommendations: Look at the packages `haven` and `readr` ``` -### 5 {-} +### 5 {.unnumbered} Read Ober's codebook and find a variable that you think is interesting. Check the distribution of that variable in your data, get a couple of statistics, and summarize it in English. ```{r} @@ -391,7 +405,7 @@ Read Ober's codebook and find a variable that you think is interesting. Check th ``` -### 6 {-} +### 6 {.unnumbered} This is day 1 and we covered a lot of material. Some of you might have found this completely new; others not so. Please click through this survey before you leave so we can adjust accordingly on the next few days. diff --git a/12_matricies-manipulation.qmd b/12_matricies-manipulation.qmd index 720c989..d797bd7 100644 --- a/12_matricies-manipulation.qmd +++ b/12_matricies-manipulation.qmd @@ -1,20 +1,23 @@ # Manipulating Vectors and Matrices^[Module originally written by Shiro Kuriwaki and Yon Soo Park] {#rmatrices} -```{r, include=FALSE, message = FALSE, warning = FALSE} +```{r} +#| include: false +#| message: false +#| warning: false library(dplyr) library(readr) library(haven) library(ggplot2) ``` -### Motivation {-} +### Motivation {.unnumbered} [Nunn and Wantchekon (2011)](https://dash.harvard.edu/bitstream/handle/1/11986331/nunn-slave-trade.pdf) -- "The Slave Trade and the Origins of Mistrust in Africa"^[[Nunn, Nathan, and Leonard Wantchekon. 2011. “The Slave Trade and the Origins of Mistrust in Africa.” American Economic Review 101(7): 3221–52.](https://dash.harvard.edu/bitstream/handle/1/11986331/nunn-slave-trade.pdf)] -- argues that across African countries, the distrust of co-ethnics fueled by the slave trade has had long-lasting effects on modern day trust in these territories. They argued that the slave trade created distrust in these societies in part because as some African groups were employed by European traders to capture their neighbors and bring them to the slave ships. Nunn and Wantchekon use a variety of statistical tools to make their case (adding controls, ordered logit, instrumental variables, falsification tests, causal mechanisms), many of which will be covered in future courses. In this module we will only touch on their first set of analysis that use Ordinary Least Squares (OLS). OLS is likely the most common application of linear algebra in the social sciences. We will cover some linear algebra, matrix manipulation, and vector manipulation from this data. -### Where are we? Where are we headed? {-} +### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -50,7 +53,9 @@ colnames(nunn_full) First, let's consider a small subset of this dataset. -```{r, include = FALSE, eval = FALSE} +```{r} +#| include: false +#| eval: false set.seed(02138) sample <- sample_n(nunn_full, 10) sample <- select(sample, trust_neighbors, exports, ln_exports, export_area, ln_export_area) @@ -93,7 +98,8 @@ You can think of data frames maybe as matrices-plus, because a column can take o Another way to think about data frames is that it is a type of list. Try the `str()` code below and notice how it is organized in slots. Each slot is a vector. They can be vectors of numbers or characters. -```{r, eval = FALSE} +```{r} +#| eval: false # enter this on your console str(cen10) ``` @@ -239,7 +245,9 @@ X2 %*% B What is this multiplication doing in terms of equations? -```{r, echo = FALSE, eval = FALSE} +```{r} +#| echo: false +#| eval: false ## FYI regression in Table 1 (without cluster SEs) form <- "trust_neighbors ~ exports + age + age2 + male + urban_dum + factor(education) + factor(occupation) + factor(religion) + factor(living_conditions) + district_ethnic_frac + frac_ethnicity_in_district + isocode" @@ -253,13 +261,15 @@ summary(lm_1_1) Let's take a look at Matrices in the context of R -```{r, message = FALSE} +```{r} +#| message: false cen10 <- read_csv("data/input/usc2010_001percent.csv") head(cen10) ``` What is the dimension of this dataframe? What does the number of rows represent? What does the number of columns represent? -```{r, message = FALSE} +```{r} +#| message: false dim(cen10) nrow(cen10) ncol(cen10) @@ -268,13 +278,15 @@ ncol(cen10) What variables does this dataset hold? What kind of information does it have? -```{r, message = FALSE} +```{r} +#| message: false colnames(cen10) ``` We can access column vectors, or vectors that contain values of variables by using the $ sign -```{r, message = FALSE} +```{r} +#| message: false head(cen10$state) head(cen10$race) @@ -283,13 +295,15 @@ head(cen10$race) We can look at a unique set of variable values by calling the unique function -```{r, message = FALSE} +```{r} +#| message: false unique(cen10$state) ``` How many different states are represented (this dataset includes DC as a state)? -```{r, message = FALSE} +```{r} +#| message: false length(unique(cen10$state)) ``` @@ -311,7 +325,8 @@ cross_tab[6, 2] But a subset of your data -- individual values-- can be considered a matrix too. -```{r, warning = FALSE} +```{r} +#| warning: false # First 20 rows of the entire data # Below two lines of code do the same thing cen10[1:20, ] @@ -380,23 +395,23 @@ s8 <- cen10 %>% filter(!state %in% c("California", "Ohio", "Nevada", "Michigan") all_equal(s7, s8) ``` -## Checkpoint {-} +## Checkpoint {.unnumbered} -### 1 {-} +### 1 {.unnumbered} Get the subset of cen10 for non-white individuals (Hint: look at the set of values for the race variable by using the unique function) ```{r} # Enter here ``` -### 2 {-} +### 2 {.unnumbered} Get the subset of cen10 for females over the age of 40 ```{r} # Enter here ``` -### 3 {-} +### 3 {.unnumbered} Get all the serial numbers for black, male individuals who don't live in Ohio or Nevada. ```{r} # Enter here @@ -404,9 +419,9 @@ Get all the serial numbers for black, male individuals who don't live in Ohio or -## Exercises {-} +## Exercises {.unnumbered} -### 1 {-} +### 1 {.unnumbered} Let \[\mathbf{A} = \left[\begin{array} @@ -426,7 +441,7 @@ Use R to write code that will create the matrix $A$, and then consecutively mul Note that R notation of matrices is different from the math notation. Simply trying `X^n` where `X` is a matrix will only take the power of each element to `n`. Instead, this problem asks you to perform matrix multiplication. -### 2 {-} +### 2 {.unnumbered} Let's apply what we learned about subsetting or filtering/selecting. Use the `nunn_full` dataset you have already loaded a) First, show all observations (rows) that have a `"male"` variable higher than 0.5 @@ -447,18 +462,18 @@ c) Lastly, show all values of `"trust_neighbors"` and `"age"` for observations ( ## Enter yourself ``` -### 3 {-} +### 3 {.unnumbered} Find a way to generate a vector of "column averages" of the matrix `X` from the Nunn and Wantchekon data in one line of code. Each entry in the vector should contain the sample average of the values in the column. So a 100 by 4 matrix should generate a length-4 matrix. -### 4 {-} +### 4 {.unnumbered} Similarly, generate a vector of "column medians". -### 5 {-} +### 5 {.unnumbered} Consider the regression that was run to generate Table 1: ```{r} diff --git a/13_functions_obj_loops.qmd b/13_functions_obj_loops.qmd index feac1f0..62c53f4 100644 --- a/13_functions_obj_loops.qmd +++ b/13_functions_obj_loops.qmd @@ -1,7 +1,7 @@ # Objects, Functions, Loops {#robjloops} -### Where are we? Where are we headed? {-} +### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -25,7 +25,9 @@ Now that we have covered some hands-on ways to use graphics, let's go into some Let's first set up -```{r, message=FALSE, warning=FALSE} +```{r} +#| message: false +#| warning: false library(dplyr) library(readr) library(haven) @@ -160,7 +162,8 @@ class(ols) Anything can be an object! Even graphs (in `ggplot`) can be assigned, re-assigned, and edited. -```{r, warning=FALSE} +```{r} +#| warning: false grp_race <- group_by(cen10, race)%>% summarize(count = n()) @@ -201,7 +204,8 @@ str(pikachu) What does a `ggplot` object look like? Very complicated, but at least you can see it: -```{r, eval = FALSE} +```{r} +#| eval: false # enter this on your console str(gg_tab) ``` @@ -276,7 +280,8 @@ Most of what we do in R is executing a function. `read_csv()`, `nrow()`, `ggplot A function is a set of instructions with specified ingredients. It takes an __input__, then __manipulates__ it -- changes it in some way -- and then returns the manipulated product. One way to see what a function actually does is to enter it without parentheses. -```{r, eval = FALSE} +```{r} +#| eval: false # enter this on your console table ``` @@ -287,7 +292,8 @@ You'll notice that functions contain other functions. _wrapper_ functions are fu ### Write your own function It's worth remembering the basic structure of a function. You create a new function, call it `my_fun` by this: -```{r, eval = F} +```{r} +#| eval: !expr F my_fun <- function() { } @@ -339,10 +345,10 @@ nw_in_state(cen10, "Massachusetts") ``` -## Checkpoint {-} +## Checkpoint {.unnumbered} -### 1 {-} +### 1 {.unnumbered} Try making your own function, `average_age_in_state`, that will give you the average age of people in a given state. ```{r} @@ -353,14 +359,14 @@ Try making your own function, `average_age_in_state`, that will give you the ave -### 2 {-} +### 2 {.unnumbered} Try making your own function, `asians_in_state`, that will give you the number of `Chinese`, `Japanese`, and `Other Asian or Pacific Islander` people in a given state. ```{r} # Enter on your own ``` -### 3 {-} +### 3 {.unnumbered} Try making your own function, 'top_10_oldest_cities', that will give you the names of cities whose population's average age is top 10 oldest. @@ -380,7 +386,8 @@ help(package = "ggplot2") To use a package, you need to do two things: (1) install it, and then (2) load it. Installing is a one-time thing -```{r, eval = FALSE} +```{r} +#| eval: false install.packages("ggplot2") ``` @@ -413,7 +420,8 @@ if (x >0) { You can wrap that whole things in a function -```{r, warning = FALSE} +```{r} +#| warning: false is_positive <- function(number) { if (number >0) { print("positive number") @@ -506,10 +514,10 @@ for (state in states_of_interest) { } } ``` -## Exercises {-} +## Exercises {.unnumbered} -### Exercise 1: Write your own function {-} +### Exercise 1: Write your own function {.unnumbered} Write your own function that makes some task of data analysis simpler. Ideally, it would be a function that helps you do either of the previous tasks in fewer lines of code. You can use the three lines of code that was provided in exercise 1 to wrap that into another function too! ```{r} # Enter yourself @@ -518,7 +526,7 @@ Write your own function that makes some task of data analysis simpler. Ideally, -### Exercise 2: Using Loops {-} +### Exercise 2: Using Loops {.unnumbered} Using a loop, create a crosstab of sex and race for each state in the set "states_of_interest" @@ -530,7 +538,7 @@ states_of_interest <- c("California", "Massachusetts", "New Hampshire", "Washing -### Exercise 3: Storing information derived within loops in a global dataframe {-} +### Exercise 3: Storing information derived within loops in a global dataframe {.unnumbered} Recall the following nested loop ```{r} diff --git a/14_visualization.qmd b/14_visualization.qmd index ca73b4b..e489c99 100644 --- a/14_visualization.qmd +++ b/14_visualization.qmd @@ -1,7 +1,10 @@ # Visualization^[Module originally written by Shiro Kuriwaki] {#dataviz} -```{r, include=FALSE, message=FALSE, warning=FALSE} +```{r} +#| include: false +#| message: false +#| warning: false library(dplyr) library(readr) library(ggplot2) @@ -10,7 +13,7 @@ library(scales) ``` -### Motivation: The Law of the Census {-} +### Motivation: The Law of the Census {.unnumbered} In this module, let's visualize some cross-sectional stats with an actual Census. Then, we'll do an example on time trends with Supreme Court ideal points. @@ -22,7 +25,7 @@ Time series data is a common form of data in social science data, and there is g -### Where are we? Where are we headed? {-} +### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -39,7 +42,7 @@ Today we'll cover: * Visualization * A bit of data wrangling -### Check your understanding {-} +### Check your understanding {.unnumbered} * How do you make a barplot, in base-R and in ggplot? * How do you add layers to a ggplot? @@ -57,7 +60,8 @@ Today we'll cover: First, the census. Read in a subset of the 2010 Census that we looked at earlier. This time, it is in Rds form. -```{r, message=FALSE} +```{r} +#| message: false cen10 <- readRDS("data/input/usc2010_001percent.Rds") ``` @@ -145,7 +149,8 @@ Although the tutorial covered making scatter plots as the first cut, often data For this example, let's group and count first like we just did. But assign it to a new object. -```{r, fig.fullwidth = TRUE} +```{r} +#| fig.fullwidth: true grp_race <- count(cen10, race) ``` @@ -159,7 +164,8 @@ We will now plot this grouped set of numbers. Recall that the `ggplot()` functio What is the right geometry layer to make a barplot? Turns out: -```{r, fig.fullwidth = TRUE} +```{r} +#| fig.fullwidth: true ggplot(data = grp_race, aes(x = race, y = n)) + geom_col() ``` @@ -169,7 +175,8 @@ ggplot(data = grp_race, aes(x = race, y = n)) + geom_col() Adjusting your graphics to make the point clear is an important skill. Here is a base-R example of showing the same numbers but with a different design, in a way that aims to maximize the "data-to-ink ratio". -```{r, fig.fullwidth = TRUE} +```{r} +#| fig.fullwidth: true par(oma = c(1, 11, 1, 1)) barplot(sort(table(cen10$race)), # sort numbers horiz = TRUE, # flip @@ -213,7 +220,8 @@ xtab_race_state Another function to make a cross-tab is the `xtabs` command, which uses formula notation. -```{r, eval = FALSE} +```{r} +#| eval: false xtabs(~ state + race, cen10) ``` @@ -240,7 +248,8 @@ grp_race_state <- cen10 %>% Can you tell from the code what `grp_race_state` will look like? -```{r, eval = FALSE} +```{r} +#| eval: false # run on your own grp_race_state ``` @@ -248,7 +257,8 @@ grp_race_state Now, we want to tell `ggplot2` something like the following: I want bars by state, where heights indicate racial groups. Each bar should be colored by the race. With some googling, you will get something like this: -```{r, fig.height = 8} +```{r} +#| fig-height: 8 ggplot(data = grp_race_state, aes(x = state, y = n, fill = race)) + geom_col(position = "fill") + # the position is determined by the fill ae scale_fill_brewer(name = "Census Race", palette = "OrRd", direction = -1) + # choose palette @@ -270,7 +280,8 @@ The Census does not track individuals over time. So let's take up another exampl This data is adapted from the estimates of Martin and Quinn on their website .^[This exercise inspired from Princeton's R Camp Assignment.] -```{r, message=FALSE} +```{r} +#| message: false justice <- read_csv("data/input/justices_court-median.csv") ``` @@ -306,10 +317,10 @@ As social scientists, we should also not forget to ask ourselves whether these n -## Exercises {-} +## Exercises {.unnumbered} In the time remaining, try the following exercises. Order doesn't matter. -### 1: Rural states {-} +### 1: Rural states {.unnumbered} Make a well-labelled figure that plots the proportion of the state's population (as per the census) that is 65 years or older. Each state should be visualized as a point, rather than a bar, and there should be 51 points, ordered by their value. All labels should be readable. @@ -320,7 +331,7 @@ Make a well-labelled figure that plots the proportion of the state's population ``` -### 2: The swing justice {-} +### 2: The swing justice {.unnumbered} Using the `justices_court-median.csv` dataset and building off of the plot that was given, make an improved plot by implementing as many of the following changes (which hopefully improves the graph): @@ -343,7 +354,7 @@ Using the `justices_court-median.csv` dataset and building off of the plot that ``` -### 3: Don't sort by the alphabet {-} +### 3: Don't sort by the alphabet {.unnumbered} The Figure we made that shows racial composition by state has one notable shortcoming: it orders the states alphabetically, which is not particularly useful if you want see an overall pattern, without having particular states in mind. @@ -356,6 +367,6 @@ Find a way to modify the figures so that the states are ordered by the _proporti ``` -### 4 What to show and how to show it {-} +### 4 What to show and how to show it {.unnumbered} As a student of politics our goal is not necessarily to make pretty pictures, but rather make pictures that tell us something about politics, government, or society. If you could augment either the census dataset or the justices dataset in some way, what would be an substantively significant thing to show as a graphic? diff --git a/15_project-dempeace.qmd b/15_project-dempeace.qmd index 8a4967b..1b37249 100644 --- a/15_project-dempeace.qmd +++ b/15_project-dempeace.qmd @@ -1,13 +1,13 @@ # Joins and Merges, Wide and Long^[Module originally written by Shiro Kuriwaki, Connor Jerzak, and Yon Soo Park] {#dempeace} -### Motivation {-} +### Motivation {.unnumbered} The "Democratic Peace" is one of the most widely discussed propositions in political science, covering the fields of International Relations and Comparative Politics, with insights to domestic politics of democracies (e.g. American Politics). The one-sentence idea is that democracies do not fight with each other. There have been much theoretical debate -- for example in earlier work, [Oneal and Russet (1999)](https://blackboard.angelo.edu/bbcswebdav/institution/LFA/CSS/Course%20Material/SEC6302/Readings/Lesson_3/Oneal-Russett.pdf) argue that the democratic peace is not due to the hegemony of strong democracies like the U.S. and attempt to distinguish between realist and what they call Kantian propositions (e.g. democratic governance, international organizations)^[[The Kantian Peace: The Pacific Benefits of Democracy, Interdependence, and International Organizations, 1885-1992. _World Politics_ 52(1):1-37](https://blackboard.angelo.edu/bbcswebdav/institution/LFA/CSS/Course%20Material/SEC6302/Readings/Lesson_3/Oneal-Russett.pdf)]. An empirical demonstration of the democratic peace is also a good example of a __Time Series Cross Sectional__ (or panel) dataset, where the same units (in this case countries) are observed repeatedly for multiple time periods. Experience in assembling and analyzing a TSCS dataset will prepare you for any future research in this area. -## Where are we? Where are we headed? {-} +## Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -26,7 +26,9 @@ Today you will work on your own, but feel free to ask a fellow classmate nearby ## Setting up -```{r, message=FALSE, warning=FALSE} +```{r} +#| message: false +#| warning: false library(dplyr) library(tidyr) library(readr) @@ -56,7 +58,9 @@ Most projects you do will start with downloading data from elsewhere. For this t ## Example with 2 Datasets Let's read in a sample dataset. -```{r, warning = FALSE, message = FALSE} +```{r} +#| warning: false +#| message: false polity <- read_csv("data/input/sample_polity.csv") mid <- read_csv("data/input/sample_mid.csv") ``` @@ -139,7 +143,7 @@ select(p_m_wide, 1:10) Try building a panel that would be useful in answering the Democratic Peace Question, perhaps in these steps. -### Task 1: Data Input and Standardization {-} +### Task 1: Data Input and Standardization {.unnumbered} Often, files we need are saved in the `.xls` or `xlsx` format. It is possible to read these files directly into `R`, but experience suggests that this process is slower than converting them first to `.csv` format and reading them in as `.csv` files. @@ -152,7 +156,7 @@ Often, files we need are saved in the `.xls` or `xlsx` format. It is possible to ``` -### Task 2: Data Merging {-} +### Task 2: Data Merging {.unnumbered} We will use data to test a version of the Democratic Peace Thesis (DPS). Democracies are said to go to war less because the leaders who wage wars are accountable to voters who have to bear the costs of war. Are democracies less likely to engage in militarized interstate disputes? To start, let's download and merge some data. @@ -167,7 +171,7 @@ To start, let's download and merge some data. ``` -### Task 3: Tabulations and Visualization {-} +### Task 3: Tabulations and Visualization {.unnumbered} 1. Calculate the mean Polity2 score by year. Plot the result. Use graphical indicators of your choosing to show where key events fall in this timeline (such as 1914, 1929, 1939, 1989, 2008). Speculate on why the behavior from 1800 to 1920 seems to be qualitatively different than behavior afterwards. 2. Do the same but only among state-years that were invovled in a MID. Plot this line together with your results from 1. diff --git a/16_simulation.qmd b/16_simulation.qmd index 51b20a1..7926e21 100644 --- a/16_simulation.qmd +++ b/16_simulation.qmd @@ -1,11 +1,12 @@ # Simulation^[Module originally written by Connor Jerzak and Shiro Kuriwaki] {#simulation} -```{r, include=FALSE} +```{r} +#| include: false library(dplyr) ``` -### Motivation: Simulation as an Analytical Tool {-} +### Motivation: Simulation as an Analytical Tool {.unnumbered} An increasing amount of political science contributions now include a simulation. @@ -22,7 +23,7 @@ Statistical methods also incorporate simulation: -### Where are we? Where are we headed? {-} +### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -36,7 +37,7 @@ Up till now, you should have covered: In this module, we will start to work with generating data within R, from thin air, as it were. Doing simulation also strengthens your understanding of Probability (Section \@ref{probability}). -### Check your Understanding {-} +### Check your Understanding {.unnumbered} * What does the `sample()` function do? * What does `runif()` stand for? @@ -79,7 +80,8 @@ sample(x = 1:10, size = 10, replace = TRUE) ## any number can appear more than o ``` It follows then that you cannot sample without replacement a sample that is larger than the pool. -```{r, error = TRUE} +```{r} +#| error: true sample(x = 1:10, size = 100, replace = FALSE) ``` @@ -126,7 +128,7 @@ As a side-note, these functions have very practical uses for any type of data an ## Random numbers from specific distributions -### `rbinom()` {-} +### `rbinom()` {.unnumbered} `rbinom` builds upon `sample` as a tool to help you answer the question -- what is the _total number of successes_ I would get if I sampled a binary (Bernoulli) result from a test with `size` number of trials each, with a event-wise probability of `prob`. The first argument `n` asks me how many such numbers I want. For example, I want to know how many Heads I would get if I flipped a fair coin 100 times. @@ -139,7 +141,7 @@ Now imagine this I wanted to do this experiment 10 times, which would require I rbinom(n = 10, size = 100, prob = 0.5) ``` -### `runif()` {-} +### `runif()` {.unnumbered} `runif` also simulates a stochastic scheme where each event has equal probability of getting chosen like `sample`, but is a continuous rather than discrete system. We will cover this more in the next math module. The intuition to emphasize here is that one can generate potentially infinite amounts (size `n`) of noise that is a essentially random @@ -150,7 +152,7 @@ runif(n = 5) ``` -### `rnorm()` {-} +### `rnorm()` {.unnumbered} `rnorm` is also a continuous distribution, but draws from a Normal distribution -- perhaps the most important distribution in statistics. It runs the same way as `runif` @@ -198,9 +200,9 @@ runif(n = 10) -## Exercises {-} +## Exercises {.unnumbered} -### Census Sampling {-} +### Census Sampling {.unnumbered} What can we learn from surveys of populations, and how wrong do we get if our sampling is biased?^[This example is inspired from [Meng, Xiao-Li (2018). Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election. _Annals of Applied Statistics_ 12:2, 685–726. doi:10.1214/18-AOAS1161SF.](https://statistics.fas.harvard.edu/files/statistics-2/files/statistical_paradises_and_paradoxes.pdf)] Suppose we want to estimate the proportion of U.S. residents who are non-white (`race != "White"`). In reality, we do not have any population dataset to utilize and so we _only see the sample survey_. Here, however, to understand how sampling works, let's conveniently use the Census extract in some cases and pretend we didn't in others. @@ -258,7 +260,7 @@ You can do this by creating a variable, e.g. `propensity`, that is 0.9 for non-W -### Conditional Proportions {-} +### Conditional Proportions {.unnumbered} This example is not on simulation, but is meant to reinforce some of the probability discussion from math lecture. @@ -297,7 +299,7 @@ In addition to some standard demographic questions, we will focus on one called -### The Birthday problem {-} +### The Birthday problem {.unnumbered} Write code that will answer the well-known birthday problem via simulation.^[This exercise draws from Imai (2017)] diff --git a/17_non-wysiwyg.qmd b/17_non-wysiwyg.qmd index 60934f6..ae9ac5a 100644 --- a/17_non-wysiwyg.qmd +++ b/17_non-wysiwyg.qmd @@ -1,7 +1,7 @@ # LaTeX and markdown^[Module originally written by Shiro Kuriwaki] {#nonwysiwyg} -### Where are we? Where are we headed? {-} +### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -25,7 +25,7 @@ command-line are a basic set of tools that you may have to use from time to time -### Check your understanding {-} +### Check your understanding {.unnumbered} Check if you have an idea of how you might code the following tasks: @@ -120,7 +120,7 @@ LaTeX is a typesetting program. You'd engage with LaTeX much like you engage wit 3. Let's discuss the default template. 4. Make a new document, and set it as your main document. Then type in the Minimal Working Example (MWE): -```{bash, eval = FALSE} +```tex \documentclass{article} \begin{document} Hello World @@ -134,7 +134,7 @@ LaTeX is a very stable system, and few changes to it have been made since the 19 1. Open a plain text editor. Then type in the MWE -```{bash, eval = FALSE} +```tex \documentclass{article} \begin{document} Hello World @@ -192,7 +192,7 @@ For some LaTeX commands you might need to load a separate package that someone e where `package` is the name of the package and `options` are options specific to the package. -### Further Guides {-} +### Further Guides {.unnumbered} For a more comprehensive listing of LaTeX commands, Mayya Komisarchik has a great tutorial set of folders: @@ -206,7 +206,7 @@ BibTeX is a reference system for bibliographical tests. We have a `.bib` file se ### what is a `.bib` file? For example, here is the Nunn and Wantchekon article entry in `.bib` form. -```{} +```tex @article{nunn2011slave, title={The Slave Trade and the Origins of Mistrust in Africa}, author={Nunn, Nathan and Wantchekon, Leonard}, @@ -246,7 +246,7 @@ For example, Gary's BibTeX file is here: -### Your Feedback Matters {-} +### Your Feedback Matters {.unnumbered} _Please tell us how we can improve the Prefresher_: The Prefresher is a work in progress, with material mainly driven by graduate students. Please tell us how we should change (or not change) each of its elements: diff --git a/18_text.qmd b/18_text.qmd index fa9558a..7e6a671 100644 --- a/18_text.qmd +++ b/18_text.qmd @@ -2,7 +2,8 @@ -```{r, include=FALSE} +```{r} +#| include: false library(dplyr) library(readr) library(scales) @@ -10,7 +11,7 @@ library(forcats) library(ggplot2) ``` -## Where are we? Where are we headed? {-} +## Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -257,14 +258,16 @@ DocVec <- c(doc1, doc2, doc3) Now we can use utility functions in the `tm` package: -```{R, eval = FALSE} +```{R} +#| eval: false library(tm) DocCorpus <- Corpus(VectorSource(DocVec) ) DTM1 <- inspect( DocumentTermMatrix(DocCorpus) ) ``` Consider the effect of different "pre-processing" choices on the resulting DTM! -```{r, eval = FALSE} +```{r} +#| eval: false DocVec <- tolower(DocVec) DocVec <- gsub(DocVec, pattern ="[[:punct:]]", replace = " ") DocVec <- gsub(DocVec, pattern ="[[:cntrl:]]", replace = " ") @@ -282,9 +285,9 @@ Stemming is the process of reducing inflected/derived words to their word stem o 3. stringr -- Useful for string parsing. -## Exercises {-} +## Exercises {.unnumbered} -### 1 {-} +### 1 {.unnumbered} Figure out why this command does what it does: @@ -294,7 +297,7 @@ sprintf("%s of spontaneous events are %s in the mind. "15.03322123", "puzzles", 15.03322123) ``` -### 2 {-} +### 2 {.unnumbered} Why does this command not work? ```{r} @@ -302,7 +305,7 @@ try(sprintf("%s of spontaneous events are %s in the mind. Really, %.2f?", "15.03322123", "puzzles", "15.03322123" ), TRUE) ``` -### 3 {-} +### 3 {.unnumbered} Using `grepl`, these materials, Google, and your friends, describe what the following command does. What changes when `value = FALSE`? @@ -313,7 +316,7 @@ grep('\'', -### 4 {-} +### 4 {.unnumbered} Write code to automatically extract the file names that DO end start with presidential and DO end in .pdf @@ -329,7 +332,7 @@ my_string <- c("legislative1_term1.png", "presidential1_term2.pdf") ``` -### 5 {-} +### 5 {.unnumbered} Using the same string as in the above, write code to automatically extract the file names that end in .pdf and that contain the text `term2`. ```{r} @@ -337,7 +340,7 @@ Using the same string as in the above, write code to automatically extract the ``` -### 6 {-} +### 6 {.unnumbered} Combine these two strings into a single string separated by a "-". Desired output: "The carbonyl group in aldehydes and ketones is an oxygen analog of the carbon–carbon double bond." @@ -348,7 +351,7 @@ string2 <- "–carbon double bond." ``` -### 7 {-} +### 7 {.unnumbered} Challenge problem! Download this webpage diff --git a/21_solutions-warmup.qmd b/21_solutions-warmup.qmd index a527ec1..452f9f2 100644 --- a/21_solutions-warmup.qmd +++ b/21_solutions-warmup.qmd @@ -1,12 +1,12 @@ -# (PART) Solutions {-} +# (PART) Solutions {.unnumbered} -# Solutions to Warmup Questions {-} +# Solutions to Warmup Questions {.unnumbered} -## Linear Algebra {-} +## Linear Algebra {.unnumbered} -### Vectors {-} +### Vectors {.unnumbered} Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pmatrix} 4\\5\\6 \end{pmatrix}$, and the scalar $c = 2$. @@ -47,7 +47,7 @@ i.e., a linear combination of these three vectors that would amount to zero exis If you are having trouble with these problems, please review Section \@ref(linearindependence). -### Matrices {-} +### Matrices {.unnumbered} \[{\bf A}=\begin{pmatrix} 7 & 5 & 1 \\ @@ -105,9 +105,9 @@ If you are having trouble with these problems, please review Section \@ref(matri -## Operations {-} +## Operations {.unnumbered} -### Summation {-} +### Summation {.unnumbered} Simplify the following @@ -118,7 +118,7 @@ Simplify the following 3. \(\sum\limits_{i= 1}^4 (3k + i + 2) = 3\sum\limits_{i= 1}^4k + \sum\limits_{i= 1}^4i + \sum\limits_{i= 1}^42 = 12k + 10 + 8 = 12k + 18\) -### Products {-} +### Products {.unnumbered} 1. \(\prod\limits_{i= 1}^3 i = 1\cdot 2\cdot 3 = 6\) @@ -128,7 +128,7 @@ Simplify the following To review this material, please see Section \@ref(sum-notation). -### Logs and exponents {-} +### Logs and exponents {.unnumbered} Simplify the following @@ -147,7 +147,7 @@ To review this material, please see Section \@ref(logexponents) -## Limits {-} +## Limits {.unnumbered} Find the limit of the following. @@ -158,7 +158,7 @@ Find the limit of the following. To review this material please see Section \@ref(limitsfun) -## Calculus {-} +## Calculus {.unnumbered} For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\frac{d}{dx}f(x)$ @@ -171,7 +171,7 @@ For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\fra For a review, please see Section \@ref(derivintro) - \@ref(derivpoly) -## Optimization {-} +## Optimization {.unnumbered} For each of the followng functions $f(x)$, does a maximum and minimum exist in the domain $x \in \mathbf{R}$? If so, for what are those values and for which values of $x$? @@ -182,7 +182,7 @@ For each of the followng functions $f(x)$, does a maximum and minimum exist in t If you are stuck, please try sketching out a picture of each of the functions. -## Probability {-} +## Probability {.unnumbered} 1. If there are 12 cards, numbered 1 to 12, and 4 cards are chosen, $\binom{12}{4} = \frac{12\cdot 11\cdot 10\cdot 9}{4!} = 495$ possible hands exist (unordered, without replacement) . 2. Let $A = \{1,3,5,7,8\}$ and $B = \{2,4,7,8,12,13\}$. Then $A \cup B = \{1, 2, 3, 4, 5, 7, 8, 12, 13\}$, $A \cap B = \{7, 8\}$? If $A$ is a subset of the Sample Space $S = \{1,2,3,4,5,6,7,8,9,10\}$, then the complement $A^C = \{2, 4, 6, 9, 10\}$ diff --git a/23_solution_programming.qmd b/23_solution_programming.qmd index aad7e66..a17fb69 100644 --- a/23_solution_programming.qmd +++ b/23_solution_programming.qmd @@ -1,6 +1,7 @@ -# Suggested Programming Solutions {-} +# Suggested Programming Solutions {.unnumbered} -```{r, message = FALSE} +```{r} +#| message: false library(dplyr) library(readr) library(ggplot2) @@ -14,7 +15,7 @@ library(scales) ## Chapter \@ref(dataviz): Visualization -### 1 State Proportions {-} +### 1 State Proportions {.unnumbered} ```{r} cen10 <- readRDS("data/input/usc2010_001percent.Rds") @@ -49,10 +50,11 @@ ggplot(grp_st, aes(x = state, y = prop)) + -### 2 Swing Justice {-} +### 2 Swing Justice {.unnumbered} -```{r, message = FALSE} +```{r} +#| message: false justices <- read_csv("data/input/justices_court-median.csv") ``` @@ -93,13 +95,14 @@ ggplot(df_indicator, aes(x = term, y = idealpt, group = justice_id)) + ## Chapter \@ref(robjloops): Objects and Loops -```{r, message=FALSE} +```{r} +#| message: false cen10 <- read_csv("data/input/usc2010_001percent.csv") sample_acs <- read_csv("data/input/acs2015_1percent.csv") ``` -### Checkpoint #3 {-} +### Checkpoint #3 {.unnumbered} ```{r} @@ -112,7 +115,7 @@ cen10 %>% -### Exercise 2 {-} +### Exercise 2 {.unnumbered} ```{r} @@ -129,7 +132,7 @@ for (state_i in states_of_interest) { -### Exercise 3 {-} +### Exercise 3 {.unnumbered} ```{r} @@ -156,9 +159,10 @@ for (state in states_of_interest) { ## Chapter \@ref(dempeace): Demoratic Peace Project -### Task 1: Data Input and Standardization {-} +### Task 1: Data Input and Standardization {.unnumbered} -```{r, eval = FALSE} +```{r} +#| eval: false mid_b <- read_csv("data/input/MIDB_4.2.csv") polity <- read_excel("data/input/p4v2017.xls") @@ -166,9 +170,10 @@ polity <- read_excel("data/input/p4v2017.xls") -### Task 2: Data Merging {-} +### Task 2: Data Merging {.unnumbered} -```{r, eval = FALSE} +```{r} +#| eval: false mid_y_by_y <- data_frame(ccode = numeric(), year = numeric(), dispute = numeric()) @@ -186,10 +191,11 @@ merged_mid_polity <- left_join(polity, ``` -### Task 3: Tabulations and Visualization {-} +### Task 3: Tabulations and Visualization {.unnumbered} -```{r, eval = FALSE} +```{r} +#| eval: false #don't include the -88, -77, -66 values in calculating the mean of polity mean_polity_by_year <- merged_mid_polity %>% group_by(year) %>% summarise(mean_polity = mean(polity[which(polity <11 & polity > -11)])) @@ -236,7 +242,8 @@ mean(samp$race != "White") ``` -```{r, eval = FALSE} +```{r} +#| eval: false ests <- c() set.seed(1669482) @@ -255,7 +262,8 @@ mean(ests) pop_with_prop <- mutate(pop, propensity = ifelse(race != "White", 0.9, 1)) ``` -```{r, eval = FALSE} +```{r} +#| eval: false ests <- c() set.seed(1669482) @@ -268,7 +276,8 @@ mean(ests) ``` -```{r, eval = FALSE} +```{r} +#| eval: false ests <- c() set.seed(1669482) diff --git a/R_exercises/04_visualization.R b/R_exercises/04_visualization.R index 17ed790..86c7b53 100644 --- a/R_exercises/04_visualization.R +++ b/R_exercises/04_visualization.R @@ -8,7 +8,7 @@ library(scales) -### Where are we? Where are we headed? {-} +### Where are we? Where are we headed? {.unnumbered} # Up till now, you should have covered: # @@ -342,4 +342,4 @@ ggplot(justice, aes(x = term, y = idealpt)) + - \ No newline at end of file + diff --git a/R_exercises/05_project-dempeace.R b/R_exercises/05_project-dempeace.R index 369a535..fdec6db 100644 --- a/R_exercises/05_project-dempeace.R +++ b/R_exercises/05_project-dempeace.R @@ -182,7 +182,7 @@ p_m_wide <- pivot_wider(p_m, # Merge data here: -### Task 3: Tabulations and Visualization {-} +### Task 3: Tabulations and Visualization {.unnumbered} # 1.Calculate the mean Polity2 score by year. Plot the result. Use graphical indicators of # your choosing to show where key events fall in this timeline (such as 1914, 1929, 1939, 1989, 2008). diff --git a/_quarto.yml b/_quarto.yml index 8887b98..2e5ef87 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -1,7 +1,17 @@ -book_filename: "prefresher" -repo: https://github.com/IQSS/prefresher/ +project: + type: book + +book: + title: "Math Prefresher for Political Scientists" + site-url: https://iqss.github.io/prefresher/ + chapters: + - index.qmd + - 01_warmup.qmd + - 02_functions.qmd + output_dir: _book delete_merged_file: true + language: ui: chapter_name: "Chapter " diff --git a/index.qmd b/index.qmd index 6f9af30..90f1dc3 100644 --- a/index.qmd +++ b/index.qmd @@ -1,7 +1,7 @@ --- title: "Math Prefresher for Political Scientists" author: "" -date: "`r format(Sys.Date(), '%B %Y')`" +date: today description: "Text for Harvard Department of Government Math Prefresher" github-repo: "IQSS/prefresher" site: "bookdown::bookdown_site" @@ -11,7 +11,7 @@ biblio-style: apalike link-citations: true cover-image: "./images/logo.png" --- -# About this Booklet {-} +# About this Booklet {.unnumbered} The [__Harvard Gov Prefresher__](https://projects.iq.harvard.edu/prefresher) is held each year in August. All relevant information is on our website, including the day-to-day schedule. The 2023 Prefresher instructors are [Sooahn Shin](http://sooahnshin.com/) and [María Ballesteros](https://scholar.harvard.edu/mariaballesteros/home), and the faculty sponsor is [Gary King](https://gking.harvard.edu). @@ -20,14 +20,14 @@ This booklet serves as the text for the Prefresher, available as a [webpage](htt For information about the role of the prefresher (or "math camp") as a introduction to graduate school, you may also be interested in ["The Math Prefresher and The Collective Future of Political Science Graduate Training"](https://gking.harvard.edu/prefresher), in _PS: Political Science & Politics_, by Gary King, Shiro Kuriwaki, and Yon Soo Park. -### Authors and Contributors {-} +### Authors and Contributors {.unnumbered} - Past Authors and Instructors: Curt Signorino 1996-1997; Ken Scheve 1997-1998; Eric Dickson 1998-2000; Orit Kedar 1999; James Fowler 2000-2001; Kosuke Imai 2001-2002; Jacob Kline 2002; Dan Epstein 2003; Ben Ansell 2003-2004; Ryan Moore 2004-2005; Mike Kellermann 2005-2006; Ellie Powell 2006-2007; Jen Katkin 2007-2008; Patrick Lam 2008-2009; Viridiana Rios 2009-2010; Jennifer Pan 2010-2011; Konstantin Kashin 2011-2012; Soledad Prillaman 2013; Stephen Pettigrew 2013-2014; Anton Strezhnev 2014-2015; Mayya Komisarchik 2015-2016; Connor Jerzak 2016-2017; Shiro Kuriwaki 2017-2018; Yon Soo Park 2018; Meg Schwenzfeier 2019; Shannon Parker 2019; Laura Royden 2020-2021; Hunter Rendleman 2020-2021; Christopher T. Kenny 2022; Jialu Li 2022; Sooahn Shin 2023; María Ballesteros 2023. - Repository Maintainer: Christopher T. Kenny ([christopherkenny](https://github.com/christopherkenny)) - Contributors: Thanks to Juan Dodyk ([juandodyk](https://github.com/juandodyk)), Hunter Rendleman ([hrendleman](https://github.com/hrendleman)), and Tyler Simko ([tylersimko](https://github.com/tylersimko)) for contributing to the booklet for corrections and improvements as students. -### Contributing {-} +### Contributing {.unnumbered} We transitioned the booklet into a bookdown [github repository](https://github.com/IQSS/prefresher) in 2018. As we update this version, we appreciate any bug reports or fixes appreciated. From 03f4428b8e61c1fa5e739c475b3af946064a0807 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 16:22:29 -0400 Subject: [PATCH 03/34] fix index and 01-03 --- 01_warmup.qmd | 31 ++++---- 02_functions.qmd | 190 +++++++++++++++++++++++------------------------ 03_limits.qmd | 34 +++++---- _quarto.yml | 9 ++- index.qmd | 4 +- 5 files changed, 136 insertions(+), 132 deletions(-) diff --git a/01_warmup.qmd b/01_warmup.qmd index a0d53e4..e34f01c 100644 --- a/01_warmup.qmd +++ b/01_warmup.qmd @@ -17,7 +17,7 @@ Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pm 2. $cv$ 3. $u \cdot v$ -If you are having trouble with these problems, please review Section \@ref(vector-def) "Working with Vectors" in Chapter \@ref(linearalgebra). +If you are having trouble with these problems, please review Section \@ref(vector-def) "Working with Vectors" in Chapter @linearalgebra. Are the following sets of vectors linearly independent? @@ -31,17 +31,18 @@ Are the following sets of vectors linearly independent? 3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ (this requires some guesswork) -If you are having trouble with these problems, please review Section \@ref(linearindependence). +If you are having trouble with these problems, please review Section @linearindependence. ### Matrices {.unnumbered} -\[{\bf A}=\begin{pmatrix} +$${\bf A}=\begin{pmatrix} 7 & 5 & 1 \\ 11 & 9 & 3 \\ 2 & 14 & 21 \\ 4 & 1 & 5 - \end{pmatrix}\] + \end{pmatrix}$$ + What is the dimensionality of matrix ${\bf A}$? @@ -50,23 +51,23 @@ What is the element $a_{23}$ of ${\bf A}$? Given that -\[{\bf B}=\begin{pmatrix} +$${\bf B}=\begin{pmatrix} 1 & 2 & 8 \\ 3 & 9 & 11 \\ 4 & 7 & 5 \\ 5 & 1 & 9 - \end{pmatrix}\] + \end{pmatrix}$$ What is ${\bf A}$ + ${\bf B}$? Given that -\[{\bf C}=\begin{pmatrix} +$${\bf C}=\begin{pmatrix} 1 & 2 & 8 \\ 3 & 9 & 11 \\ 4 & 7 & 5 \\ - \end{pmatrix}\] + \end{pmatrix}$$ What is ${\bf A}$ + ${\bf C}$? @@ -74,11 +75,11 @@ What is ${\bf A}$ + ${\bf C}$? Given that -\[c = 2\] +$$c = 2$$ What is $c$${\bf A}$? -If you are having trouble with these problems, please review Section \@ref(matrixbasics). +If you are having trouble with these problems, please review Section @matrixbasics. @@ -88,7 +89,7 @@ If you are having trouble with these problems, please review Section \@ref(matri Simplify the following -1. \(\sum\limits_{i = 1}^3 i\) +1. $$\sum\limits_{i = 1}^3 i$$ 2. \(\sum\limits_{k = 1}^3(3k + 2)\) @@ -120,7 +121,7 @@ Simplify the following 9. \(e^1\) 10. \(\log e^2\) -To review this material, please see Section \@ref(logexponents) +To review this material, please see Section @logexponents @@ -132,7 +133,7 @@ Find the limit of the following. 2. $\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)}$ 3. $\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2}$ -To review this material please see Section \@ref(limitsfun) +To review this material please see Section @limitsfun ## Calculus {.unnumbered} @@ -146,7 +147,7 @@ For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\fra 5. $f(x)=3x^2+2x^{1/3}$ 6. $f(x)=(x^3)(2x^4)$ -For a review, please see Section \@ref(derivintro) - \@ref(derivpoly) +For a review, please see Section @derivintro - @derivpoly ## Optimization {.unnumbered} @@ -167,5 +168,5 @@ If you are stuck, please try sketching out a picture of each of the functions. 4. If we roll two fair dice, what is the probability that their sum would be 12? -For a review, please see Sections \@ref(setoper) - \@ref(probdef). +For a review, please see Sections @setoper - @probdef. diff --git a/02_functions.qmd b/02_functions.qmd index 5faebbe..8212df6 100644 --- a/02_functions.qmd +++ b/02_functions.qmd @@ -1,6 +1,6 @@ -# (PART) Math {.unnumbered} - -# Functions and Operations +--- +title: Functions and Operations +--- __Topics__ Dimensionality; @@ -25,13 +25,13 @@ For this we use the summation operator $\sum$ and the product operator $\prod$. __Summation:__ -\[\sum\limits_{i=1}^{100} x_i = x_1+x_2+x_3+\cdots+x_{100}\] +$$\sum\limits_{i=1}^{100} x_i = x_1+x_2+x_3+\cdots+x_{100}$$ The bottom of the $\sum$ symbol indicates an index (here, $i$), and its start value $1$. At the top is where the index ends. The notion of "addition" is part of the $\sum$ symbol. The content to the right of the summation is the meat of what we add. While you can pick your favorite index, start, and end values, the content must also have the index. -* \(\sum\limits_{i=1}^n c x_i = c \sum\limits_{i=1}^n x_i \) -* \(\sum\limits_{i=1}^n (x_i + y_i) = \sum\limits_{i=1}^n x_i + \sum\limits_{i=1}^n y_i\) -* \(\sum\limits_{i=1}^n c = n c \) +- $\sum\limits_{i=1}^n c x_i = c \sum\limits_{i=1}^n x_i$ +- $\sum\limits_{i=1}^n (x_i + y_i) = \sum\limits_{i=1}^n x_i + \sum\limits_{i=1}^n y_i$ +- $\sum\limits_{i=1}^n c = n c$ __Product:__ @@ -40,10 +40,10 @@ $$\prod\limits_{i=1}^n x_i = x_1 x_2 x_3 \cdots x_n$$ Properties: -* $\prod\limits_{i=1}^n c x_i = c^n \prod\limits_{i=1}^n x_i$ -* $\prod\limits_{i=k}^n c x_i = c^{n-k+1} \prod\limits_{i=k}^n x_i$ -* $\prod\limits_{i=1}^n (x_i + y_i) =$ a total mess -* $\prod\limits_{i=1}^n c = c^n$ +- $\prod\limits_{i=1}^n c x_i = c^n \prod\limits_{i=1}^n x_i$ +- $\prod\limits_{i=k}^n c x_i = c^{n-k+1} \prod\limits_{i=k}^n x_i$ +- $\prod\limits_{i=1}^n (x_i + y_i) =$ a total mess +- $\prod\limits_{i=1}^n c = c^n$ @@ -55,15 +55,14 @@ $$x! = x\cdot (x-1) \cdot (x-2) \cdots (1)$$ __Modulo:__ Tells you the remainder when you divide the first number by the second. -* $17 \mod 3 = 2$ -* $100 \ \% \ 30 = 10$ +- $17\mod3 = 2$ +- $100 \ \% \ 30 = 10$ -```{example} -#| name: Operators -#| label: operators +:::{#exm-operators} +## Operators 1. $\sum\limits_{i=1}^{5} i =$ @@ -73,13 +72,12 @@ __Modulo:__ Tells you the remainder when you divide the first number by the seco 4. $4! =$ - +::: -``` +:::{#exr-operators1} -```{exercise} -#| name: Operators -#| label: operators1 +## Operators + Let $x_1 = 4, x_2 = 3, x_3 = 7, x_4 = 11, x_5 = 2$ 1. $\sum\limits_{i=1}^{3} (7)x_i$ @@ -90,7 +88,7 @@ Let $x_1 = 4, x_2 = 3, x_3 = 7, x_4 = 11, x_5 = 2$ -``` +::: ## Introduction to Functions @@ -100,36 +98,36 @@ A __function__ (in ${\bf R}^1$) is a mapping, or transformation, that relates me __Dimensionality__: ${\bf R}^1$ is the set of all real numbers extending from $-\infty$ to $+\infty$ --- i.e., the real number line. ${\bf R}^n$ is an $n$-dimensional space, where each of the $n$ axes extends from $-\infty$ to $+\infty$. -* ${\bf R}^1$ is a one dimensional line. -* ${\bf R}^2$ is a two dimensional plane. -* ${\bf R}^3$ is a three dimensional space. +- ${\bf R}^1$ is a one dimensional line. +- ${\bf R}^2$ is a two dimensional plane. +- ${\bf R}^3$ is a three dimensional space. Points in ${\bf R}^n$ are ordered $n$-tuples (just means an combination of $n$ elements where order matters), where each element of the $n$-tuple represents the coordinate along that dimension. For example: -* ${\bf R}^1$: (3) -* ${\bf R}^2$: (-15, 5) -* ${\bf R}^3$: (86, 4, 0) +- ${\bf R}^1$: (3) +- ${\bf R}^2$: (-15, 5) +- ${\bf R}^3$: (86, 4, 0) Examples of mapping notation: Function of one variable: $f:{\bf R}^1\to{\bf R}^1$ -* $f(x)=x+1$. For each $x$ in ${\bf R}^1$, $f(x)$ assigns the number $x+1$. +- $f(x)=x+1$. For each $x$ in ${\bf R}^1$, $f(x)$ assigns the number $x+1$. Function of two variables: $f: {\bf R}^2\to{\bf R}^1$. -* $f(x,y)=x^2+y^2$. For each ordered pair $(x,y)$ in ${\bf R}^2$, $f(x,y)$ assigns the number $x^2+y^2$. +- $f(x,y)=x^2+y^2$. For each ordered pair $(x,y)$ in ${\bf R}^2$, $f(x,y)$ assigns the number $x^2+y^2$. We often use variable $x$ as input and another $y$ as output, e.g. $y=x+1$ -```{example} -#| name: Functions -#| label: functions +:::{#exm-functions} + +## Functions For each of the following, state whether they are one-to-one or many-to-one functions. @@ -139,12 +137,12 @@ For each of the following, state whether they are one-to-one or many-to-one func -``` +::: -```{exercise} -#| name: Functions -#| label: functions1 +:::{#exr-functions1} + +## Functions For each of the following, state whether they are one-to-one or many-to-one functions. @@ -156,17 +154,17 @@ For each of the following, state whether they are one-to-one or many-to-one func -``` +::: Some functions are defined only on proper subsets of ${\bf R}^n$. -* __Domain__: the set of numbers in $X$ at which $f(x)$ is defined. -* __Range__: elements of $Y$ assigned by $f(x)$ to elements of $X$, or $$f(X)=\{ y : y=f(x), x\in X\}$$ +- __Domain__: the set of numbers in $X$ at which $f(x)$ is defined. +- __Range__: elements of $Y$ assigned by $f(x)$ to elements of $X$, or $$f(X)=\{ y : y=f(x), x\in X\}$$ Most often used when talking about a function $f:{\bf R}^1\to{\bf R}^1$. -* __Image__: same as range, but more often used when talking about a function $f:{\bf R}^n\to{\bf R}^1$. +- __Image__: same as range, but more often used when talking about a function $f:{\bf R}^n\to{\bf R}^1$. Some General Types of Functions @@ -207,11 +205,11 @@ __Common Bases__: The two most common logarithms are base 10 and base $e$. __Properties of exponential functions:__ -* $a^x a^y = a^{x+y}$ -* $a^{-x} = 1/a^x$ -* $a^x/a^y = a^{x-y}$ -* $(a^x)^y = a^{x y}$ -* $a^0 = 1$ +- $a^x a^y = a^{x+y}$ +- $a^{-x} = 1/a^x$ +- $a^x/a^y = a^{x-y}$ +- $(a^x)^y = a^{x y}$ +- $a^0 = 1$ __Properties of logarithmic functions__ (any base): @@ -221,11 +219,11 @@ Generally, when statisticians or social scientists write $\log(x)$ they mean $\l $$\log_a(a^x)=x$$ and $$a^{\log_a(x)}=x$$ -* $\log(x y)=\log(x)+\log(y)$ -* $\log(x^y)=y\log(x)$ -* $\log(1/x)=\log(x^{-1})=-\log(x)$ -* $\log(x/y)=\log(x\cdot y^{-1})=\log(x)+\log(y^{-1})=\log(x)-\log(y)$ -* $\log(1)=\log(e^0)=0$ +- $\log(x y)=\log(x)+\log(y)$ +- $\log(x^y)=y\log(x)$ +- $\log(1/x)=\log(x^{-1})=-\log(x)$ +- $\log(x/y)=\log(x\cdot y^{-1})=\log(x)+\log(y^{-1})=\log(x)-\log(y)$ +- $\log(1)=\log(e^0)=0$ __Change of Base Formula__: Use the change of base formula to switch bases as necessary: @@ -255,9 +253,9 @@ Therefore, you can see that the log of a product is equal to the sum of the logs -```{example} -#| name: Logarithmic Functions -#| label: log +:::{#exm-log} + +## Logarithmic Functions Evaluate each of the following logarithms @@ -272,13 +270,12 @@ Simplify the following logarithm. By "simplify", we actually really mean - use a -``` +::: -```{exercise} -#| name: Logarithmic Functions -#| label: log1 +:::{#exr-log1} +## Logarithmic Functions Evaluate each of the following logarithms @@ -294,30 +291,29 @@ Simplify each of the following logarithms. By "simplify", we actually really mea -``` +::: ## Graphing Functions What can a graph tell you about a function? -* Is the function increasing or decreasing? Over what part of the domain? -* How ``fast" does it increase or decrease? -* Are there global or local maxima and minima? Where? -* Are there inflection points? -* Is the function continuous? -* Is the function differentiable? -* Does the function tend to some limit? -* Other questions related to the substance of the problem at hand. +- Is the function increasing or decreasing? Over what part of the domain? +- How ``fast" does it increase or decrease? +- Are there global or local maxima and minima? Where? +- Are there inflection points? +- Is the function continuous? +- Is the function differentiable? +- Does the function tend to some limit? +- Other questions related to the substance of the problem at hand. ## Solving for Variables and Finding Roots Sometimes we're given a function $y=f(x)$ and we want to find how $x$ varies as a function of $y$. Use algebra to move $x$ to the left hand side (LHS) of the equation and so that the right hand side (RHS) is only a function of $y$. -```{example} -#| name: Solving for Variables -#| label: solvevar +:::{#exm-solvevar} +## Solving for Variables Solve for x: @@ -328,7 +324,7 @@ Solve for x: -``` +::: Solving for variables is especially important when we want to find the __roots__ of an equation: those values of variables that cause an equation to equal zero. Especially important in finding equilibria and in doing maximum likelihood estimation. @@ -342,9 +338,9 @@ __Quadratic Formula:__ For quadratic equations $ax^2+bx+c=0$, use the quadratic -```{exercise} -#| name: Finding Roots -#| label: solvevar1 +:::{#exr-solvevar1} + +## Finding Roots Solve for x: @@ -356,49 +352,49 @@ Solve for x: -``` +::: ## Sets __Interior Point__: The point $\bf x$ is an interior point of the set $S$ if $\bf x$ is in $S$ and if there is some $\epsilon$-ball around $\bf x$ that contains only points in $S$. The __interior__ of $S$ is the collection of all interior points in $S$. The interior can also be defined as the union of all open sets in $S$. -* If the set $S$ is circular, the interior points are everything inside of the circle, but not on the circle's rim. -* Example: The interior of the set $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2< 4 \}$ . +- If the set $S$ is circular, the interior points are everything inside of the circle, but not on the circle's rim. +- Example: The interior of the set $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2< 4 \}$ . __Boundary Point__: The point $\bf x$ is a boundary point of the set $S$ if every $\epsilon$-ball around $\bf x$ contains both points that are in $S$ and points that are outside $S$. The __boundary__ is the collection of all boundary points. -* If the set $S$ is circular, the boundary points are everything on the circle's rim. -* Example: The boundary of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 = 4 \}$. +- If the set $S$ is circular, the boundary points are everything on the circle's rim. +- Example: The boundary of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 = 4 \}$. __Open__: A set $S$ is open if for each point $\bf x$ in $S$, there exists an open $\epsilon$-ball around $\bf x$ completely contained in $S$. -* If the set $S$ is circular and open, the points contained within the set get infinitely close to the circle's rim, but do not touch it. -* Example: $\{ (x,y) : x^2+y^2<4 \}$ +- If the set $S$ is circular and open, the points contained within the set get infinitely close to the circle's rim, but do not touch it. +- Example: $\{ (x,y) : x^2+y^2<4 \}$ __Closed__: A set $S$ is closed if it contains all of its boundary points. -* Alternatively: A set is closed if its complement is open. -* If the set $S$ is circular and closed, the set contains all points within the rim as well as the rim itself. -* Example: $\{ (x,y) : x^2+y^2\le 4 \}$ -* Note: a set may be neither open nor closed. Example: $\{ (x,y) : 2 < x^2+y^2\le 4 \}$ +- Alternatively: A set is closed if its complement is open. +- If the set $S$ is circular and closed, the set contains all points within the rim as well as the rim itself. +- Example: $\{ (x,y) : x^2+y^2\le 4 \}$ +- Note: a set may be neither open nor closed. Example: $\{ (x,y) : 2 < x^2+y^2\le 4 \}$ __Complement__: The complement of set $S$ is everything outside of $S$. -* If the set $S$ is circular, the complement of $S$ is everything outside of the circle. -* Example: The complement of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 > 4 \}$. +- If the set $S$ is circular, the complement of $S$ is everything outside of the circle. +- Example: The complement of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 > 4 \}$. __Empty__: The empty (or null) set is a unique set that has no elements, denoted by \{\} or $\emptyset$. -* The empty set is an example of a set that is open and closed, or a "clopen" set. -* Examples: The set of squares with 5 sides; the set of countries south of the South Pole. +- The empty set is an example of a set that is open and closed, or a "clopen" set. +- Examples: The set of squares with 5 sides; the set of countries south of the South Pole. ## Answers to Examples and Exercises {.unnumbered} -Answer to Example \@ref(exm:operators): +Answer to Example @exm-operators: 1. 1 + 2 + 3 + 4 + 5 = 15 @@ -408,7 +404,7 @@ Answer to Example \@ref(exm:operators): 4. 4 * 3 * 2 * 1 = 24 -Answer to Exercise \@ref(exr:operators1): +Answer to Exercise @exr-operators1: 1. 7(4 + 3 + 7) = 98 @@ -417,19 +413,19 @@ Answer to Exercise \@ref(exr:operators1): 3. $2^3(7)(11)(2)$ = 1232 -Answer to Example \@ref(exm:functions): +Answer to Example @exm-functions: 1. one-to-one 2. many-to-one -Answer to Exercise \@ref(exr:functions1): +Answer to Exercise @exr-functions1: 1. many-to-one 2. one-to-one -Answer to Example \@ref(exm:log): +Answer to Example @exm-log: 1. 2 @@ -438,7 +434,7 @@ Answer to Example \@ref(exm:log): 3. $3\log_4(x) + 5\log_4(y)$ -Answer to Exercise \@ref(exr:log1): +Answer to Exercise @exr-log1: 1. 3 @@ -446,14 +442,14 @@ Answer to Exercise \@ref(exr:log1): 3. $\frac{1}{2}(\ln{x} + \ln{y})$ -Answer to Example \@ref(exm:solvevar): +Answer to Example @exm-solvevar: 1. $y=3x+2 \quad\Longrightarrow\quad -3x=2-y \quad\Longrightarrow\quad 3x=y-2 \quad\Longrightarrow\quad x=\frac{1}{3}(y-2)$ 2. $x = \ln{y}$ -Answer to Exercise \@ref(exr:solvevar1): +Answer to Exercise @exr-solvevar1: 1. $\frac{-2}{3}$ diff --git a/03_limits.qmd b/03_limits.qmd index 2e7de2b..6fbee3d 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -16,17 +16,18 @@ Solving limits, i.e. finding out the value of functions as its input moves clos Perhaps the most important theorem in statistics is the Central Limit Theorem, -```{theorem} -#| label: clt-lim -#| name: Central Limit Theorem (i.i.d. case) +:::{#thm-clt-lim} + +## Central Limit Theorem (i.i.d. case) + For any series of independent and identically distributed random variables $X_1, X_2, \cdots$, we know the distribution of its sum even if we do not know the distribution of $X$. The distribution of the sum is a Normal distribution. -\[\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),\] +$$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),$$ where $\mu$ is the mean of $X$ and $\sigma$ is the standard deviation of $X$. The arrow is read as "converges in distribution to". $\text{Normal}(0, 1)$ indicates a Normal Distribution with mean 0 and variance 1. That is, the limit of the distribution of the lefthand side is the distribution of the righthand side. -``` +::: The sign of a limit is the arrow "$\rightarrow$". Although we have not yet covered probability (in Section \@ref(probability-theory)) so we have not described what distributions and random variables are, it is worth foreshadowing the Central Limit Theorem. The Central Limit Theorem is powerful because it gives us a _guarantee_ of what would happen if $n \rightarrow \infty$, which in this case means we collected more data. @@ -35,16 +36,17 @@ The sign of a limit is the arrow "$\rightarrow$". Although we have not yet cover A finding that perhaps rivals the Central Limit Theorem is the Law of Large Numbers: -```{theorem} -#| label: lln-lim -#| name: (Weak) Law of Large Numbers +:::{#thm-lln-lim} + +## (Weak) Law of Large Numbers + For any draw of identically distributed independent variables with mean $\mu$, the sample average after $n$ draws, $\bar{X}_n$, converges in probability to the true mean as $n \rightarrow \infty$: -\[\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0\] +$$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ A shorthand of which is \(\bar{X}_n \xrightarrow{p} \mu\), where the arrow is read as "converges in probability to". -``` +::: Intuitively, the more data, the more accurate is your guess. For example, the Figure \@ref(fig:llnsim) shows how the sample average from many coin tosses converges to the true value : 0.5. @@ -75,7 +77,7 @@ ggplot(tibble(n = 1:n, estimate = means), aes(x = n, y = estimate)) + We need a couple of steps until we get to limit theorems in probability. First we will introduce a "sequence", then we will think about the limit of a sequence, then we will think about the limit of a _function_. -A **sequence** \[\{x_n\}=\{x_1, x_2, x_3, \ldots, x_n\}\] is an ordered set of real numbers, where $x_1$ is the first term in the sequence and $y_n$ is the $n$th term. Generally, a sequence is infinite, that is it extends to $n=\infty$. We can also write the sequence as \[\{x_n\}^\infty_{n=1}\] +A **sequence** $$\{x_n\}=\{x_1, x_2, x_3, \ldots, x_n\}$$ is an ordered set of real numbers, where $x_1$ is the first term in the sequence and $y_n$ is the $n$th term. Generally, a sequence is infinite, that is it extends to $n=\infty$. We can also write the sequence as $$\{x_n\}^\infty_{n=1}$$ where the subscript and superscript are read together as "from 1 to infinity." @@ -127,7 +129,7 @@ The notion of "converging to a limit" is the behavior of the points in Example \ ```{definition} -The sequence $\{y_n\}$ has the limit $L$, which we write as \[\lim\limits_{n \to \infty} y_n =L\], if for any $\epsilon>0$ there is an integer $N$ (which depends on $\epsilon$) with the property that $|y_n -L|<\epsilon$ for each $n>N$. $\{y_n\}$ is said to converge to $L$. If the above does not hold, then $\{y_n\}$ diverges. +The sequence $\{y_n\}$ has the limit $L$, which we write as $$\lim\limits_{n \to \infty} y_n =L$$, if for any $\epsilon>0$ there is an integer $N$ (which depends on $\epsilon$) with the property that $|y_n -L|<\epsilon$ for each $n>N$. $\{y_n\}$ is said to converge to $L$. If the above does not hold, then $\{y_n\}$ diverges. ``` We can also express the behavior of a sequence as bounded or not: @@ -152,13 +154,13 @@ It is nice for a sequence to converge in limit. We want to know if complex-looki ```{example} #| name: Simplifying a Fraction into Sums Find the limit of -\[\lim_{n\to \infty} \frac{n + 3}{n},\] +$$\lim_{n\to \infty} \frac{n + 3}{n},$$ ``` ```{solution} At first glance, $n + 3$ and $n$ both grow to $\infty$, so it looks like we need to divide infinity by infinity. However, we can express this fraction as a sum, then the limits apply separately: -\[\lim_{n\to \infty} \frac{n + 3}{n} = \lim_{n\to \infty} \left(1 + \frac{3}{n}\right) = \underbrace{\lim_{n\to \infty}1}_{1} + \underbrace{\lim_{n\to \infty}\left(\frac{3}{n}\right)}_{0}\] +$$\lim_{n\to \infty} \frac{n + 3}{n} = \lim_{n\to \infty} \left(1 + \frac{3}{n}\right) = \underbrace{\lim_{n\to \infty}1}_{1} + \underbrace{\lim_{n\to \infty}\left(\frac{3}{n}\right)}_{0}$$ so, the limit is actually 1. ``` @@ -224,7 +226,7 @@ The function can be thought of as a more general or "smooth" version of sequence Find the limit of -\[\lim_{x\to\infty} \frac{(x^4 +3x−99)(2−x^5)}{(18x^7 +9x^6 −3x^2 −1)(x+1)}\] +$$\lim_{x\to\infty} \frac{(x^4 +3x−99)(2−x^5)}{(18x^7 +9x^6 −3x^2 −1)(x+1)}$$ ``` @@ -356,7 +358,7 @@ Some properties of continuous functions: #| label: discontdraw #| name: Limit when Denominator converges to 0 -Let \[f(x) = \frac{x^2 + 2x}{x}.\] +Let $$f(x) = \frac{x^2 + 2x}{x}.$$ 1. Graph the function. Is it defined everywhere? 2. What is the functions limit at $x \rightarrow 0$? diff --git a/_quarto.yml b/_quarto.yml index 2e5ef87..eee1e00 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -1,5 +1,6 @@ project: type: book + output-dir: _book book: title: "Math Prefresher for Political Scientists" @@ -7,11 +8,13 @@ book: chapters: - index.qmd - 01_warmup.qmd - - 02_functions.qmd + - part: Math + chapters: + - 02_functions.qmd + - 03_limits.qmd -output_dir: _book -delete_merged_file: true +delete_merged_file: true language: ui: chapter_name: "Chapter " diff --git a/index.qmd b/index.qmd index 90f1dc3..27b7012 100644 --- a/index.qmd +++ b/index.qmd @@ -1,5 +1,4 @@ --- -title: "Math Prefresher for Political Scientists" author: "" date: today description: "Text for Harvard Department of Government Math Prefresher" @@ -11,6 +10,7 @@ biblio-style: apalike link-citations: true cover-image: "./images/logo.png" --- + # About this Booklet {.unnumbered} The [__Harvard Gov Prefresher__](https://projects.iq.harvard.edu/prefresher) is held each year in August. All relevant information is on our website, including the day-to-day schedule. The 2023 Prefresher instructors are [Sooahn Shin](http://sooahnshin.com/) and [María Ballesteros](https://scholar.harvard.edu/mariaballesteros/home), and the faculty sponsor is [Gary King](https://gking.harvard.edu). @@ -24,6 +24,8 @@ For information about the role of the prefresher (or "math camp") as a introduct - Past Authors and Instructors: Curt Signorino 1996-1997; Ken Scheve 1997-1998; Eric Dickson 1998-2000; Orit Kedar 1999; James Fowler 2000-2001; Kosuke Imai 2001-2002; Jacob Kline 2002; Dan Epstein 2003; Ben Ansell 2003-2004; Ryan Moore 2004-2005; Mike Kellermann 2005-2006; Ellie Powell 2006-2007; Jen Katkin 2007-2008; Patrick Lam 2008-2009; Viridiana Rios 2009-2010; Jennifer Pan 2010-2011; Konstantin Kashin 2011-2012; Soledad Prillaman 2013; Stephen Pettigrew 2013-2014; Anton Strezhnev 2014-2015; Mayya Komisarchik 2015-2016; Connor Jerzak 2016-2017; Shiro Kuriwaki 2017-2018; Yon Soo Park 2018; Meg Schwenzfeier 2019; Shannon Parker 2019; Laura Royden 2020-2021; Hunter Rendleman 2020-2021; Christopher T. Kenny 2022; Jialu Li 2022; Sooahn Shin 2023; María Ballesteros 2023. - Repository Maintainer: Christopher T. Kenny ([christopherkenny](https://github.com/christopherkenny)) +- Past Repository Maintainers: + - Shiro Kurikwaki ([kuriwaki](https://github.com/kuriwaki)) 2018-2023. - Contributors: Thanks to Juan Dodyk ([juandodyk](https://github.com/juandodyk)), Hunter Rendleman ([hrendleman](https://github.com/hrendleman)), and Tyler Simko ([tylersimko](https://github.com/tylersimko)) for contributing to the booklet for corrections and improvements as students. From b3b661979bcdd632b10a81be4a09ddf8af4cac23 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 16:47:18 -0400 Subject: [PATCH 04/34] more 1-3 cleaning needed --- 01_warmup.qmd | 14 +++--- 03_limits.qmd | 133 ++++++++++++++++++++++++++------------------------ _quarto.yml | 8 +++ index.qmd | 9 ---- 4 files changed, 84 insertions(+), 80 deletions(-) diff --git a/01_warmup.qmd b/01_warmup.qmd index e34f01c..ea9c7e3 100644 --- a/01_warmup.qmd +++ b/01_warmup.qmd @@ -17,7 +17,7 @@ Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pm 2. $cv$ 3. $u \cdot v$ -If you are having trouble with these problems, please review Section \@ref(vector-def) "Working with Vectors" in Chapter @linearalgebra. +If you are having trouble with these problems, please review Section @sec-vector-def "Working with Vectors" in Chapter @sec-linearalgebra. Are the following sets of vectors linearly independent? @@ -31,7 +31,7 @@ Are the following sets of vectors linearly independent? 3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ (this requires some guesswork) -If you are having trouble with these problems, please review Section @linearindependence. +If you are having trouble with these problems, please review Section @sec-linearindependence. ### Matrices {.unnumbered} @@ -79,7 +79,7 @@ $$c = 2$$ What is $c$${\bf A}$? -If you are having trouble with these problems, please review Section @matrixbasics. +If you are having trouble with these problems, please review Section @sec-matrixbasics. @@ -121,7 +121,7 @@ Simplify the following 9. \(e^1\) 10. \(\log e^2\) -To review this material, please see Section @logexponents +To review this material, please see Section @sec-logexponents @@ -133,7 +133,7 @@ Find the limit of the following. 2. $\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)}$ 3. $\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2}$ -To review this material please see Section @limitsfun +To review this material please see Section @sec-limitsfun ## Calculus {.unnumbered} @@ -147,7 +147,7 @@ For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\fra 5. $f(x)=3x^2+2x^{1/3}$ 6. $f(x)=(x^3)(2x^4)$ -For a review, please see Section @derivintro - @derivpoly +For a review, please see Section @sec-derivintro - @sec-derivpoly ## Optimization {.unnumbered} @@ -168,5 +168,5 @@ If you are stuck, please try sketching out a picture of each of the functions. 4. If we roll two fair dice, what is the probability that their sum would be 12? -For a review, please see Sections @setoper - @probdef. +For a review, please see Sections @sec-setoper - @sec-probdef. diff --git a/03_limits.qmd b/03_limits.qmd index 6fbee3d..dbdf0f6 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -29,7 +29,7 @@ where $\mu$ is the mean of $X$ and $\sigma$ is the standard deviation of $X$. Th That is, the limit of the distribution of the lefthand side is the distribution of the righthand side. ::: -The sign of a limit is the arrow "$\rightarrow$". Although we have not yet covered probability (in Section \@ref(probability-theory)) so we have not described what distributions and random variables are, it is worth foreshadowing the Central Limit Theorem. The Central Limit Theorem is powerful because it gives us a _guarantee_ of what would happen if $n \rightarrow \infty$, which in this case means we collected more data. +The sign of a limit is the arrow "$\rightarrow$". Although we have not yet covered probability (in Section @probability-theory) so we have not described what distributions and random variables are, it is worth foreshadowing the Central Limit Theorem. The Central Limit Theorem is powerful because it gives us a _guarantee_ of what would happen if $n \rightarrow \infty$, which in this case means we collected more data. ## Example: The Law of Large Numbers {.unnumbered} @@ -49,10 +49,10 @@ A shorthand of which is \(\bar{X}_n \xrightarrow{p} \mu\), where the arrow is r ::: -Intuitively, the more data, the more accurate is your guess. For example, the Figure \@ref(fig:llnsim) shows how the sample average from many coin tosses converges to the true value : 0.5. +Intuitively, the more data, the more accurate is your guess. For example, the Figure @fig-llnsim shows how the sample average from many coin tosses converges to the true value : 0.5. ```{r} -#| label: llnsim +#| label: fig-llnsim #| echo: false #| fig-cap: As the number of coin tosses goes to infinity, the average probabiity of #| heads converges to 0.5 @@ -82,24 +82,23 @@ A **sequence** $$\{x_n\}=\{x_1, x_2, x_3, \ldots, x_n\}$$ is an ordered set of r where the subscript and superscript are read together as "from 1 to infinity." -```{example} -#| label: seqbehav -#| name: Sequences +:::{#exm-seqbehav} + +## Sequences How do these sequences behave? 1. $\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\}$ 2. $\{B_n\}=\left\{\frac{n^2+1}{n} \right\}$ 3. $\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\}$ - - -``` + +::: -We find the sequence by simply "plugging in" the integers into each $n$. The important thing is to get a sense of how these numbers are going to change. Example 1's numbers seem to come closer and closer to 2, but will it ever surpass 2? Example 2's numbers are also increasing each time, but will it hit a limit? What is the pattern in Example 3? Graphing helps you make this point more clearly. See the sequence of $n = 1, ...20$ for each of the three examples in Figure \@ref(fig:seqabc). +We find the sequence by simply "plugging in" the integers into each $n$. The important thing is to get a sense of how these numbers are going to change. Example 1's numbers seem to come closer and closer to 2, but will it ever surpass 2? Example 2's numbers are also increasing each time, but will it hit a limit? What is the pattern in Example 3? Graphing helps you make this point more clearly. See the sequence of $n = 1, ...20$ for each of the three examples in Figure @fig-seqabc. ```{r} -#| label: seqabc +#| label: fig-seqabc #| echo: false #| fig-cap: Behavior of Some Sequences seq <- 1:20 @@ -121,16 +120,16 @@ gA + gB + gC + plot_layout(nrow = 1) ## The Limit of a Sequence -The notion of "converging to a limit" is the behavior of the points in Example \@ref(exm:seqbehav). In some sense, that's the counterfactual we want to know. What happens as $n\rightarrow \infty$? +The notion of "converging to a limit" is the behavior of the points in Example @exm-seqbehav. In some sense, that's the counterfactual we want to know. What happens as $n\rightarrow \infty$? 1. Sequences like 1 above that converge to a limit. 2. Sequences like 2 above that increase without bound. 3. Sequences like 3 above that neither converge nor increase without bound --- alternating over the number line. -```{definition} +:::{#def-limseq} The sequence $\{y_n\}$ has the limit $L$, which we write as $$\lim\limits_{n \to \infty} y_n =L$$, if for any $\epsilon>0$ there is an integer $N$ (which depends on $\epsilon$) with the property that $|y_n -L|<\epsilon$ for each $n>N$. $\{y_n\}$ is said to converge to $L$. If the above does not hold, then $\{y_n\}$ diverges. -``` +::: We can also express the behavior of a sequence as bounded or not: @@ -151,30 +150,33 @@ This looks reasonable enough. The harder question, obviously is when the parts o It is nice for a sequence to converge in limit. We want to know if complex-looking sequences converge or not. The name of the game here is to break that complex sequence up into sums of simple fractions where $n$ only appears in the denominator: $\frac{1}{n}, \frac{1}{n^2}$, and so on. Each of these will converge to 0, because the denominator gets larger and larger. Then, because of the properties above, we can then find the final sequence. -```{example} -#| name: Simplifying a Fraction into Sums +:::{#exm-simplfrac} + +## Simplifying a Fraction into Sums + Find the limit of $$\lim_{n\to \infty} \frac{n + 3}{n},$$ -``` +::: + +:::{#sol-simplfrac} -```{solution} At first glance, $n + 3$ and $n$ both grow to $\infty$, so it looks like we need to divide infinity by infinity. However, we can express this fraction as a sum, then the limits apply separately: $$\lim_{n\to \infty} \frac{n + 3}{n} = \lim_{n\to \infty} \left(1 + \frac{3}{n}\right) = \underbrace{\lim_{n\to \infty}1}_{1} + \underbrace{\lim_{n\to \infty}\left(\frac{3}{n}\right)}_{0}$$ so, the limit is actually 1. -``` +::: After some practice, the key to intuition is whether one part of the fraction grows "faster" than another. If the denominator grows faster to infinity than the numerator, then the fraction will converge to 0, even if the numerator will also increase to infinity. In a sense, limits show how not all infinities are the same. -```{exercise} -#| label: limseq2 +:::{#exr-limseq2} + Find the following limits of sequences, then explain in English the intuition for why that is the case. 1. \(\lim\limits_{n\to\infty} \frac{2n}{n^2 + 1}\) 2. \(\lim\limits_{n\to\infty} (n^3 - 100n^2)\) -``` +::: @@ -187,10 +189,12 @@ A function $f$ is a compact representation of some behavior we care about. Like For a limit $L$ to exist, the function $f(x)$ must approach $L$ from both the left (increasing) and the right (decreasing). -```{definition} -#| name: Limit of a function +:::{#def-limfunc} + +## Limit of a function + Let $f(x)$ be defined at each point in some open interval containing the point $c$. Then $L$ equals $\lim\limits_{x \to c} f(x)$ if for any (small positive) number $\epsilon$, there exists a corresponding number $\delta>0$ such that if $0<|x-c|<\delta$, then $|f(x)-L|<\epsilon$. -``` +::: A neat, if subtle result is that $f(x)$ does not necessarily have to be defined at $c$ for $\lim\limits_{x \to c}$ to exist. @@ -204,9 +208,10 @@ Properties: Let $f$ and $g$ be functions with $\lim\limits_{x \to c} f(x)=k$ and Simple limits of functions can be solved as we did limits of sequences. Just be careful which part of the function is changing. -```{example} -#| label: limfun1 -#| name: Limits of Functions +:::{#exm-limfun1} + +## Limits of Functions + Find the limit of the following functions. 1. \(\lim_{x \to c} k\) @@ -214,34 +219,34 @@ Find the limit of the following functions. 1. \(\lim_{x\to 2} (2x-3)\) 1. \(\lim_{x \to c} x^n\) -``` +::: Limits can get more complex in roughly two ways. First, the functions may become large polynomials with many moving pieces. Second,the functions may become discontinuous. The function can be thought of as a more general or "smooth" version of sequences. For example, -```{exercise} -#| label: limfunmax -#| name: Limits of a Fraction of Functions +:::{#exr-limfunmax} + +## Limits of a Fraction of Functions Find the limit of $$\lim_{x\to\infty} \frac{(x^4 +3x−99)(2−x^5)}{(18x^7 +9x^6 −3x^2 −1)(x+1)}$$ -``` +::: Now, the functions will become a bit more complex: -```{exercise} -#| label: discontlim +:::{#exr-discontlim} + Solve the following limits of functions 1. $\lim\limits_{x\to 0} |x|$ 2. $\lim\limits_{x\to 0} \left(1+\frac{1}{x^2}\right)$ -``` +::: @@ -282,16 +287,16 @@ fx1 + fx2 + plot_layout(nrow = 1) To repeat a finding from the limits of functions: $f(x)$ does not necessarily have to be defined at $c$ for $\lim\limits_{x \to c}$ to exist. Functions that have breaks in their lines are called discontinuous. Functions that have no breaks are called continuous. Continuity is a concept that is more fundamental to, but related to that of "differentiability", which we will cover next in calculus. -```{definition} -#| name: Continuity +:::{#def-continuity} +## Continuity Suppose that the domain of the function $f$ includes an open interval containing the point $c$. Then $f$ is continuous at $c$ if $\lim\limits_{x \to c} f(x)$ exists and if $\lim\limits_{x \to c} f(x)=f(c)$. Further, $f$ is continuous on an open interval $(a,b)$ if it is continuous at each point in the interval. -``` +::: To prove that a function is continuous for all points is beyond this practical introduction to math, but the general intuition can be grasped by graphing. -```{example} -#| label: contdiscont -#| name: Continuous and Discontinuous Functions +:::{#exm-contdiscont} + +## Continuous and Discontinuous Functions For each function, determine if it is continuous or discontinuous. @@ -302,11 +307,11 @@ For each function, determine if it is continuous or discontinuous. The floor is the smaller of the two integers bounding a number. So $\text{floor}(x = 2.999) = 2$, $\text{floor}(x = 2.0001) = 2$, and $\text{floor}(x = 2) = 2.$ -``` +::: -```{solution} -In Figure \@ref(fig:fig-contdiscont), we can see that the first two functions are continuous, and the next two are discontinuous. $f(x) = 1 + \frac{1}{x^2}$ is discontinuous at $x= 0$, and $f(x) = \text{floor}(x)$ is discontinuous at each whole number. -``` +:::{#sol-contdiscont} +In Figure @fig-contdiscont, we can see that the first two functions are continuous, and the next two are discontinuous. $f(x) = 1 + \frac{1}{x^2}$ is discontinuous at $x= 0$, and $f(x) = \text{floor}(x)$ is discontinuous at each whole number. +::: ```{r} @@ -354,9 +359,9 @@ Some properties of continuous functions: -```{exercise} -#| label: discontdraw -#| name: Limit when Denominator converges to 0 +:::{#exr-discontdraw} + +## Limit when Denominator converges to 0 Let $$f(x) = \frac{x^2 + 2x}{x}.$$ @@ -364,16 +369,16 @@ Let $$f(x) = \frac{x^2 + 2x}{x}.$$ 2. What is the functions limit at $x \rightarrow 0$? -``` +::: ## Answers to Examples {.unnumbered} -Example \@ref(exm:seqbehav) +Example @exm-seqbehav -```{solution} +:::{#sol-seqbehav} 1. $\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\} = \left\{1, \frac{7}{4}, \frac{17}{9}, \frac{31}{16}, \frac{49}{25}, \ldots\right\} = 2$ @@ -381,19 +386,19 @@ Example \@ref(exm:seqbehav) 3. $\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\} = \left\{0, \frac{1}{2}, -\frac{2}{3}, \frac{3}{4}, -\frac{4}{5}\right\}$ -``` +::: -Exercise \@ref(exr:limseq2) +Exercise @exr-limseq2 ```{r} ``` -Example \@ref(exm:limfun1) +Example @exm-limfun1 -```{solution} +:::{#sol-limfun1} 1. $k$ @@ -402,14 +407,14 @@ Example \@ref(exm:limfun1) 4. \(\lim_{x \to c} x^n = \lim\limits_{x \to c} x \cdots[\lim\limits_{x \to c} x] = c\cdots c =c^n\) -``` +::: -Exercise \@ref(exr:limfunmax) +Exercise @exr-limfunmax -```{solution} +:::{#sol-} Although this function seems large, the thing our eyes should focus on is where the highest order polynomial remains. That will grow the fastest, so if the highest order term is on the denominator, the fraction will converge to 0, if it is on the numerator it will converge to negative infinity. Previewing the multiplication by hand, we can see that the $-x^9$ on the numerator will be the largest power. So the answer will be $-\infty$. We can also confirm this by writing out fractions: \begin{align*} @@ -418,19 +423,19 @@ Although this function seems large, the thing our eyes should focus on is where =& 1 \times \lim_{-x\to\infty} \frac{x}{18} \end{align*} -``` +::: -Exercise \@ref(exr:discontdraw) +Exercise @exr-discontdraw -```{solution} +:::{#sol-discontdraw} -See Figure \@ref(fig:fig-hole-0). +See Figure @fig-hole-0. Divide each part by $x$, and we get $x + \frac{2}{x}$ on the numerator, $1$ on the denominator. So, without worrying about a function being not defined, we can say $\lim_{x\to 0}f(x) = 0$. -``` +::: ```{r} diff --git a/_quarto.yml b/_quarto.yml index eee1e00..29204f2 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -2,8 +2,10 @@ project: type: book output-dir: _book + book: title: "Math Prefresher for Political Scientists" + description: "Text for Harvard Department of Government Math Prefresher" site-url: https://iqss.github.io/prefresher/ chapters: - index.qmd @@ -18,3 +20,9 @@ delete_merged_file: true language: ui: chapter_name: "Chapter " + +github-repo: "IQSS/prefresher" +geometry: "margin=1.5in" +biblio-style: apalike +link-citations: true +cover-image: "./images/logo.png" diff --git a/index.qmd b/index.qmd index 27b7012..0cef19e 100644 --- a/index.qmd +++ b/index.qmd @@ -1,14 +1,5 @@ --- -author: "" date: today -description: "Text for Harvard Department of Government Math Prefresher" -github-repo: "IQSS/prefresher" -site: "bookdown::bookdown_site" -documentclass: book -geometry: "margin=1.5in" -biblio-style: apalike -link-citations: true -cover-image: "./images/logo.png" --- # About this Booklet {.unnumbered} From 99be3ccfb597737d9da31ec9b821278504de5f14 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 17:54:14 -0400 Subject: [PATCH 05/34] latex for quarto --- 04_calculus.qmd | 114 +++++++++++++++++----------------- 05_optimization.qmd | 34 +++++----- 06_probability.qmd | 92 +++++++++++++-------------- 07_linear-algebra.qmd | 62 +++++++++--------- 12_matricies-manipulation.qmd | 4 +- 17_non-wysiwyg.qmd | 4 +- 17_non-wysiwyg.quarto_ipynb | 6 +- 21_solutions-warmup.qmd | 30 ++++----- 8 files changed, 173 insertions(+), 173 deletions(-) diff --git a/04_calculus.qmd b/04_calculus.qmd index 2a69850..f0ac59c 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -18,17 +18,17 @@ The average of a quantity is a type of weighted mean, where the potential values If $X$ is a continuous random variable, its expected value $E(X)$ -- the center of mass -- is given by -\[E(X) = \int^{\infty}_{-\infty}x f(x) dx\] +$$E(X) = \int^{\infty}_{-\infty}x f(x) dx$$ where $f(x)$ is the probability density function of $X$. This is a continuous version of the case where $X$ is discrete, in which case -\[E(X) = \sum^\infty_{j=1} x_j P(X = x_j)\] +$$E(X) = \sum^\infty_{j=1} x_j P(X = x_j)$$ even more concretely, if the potential values of $X$ are finite, then we can write out the expected value as a weighted mean, where the weights is the probability that the value occurs. -\[E(X) = \large \sum_{x} \quad\left( \underbrace{x}_{\text{value}}\cdot \underbrace{P(X = x)}_{\text{weight, or PMF}}\right)\] +$$E(X) = \large \sum_{x} \quad\left( \underbrace{x}_{\text{value}}\cdot \underbrace{P(X = x)}_{\text{weight, or PMF}}\right)$$ ## Derivatives {#derivintro} @@ -40,8 +40,8 @@ The derivative of $f$ at $x$ is its rate of change at $x$: how much $f(x)$ chang #| name: Derivative Let $f$ be a function whose domain includes an open interval containing the point $x$. The derivative of $f$ at $x$ is given by -\[\frac{d}{dx}f(x) =\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x} = \lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h} -\] +$$\frac{d}{dx}f(x) =\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x} = \lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h} +$$ There are a two main ways to denote a derivate: @@ -91,20 +91,20 @@ If $f'(x)$ exists at a point $x_0$, then $f$ is said to be __differentiable__ at ### Properties of derivatives {.unnumbered} Suppose that $f$ and $g$ are differentiable at $x$ and that $\alpha$ is a constant. Then the functions $f\pm g$, $\alpha f$, $f g$, and $f/g$ (provided $g(x)\ne 0$) are also differentiable at $x$. Additionally, -__Constant rule:__ \[\left[k f(x)\right]' = k f'(x)\] +__Constant rule:__ $$\left[k f(x)\right]' = k f'(x)$$ -__Sum rule:__ \[\left[f(x)\pm g(x)\right]' = f'(x)\pm g'(x)\] +__Sum rule:__ $$\left[f(x)\pm g(x)\right]' = f'(x)\pm g'(x)$$ With a bit more algebra, we can apply the definition of derivatives to get a formula for of the derivative of a product and a derivative of a quotient. -__Product rule:__ \[\left[f(x)g(x)\right]^\prime = f^\prime(x)g(x)+f(x)g^\prime(x)\] +__Product rule:__ $$\left[f(x)g(x)\right]^\prime = f^\prime(x)g(x)+f(x)g^\prime(x)$$ -__Quotient rule:__ \[\left[f(x)/g(x)\right]^\prime = \frac{f^\prime(x)g(x) - f(x)g^\prime(x)}{[g(x)]^2}, ~g(x)\neq 0\] +__Quotient rule:__ $$\left[f(x)/g(x)\right]^\prime = \frac{f^\prime(x)g(x) - f(x)g^\prime(x)}{[g(x)]^2}, ~g(x)\neq 0$$ Finally, one way to think of the power of derivatives is that it takes a function a notch down in complexity. The power rule applies to any higher-order function: -__Power rule:__ \[\left[x^k\right]^\prime = k x^{k-1}\] +__Power rule:__ $$\left[x^k\right]^\prime = k x^{k-1}$$ For any real number $k$ (that is, both whole numbers and fractions). The power rule is proved __by induction__, a neat method of proof used in many fundamental applications to prove that a general statement holds for every possible case, even if there are countably infinite cases. We'll show a simple case where $k$ is an integer here. @@ -114,17 +114,17 @@ For any real number $k$ (that is, both whole numbers and fractions). The power r We would like to prove that -\[\left[x^k\right]^\prime = k x^{k-1}\] +$$\left[x^k\right]^\prime = k x^{k-1}$$ for any integer $k$. First, consider the first case (the base case) of $k = 1$. We can show by the definition of derivatives (setting $f(x) = x^1 = 1$) that -\[[x^1]^\prime = \lim_{h \rightarrow 0}\frac{(x + h) - x}{(x + h) - x}= 1.\] +$$[x^1]^\prime = \lim_{h \rightarrow 0}\frac{(x + h) - x}{(x + h) - x}= 1.$$ Because $1$ is also expressed as $1 x^{1- 1}$, the statement we want to prove holds for the case $k =1$. Now, \emph{assume} that the statement holds for some integer $m$. That is, assume -\[\left[x^m\right]^\prime = m x^{m-1}\] +$$\left[x^m\right]^\prime = m x^{m-1}$$ Then, for the case $m + 1$, using the product rule above, we can simplify @@ -176,12 +176,12 @@ For each of the following functions, find the first-order derivative $f^\prime(x The first derivative is applying the definition of derivatives on the function, and it can be expressed as -\[f'(x), ~~ y', ~~ \frac{d}{dx}f(x), ~~ \frac{dy}{dx}\] +$$f'(x), ~~ y', ~~ \frac{d}{dx}f(x), ~~ \frac{dy}{dx}$$ We can keep applying the differentiation process to functions that are themselves derivatives. The derivative of $f'(x)$ with respect to $x$, would then be $$f''(x)=\lim\limits_{h\to 0}\frac{f'(x+h)-f'(x)}{h}$$ and we can therefore call it the __Second derivative:__ -\[f''(x), ~~ y'', ~~ \frac{d^2}{dx^2}f(x), ~~ \frac{d^2y}{dx^2}\] +$$f''(x), ~~ y'', ~~ \frac{d^2}{dx^2}f(x), ~~ \frac{d^2y}{dx^2}$$ Similarly, the derivative of $f''(x)$ would be called the third derivative and is denoted $f'''(x)$. And by extension, the __nth derivative__ is expressed as $\frac{d^n}{dx^n}f(x)$, $\frac{d^ny}{dx^n}$. @@ -206,16 +206,16 @@ Earlier, in Section \@ref(derivintro), we said that if a function differentiable As useful as the above rules are, many functions you'll see won't fit neatly in each case immediately. Instead, they will be functions of functions. For example, the difference between $x^2 + 1^2$ and $(x^2 + 1)^2$ may look trivial, but the sum rule can be easily applied to the former, while it's actually not obvious what do with the latter. -__Composite functions__ are formed by substituting one function into another and are denoted by \[(f\circ g)(x)=f[g(x)].\] To form $f[g(x)]$, the range of $g$ must be contained (at least in part) within the domain of $f$. The domain of $f\circ g$ consists of all the points in the domain of $g$ for which $g(x)$ is in the domain of $f$. +__Composite functions__ are formed by substituting one function into another and are denoted by $$(f\circ g)(x)=f[g(x)].$$ To form $f[g(x)]$, the range of $g$ must be contained (at least in part) within the domain of $f$. The domain of $f\circ g$ consists of all the points in the domain of $g$ for which $g(x)$ is in the domain of $f$. ```{example} Let $f(x)=\log x$ for $0 0\] +$$\left(e^{g(x)}\right)^\prime = e^{g(x)} \cdot g^\prime(x)$$ +$$\left(\log g(x)\right)^\prime = \frac{g^\prime(x)}{g(x)}, ~~\text{if}~~ g(x) > 0$$ ``` @@ -399,7 +399,7 @@ y &= a^x \\ Then in the special case where $a = e$, -\[(e^x)^\prime = (e^x)\] +$$(e^x)^\prime = (e^x)$$ @@ -414,7 +414,7 @@ Suppose we have a function $f$ now of two (or more) variables and we want to det __Partial Derivative__: Let $f$ be a function of the variables $(x_1,\ldots,x_n)$. The partial derivative of $f$ with respect to $x_i$ is -\[\frac{\partial f}{\partial x_i} (x_1,\ldots,x_n) = \lim\limits_{h\to 0} \frac{f(x_1,\ldots,x_i+h,\ldots,x_n)-f(x_1,\ldots,x_i,\ldots,x_n)}{h}\] +$$\frac{\partial f}{\partial x_i} (x_1,\ldots,x_n) = \lim\limits_{h\to 0} \frac{f(x_1,\ldots,x_i+h,\ldots,x_n)-f(x_1,\ldots,x_i,\ldots,x_n)}{h}$$ Only the $i$th variable changes --- the others are treated as constants. @@ -470,14 +470,14 @@ Specifically, a Taylor series of a real or complex function $f(x)$ that is infi __Taylor Approximation__: We can often approximate the curvature of a function $f(x)$ at point $a$ using a 2nd order Taylor polynomial around point $a$: -\[f(x) = f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 -+ R_2\] +$$f(x) = f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 ++ R_2$$ $R_2$ is the remainder (R for remainder, 2 for the fact that we took two derivatives) and often treated as negligible, giving us: -\[f(x) \approx f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2} (x-a)^2\] +$$f(x) \approx f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2} (x-a)^2$$ The more derivatives that are added, the smaller the remainder $R$ and the more accurate the approximation. Proofs involving limits guarantee that the remainder converges to 0 as the order of derivation increases. @@ -493,7 +493,7 @@ So far, we've been interested in finding the derivative $f=F'$ of a function $F$ #| name: Antiderivative The antiverivative of a function $f(x)$ is a differentiable function $F$ whose derivative is $f$. -\[F^\prime = f.\] +$$F^\prime = f.$$ ``` @@ -521,7 +521,7 @@ We know from derivatives how to manipulate $F$ to get $f$. But how do you expres #| name: Indefinite Integral The indefinite integral of $f(x)$ is written -\[\int f(x) dx \] +$$\int f(x) dx $$ and is equal to the antiderivative of $f$. @@ -532,7 +532,7 @@ and is equal to the antiderivative of $f$. Draw the function $f(x)$ and its indefinite integral, $\int\limits f(x) dx$ -\[f(x) = (x^2-4)\] +$$f(x) = (x^2-4)$$ ``` @@ -642,7 +642,7 @@ This is how we define the "Definite" Integral: If for a given function $f$ the Riemann sum approaches a limit as $\Delta x \to 0$, then that limit is called the Riemann integral of $f$ from $a$ to $b$. We express this with the $\int$, symbol, and write $$\int\limits_a^b f(x) dx= \lim\limits_{\Delta x\to 0} \sum\limits_{i=1}^n f(x_i)\Delta x$$ The most straightforward of a definite integral is the definite integral. That is, we read -\[\int\limits_a^b f(x) dx\] as the definite integral of $f$ from $a$ to $b$ and we defined as the area under the "curve" $f(x)$ from point $x=a$ to $x=b$. +$$\int\limits_a^b f(x) dx$$ as the definite integral of $f$ from $a$ to $b$ and we defined as the area under the "curve" $f(x)$ from point $x=a$ to $x=b$. ``` The fundamental theorem of calculus shows us that this sum is, in fact, the antiderivative. @@ -650,9 +650,9 @@ The fundamental theorem of calculus shows us that this sum is, in fact, the anti ```{theorem} #| name: First Fundamental Theorem of Calculus Let the function $f$ be bounded on $[a,b]$ and continuous on $(a,b)$. Then, suggestively, use the symbol $F(x)$ to denote the definite integral from $a$ to $x$: -\[F(x)=\int\limits_a^x f(t)dt, \quad a\le x\le b\] +$$F(x)=\int\limits_a^x f(t)dt, \quad a\le x\le b$$ -Then $F(x)$ has a derivative at each point in $(a,b)$ and \[F^\prime(x)=f(x), \quad a 0 & \Rightarrow & \text{Convex} -\end{array}\] +\end{array}$$ ### Quadratic Forms {.unnumbered} @@ -283,7 +283,7 @@ which can be written in matrix terms: One variable -\[Q(\mathbf{x}) = x_1^\top a_{11} x_1\] +$$Q(\mathbf{x}) = x_1^\top a_{11} x_1$$ N variables: \begin{align*} @@ -333,9 +333,9 @@ In a derivative, we only took the derivative with respect to one variable at a t Given a function $f(\textbf{x})$ in $n$ variables, the gradient $\nabla f(\mathbf{x})$ (the greek letter nabla ) is a column vector, where the $i$th element is the partial derivative of $f(\textbf{x})$ with respect to $x_i$: -\[\nabla f(\mathbf{x}) = \begin{pmatrix} +$$\nabla f(\mathbf{x}) = \begin{pmatrix} \frac{\partial f(\mathbf{x})}{\partial x_1}\\ \frac{\partial f(\mathbf{x})}{\partial x_2}\\ - \vdots \\ \frac{\partial f(\mathbf{x})}{\partial x_n} \end{pmatrix}\] + \vdots \\ \frac{\partial f(\mathbf{x})}{\partial x_n} \end{pmatrix}$$ ``` @@ -377,7 +377,7 @@ Critical Point $\mathbf{x}^* =$ \end{align*} -So \[\mathbf{x}^* = (1,0)\] +So $$\mathbf{x}^* = (1,0)$$ ``` @@ -395,14 +395,14 @@ Given a function $f(\mathbf{x})$ in $n$ variables, the hessian $\mathbf{H(x)}$ i an $n\times n$ matrix, where the $(i,j)$th element is the second order partial derivative of $f(\mathbf{x})$ with respect to $x_i$ and $x_j$: -\[\mathbf{H(x)}=\begin{pmatrix} +$$\mathbf{H(x)}=\begin{pmatrix} \frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}&\frac{\partial^2f(\mathbf{x})}{\partial x_1 \partial x_2}& -\cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_1 \partial x_n}\\[9pt] +\cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_1 \partial x_n}\$$9pt] \frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_2^2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_n}\\ -\vdots & \vdots & \ddots & \vdots \\[3pt] +\vdots & \vdots & \ddots & \vdots \$$3pt] \frac{\partial^2 f(\mathbf{x})}{\partial x_n \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_n \partial x_2}& -\cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_n^2}\end{pmatrix}\] +\cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_n^2}\end{pmatrix}$$ ``` @@ -662,7 +662,7 @@ $$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ 3. Solve the system of equations: Using the first two partials, we see that $\lambda = -2x_1$ and $\lambda = -4x_2$ Set these equal to see that $x_1 = 2x_2$. - Using the third partial and the above equality, $4 = 2x_2 + x_2$ from which we get \[x_2^* = 4/3, x_1^* = 8/3, \lambda = -16/3\] + Using the third partial and the above equality, $4 = 2x_2 + x_2$ from which we get $$x_2^* = 4/3, x_1^* = 8/3, \lambda = -16/3$$ 4. Therefore, the only critical point is $x_1^* = \frac{8}{3}$ and $x_2^* = \frac{4}{3}$ 5. This gives $f(\frac{8}{3}, \frac{4}{3}) = -\frac{96}{9}$, which is less than the unconstrained optimum $f(0,0) = 0$ @@ -880,7 +880,7 @@ In a two-dimensional set-up, this means we must check the following cases: ```{example} #| name: Kuhn-Tucker with two variables Solve the following optimization problem with inequality constraints -\[\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)\] +$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)$$ \begin{align*} \text{ s.t. } diff --git a/06_probability.qmd b/06_probability.qmd index c11197c..dc07e38 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -104,8 +104,8 @@ __Empty Set__: a set with no elements. $S = \{\}$. It is denoted by the symbol Set operations: -1. __Union__: The union of two sets $A$ and $B$, $A \cup B$, is the set containing all of the elements in $A$ or $B$. \[A_1 \cup A_2 \cup \cdots \cup A_n = \bigcup_{i=1}^n A_i\] -2. __Intersection__: The intersection of sets $A$ and $B$, $A \cap B$, is the set containing all of the elements in both $A$ and $B$. \[A_1 \cap A_2 \cap \cdots \cap A_n = \bigcap_{i=1}^n A_i\] +1. __Union__: The union of two sets $A$ and $B$, $A \cup B$, is the set containing all of the elements in $A$ or $B$. $$A_1 \cup A_2 \cup \cdots \cup A_n = \bigcup_{i=1}^n A_i$$ +2. __Intersection__: The intersection of sets $A$ and $B$, $A \cap B$, is the set containing all of the elements in both $A$ and $B$. $$A_1 \cap A_2 \cap \cdots \cap A_n = \bigcap_{i=1}^n A_i$$ 3. __Complement__: If set $A$ is a subset of $S$, then the complement of $A$, denoted $A^C$, is the set containing all of the elements in $S$ that are not in $A$. @@ -195,8 +195,8 @@ The axioms of probability make sure that the separate events add up in terms of 1. For any event $A$, $P(A)\ge 0$. 2. $P(S)=1$ -3. The Countable Additivity Axiom: For any sequence of _disjoint_ (mutually exclusive) events $A_1,A_2,\ldots$ (of which there may be infinitely many), \[P\left( \bigcup\limits_{i=1}^k -A_i\right)=\sum\limits_{i=1}^k P(A_i)\] +3. The Countable Additivity Axiom: For any sequence of _disjoint_ (mutually exclusive) events $A_1,A_2,\ldots$ (of which there may be infinitely many), $$P\left( \bigcup\limits_{i=1}^k +A_i\right)=\sum\limits_{i=1}^k P(A_i)$$ The last axiom is an extension of a union to infinite sets. When there are only two events in the space, it boils down to: @@ -266,7 +266,7 @@ Suppose you had a pair of four-sided dice. You sum the results from a single tos __Conditional Probability__: The conditional probability $P(A|B)$ of an event $A$ is the probability of $A$, given that another event $B$ has occurred. Conditional probability allows for the inclusion of other information into the calculation of the probability of an event. It is calculated as -\[P(A|B)=\frac{P(A\cap B)}{P(B)}\] +$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$ Note that conditional probabilities are probabilities and must also follow the Kolmagorov axioms of probability. @@ -311,7 +311,7 @@ You could rearrange the fraction to highlight how a joint probability could be e The probability of the intersection of two events $A$ and $B$ is $P(A\cap B)=P(A)P(B|A)=P(B)P(A|B)$ which follows directly from the definition of conditional probability. More generally, -\[P(A_1\cap \cdots\cap A_k) = P(A_k| A_{k-1}\cap \cdots \cap A_1)\times P(A_{k-1}|A_{k-2}\cap \cdots A_1) \times \ldots \times P(A_2|A_1)\times P(A_1)\] +$$P(A_1\cap \cdots\cap A_k) = P(A_k| A_{k-1}\cap \cdots \cap A_1)\times P(A_{k-1}|A_{k-2}\cap \cdots A_1) \times \ldots \times P(A_2|A_1)\times P(A_1)$$ Sometimes it is easier to calculate these conditional probabilities and sum them than it is to calculate $P(A)$ directly. @@ -321,7 +321,7 @@ Sometimes it is easier to calculate these conditional probabilities and sum them ```{definition} #| name: Law of Total Probability -Let $S$ be the sample space of some experiment and let the disjoint $k$ events $B_1,\ldots,B_k$ partition $S$, such that $P(B_1\cup ... \cup B_k) = P(S) = 1$. If $A$ is some other event in $S$, then the events $A\cap B_1, A\cap B_2, \ldots, A\cap B_k$ will form a partition of $A$ and we can write $A$ as \[A=(A\cap B_1)\cup\cdots\cup (A\cap B_k)\]. +Let $S$ be the sample space of some experiment and let the disjoint $k$ events $B_1,\ldots,B_k$ partition $S$, such that $P(B_1\cup ... \cup B_k) = P(S) = 1$. If $A$ is some other event in $S$, then the events $A\cap B_1, A\cap B_2, \ldots, A\cap B_k$ will form a partition of $A$ and we can write $A$ as $$A=(A\cap B_1)\cup\cdots\cup (A\cap B_k)$$. Since the $k$ events are disjoint, @@ -335,10 +335,10 @@ P(A)&=&\sum\limits_{i=1}^k P(A \cap B_i)\\ __Bayes Rule__: Assume that events $B_1,\ldots,B_k$ form a partition of the space $S$. Then by the Law of Total Probability -\[P(B_j|A)= \frac{P(A \cap B_j)} {P(A)} = \frac{P(B_j) P(A|B_j)}{\sum\limits_{i=1}^k P(B_i)P(A|B_i)}\] +$$P(B_j|A)= \frac{P(A \cap B_j)} {P(A)} = \frac{P(B_j) P(A|B_j)}{\sum\limits_{i=1}^k P(B_i)P(A|B_i)}$$ If there are only two states of $B$, then this is just -\[P(B_1|A)=\frac{P(B_1)P(A|B_1)} {P(B_1)P(A|B_1)+P(B_2)P(A|B_2)}\] +$$P(B_1|A)=\frac{P(B_1)P(A|B_1)} {P(B_1)P(A|B_1)+P(B_2)P(A|B_2)}$$ Bayes' rule determines the posterior probability of a state $P(B_j|A)$ by calculating the probability $P(A \cap B_j)$ that both the event $A$ and the state $B_j$ will occur and dividing it by the probability that the event will occur regardless of the state (by summing across all $B_i$). The states could be something like Normal/Defective, Healthy/Diseased, Republican/Democrat/Independent, etc. The event on which one conditions could be something like a sampling from a batch of components, a test for a disease, or a question about a policy position. @@ -505,7 +505,7 @@ Example: For a fair six-sided die, there is an equal probability of rolling any -In a discrete random variable, __cumulative density function__ (Also referred to simply as the "cumulative distribution" or previously as the "density function"), $F(x)$ or $P(X\le x)$, is the probability that $X$ is less than or equal to some value $x$, or \[P(X\le x)=\sum\limits_{i\le x} p(i)\] +In a discrete random variable, __cumulative density function__ (Also referred to simply as the "cumulative distribution" or previously as the "density function"), $F(x)$ or $P(X\le x)$, is the probability that $X$ is less than or equal to some value $x$, or $$P(X\le x)=\sum\limits_{i\le x} p(i)$$ Properties a CDF must satisfy: @@ -546,8 +546,8 @@ $X$ is a continuous random variable if there exists a nonnegative function $f(x) #| name: Probability Density Function The function $f$ above is called the probability density function (pdf) of $X$ and must satisfy -\[f(x)\ge 0\] -\[\int\limits_{-\infty}^\infty f(x)dx=1\] +$$f(x)\ge 0$$ +$$\int\limits_{-\infty}^\infty f(x)dx=1$$ Note also that $P(X = x)=0$ --- i.e., the probability of any point $y$ is zero. ``` @@ -568,9 +568,9 @@ Because the probability that a continuous random variable will assume any partic ``` We can also make statements about the probability of $Y$ falling in an interval $a\le y\le b$. -\[P(a\le x\le b)=\int\limits_a^b f(x)dx\] +$$P(a\le x\le b)=\int\limits_a^b f(x)dx$$ -The PDF and CDF are linked by the integral: The CDF of the integral of the PDF: \[f(x) = F'(x)=\frac{dF(x)}{dx}\] +The PDF and CDF are linked by the integral: The CDF of the integral of the PDF: $$f(x) = F'(x)=\frac{dF(x)}{dx}$$ ```{example} @@ -590,18 +590,18 @@ __Joint Probability Distribution__: If both $X$ and $Y$ are random variable, the Discrete: -\[p(x, y) = P(X = x, Y = y)\] +$$p(x, y) = P(X = x, Y = y)$$ -such that $p(x,y) \in [0,1]$ and \[\sum\sum p(x,y) = 1\] +such that $p(x,y) \in [0,1]$ and $$\sum\sum p(x,y) = 1$$ Continuous: -\[f(x,y);P((X,Y) \in A) = \int\!\!\!\int_A f(x,y)dx dy \] +$$f(x,y);P((X,Y) \in A) = \int\!\!\!\int_A f(x,y)dx dy $$ s.t. $f(x,y)\ge 0$ and -\[\int_{-\infty}^\infty\int_{-\infty}^\infty f(x,y)dxdy = 1\] +$$\int_{-\infty}^\infty\int_{-\infty}^\infty f(x,y)dxdy = 1$$ If X and Y are independent, then $P(X=x,Y=y) = P(X=x)P(Y=y)$ and $f(x,y) = f(x)f(y)$ @@ -645,7 +645,7 @@ We often want to summarize some characteristics of the distribution of a random ```{definition} #| name: Expectation of a Discrete Random Variable -The expected value of a discrete random variable $Y$ is \[E(Y)=\sum\limits_{y} y P(Y=y)= \sum\limits_{y} y p(y)\] +The expected value of a discrete random variable $Y$ is $$E(Y)=\sum\limits_{y} y P(Y=y)= \sum\limits_{y} y p(y)$$ In words, it is the weighted average of all possible values of $Y$, weighted by the probability that $y$ occurs. It is not necessarily the number we would expect $Y$ to take on, but the average value of $Y$ after a large number of repetitions of an experiment. ``` @@ -668,7 +668,7 @@ value of a continuous random variable is similar in concept to that of the discrete random variable, except that instead of summing using probabilities as weights, we integrate using the density to weight. Hence, the expected value of the continuous variable $Y$ is defined by -\[E(Y)=\int\limits_{y} y f(y) dy\] +$$E(Y)=\int\limits_{y} y f(y) dy$$ ```{example} @@ -690,25 +690,25 @@ Remember: An Expected Value is a type of weighted average. We can extend this to If $Y$ is Discrete with PMF $p(y)$, -\[E[g(Y)]=\sum\limits_y g(y)p(y)\] +$$E[g(Y)]=\sum\limits_y g(y)p(y)$$ If $Y$ is Continuous with PDF $f(y)$, -\[E[g(Y)]=\int\limits_{-\infty}^\infty g(y)f(y)dy\] +$$E[g(Y)]=\int\limits_{-\infty}^\infty g(y)f(y)dy$$ ### Properties of Expected Values {.unnumbered} Dealing with Expectations is easier when the thing inside is a sum. The intuition behind this that Expectation is an integral, which is a type of sum. -1. Expectation of a constant is a constant \[E(c)=c\] -2. Constants come out \[E(c g(Y))= c E(g(Y))\] -4. Expectation is Linear \[E(g(Y_1) + \cdots + g(Y_n))=E(g(Y_1)) +\cdots+E(g(Y_n)),\] regardless of independence -2. Expected Value of Expected Values: \[E(E(Y)) = E(Y)\] (because the expected value of a random variable is a constant) +1. Expectation of a constant is a constant $$E(c)=c$$ +2. Constants come out $$E(c g(Y))= c E(g(Y))$$ +4. Expectation is Linear $$E(g(Y_1) + \cdots + g(Y_n))=E(g(Y_1)) +\cdots+E(g(Y_n)),$$ regardless of independence +2. Expected Value of Expected Values: $$E(E(Y)) = E(Y)$$ (because the expected value of a random variable is a constant) Finally, if $X$ and $Y$ are independent, even products are easy: -\[E(XY) = E(X)E(Y)\] +$$E(XY) = E(X)E(Y)$$ @@ -730,10 +730,10 @@ We can also look at other summaries of the distribution, which build on the idea #| name: Variance The Variance of a Random Variable $Y$ is - \[\text{Var}(Y) = E[(Y - E(Y))^2] = E(Y^2)-[E(Y)]^2\] + $$\text{Var}(Y) = E[(Y - E(Y))^2] = E(Y^2)-[E(Y)]^2$$ -The Standard Deviation is the square root of the variance : \[SD(Y) = \sigma_Y= \sqrt{\text{Var}(Y)}\] +The Standard Deviation is the square root of the variance : $$SD(Y) = \sigma_Y= \sqrt{\text{Var}(Y)}$$ ``` @@ -745,11 +745,11 @@ The Standard Deviation is the square root of the variance : \[SD(Y) = \sigma_Y= Given the following PMF: -\[f(x) = \begin{cases} +$$f(x) = \begin{cases} \frac{3!}{x!(3-x)!}(\frac{1}{2})^3 \quad x = 0,1,2,3\\ 0 \quad otherwise \end{cases} - \] + $$ What is $\text{Var}(x)$? @@ -767,7 +767,7 @@ __Hint:__ First calculate $E(X)$ and $E(X^2)$ The covariance measures the degree to which two random variables vary together; if the covariance between $X$ and $Y$ is positive, X tends to be larger than its mean when Y is larger than its mean. -\[\text{Cov}(X,Y) = E[(X - E(X))(Y - E(Y))] \] +$$\text{Cov}(X,Y) = E[(X - E(X))(Y - E(Y))] $$ We can also write this as \begin{align*} @@ -787,7 +787,7 @@ The Covariance is unfortunately hard to interpret in magnitude. The correlation #| name: Correlation The correlation coefficient is the covariance divided by the standard deviations of $X$ and $Y$. It is a unitless measure and always takes on values in the interval $[-1,1]$. -\[\text{Corr}(X, Y) = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}} = \frac{\text{Cov}(X,Y)}{SD(X)SD(Y)}\] +$$\text{Corr}(X, Y) = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}} = \frac{\text{Cov}(X,Y)}{SD(X)SD(Y)}$$ ``` @@ -840,11 +840,11 @@ Define the random variable $Y = X^2$. 1. Find the expectation and variance Given the following PDF: -\[f(x) = \begin{cases} +$$f(x) = \begin{cases} \frac{3}{10}(3x - x^2) \quad 0 \leq x \leq 2\\ 0 \quad otherwise \end{cases} - \] + $$ @@ -857,12 +857,12 @@ Given the following PDF: 1. Find the mean and standard deviation of random variable X. The PDF of this X is as follows: - \[f(x) = \begin{cases} + $$f(x) = \begin{cases} \frac{1}{4}x \quad 0 \leq x \leq 2\\ \frac{1}{4}(4 - x) \quad 2 \leq x \leq 4\\ 0 \quad otherwise \end{cases} - \] + $$ 2. Next, calculate $P(X < \mu - \sigma)$ Remember, $\mu$ is the mean and $\sigma$ is the standard deviation @@ -899,7 +899,7 @@ Republicans vote for Democrat-sponsored bills 2\% of the time. What is the proba #| name: Poisson Distribution A random variable $Y$ has a Poisson distribution if -\[P(Y = y)=\frac{\lambda^y}{y!}e^{-\lambda}, \quad y=0,1,2,\ldots, \quad \lambda>0\] +$$P(Y = y)=\frac{\lambda^y}{y!}e^{-\lambda}, \quad y=0,1,2,\ldots, \quad \lambda>0$$ The Poisson has the unusual feature that its expectation equals its variance: $E(Y)=\text{Var}(Y)=\lambda$. The Poisson distribution is often used to model rare event counts: counts of the number of events that occur during some unit of time. $\lambda$ is often called the "arrival rate." @@ -945,7 +945,7 @@ For $Y$ uniformly distributed over $(1,3)$, what are the following probabilities A random variable $Y$ is normally distributed with mean $E(Y)=\mu$ and variance $\text{Var}(Y)=\sigma^2$ if its density is -\[f(y)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(y-\mu)^2}{2\sigma^2}}\] +$$f(y)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(y-\mu)^2}{2\sigma^2}}$$ ``` @@ -981,7 +981,7 @@ So far, we've talked about distributions in a theoretical sense, looking at diff __Sample mean__: This is the most common measure of central tendency, calculated by summing across the observations and dividing by the number of observations. -\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i\] +$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i$$ The sample mean is an _estimate_ of the expected value of a distribution. @@ -1062,15 +1062,15 @@ We are now finally ready to revisit, with a bit more precise terms, the two pill #| name: Central Limit Theorem (i.i.d. case) Let $\{X_n\} = \{X_1, X_2, \ldots\}$ be a sequence of i.i.d. random variables with finite mean ($\mu$) and variance ($\sigma^2$). Then, the sample mean $\bar{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n}$ increasingly converges into a Normal distribution. -\[\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),\] +$$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),$$ Another way to write this as a probability statement is that for all real numbers $a$, $$P\left(\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \le a\right) \rightarrow \Phi(a)$$ -as $n\to \infty$, where \[\Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \, dx\] is the CDF of a Normal distribution with mean 0 and variance 1. +as $n\to \infty$, where $$\Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \, dx$$ is the CDF of a Normal distribution with mean 0 and variance 1. This result means that, as $n$ grows, the distribution of the sample mean $\bar X_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n)$ is approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt n}$, i.e., -\[\bar{X}_n \approx \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).\] The standard deviation of $\bar X_n$ (which is roughly a measure of the precision of $\bar X_n$ as an estimator of $\mu$) decreases at the rate $1/\sqrt{n}$, so, for example, to increase its precision by $10$ (i.e., to get one more digit right), one needs to collect $10^2=100$ times more units of data. +$$\bar{X}_n \approx \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).$$ The standard deviation of $\bar X_n$ (which is roughly a measure of the precision of $\bar X_n$ as an estimator of $\mu$) decreases at the rate $1/\sqrt{n}$, so, for example, to increase its precision by $10$ (i.e., to get one more digit right), one needs to collect $10^2=100$ times more units of data. Intuitively, this result also justifies that whenever a lot of small, independent processes somehow combine together to form the realized observations, practitioners often feel comfortable assuming Normality. @@ -1080,7 +1080,7 @@ Intuitively, this result also justifies that whenever a lot of small, independen #| name: Law of Large Numbers (LLN) For any draw of independent random variables with the same mean $\mu$, the sample average after $n$ draws, $\bar{X}_n = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)$, converges in probability to the expected value of $X$, $\mu$ as $n \rightarrow \infty$: -\[\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0\] +$$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ A shorthand of which is \(\bar{X}_n \xrightarrow{p} \mu\), where the arrow is read as "converges in probability to". @@ -1103,9 +1103,9 @@ Example. What is $\mathcal{O}( 5\exp(0.5 n) + n^2 + n / 2)$? Answer: $\exp(n)$. Why? Because, for large $n$, -\[ +$$ \frac{ 5\exp(0.5 n) + n^2 + n / 2 }{ \exp(n)} \leq \frac{ c \exp(n) }{ \exp(n)} = c. -\] +$$ whenever $n > 4$ and where $c = 1$. \begin{comment} diff --git a/07_linear-algebra.qmd b/07_linear-algebra.qmd index daf551f..7ac4d62 100644 --- a/07_linear-algebra.qmd +++ b/07_linear-algebra.qmd @@ -126,12 +126,12 @@ Are the following sets of vectors linearly independent? __Matrix__: A matrix is an array of real numbers arranged in $m$ rows by $n$ columns. The dimensionality of the matrix is defined as the number of rows by the number of columns, $m \times n$. -\[{\bf A}=\begin{pmatrix} +$${\bf A}=\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} - \end{pmatrix}\] + \end{pmatrix}$$ Note that you can think of vectors as special cases of matrices; a column vector of length $k$ is a $k \times 1$ matrix, while a row vector of the same length is a $1 \times k$ matrix. @@ -296,10 +296,10 @@ We are often interested in solving linear systems like -\[\begin{matrix} +$$\begin{matrix} x & - & 3y & = & -3\\ 2x & + & y & = & 8 - \end{matrix}\] + \end{matrix}$$ \begin{comment} \parbox[t]{1in}{\includegraphics[angle=270, width = 1in]{linsys.eps}} @@ -307,12 +307,12 @@ We are often interested in solving linear systems like More generally, we might have a system of $m$ equations in $n$ unknowns -\[\begin{matrix} +$$\begin{matrix} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ \vdots & & & & \vdots & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m - \end{matrix}\] + \end{matrix}$$ A __solution__ to a linear system of $m$ equations in $n$ unknowns is a set of $n$ numbers $x_1, x_2, \cdots, x_n$ that satisfy each of the $m$ equations. @@ -489,21 +489,21 @@ $\begin{matrix} Put the following system of equations into augmented matrix form. Then, using Gaussian or Gauss-Jordan elimination, solve the system of equations by putting the matrix into row echelon or reduced row echelon form. -\[ +$$ 1. \begin{cases} x + y + 2z = 2\\ 3x - 2y + z = 1\\ y - z = 3 \end{cases} - \] + $$ -\[ +$$ 2. \begin{cases} 2x + 3y - z = -8\\ x + 2y - z = 12\\ -x -4y + z = -6 \end{cases} - \] + $$ @@ -646,7 +646,7 @@ Find the inverse of the following matrix: Let's return to the matrix representation of a linear system -\[\bf{Ax} = \bf{b}\] +$$\bf{Ax} = \bf{b}$$ If $\bf{A}$ is an $n\times n$ matrix,then $\bf{Ax}=\bf{b}$ is a system of $n$ equations in $n$ unknowns. Suppose $\bf{A}$ is nonsingular. Then $\bf{A}^{-1}$ exists. To solve this system, we can multiply each side by $\bf{A}^{-1}$ and reduce it as follows: @@ -677,11 +677,11 @@ ___Hint: the linear system above can be written in the matrix form___ $\textbf{A}\textbf{z} = \textbf{b}$ -given \[\textbf{A} = \begin{pmatrix} -3&4\\2&-1 \end{pmatrix},\] +given $$\textbf{A} = \begin{pmatrix} -3&4\\2&-1 \end{pmatrix},$$ -\[\textbf{z} = \begin{pmatrix} x\\y \end{pmatrix},\] +$$\textbf{z} = \begin{pmatrix} x\\y \end{pmatrix},$$ and -\[\textbf{b} = \begin{pmatrix} 5\\-10 \end{pmatrix}\] +$$\textbf{b} = \begin{pmatrix} 5\\-10 \end{pmatrix}$$ @@ -782,10 +782,10 @@ Hence, we can examine how changes in the parameters and $b_i$ affect the solutio __Determinant Formula for the Inverse of a $2 \times 2$__: The determinant of a $2 \times 2$ matrix __A__ $\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}$ is defined as: -\[\frac{1}{\det({\bf A})} \begin{pmatrix} +$$\frac{1}{\det({\bf A})} \begin{pmatrix} d & -b\\ -c & a\\ - \end{pmatrix}\] + \end{pmatrix}$$ For example, Let's calculate the inverse of matrix A from Exercise \@ref(exr:invlinsys) using the determinant formula. @@ -793,27 +793,27 @@ For example, Let's calculate the inverse of matrix A from Exercise \@ref(exr:inv Recall, -\[A = \begin{pmatrix} +$$A = \begin{pmatrix} -3 & 4\\ 2 & -1\\ - \end{pmatrix}\] + \end{pmatrix}$$ -\[\det({\bf A}) = (-3)(-1) - (4)(2) = 3 - 8 = -5\] +$$\det({\bf A}) = (-3)(-1) - (4)(2) = 3 - 8 = -5$$ -\[\frac{1}{\det({\bf A})} \begin{pmatrix} +$$\frac{1}{\det({\bf A})} \begin{pmatrix} -1 & -4\\ -2 & -3\\ - \end{pmatrix}\] + \end{pmatrix}$$ -\[\frac{1}{-5} \begin{pmatrix} +$$\frac{1}{-5} \begin{pmatrix} -1 & -4\\ -2 & -3\\ - \end{pmatrix}\] + \end{pmatrix}$$ -\[ \begin{pmatrix} +$$ \begin{pmatrix} \frac{1}{5} & \frac{4}{5}\\ \frac{2}{5} & \frac{3}{5}\\ - \end{pmatrix}\] + \end{pmatrix}$$ ```{exercise} @@ -895,20 +895,20 @@ Answer to Exercise \@ref(exr:lineareq): There are many answers to this. Some possible simple ones are as follows: -1. One solution: \[\begin{matrix} +1. One solution: $$\begin{matrix} -x & + & y & = & 0\\ x & + & y & = & 2 - \end{matrix}\] + \end{matrix}$$ -2. No solution: \[\begin{matrix} +2. No solution: $$\begin{matrix} -x & + & y & = & 0\\ x & - & y & = & 2 - \end{matrix}\] + \end{matrix}$$ -3. Infinite solutions: \[\begin{matrix} +3. Infinite solutions: $$\begin{matrix} -x & + & y & = & 0\\ 2x & - & 2y & = & 0 - \end{matrix}\] + \end{matrix}$$ diff --git a/12_matricies-manipulation.qmd b/12_matricies-manipulation.qmd index d797bd7..ac3c56b 100644 --- a/12_matricies-manipulation.qmd +++ b/12_matricies-manipulation.qmd @@ -424,12 +424,12 @@ Get all the serial numbers for black, male individuals who don't live in Ohio or ### 1 {.unnumbered} Let -\[\mathbf{A} = \left[\begin{array} +$$\mathbf{A} = \left[\begin{array} {rrr} 0.6 & 0.2\\ 0.4 & 0.8\\ \end{array}\right] -\] +$$ Use R to write code that will create the matrix $A$, and then consecutively multiply $A$ to itself 4 times. What is the value of $A^{4}$? diff --git a/17_non-wysiwyg.qmd b/17_non-wysiwyg.qmd index ae9ac5a..32f9227 100644 --- a/17_non-wysiwyg.qmd +++ b/17_non-wysiwyg.qmd @@ -164,9 +164,9 @@ Notice that each line starts with a backslash `\` -- in LaTeX this is the symbol The following syntax at the endpoints are shorthand for math equations. ``` -\[\int x^2 dx\] +$$\int x^2 dx$$ ``` -these compile math symbols: $\displaystyle \int x^2 dx.$^[Enclosing with `$$` instead of `\[` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.] +these compile math symbols: $\displaystyle \int x^2 dx.$^[Enclosing with `$$` instead of `$$` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.] The `align` environment is useful to align your multi-line math, for example. diff --git a/17_non-wysiwyg.quarto_ipynb b/17_non-wysiwyg.quarto_ipynb index e9e3577..ad04ee2 100644 --- a/17_non-wysiwyg.quarto_ipynb +++ b/17_non-wysiwyg.quarto_ipynb @@ -182,9 +182,9 @@ "\n", "The following syntax at the endpoints are shorthand for math equations.\n", "```\n", - "\\[\\int x^2 dx\\]\n", + "\$$\\int x^2 dx\$$\n", "```\n", - "these compile math symbols: $\\displaystyle \\int x^2 dx.$^[Enclosing with `$$` instead of `\\[` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.]\n", + "these compile math symbols: $\\displaystyle \\int x^2 dx.$^[Enclosing with `$$` instead of `\$$` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.]\n", "\n", "\n", "The `align` environment is useful to align your multi-line math, for example. \n", @@ -330,4 +330,4 @@ }, "nbformat": 4, "nbformat_minor": 5 -} \ No newline at end of file +} diff --git a/21_solutions-warmup.qmd b/21_solutions-warmup.qmd index 452f9f2..346b967 100644 --- a/21_solutions-warmup.qmd +++ b/21_solutions-warmup.qmd @@ -23,7 +23,7 @@ Are the following sets of vectors linearly independent? 1. $u = \begin{pmatrix} 1\\ 2\end{pmatrix}$, $v = \begin{pmatrix} 2\\4\end{pmatrix}$ -$\leadsto$ No: \[2u = \begin{pmatrix} 2\\ 4\end{pmatrix}, v = \begin{pmatrix} 2\\ 4\end{pmatrix}\] +$\leadsto$ No: $$2u = \begin{pmatrix} 2\\ 4\end{pmatrix}, v = \begin{pmatrix} 2\\ 4\end{pmatrix}$$ so infinitely many linear combinations of $u$ and $v$ that amount to 0 exist. @@ -36,10 +36,10 @@ $\leadsto$ Yes: we cannot find linear combination of these two vectors that wou 3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ $\leadsto$ No: After playing around with some numbers, we can find that -\[-2a = \begin{pmatrix} -4\\ 2\\ -2 \end{pmatrix}, 3b = \begin{pmatrix} 9\\ -12\\ -6 \end{pmatrix}, -1c = \begin{pmatrix} -5\\ 10\\ 8 \end{pmatrix}\] +$$-2a = \begin{pmatrix} -4\\ 2\\ -2 \end{pmatrix}, 3b = \begin{pmatrix} 9\\ -12\\ -6 \end{pmatrix}, -1c = \begin{pmatrix} -5\\ 10\\ 8 \end{pmatrix}$$ So -\[-2a + 3b - c = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\] +$$-2a + 3b - c = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ i.e., a linear combination of these three vectors that would amount to zero exists. @@ -49,12 +49,12 @@ If you are having trouble with these problems, please review Section \@ref(linea ### Matrices {.unnumbered} -\[{\bf A}=\begin{pmatrix} +$${\bf A}=\begin{pmatrix} 7 & 5 & 1 \\ 11 & 9 & 3 \\ 2 & 14 & 21 \\ 4 & 1 & 5 - \end{pmatrix}\] + \end{pmatrix}$$ What is the dimensionality of matrix ${\bf A}$? 4 $\times$ 3 @@ -63,43 +63,43 @@ What is the element $a_{23}$ of ${\bf A}$? 3 Given that -\[{\bf B}=\begin{pmatrix} +$${\bf B}=\begin{pmatrix} 1 & 2 & 8 \\ 3 & 9 & 11 \\ 4 & 7 & 5 \\ 5 & 1 & 9 - \end{pmatrix}\] + \end{pmatrix}$$ -\[\mathbf{A} + \mathbf{B} = \begin{pmatrix} +$$\mathbf{A} + \mathbf{B} = \begin{pmatrix} 8 & 7 & 9 \\ 14 & 18 & 14 \\ 6 & 21 & 26 \\ 9 & 2 & 14 - \end{pmatrix}\] + \end{pmatrix}$$ Given that -\[{\bf C}=\begin{pmatrix} +$${\bf C}=\begin{pmatrix} 1 & 2 & 8 \\ 3 & 9 & 11 \\ 4 & 7 & 5 \\ - \end{pmatrix}\] + \end{pmatrix}$$ -\[\mathbf{A} + \mathbf{C} = \text{No solution, matrices non-conformable}\] +$$\mathbf{A} + \mathbf{C} = \text{No solution, matrices non-conformable}$$ Given that -\[c = 2\] +$$c = 2$$ -\[c\textbf{A} = \begin{pmatrix} +$$c\textbf{A} = \begin{pmatrix} 14 & 10 & 2 \\ 22 & 18 & 6 \\ 4 & 28 & 42 \\ 8 & 2 & 10 - \end{pmatrix}\] + \end{pmatrix}$$ If you are having trouble with these problems, please review Section \@ref(matrixbasics). From 59c87f3e0ec7c7edc989b9dd4a0333ebd06ae478 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 17:55:02 -0400 Subject: [PATCH 06/34] more latex quarto --- 01_warmup.qmd | 28 ++++++++++++++-------------- 03_limits.qmd | 18 +++++++++--------- 04_calculus.qmd | 6 +++--- 06_probability.qmd | 2 +- 21_solutions-warmup.qmd | 30 +++++++++++++++--------------- 5 files changed, 42 insertions(+), 42 deletions(-) diff --git a/01_warmup.qmd b/01_warmup.qmd index ea9c7e3..7404828 100644 --- a/01_warmup.qmd +++ b/01_warmup.qmd @@ -91,16 +91,16 @@ Simplify the following 1. $$\sum\limits_{i = 1}^3 i$$ -2. \(\sum\limits_{k = 1}^3(3k + 2)\) +2. $\sum\limits_{k = 1}^3(3k + 2)$ -3. \(\sum\limits_{i= 1}^4 (3k + i + 2)\) +3. $\sum\limits_{i= 1}^4 (3k + i + 2)$ ### Products {.unnumbered} -1. \(\prod\limits_{i= 1}^3 i\) +1. $\prod\limits_{i= 1}^3 i$ -2. \(\prod\limits_{k=1}^3(3k + 2)\) +2. $\prod\limits_{k=1}^3(3k + 2)$ To review this material, please see Section \@ref(sum-notation). @@ -110,16 +110,16 @@ To review this material, please see Section \@ref(sum-notation). Simplify the following -1. \(4^2\) -2. \(4^2 2^3\) -3. \(\log_{10}100\) -4. \(\log_{2}4\) -5. \(\log e\), where $\log$ is the natural log (also written as $\ln$) -- a log with base $e$, and $e$ is Euler's constant -6. \(e^a e^b e^c\), where $a, b, c$ are each constants -7. \(\log 0\) -8. \(e^0\) -9. \(e^1\) -10. \(\log e^2\) +1. $4^2$ +2. $4^2 2^3$ +3. $\log_{10}100$ +4. $\log_{2}4$ +5. $\log e$, where $\log$ is the natural log (also written as $\ln$) -- a log with base $e$, and $e$ is Euler's constant +6. $e^a e^b e^c$, where $a, b, c$ are each constants +7. $\log 0$ +8. $e^0$ +9. $e^1$ +10. $\log e^2$ To review this material, please see Section @sec-logexponents diff --git a/03_limits.qmd b/03_limits.qmd index dbdf0f6..13a4c4d 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -45,7 +45,7 @@ For any draw of identically distributed independent variables with mean $\mu$, t $$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ -A shorthand of which is \(\bar{X}_n \xrightarrow{p} \mu\), where the arrow is read as "converges in probability to". +A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". ::: @@ -173,8 +173,8 @@ After some practice, the key to intuition is whether one part of the fraction gr Find the following limits of sequences, then explain in English the intuition for why that is the case. - 1. \(\lim\limits_{n\to\infty} \frac{2n}{n^2 + 1}\) - 2. \(\lim\limits_{n\to\infty} (n^3 - 100n^2)\) + 1. $\lim\limits_{n\to\infty} \frac{2n}{n^2 + 1}$ + 2. $\lim\limits_{n\to\infty} (n^3 - 100n^2)$ ::: @@ -214,10 +214,10 @@ Simple limits of functions can be solved as we did limits of sequences. Just be Find the limit of the following functions. -1. \(\lim_{x \to c} k\) -1. \(\lim_{x \to c} x\) -1. \(\lim_{x\to 2} (2x-3)\) -1. \(\lim_{x \to c} x^n\) +1. $\lim_{x \to c} k$ +1. $\lim_{x \to c} x$ +1. $\lim_{x\to 2} (2x-3)$ +1. $\lim_{x \to c} x^n$ ::: @@ -403,8 +403,8 @@ Example @exm-limfun1 1. $k$ 2. $c$ - 3. \(\lim_{x\to 2} (2x-3) = 2\lim\limits_{x\to 2} x - 3\lim\limits_{x\to 2} 1 = 1\) - 4. \(\lim_{x \to c} x^n = \lim\limits_{x \to c} x \cdots[\lim\limits_{x \to c} x] = c\cdots c =c^n\) + 3. $\lim_{x\to 2} (2x-3) = 2\lim\limits_{x\to 2} x - 3\lim\limits_{x\to 2} 1 = 1$ + 4. $\lim_{x \to c} x^n = \lim\limits_{x \to c} x \cdots[\lim\limits_{x \to c} x] = c\cdots c =c^n$ ::: diff --git a/04_calculus.qmd b/04_calculus.qmd index f0ac59c..e93c2a4 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -587,9 +587,9 @@ Some common rules of integrals follow by virtue of being the inverse of a deriva #| name: Common Integration Simplify the following indefinite integrals: -* \(\int 3x^2 dx\) -* \(\int (2x+1)dx\) -* \(\int e^x e^{e^x} dx\) +* $\int 3x^2 dx$ +* $\int (2x+1)dx$ +* $\int e^x e^{e^x} dx$ ``` diff --git a/06_probability.qmd b/06_probability.qmd index dc07e38..c0e8496 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -1083,7 +1083,7 @@ For any draw of independent random variables with the same mean $\mu$, the sampl $$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ -A shorthand of which is \(\bar{X}_n \xrightarrow{p} \mu\), where the arrow is read as "converges in probability to". +A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". ``` diff --git a/21_solutions-warmup.qmd b/21_solutions-warmup.qmd index 346b967..c42f21a 100644 --- a/21_solutions-warmup.qmd +++ b/21_solutions-warmup.qmd @@ -111,18 +111,18 @@ If you are having trouble with these problems, please review Section \@ref(matri Simplify the following -1. \(\sum\limits_{i = 1}^3 i = 1 + 2+ 3 = 6\) +1. $\sum\limits_{i = 1}^3 i = 1 + 2+ 3 = 6$ -2. \(\sum\limits_{k = 1}^3(3k + 2) = 3\sum\limits_{k=1}^3k + \sum\limits_{k=1}^3 2= 3\times 6 + 3\times 2 = 24\) +2. $\sum\limits_{k = 1}^3(3k + 2) = 3\sum\limits_{k=1}^3k + \sum\limits_{k=1}^3 2= 3\times 6 + 3\times 2 = 24$ -3. \(\sum\limits_{i= 1}^4 (3k + i + 2) = 3\sum\limits_{i= 1}^4k + \sum\limits_{i= 1}^4i + \sum\limits_{i= 1}^42 = 12k + 10 + 8 = 12k + 18\) +3. $\sum\limits_{i= 1}^4 (3k + i + 2) = 3\sum\limits_{i= 1}^4k + \sum\limits_{i= 1}^4i + \sum\limits_{i= 1}^42 = 12k + 10 + 8 = 12k + 18$ ### Products {.unnumbered} -1. \(\prod\limits_{i= 1}^3 i = 1\cdot 2\cdot 3 = 6\) +1. $\prod\limits_{i= 1}^3 i = 1\cdot 2\cdot 3 = 6$ -2. \(\prod\limits_{k=1}^3(3k + 2) = (3 + 2)\cdot (6 + 2)\cdot (9 + 2) = 440\) +2. $\prod\limits_{k=1}^3(3k + 2) = (3 + 2)\cdot (6 + 2)\cdot (9 + 2) = 440$ To review this material, please see Section \@ref(sum-notation). @@ -132,16 +132,16 @@ To review this material, please see Section \@ref(sum-notation). Simplify the following -1. \(4^2 = 16\) -2. \(4^2 2^3 = 2^{2\cdot 2}2^{3} = 2^{4 + 3} = 128\) -3. \(\log_{10}100 = \log_{10}10^2 = 2\) -4. \(\log_{2}4 = \log_{2}2^2 = 2\) -5. when $\log$ is the natural log, \(\log e = \log_{e} e^1 = 1\) -6. when $a, b, c$ are each constants, \(e^a e^b e^c = e^{a + b + c}\), -7. \(\log 0 = \text{undefined}\) -- no exponentiation of anything will result in a 0. -8. \(e^0 = 1\) -- any number raised to the 0 is always 1. -9. \(e^1 = e\) -- any number raised to the 1 is always itself -10. \(\log e^2 = \log_e e^2 = 2\) +1. $4^2 = 16$ +2. $4^2 2^3 = 2^{2\cdot 2}2^{3} = 2^{4 + 3} = 128$ +3. $\log_{10}100 = \log_{10}10^2 = 2$ +4. $\log_{2}4 = \log_{2}2^2 = 2$ +5. when $\log$ is the natural log, $\log e = \log_{e} e^1 = 1$ +6. when $a, b, c$ are each constants, $e^a e^b e^c = e^{a + b + c}$, +7. $\log 0 = \text{undefined}$ -- no exponentiation of anything will result in a 0. +8. $e^0 = 1$ -- any number raised to the 0 is always 1. +9. $e^1 = e$ -- any number raised to the 1 is always itself +10. $\log e^2 = \log_e e^2 = 2$ To review this material, please see Section \@ref(logexponents) From d48724b5bdf9f755f9c34093262a61638d6044c1 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 20:58:34 -0400 Subject: [PATCH 07/34] clean up 4+5 except phantoms --- 04_calculus.qmd | 263 ++++++++++++++++++++++---------------------- 05_optimization.qmd | 196 +++++++++++++++------------------ _quarto.yml | 2 + 3 files changed, 220 insertions(+), 241 deletions(-) diff --git a/04_calculus.qmd b/04_calculus.qmd index e93c2a4..71aa825 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -36,26 +36,25 @@ $$E(X) = \large \sum_{x} \quad\left( \underbrace{x}_{\text{value}}\cdot \underbr The derivative of $f$ at $x$ is its rate of change at $x$: how much $f(x)$ changes with a change in $x$. The rate of change is a fraction --- rise over run --- but because not all lines are straight and the rise over run formula will give us different values depending on the range we examine, we need to take a limit (Section \@ref(limits-precalc)). -```{definition} -#| name: Derivative +:::{#def-derivative} +## Derivative Let $f$ be a function whose domain includes an open interval containing the point $x$. The derivative of $f$ at $x$ is given by -$$\frac{d}{dx}f(x) =\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x} = \lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h} -$$ +$$\frac{d}{dx}f(x) =\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x} = \lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ -There are a two main ways to denote a derivate: +There are a two main ways to denote a derivative: * Leibniz Notation: $\frac{d}{dx}(f(x))$ * Prime or Lagrange Notation: $f'(x)$ -``` +::: -If $f(x)$ is a straight line, the derivative is the slope. For a curve, the slope changes by the values of $x$, so the derivative is the slope of the line tangent to the curve at $x$. See, For example, Figure \@ref(fig:derivsimple). +If $f(x)$ is a straight line, the derivative is the slope. For a curve, the slope changes by the values of $x$, so the derivative is the slope of the line tangent to the curve at $x$. See, For example, Figure @fig-derivsimple. ```{r} -#| label: derivsimple +#| label: fig-derivsimple #| echo: false #| fig-cap: The Derivative as a Slope range <- tibble::tibble(x = c(-3, 3)) @@ -108,9 +107,8 @@ __Power rule:__ $$\left[x^k\right]^\prime = k x^{k-1}$$ For any real number $k$ (that is, both whole numbers and fractions). The power rule is proved __by induction__, a neat method of proof used in many fundamental applications to prove that a general statement holds for every possible case, even if there are countably infinite cases. We'll show a simple case where $k$ is an integer here. -```{proof} -#| label: pwrinduct -#| name: Proof of Power Rule by Induction +:::{.proof} +## Proof of Power Rule by Induction We would like to prove that @@ -141,7 +139,7 @@ Therefore, the rule holds for the case $k = m + 1$ once we have assumed it holds To show that it holds for real fractions as well, we can prove expressing that exponent by a fraction of two integers. -``` +::: @@ -150,9 +148,9 @@ To show that it holds for real fractions as well, we can prove expressing that e These "rules" become apparent by applying the definition of the derivative above to each of the things to be "derived", but these come up so frequently that it is best to repeat until it is muscle memory. -```{exercise} -#| label: introderivatives -#| name: Derivative of Polynomials +:::{#exr-introderivatives} + +## Derivative of Polynomials For each of the following functions, find the first-order derivative $f^\prime(x)$. @@ -169,7 +167,7 @@ For each of the following functions, find the first-order derivative $f^\prime(x 1. $f(x)=\frac{x^2+1}{x^2-1}$ -``` +::: ## Higher-Order Derivatives (Derivatives of Derivatives of Derivatives) {#derivpoly} @@ -188,8 +186,8 @@ Similarly, the derivative of $f''(x)$ would be called the third derivative and i -```{example} -#| name: Succession of Derivatives +:::{#exm-succession} +## Succession of Derivatives \begin{align*} f(x) &=x^3\\ f^{\prime}(x) &=3x^2\\ @@ -197,9 +195,9 @@ f^{\prime\prime}(x) &=6x \\ f^{\prime\prime\prime}(x) &=6\\ f^{\prime\prime\prime\prime}(x) &=0\\ \end{align*} -``` +::: -Earlier, in Section \@ref(derivintro), we said that if a function differentiable at a given point, then it must be continuous. Further, if $f'(x)$ is itself continuous, then $f(x)$ is called continuously differentiable. All of this matters because many of our findings about optimization (Section \@ref(optim)) rely on differentiation, and so we want our function to be differentiable in as many layers. A function that is continuously differentiable infinitly is called "smooth". Some examples: $f(x) = x^2$, $f(x) = e^x$. +Earlier, in Section @sec-derivintro, we said that if a function differentiable at a given point, then it must be continuous. Further, if $f'(x)$ is itself continuous, then $f(x)$ is called continuously differentiable. All of this matters because many of our findings about optimization (Section @sec-optim) rely on differentiation, and so we want our function to be differentiable in as many layers. A function that is continuously differentiable infinitly is called "smooth". Some examples: $f(x) = x^2$, $f(x) = e^x$. ## Composite Functions and the Chain Rule @@ -208,7 +206,7 @@ As useful as the above rules are, many functions you'll see won't fit neatly in __Composite functions__ are formed by substituting one function into another and are denoted by $$(f\circ g)(x)=f[g(x)].$$ To form $f[g(x)]$, the range of $g$ must be contained (at least in part) within the domain of $f$. The domain of $f\circ g$ consists of all the points in the domain of $g$ for which $g(x)$ is in the domain of $f$. -```{example} +:::{#exm-composite} Let $f(x)=\log x$ for $0 0$$ -``` +::: @@ -282,9 +280,9 @@ We will relegate the proofs to small excerpts. ### Derivatives of natural exponential function ($e$) {.unnumbered} -To repeat the main rule in Theorem \@ref(thm:derivexplog), the intuition is that +To repeat the main rule in Theorem @thm-derivexplog, the intuition is that -1. Derivative of $e^x$ is itself: $\frac{d}{dx}e^x = e^x$ (See Figure \@ref(fig:fig-derivexponent)) +1. Derivative of $e^x$ is itself: $\frac{d}{dx}e^x = e^x$ (See Figure @fig-derivexponent.) 2. Same thing if there were a constant in front: $\frac{d}{dx}\alpha e^x = \alpha e^x$ 3. Same thing no matter how many derivatives there are in front: $\frac{d^n}{dx^n} \alpha e^x = \alpha e^x$ 4. Chain Rule: When the exponent is a function of $x$, remember to take derivative of that function and add to product. $\frac{d}{dx}e^{g(x)}= e^{g(x)} g^\prime(x)$ @@ -310,16 +308,16 @@ fprimex <- fx0 + stat_function(fun = function(x) exp(x), size = 0.5) + fx + fprimex + plot_layout(nrow = 1) ``` -```{example} -#| label: exmderivexp -#| name: Derivative of exponents +:::{#exm-exmderivexp} + +## Derivative of exponents Find the derivative for the following. 1. $f(x)=e^{-3x}$ 2. $f(x)=e^{x^2}$ 3. $f(x)=(x-1)e^x$ -``` +::: @@ -327,7 +325,7 @@ Find the derivative for the following. The natural log is the mirror image of the natural exponent and has mirroring properties, again, to repeat the theorem, -1. log prime x is one over x: $\frac{d}{dx} \log x = \frac{1}{x}$ (Figure \@ref(fig:fig-derivlog)) +1. log prime x is one over x: $\frac{d}{dx} \log x = \frac{1}{x}$ (Figure @fig-derivlog.) 2. Exponents become multiplicative constants: $\frac{d}{dx} \log x^k = \frac{d}{dx} k \log x = \frac{k}{x}$ 3. Chain rule again: $\frac{d}{dx} \log u(x) = \frac{u'(x)}{u(x)}\quad$ 4. For any positive base $b$, $\frac{d}{dx} b^x = (\log b)\left(b^x\right)$. @@ -355,9 +353,10 @@ fprimex <- fx0 + stat_function(fun = function(x) ifelse(x <= 0, NA, 1/x), size = fx + fprimex + plot_layout(nrow = 1) ``` -```{example} -#| label: exmderivlog -#| name: Derivative of logs +:::{#exm-exmderivlog} + +## Derivative of logs + Find $dy/dx$ for the following. 1. $f(x)=\log(x^2+9)$ @@ -365,10 +364,7 @@ Find $dy/dx$ for the following. 3. $f(x)=(\log x)^2$ 4. $f(x)=\log e^x$ -``` - - - +::: ### Outline of Proof {.unnumbered} @@ -421,8 +417,8 @@ Only the $i$th variable changes --- the others are treated as constants. We can take higher-order partial derivatives, like we did with functions of a single variable, except now the higher-order partials can be with respect to multiple variables. -```{example} -#| name: More than one type of partial +:::{#exm-partialtypes} +## More than one type of partial Notice that you can take partials with regard to different variables. Suppose $f(x,y)=x^2+y^2$. Then @@ -433,10 +429,10 @@ Suppose $f(x,y)=x^2+y^2$. Then \frac{\partial^2 f}{\partial x^2}(x,y) &=\\ \frac{\partial^2 f}{\partial x \partial y}(x,y) &= \end{align*} -``` +::: -```{exercise} +:::{#exr-partialderivs} Let $f(x,y)=x^3 y^4 +e^x -\log y$. What are the following partial derivaitves? \begin{align*} @@ -447,7 +443,7 @@ Let $f(x,y)=x^3 y^4 +e^x -\log y$. What are the following partial derivaitves? \end{align*} -``` +::: @@ -489,13 +485,13 @@ The more derivatives that are added, the smaller the remainder $R$ and the more So far, we've been interested in finding the derivative $f=F'$ of a function $F$. However, sometimes we're interested in exactly the reverse: finding the function $F$ for which $f$ is its derivative. We refer to $F$ as the antiderivative of $f$. -```{definition} -#| name: Antiderivative +:::{#def-antiderivative} +## Antiderivative The antiverivative of a function $f(x)$ is a differentiable function $F$ whose derivative is $f$. $$F^\prime = f.$$ -``` +::: Another way to describe is through the inverse formula. Let $DF$ be the derivative of $F$. And let $DF(x)$ be the derivative of $F$ evaluated at $x$. Then the antiderivative is denoted by $D^{-1}$ (i.e., the inverse derivative). If $DF=f$, then $F=D^{-1}f$. @@ -503,50 +499,50 @@ Another way to describe is through the inverse formula. Let $DF$ be the derivati This definition bolsters the main takeaway about integrals and derivatives: They are inverses of each other. -```{exercise} -#| name: Antiderivative +:::{#exr-antiderivatives} +## Antiderivative Find the antiderivative of the following: 1. $f(x) = \frac{1}{x^2}$ 2. $f(x) = 3e^{3x}$ -``` +::: We know from derivatives how to manipulate $F$ to get $f$. But how do you express the procedure to manipulate $f$ to get $F$? For that, we need a new symbol, which we will call indefinite integration. -```{definition} -#| name: Indefinite Integral +:::{#def-integral} +## Indefinite Integral The indefinite integral of $f(x)$ is written $$\int f(x) dx $$ and is equal to the antiderivative of $f$. -``` +::: -```{example} +:::{#exm-integralc} Draw the function $f(x)$ and its indefinite integral, $\int\limits f(x) dx$ $$f(x) = (x^2-4)$$ -``` +::: -```{solution} +:::{#sol-integralc} The Indefinite Integral of the function $f(x) = (x^2-4)$ can, for example, be $F(x) = \frac{1}{3}x^3 - 4x.$ But it can also be $F(x) = \frac{1}{3}x^3 - 4x + 1$, because the constant 1 disappears when taking the derivative. -``` +::: -Some of these functions are plotted in the bottom panel of Figure \@ref(fig:integralc) as dotted lines. +Some of these functions are plotted in the bottom panel of Figure @fig-integralc as dotted lines. ```{r} -#| label: integralc +#| label: fig-integralc #| echo: false #| fig-cap: The Many Indefinite Integrals of a Function range1 <- tibble(x = c(-4, 4)) @@ -583,8 +579,8 @@ Some common rules of integrals follow by virtue of being the inverse of a deriva 1. Remember the derivative of a log of a function: $\int \frac{f^\prime(x)}{f(x)}dx=\log f(x) + c$ -```{example} -#| name: Common Integration +:::{#exm-common-integration} +## Common Integration Simplify the following indefinite integrals: * $\int 3x^2 dx$ @@ -592,7 +588,7 @@ Simplify the following indefinite integrals: * $\int e^x e^{e^x} dx$ -``` +::: @@ -603,7 +599,7 @@ If there is a indefinite integral, there _must_ be a definite integral. Indeed t Suppose we want to determine the area $A(R)$ of a region $R$ defined by a curve $f(x)$ and some interval $a\le x \le b$. ```{r} -#| label: defintfig +#| label: fig-defintfig #| echo: false #| fig-cap: The Riemann Integral as a Sum of Evaluations f3 <- function(x) -15*(x - 5) + (x - 5)^3 + 50 @@ -633,59 +629,59 @@ One way to calculate the area would be to divide the interval $a\le x\le b$ into As we decrease the size of the subintervals $\Delta x$, making the rectangles "thinner," we would expect our approximation of the area of the region to become closer to the true area. This allows us to express the area as a limit of a series: $$A(R)=\lim\limits_{\Delta x\to 0}\sum\limits_{i=1}^n f(x_i)\Delta x$$ -Figure \@ref(fig:defintfig) shows that illustration. The curve depicted is $f(x) = -15(x - 5) + (x - 5)^3 + 50.$ We want approximate the area under the curve between the $x$ values of 0 and 10. We can do this in blocks of arbitrary width, where the sum of rectangles (the area of which is width times $f(x)$ evaluated at the midpoint of the bar) shows the Riemann Sum. As the width of the bars $\Delta x$ becomes smaller, the better the estimate of $A(R)$. +Figure @fig-defintfig shows that illustration. The curve depicted is $f(x) = -15(x - 5) + (x - 5)^3 + 50.$ We want approximate the area under the curve between the $x$ values of 0 and 10. We can do this in blocks of arbitrary width, where the sum of rectangles (the area of which is width times $f(x)$ evaluated at the midpoint of the bar) shows the Riemann Sum. As the width of the bars $\Delta x$ becomes smaller, the better the estimate of $A(R)$. This is how we define the "Definite" Integral: -```{definition} -#| name: The Definite Integral (Riemann) +:::{#def-defintegral} +## The Definite Integral (Riemann) If for a given function $f$ the Riemann sum approaches a limit as $\Delta x \to 0$, then that limit is called the Riemann integral of $f$ from $a$ to $b$. We express this with the $\int$, symbol, and write $$\int\limits_a^b f(x) dx= \lim\limits_{\Delta x\to 0} \sum\limits_{i=1}^n f(x_i)\Delta x$$ The most straightforward of a definite integral is the definite integral. That is, we read $$\int\limits_a^b f(x) dx$$ as the definite integral of $f$ from $a$ to $b$ and we defined as the area under the "curve" $f(x)$ from point $x=a$ to $x=b$. -``` +::: The fundamental theorem of calculus shows us that this sum is, in fact, the antiderivative. -```{theorem} -#| name: First Fundamental Theorem of Calculus +:::{#thm-fftc} +## First Fundamental Theorem of Calculus Let the function $f$ be bounded on $[a,b]$ and continuous on $(a,b)$. Then, suggestively, use the symbol $F(x)$ to denote the definite integral from $a$ to $x$: $$F(x)=\int\limits_a^x f(t)dt, \quad a\le x\le b$$ Then $F(x)$ has a derivative at each point in $(a,b)$ and $$F^\prime(x)=f(x), \quad a af(x_1) + (1-a)f(x_2)$$ \textit{Any} line connecting two points on a concave function will lie \textit{below} the function. -``` + +::: ```{r} @@ -215,12 +219,12 @@ fx1 + fx2 + plot_layout(nrow = 1) ``` -```{definition} -#| name: Convex Function +:::{#def-convex} +## Convex Function Convex: A function f is strictly convex over the set S \underline{if} $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) < af(x_1) + (1-a)f(x_2)$$ Any line connecting two points on a convex function will lie above the function. -``` +::: \begin{comment} @@ -229,23 +233,25 @@ Convex: A function f is strictly convex over the set S \underline{if} $\forall x Sometimes, concavity and convexity are strict of a requirement. For most purposes of getting solutions, what we call quasi-concavity is enough. -```{definition} -#| name: Quasiconcave Function +:::{#def-quasiconcave} +## Quasiconcave Function A function f is quasiconcave over the set S if $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) \ge \min(f(x_1),f(x_2))$$ No matter what two points you select, the \textit{lowest} valued point will always be an end point. -``` +::: \begin{comment} \parbox{2in}{\includegraphics[scale=.4]{Quasiconcave.pdf}} \end{comment} -```{definition} -#| name: Quasiconvex +:::{#def-quasiconvex} + +## Quasiconvex A function f is quasiconvex over the set $S$ if $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) \le \max(f(x_1),f(x_2))$$ No matter what two points you select, the \textit{highest} valued point will always be an end point. -``` + +::: \begin{comment} \parbox{1.8in}{\includegraphics[scale=.4]{Quasiconvex.pdf}} @@ -328,8 +334,8 @@ When we examined functions of one variable $x$, we found critical points by taki In a derivative, we only took the derivative with respect to one variable at a time. When we take the derivative separately with respect to all variables in the elements of $\mathbf{x}$ and then express the result as a vector, we use the term Gradient and Hessian. -```{definition} -#| name: Gradient +:::{#def-gradient} +## Gradient Given a function $f(\textbf{x})$ in $n$ variables, the gradient $\nabla f(\mathbf{x})$ (the greek letter nabla ) is a column vector, where the $i$th element is the partial derivative of $f(\textbf{x})$ with respect to $x_i$: @@ -337,28 +343,28 @@ $$\nabla f(\mathbf{x}) = \begin{pmatrix} \frac{\partial f(\mathbf{x})}{\partial x_1}\\ \frac{\partial f(\mathbf{x})}{\partial x_2}\\ \vdots \\ \frac{\partial f(\mathbf{x})}{\partial x_n} \end{pmatrix}$$ -``` +::: Before we know whether a point is a maxima or minima, if it meets the FOC it is a "Critical Point". -```{definition} -#| name: Critical Point +:::{#def-criticalpoint} +## Critical Point $\mathbf{x}^*$ is a critical point if and only if $\nabla f(\mathbf{x}^*)=0$. If the partial derivative of f(x) with respect to $x^*$ is 0, then $\mathbf{x}^*$ is a critical point. To solve for $\mathbf{x}^*$, find the gradient, set each element equal to 0, and solve the system of equations. $$\mathbf{x}^* = \begin{pmatrix} x_1^*\\x_2^*\\ \vdots \\ x_n^*\end{pmatrix}$$ -``` +::: -```{example} +:::{#exm-gradcp} Example: Given a function $f(\mathbf{x})=(x_1-1)^2+x_2^2+1$, find the (1) Gradient and (2) Critical point of $f(\mathbf{x})$. -``` +::: -```{solution} +:::{#sol-gradcp} Gradient \begin{align*} @@ -380,7 +386,7 @@ Critical Point $\mathbf{x}^* =$ So $$\mathbf{x}^* = (1,0)$$ -``` +::: @@ -388,23 +394,22 @@ So $$\mathbf{x}^* = (1,0)$$ ### Second Order Conditions {.unnumbered} When we found a critical point for a function of one variable, we used the second derivative as a indicator of the curvature at the point in order to determine whether the point was a min, max, or saddle (second derivative test of concavity). For functions of $n$ variables, we use _second order partial derivatives_ as an indicator of curvature. -```{definition} -#| name: Hessian +:::{#def-hessian} + +## Hessian Given a function $f(\mathbf{x})$ in $n$ variables, the hessian $\mathbf{H(x)}$ is an $n\times n$ matrix, where the $(i,j)$th element is the second order partial derivative of $f(\mathbf{x})$ with respect to $x_i$ and $x_j$: -$$\mathbf{H(x)}=\begin{pmatrix} -\frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}&\frac{\partial^2f(\mathbf{x})}{\partial x_1 \partial x_2}& -\cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_1 \partial x_n}\$$9pt] +$$\mathbf{H(x)}=\begin{pmatrix} \frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}&\frac{\partial^2f(\mathbf{x})}{\partial x_1 \partial x_2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_1 \partial x_n}\$$9pt] \frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_2^2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_n}\\ \vdots & \vdots & \ddots & \vdots \$$3pt] \frac{\partial^2 f(\mathbf{x})}{\partial x_n \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_n \partial x_2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_n^2}\end{pmatrix}$$ -``` +::: @@ -426,17 +431,17 @@ Given a function $f(\mathbf{x})$ and a point $\mathbf{x}^*$ such that $\nabla f( -```{example} -#| name: Max and min with two dimensions +:::{#exm-maxmin2d} +## Max and min with two dimensions We found that the only critical point of $f(\mathbf{x})=(x_1-1)^2+x_2^2+1$ is at $\mathbf{x}^*=(1,0)$. Is it a min, max, or saddle point? -``` +::: -```{solution} +:::{#sol-maxmin2d} The Hessian is \begin{align*} @@ -458,7 +463,7 @@ Note: Alternate check of definiteness. Is $\mathbf{H(x^*)} \geq \leq 0 \quad \fo For any $\mathbf{x}\ne 0$, $2(x_1^2+x_2^2)>0$, so the Hessian is positive definite and $\mathbf{x}^*$ is a strict local minimum. -``` +::: @@ -501,9 +506,11 @@ is globally concave. So $x=0$ is a global min of $f_1(x)$ and a global max of $f_2(x)$. -```{exercise} +:::{#exr-maxmin} + Given $f(\mathbf{x})=x_1^3-x_2^3+9x_1x_2$, find any maxima or minima. -``` + +::: \begin{enumerate} @@ -640,15 +647,15 @@ We can then solve this system of equations, because there are $n+k$ equations an **Second-order Conditions and Unconstrained Optimization:** There may be more than one critical point, i.e. we need to verify that the critical point we find is a maximum/minimum. Similar to unconstrained optimization, we can do this by checking the second-order conditions. -```{example} -#| name: Constrained optimization with two goods and a budget constraint +:::{#exm-constropt} +## Constrained optimization with two goods and a budget constraint Find the constrained optimization of $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 = 4$$ -``` +::: -```{solution} +:::{#sol-constropt} 1. Begin by writing the Lagrangian: $$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ @@ -668,7 +675,7 @@ $$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ 5. This gives $f(\frac{8}{3}, \frac{4}{3}) = -\frac{96}{9}$, which is less than the unconstrained optimum $f(0,0) = 0$ -``` +::: Notice that when we take the partial derivative of L with respect to the Lagrangian multiplier and set it equal to 0, we return exactly our constraint! This is why signs matter. @@ -756,7 +763,7 @@ Looking at the values of $f(x_1,x_2)$ at the critical points, we see that $f(x_1 -```{exercise} +:::{#exr-critical-points-constrained-optimization} Example: Find the critical points for the following constrained optimization: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } \begin{array}{l} @@ -764,12 +771,12 @@ x_1 + x_2 \le 4\\ x_1 \ge 0\\ x_2 \ge 0 \end{array}$$ -``` +::: 1. Rewrite with the slack variables: -$$\phantom{max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } +$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. \begin{array}{l} x_1 + x_2 \le 4 - s_1^2\\ -x_1 \le 0 - s_2^2\\ @@ -790,7 +797,7 @@ $\frac{\partial L}{\partial s_1} = 2s_1\lambda_1 = 0$\\ $\frac{\partial L}{\partial s_2} = 2s_2\lambda_2 = 0$\\ $\frac{\partial L}{\partial s_3} = 2s_3\lambda_3 = 0$} -4. Consider all ways that the complementary slackness conditions are solved: +3. Consider all ways that the complementary slackness conditions are solved: \begin{center} \begin{tabular}{|l|cccccccc|c|} \hline @@ -810,7 +817,7 @@ $s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{\phantom{No solution}} \phantom{This shows that there are four critical points: $(0,0)$, $(4,0)$, $(0,4)$, and $(\frac{8}{3},\frac{4}{3})$} -5. Find maximum: \phantom{Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem.} +4. Find maximum: \phantom{Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem.} @@ -877,8 +884,8 @@ In a two-dimensional set-up, this means we must check the following cases: -```{example} -#| name: Kuhn-Tucker with two variables +:::{#exm-k-t-2} +## Kuhn-Tucker with two variables Solve the following optimization problem with inequality constraints $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)$$ @@ -891,7 +898,7 @@ $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)$$ \end{cases} \end{align*} -``` +::: 1. Write the Lagrangian: @@ -939,8 +946,8 @@ Three of the critical points violate the requirement that $\lambda \geq 0$, so t -```{exercise} -#| name: Kuhn-Tucker with logs +:::{#exr-ktlogs} +## Kuhn-Tucker with logs Solve the constrained optimization problem, @@ -950,51 +957,34 @@ x_1 + 2x_2 \leq 4\\ x_1 \geq 0\\ x_2 \geq 0 \end{array}$$ -``` +::: -\begin{enumerate} -\item Write the Lagrangian: -$$\phantom{L(x_1, x_2, \lambda) = \frac{1}{3}\log(x_1+1) + \frac{2}{3}\log(x_2+1) - \lambda(x_1 + 2x_2 - 4)}$$ +1. Write the Lagrangian: +$$L(x_1, x_2, \lambda) = \frac{1}{3}\log(x_1+1) + \frac{2}{3}\log(x_2+1) - \lambda(x_1 + 2x_2 - 4)$$ + +2. Find the First Order Conditions: + +Kuhn-Tucker Conditions -\item Find the First Order Conditions:\\ -Kuhn-Tucker Conditions\\ -\phantom{$\frac{\partial L}{\partial x_1} = \frac{1}{3(x_1+1)} - \lambda \leq 0$\\ +$\frac{\partial L}{\partial x_1} = \frac{1}{3(x_1+1)} - \lambda \leq 0$\\ $\frac{\partial L}{\partial x_2} = \frac{2}{3(x_2+1)} - \lambda \leq 0$\\ -$\frac{\partial L}{\partial \lambda} = -(x_1 + 2x_2 - 4) \geq 0$}\\ -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] +$\frac{\partial L}{\partial \lambda} = -(x_1 + 2x_2 - 4) \geq 0$\\ -Complementary Slackness Conditions\\ -\phantom{$x_1\frac{\partial L}{\partial x_2} = x_1(\frac{1}{3(x_1+1)} - \lambda) = 0$\\ +Complementary Slackness Conditions + +$x_1\frac{\partial L}{\partial x_2} = x_1(\frac{1}{3(x_1+1)} - \lambda) = 0$\\ $x_2\frac{\partial L}{\partial x_2} = x_2(\frac{2}{3(x_2+1)} - \lambda) = 0$\\ -$\lambda\frac{\partial L}{\partial \lambda} = -\lambda(x_1 + 2x_2 - 4) = 0$}\\ -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] - -Non-negativity Conditions\\ -\phantom{$x_1 \geq 0$\\ +$\lambda\frac{\partial L}{\partial \lambda} = -\lambda(x_1 + 2x_2 - 4) = 0$\\ + +Non-negativity Conditions + +$x_1 \geq 0$\\ $x_2 \geq 0$\\ -$\lambda \geq $0}\\ -\item[] -\item[] -\item[] +$\lambda \geq $0\\ -\item Consider all border and interior cases: +3. Consider all border and interior cases: \begin{center} \begin{tabular}{|l|ccc|c|} \hline @@ -1008,14 +998,9 @@ $x_1 \neq 0, x_2 \neq 0$ & & \phantom{$\frac{4}{3}$} & \phantom{$\frac{4}{3}$} \end{tabular} \end{center} -\item[] -\item[] -\item[] -\item[] +4. Find Maximum: -\item Find Maximum:\\ \phantom{Three of the critical points violate the constraints, so the point $(\frac{4}{3},\frac{4}{3})$ is the maximum.}\\ -\end{enumerate} @@ -1027,11 +1012,8 @@ The Hessian is used in a Taylor polynomial approximation to $f(\mathbf{x})$ and 1. The second order Taylor polynomial about the critical point ${\bf x}^*$ is - $$f({\bf x}^*+\bf h)=f({\bf x}^*)+\nabla f({\bf x}^*) \bf h +\frac{1}{2} \bf h^\top -{\bf H(x^*)} \bf h + R(\bf h)$$ + $$f({\bf x}^*+\bf h)=f({\bf x}^*)+\nabla f({\bf x}^*) \bf h +\frac{1}{2} \bf h^\top {\bf H(x^*)} \bf h + R(\bf h)$$ 2. Since we're looking at a critical point, $\nabla f({\bf x}^*)=0$; and for small $\bf h$, $R(\bf h)$ is negligible. Rearranging, we get -$$f({\bf x}^*+\bf h)-f({\bf x}^*)\approx \frac{1}{2} \bf h^\top {\bf H(x^*)} -\bf h $$ -3. The Righthand side here is a quadratic form and we can determine the definiteness of $\bf -H(x^*)$. +$$f({\bf x}^*+\bf h)-f({\bf x}^*)\approx \frac{1}{2} \bf h^\top {\bf H(x^*)} \bf h $$ +3. The Righthand side here is a quadratic form and we can determine the definiteness of $\bf H(x^*)$. diff --git a/_quarto.yml b/_quarto.yml index 29204f2..d3bd31c 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -14,6 +14,8 @@ book: chapters: - 02_functions.qmd - 03_limits.qmd + - 04_calculus.qmd + - 05_optimization.qmd delete_merged_file: true From 33dce0146a0c851a970760106188f48bb6e9e80c Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 21:11:30 -0400 Subject: [PATCH 08/34] phantoms sent back to opera --- 05_optimization.qmd | 93 ++++++++++++++++++++++----------------------- 1 file changed, 46 insertions(+), 47 deletions(-) diff --git a/05_optimization.qmd b/05_optimization.qmd index c74b31b..ab4cd02 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -402,10 +402,10 @@ Given a function $f(\mathbf{x})$ in $n$ variables, the hessian $\mathbf{H(x)}$ i an $n\times n$ matrix, where the $(i,j)$th element is the second order partial derivative of $f(\mathbf{x})$ with respect to $x_i$ and $x_j$: -$$\mathbf{H(x)}=\begin{pmatrix} \frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}&\frac{\partial^2f(\mathbf{x})}{\partial x_1 \partial x_2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_1 \partial x_n}\$$9pt] +$$\mathbf{H(x)}=\begin{pmatrix} \frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}&\frac{\partial^2f(\mathbf{x})}{\partial x_1 \partial x_2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_2^2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_n}\\ -\vdots & \vdots & \ddots & \vdots \$$3pt] +\vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f(\mathbf{x})}{\partial x_n \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_n \partial x_2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_n^2}\end{pmatrix}$$ @@ -512,66 +512,66 @@ Given $f(\mathbf{x})=x_1^3-x_2^3+9x_1x_2$, find any maxima or minima. ::: - +```{=latex} \begin{enumerate} \item First order conditions. \begin{enumerate} \item Gradient $\nabla f(\mathbf{x}) = $ - $$\phantom{\begin{pmatrix} \frac{\partial f}{\partial x_1} \\ + $$\begin{pmatrix} \frac{\partial f}{\partial x_1} \\ \frac{\partial f}{\partial x_2}\end{pmatrix} = - \begin{pmatrix} 3x_1^2+9x_2 \\ -3x_2^2+9x_1 \end{pmatrix}}$$ + \begin{pmatrix} 3x_1^2+9x_2 \\ -3x_2^2+9x_1 \end{pmatrix}$$ \item Critical Points $\mathbf{x^*} =$\\ - \phantom{Set the gradient equal to zero and solve for $x_1$ and + Set the gradient equal to zero and solve for $x_1$ and $x_2$.We have two equations and two unknowns. Solving for $x_1$ and $x_2$, we get two critical points: $\mathbf{x}_1^*=(0,0)$ and - $\mathbf{x}_2^*=(3,-3)$.} - $$\phantom{3x_1^2 + 9x_2 = 0 \quad \Rightarrow \quad 9x_2 = - -3x_1^2 \quad \Rightarrow \quad x_2 = -\frac{1}{3}x_1^2}$$ - $$\phantom{-3x_2^2 + 9x_1 = 0 \quad \Rightarrow \quad - -3(-\frac{1} {3}x_1^2)^2 + 9x_1 = 0}$$ - $$\phantom{\Rightarrow \quad -\frac{1}{3}x_1^4 + $\mathbf{x}_2^*=(3,-3)$. + $$3x_1^2 + 9x_2 = 0 \quad \Rightarrow \quad 9x_2 = + -3x_1^2 \quad \Rightarrow \quad x_2 = -\frac{1}{3}x_1^2$$ + $$-3x_2^2 + 9x_1 = 0 \quad \Rightarrow \quad + -3(-\frac{1} {3}x_1^2)^2 + 9x_1 = 0$$ + $$\Rightarrow \quad -\frac{1}{3}x_1^4 + 9x_1 = 0 \quad \Rightarrow \quad x_1^3 = 27x_1 \quad - \Rightarrow \quad x_1 = 3}$$ - $$\phantom{3(3)^2 + 9x_2 = 0 \quad \Rightarrow \quad x_2 = -3}$$ + \Rightarrow \quad x_1 = 3$$ + $$3(3)^2 + 9x_2 = 0 \quad \Rightarrow \quad x_2 = -3$$ \end{enumerate} \item Second order conditions. \begin{enumerate} \item Hessian $\mathbf{H(x)} = $ - $$\phantom{\begin{pmatrix} 6x_1&9\\9&-6x_2 \end{pmatrix}}$$ + $$\begin{pmatrix} 6x_1&9\\9&-6x_2 \end{pmatrix}$$ \item Hessian $\mathbf{H(x_1^*)} = $ - $$\phantom{\begin{pmatrix} 0&9\\9&0\end{pmatrix}}$$ + $$\begin{pmatrix} 0&9\\9&0\end{pmatrix}$$ \item Leading principal minors of $\mathbf{H(x_1^*)} = $ - $$\phantom{M_1=0; M_2=-81}$$\\ + $$M_1=0; M_2=-81$$\\ \item Definiteness of $\mathbf{H(x_1^*)}$?\\ - \phantom{$\mathbf{H(x_1^*)}$ is indefinite}\\ + $\mathbf{H(x_1^*)}$ is indefinite\\ \item Maxima, Minima, or Saddle Point for $\mathbf{x_1^*}$?\\ - \phantom{Since $\mathbf{H(x_1^*)}$ is indefinite, $\mathbf{x}_1^*=(0,0)$ - is a saddle point.}\\ + Since $\mathbf{H(x_1^*)}$ is indefinite, $\mathbf{x}_1^*=(0,0)$ + is a saddle point.\\ \item Hessian $\mathbf{H(x_2^*)} = $ - $$\phantom{\begin{pmatrix} 18&9\\9&18\end{pmatrix}}$$ + $$\begin{pmatrix} 18&9\\9&18\end{pmatrix}$$ \item Leading principal minors of $\mathbf{H(x_2^*)} = $ - $$\phantom{M_1=18; M_2=243}$$\\ + $$M_1=18; M_2=243$$\\ \item Definiteness of $\mathbf{H(x_2^*)}$?\\ - \phantom{$\mathbf{H(x_2^*)}$ is positive definite}\\ + $\mathbf{H(x_2^*)}$ is positive definite\\ \item Maxima, Minima, or Saddle Point for $\mathbf{x}_2^*$?\\ - \phantom{Since $\mathbf{H(x_2^*)}$ is positive definite, $\mathbf{x}_1^*=(3,-3)$ is a strict local minimum}\\ + Since $\mathbf{H(x_2^*)}$ is positive definite, $\mathbf{x}_1^*=(3,-3)$ is a strict local minimum\\ \end{enumerate} \item Global concavity/convexity. \begin{enumerate} \item Is f(x) globally concave/convex?\\ - \phantom{No. In evaluating the Hessians for $\mathbf{x}_1^*$ + No. In evaluating the Hessians for $\mathbf{x}_1^*$ and $\mathbf{x}_2^*$ we saw that the Hessian is not positive - semidefinite at x $=$ (0,0).}\\ + semidefinite at x $=$ (0,0).\\ \item Are any $\mathbf{x^*}$ global minima or maxima?\\ - \phantom{No. Since the function is not globally concave/convex, + No. Since the function is not globally concave/convex, we can't infer that $\mathbf{x}_2^*=(3,-3)$ is a global minimum. In fact, if we set $x_1=0$, the $f(\mathbf{x})=-x_2^3$, which will - go to $-\infty$ as $x_2\to \infty$.}\\ + go to $-\infty$ as $x_2\to \infty$.\\ \end{enumerate} \end{enumerate} - +``` ## Constrained Optimization @@ -784,10 +784,9 @@ x_1 + x_2 \le 4 - s_1^2\\ \end{array}}$$ 2. Write the Lagrangian: -$$\phantom{L(x_1, x_2, \lambda_1, \lambda_2, \lambda_3, s_1, s_2, s_3) = -(x_1^2 + 2x_2^2) - \lambda_1(x_1 + x_2 + s_1^2 - 4) - \lambda_2(-x_1 + s_2^2) - \lambda_3(-x_2 + s_3^2)}$$ +$$L(x_1, x_2, \lambda_1, \lambda_2, \lambda_3, s_1, s_2, s_3) = -(x_1^2 + 2x_2^2) - \lambda_1(x_1 + x_2 + s_1^2 - 4) - \lambda_2(-x_1 + s_2^2) - \lambda_3(-x_2 + s_3^2)$$ 3. Take the partial derivatives and set equal to zero: - \phantom{ $\frac{\partial L}{\partial x_1} = -2x_1 - \lambda_1 + \lambda_2 = 0$\\ $\frac{\partial L}{\partial x_2} = -4x_2 - \lambda_1 + \lambda_3 = 0$\\ $\frac{\partial L}{\partial \lambda_1} = -(x_1 + x_2 + s_1^2 - 4) = 0$\\ @@ -795,7 +794,7 @@ $\frac{\partial L}{\partial \lambda_2} = -(-x_1 + s_2^2) = 0$\\ $\frac{\partial L}{\partial \lambda_3} = -(-x_2 + s_3^2) = 0$\\ $\frac{\partial L}{\partial s_1} = 2s_1\lambda_1 = 0$\\ $\frac{\partial L}{\partial s_2} = 2s_2\lambda_2 = 0$\\ -$\frac{\partial L}{\partial s_3} = 2s_3\lambda_3 = 0$} +$\frac{\partial L}{\partial s_3} = 2s_3\lambda_3 = 0$ 3. Consider all ways that the complementary slackness conditions are solved: \begin{center} @@ -803,21 +802,21 @@ $\frac{\partial L}{\partial s_3} = 2s_3\lambda_3 = 0$} \hline Hypothesis & $s_1$ & $s_2$ & $s_3$ & $\lambda_1$ & $\lambda_2$ & $\lambda_3$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ \hline -$s_1 = s_2 = s_3 = 0$ & \multicolumn{8}{l|}{\phantom{No solution}} & \\ -$s_1 \neq 0, s_2 = s_3 = 0$ & \phantom{2} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0}\\ -$s_2 \neq 0, s_1 = s_3 = 0$ & \phantom{0} & \phantom{2} & \phantom{0} & \phantom{-8} & \phantom{0} & \phantom{-8} & \phantom{4} & \phantom{0} & \phantom{-16}\\ -$s_3 \neq 0, s_1 = s_2 = 0$ & \phantom{0} & \phantom{0} & \phantom{2} & \phantom{-16} & \phantom{-16} & \phantom{0} & \phantom{0} & \phantom{4} & \phantom{-32}\\ -$s_1 \neq 0, s_2 \neq 0, s_3 = 0$ &\multicolumn{8}{l|}{\phantom{No solution}} & \\ -$s_1 \neq 0, s_3 \neq 0, s_2 = 0$ &\multicolumn{8}{l|}{\phantom{No solution}} & \\ -$s_2 \neq 0, s_3 \neq 0, s_1 = 0$ &\phantom{0} & \phantom{$\sqrt{\frac{8}{3}}$} & \phantom{$\sqrt{\frac{4}{3}}$} & \phantom{$-\frac{16}{3}$} & \phantom{0} & \phantom{0} & \phantom{$\frac{8}{3}$}& \phantom{$\frac{4}{3}$} & \phantom{$-\frac{32}{3}$}\\ -$s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{\phantom{No solution}}& \\ +$s_1 = s_2 = s_3 = 0$ & \multicolumn{8}{l|}{No solution} & \\ +$s_1 \neq 0, s_2 = s_3 = 0$ & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ +$s_2 \neq 0, s_1 = s_3 = 0$ & 0 & 2 & 0 & -8 & 0 & -8 & 4 & 0 & -16\\ +$s_3 \neq 0, s_1 = s_2 = 0$ & 0 & 0 & 2 & -16 & -16 & 0 & 0 & 4 & -32\\ +$s_1 \neq 0, s_2 \neq 0, s_3 = 0$ &\multicolumn{8}{l|}{No solution} & \\ +$s_1 \neq 0, s_3 \neq 0, s_2 = 0$ &\multicolumn{8}{l|}{No solution} & \\ +$s_2 \neq 0, s_3 \neq 0, s_1 = 0$ &0 & $\sqrt{\frac{8}{3}}$ & $\sqrt{\frac{4}{3}}$ & $-\frac{16}{3}$ & 0 & 0 & $\frac{8}{3}$& $\frac{4}{3}$ & $-\frac{32}{3}$\\ +$s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{No solution}& \\ \hline \end{tabular} \end{center} -\phantom{This shows that there are four critical points: $(0,0)$, $(4,0)$, $(0,4)$, and $(\frac{8}{3},\frac{4}{3})$} +This shows that there are four critical points: $(0,0)$, $(4,0)$, $(0,4)$, and $(\frac{8}{3},\frac{4}{3})$ -4. Find maximum: \phantom{Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem.} +4. Find maximum: Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem. @@ -990,17 +989,17 @@ $\lambda \geq $0\\ \hline Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ \hline -$x_1 = 0, x_2 = 0$ &\multicolumn{3}{l|}{\phantom{No Solution}}& \\ -$x_1 = 0, x_2 \neq 0$ &\multicolumn{3}{l|}{\phantom{No Solution}}& \\ -$x_1 \neq 0, x_2 = 0$ & \multicolumn{3}{l|}{\phantom{No Solution}}& \\ -$x_1 \neq 0, x_2 \neq 0$ & & \phantom{$\frac{4}{3}$} & \phantom{$\frac{4}{3}$} & \phantom{$\log\frac{7}{3}$}\\ +$x_1 = 0, x_2 = 0$ &\multicolumn{3}{l|}{No Solution}& \\ +$x_1 = 0, x_2 \neq 0$ &\multicolumn{3}{l|}{No Solution}& \\ +$x_1 \neq 0, x_2 = 0$ & \multicolumn{3}{l|}{No Solution}& \\ +$x_1 \neq 0, x_2 \neq 0$ & & $\frac{4}{3}$ & $\frac{4}{3}$ & $\log\frac{7}{3}$\\ \hline \end{tabular} \end{center} 4. Find Maximum: -\phantom{Three of the critical points violate the constraints, so the point $(\frac{4}{3},\frac{4}{3})$ is the maximum.}\\ +Three of the critical points violate the constraints, so the point $(\frac{4}{3},\frac{4}{3})$ is the maximum.\\ From 7c281c2ff67b058d5e23e6c9ea0ac51b80bbdcbe Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 21:16:34 -0400 Subject: [PATCH 09/34] remove whitespace --- 01_warmup.qmd | 21 ---- 02_functions.qmd | 51 ---------- 03_limits.qmd | 50 ---------- 04_calculus.qmd | 118 ----------------------- 05_optimization.qmd | 91 ------------------ 06_probability.qmd | 174 ---------------------------------- 07_linear-algebra.qmd | 126 ------------------------ 11_data-handling_counting.qmd | 43 --------- 12_matricies-manipulation.qmd | 46 --------- 13_functions_obj_loops.qmd | 50 ---------- 14_visualization.qmd | 49 ---------- 15_project-dempeace.qmd | 27 ------ 16_simulation.qmd | 47 --------- 17_non-wysiwyg.qmd | 25 ----- 18_text.qmd | 15 --- 19_command-line_git.qmd | 13 --- 21_solutions-warmup.qmd | 24 ----- 23_solution_programming.qmd | 35 ------- index.qmd | 2 - 19 files changed, 1007 deletions(-) diff --git a/01_warmup.qmd b/01_warmup.qmd index 7404828..b7deff2 100644 --- a/01_warmup.qmd +++ b/01_warmup.qmd @@ -1,12 +1,9 @@ # Pre-Prefresher Exercises {.unnumbered} - Before our first meeting, please try solving these questions. They are a sample of the very beginning of each math section. We have provided links to the parts of the book you can read if the concepts are new to you. The goal of this "pre"-prefresher assignment is not to intimidate you but to set common expectations so you can make the most out of the actual Prefresher. Even if you do not understand some or all of these questions after skimming through the linked sections, your effort will pay off and you will be better prepared for the math prefresher. We are also open to adjusting these expectations based on feedback (this class is for _you_), so please do not hesitate to write to the instructors for feedback. - - ## Linear Algebra {.unnumbered} ### Vectors {.unnumbered} @@ -19,21 +16,16 @@ Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pm If you are having trouble with these problems, please review Section @sec-vector-def "Working with Vectors" in Chapter @sec-linearalgebra. - Are the following sets of vectors linearly independent? 1. $u = \begin{pmatrix} 1\\ 2\end{pmatrix}$, $v = \begin{pmatrix} 2\\4\end{pmatrix}$ - 2. $u = \begin{pmatrix} 1\\ 2\\ 5 \end{pmatrix}$, $v = \begin{pmatrix} 3\\ 7\\ 9 \end{pmatrix}$ - 3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ (this requires some guesswork) - If you are having trouble with these problems, please review Section @sec-linearindependence. - ### Matrices {.unnumbered} $${\bf A}=\begin{pmatrix} @@ -44,7 +36,6 @@ $${\bf A}=\begin{pmatrix} \end{pmatrix}$$ - What is the dimensionality of matrix ${\bf A}$? What is the element $a_{23}$ of ${\bf A}$? @@ -60,7 +51,6 @@ $${\bf B}=\begin{pmatrix} What is ${\bf A}$ + ${\bf B}$? - Given that $${\bf C}=\begin{pmatrix} @@ -71,8 +61,6 @@ $${\bf C}=\begin{pmatrix} What is ${\bf A}$ + ${\bf C}$? - - Given that $$c = 2$$ @@ -81,8 +69,6 @@ What is $c$${\bf A}$? If you are having trouble with these problems, please review Section @sec-matrixbasics. - - ## Operations {.unnumbered} ### Summation {.unnumbered} @@ -95,17 +81,14 @@ Simplify the following 3. $\sum\limits_{i= 1}^4 (3k + i + 2)$ - ### Products {.unnumbered} 1. $\prod\limits_{i= 1}^3 i$ 2. $\prod\limits_{k=1}^3(3k + 2)$ - To review this material, please see Section \@ref(sum-notation). - ### Logs and exponents {.unnumbered} Simplify the following @@ -123,8 +106,6 @@ Simplify the following To review this material, please see Section @sec-logexponents - - ## Limits {.unnumbered} Find the limit of the following. @@ -135,7 +116,6 @@ Find the limit of the following. To review this material please see Section @sec-limitsfun - ## Calculus {.unnumbered} For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\frac{d}{dx}f(x)$ @@ -167,6 +147,5 @@ If you are stuck, please try sketching out a picture of each of the functions. 3. If we roll two fair dice, what is the probability that their sum would be 11? 4. If we roll two fair dice, what is the probability that their sum would be 12? - For a review, please see Sections @sec-setoper - @sec-probdef. diff --git a/02_functions.qmd b/02_functions.qmd index 8212df6..ed2d20c 100644 --- a/02_functions.qmd +++ b/02_functions.qmd @@ -16,7 +16,6 @@ __Topics__ Limit of a Function; Continuity; Sets, Sets, and More Sets. - ## Summation Operators $\sum$ and $\prod$ {#sum-notation} Addition (+), Subtraction (-), multiplication and division are basic operations of arithmetic -- combining numbers. In statistics and calculus, we want to add a _sequence_ of numbers that can be expressed as a pattern without needing to write down all its components. For example, how would we express the sum of all numbers from 1 to 100 without writing a hundred numbers? @@ -33,7 +32,6 @@ The bottom of the $\sum$ symbol indicates an index (here, $i$), and its start va - $\sum\limits_{i=1}^n (x_i + y_i) = \sum\limits_{i=1}^n x_i + \sum\limits_{i=1}^n y_i$ - $\sum\limits_{i=1}^n c = n c$ - __Product:__ $$\prod\limits_{i=1}^n x_i = x_1 x_2 x_3 \cdots x_n$$ @@ -45,8 +43,6 @@ Properties: - $\prod\limits_{i=1}^n (x_i + y_i) =$ a total mess - $\prod\limits_{i=1}^n c = c^n$ - - Other Useful Functions __Factorials!:__ @@ -58,8 +54,6 @@ __Modulo:__ Tells you the remainder when you divide the first number by the seco - $17\mod3 = 2$ - $100 \ \% \ 30 = 10$ - - :::{#exm-operators} ## Operators @@ -90,12 +84,10 @@ Let $x_1 = 4, x_2 = 3, x_3 = 7, x_4 = 11, x_5 = 2$ ::: - ## Introduction to Functions A __function__ (in ${\bf R}^1$) is a mapping, or transformation, that relates members of one set to members of another set. For instance, if you have two sets: set $A$ and set $B$, a function from $A$ to $B$ maps every value $a$ in set $A$ such that $f(a) \in B$. Functions can be "many-to-one", where many values or combinations of values from set $A$ produce a single output in set $B$, or they can be "one-to-one", where each value in set $A$ corresponds to a single value in set $B$. A function by definition has a single function value for each element of its domain. This means, there cannot be "one-to-many" mapping. - __Dimensionality__: ${\bf R}^1$ is the set of all real numbers extending from $-\infty$ to $+\infty$ --- i.e., the real number line. ${\bf R}^n$ is an $n$-dimensional space, where each of the $n$ axes extends from $-\infty$ to $+\infty$. - ${\bf R}^1$ is a one dimensional line. @@ -104,7 +96,6 @@ __Dimensionality__: ${\bf R}^1$ is the set of all real numbers extending from $- Points in ${\bf R}^n$ are ordered $n$-tuples (just means an combination of $n$ elements where order matters), where each element of the $n$-tuple represents the coordinate along that dimension. - For example: - ${\bf R}^1$: (3) @@ -121,10 +112,8 @@ Function of two variables: $f: {\bf R}^2\to{\bf R}^1$. - $f(x,y)=x^2+y^2$. For each ordered pair $(x,y)$ in ${\bf R}^2$, $f(x,y)$ assigns the number $x^2+y^2$. - We often use variable $x$ as input and another $y$ as output, e.g. $y=x+1$ - :::{#exm-functions} ## Functions @@ -135,30 +124,21 @@ For each of the following, state whether they are one-to-one or many-to-one func 2. For $x \in [-\infty, \infty]$, $f: x \rightarrow x^2$. - - ::: - :::{#exr-functions1} ## Functions For each of the following, state whether they are one-to-one or many-to-one functions. - 1. For $x \in [-3, \infty]$, $f: x \rightarrow x^2$. - 2. For $x \in [0, \infty]$, $f: x \rightarrow \sqrt{x}$ - ::: - - - Some functions are defined only on proper subsets of ${\bf R}^n$. - __Domain__: the set of numbers in $X$ at which $f(x)$ is defined. @@ -169,7 +149,6 @@ Some functions are defined only on proper subsets of ${\bf R}^n$. Some General Types of Functions - __Monomials__: $f(x)=a x^k$ $a$ is the coefficient. $k$ is the degree. @@ -183,10 +162,8 @@ Examples: $y=-\frac{1}{2}x^3+x^2$, $y=3x+5$ The degree of a polynomial is the highest degree of its monomial terms. Also, it's often a good idea to write polynomials with terms in decreasing degree. - __Exponential Functions__: Example: $y=2^x$ - ## $\log$ and $\exp$ {#logexponents} __Relationship of logarithmic and exponential functions__: @@ -211,7 +188,6 @@ __Properties of exponential functions:__ - $(a^x)^y = a^{x y}$ - $a^0 = 1$ - __Properties of logarithmic functions__ (any base): Generally, when statisticians or social scientists write $\log(x)$ they mean $\log_e(x)$. In other words: $\log_e(x) \equiv \ln(x) \equiv \log(x)$ @@ -225,15 +201,11 @@ $$a^{\log_a(x)}=x$$ - $\log(x/y)=\log(x\cdot y^{-1})=\log(x)+\log(y^{-1})=\log(x)-\log(y)$ - $\log(1)=\log(e^0)=0$ - __Change of Base Formula__: Use the change of base formula to switch bases as necessary: $$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$$ Example: $$\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$$ - - - You can use logs to go between sum and product notation. This will be particularly important when you're learning maximum likelihood estimation. \begin{eqnarray*} @@ -251,8 +223,6 @@ Therefore, you can see that the log of a product is equal to the sum of the logs &=& n \log(c) + \sum\limits_{i=1}^n \log (x_i)\\ \end{eqnarray*} - - :::{#exm-log} ## Logarithmic Functions @@ -267,12 +237,8 @@ Simplify the following logarithm. By "simplify", we actually really mean - use a 3. $\log_4(x^3y^5)$ - - - ::: - :::{#exr-log1} ## Logarithmic Functions @@ -281,19 +247,15 @@ Evaluate each of the following logarithms 1. $\log_\frac{3}{2}(\frac{27}{8})$ - Simplify each of the following logarithms. By "simplify", we actually really mean - use as many of the logarithmic properties as you can. 2. $\log(\frac{x^9y^5}{z^3})$ - 3. $\ln{\sqrt{xy}}$ - ::: - ## Graphing Functions What can a graph tell you about a function? @@ -306,7 +268,6 @@ What can a graph tell you about a function? - Does the function tend to some limit? - Other questions related to the substance of the problem at hand. - ## Solving for Variables and Finding Roots Sometimes we're given a function $y=f(x)$ and we want to find how $x$ varies as a function of $y$. Use algebra to move $x$ to the left hand side (LHS) of the equation and so that the right hand side (RHS) is only a function of $y$. @@ -319,14 +280,12 @@ Solve for x: 1. $y=3x+2$ - 2. $y=e^x$ ::: - Solving for variables is especially important when we want to find the __roots__ of an equation: those values of variables that cause an equation to equal zero. Especially important in finding equilibria and in doing maximum likelihood estimation. Procedure: Given $y=f(x)$, set $f(x)=0$. Solve for $x$. @@ -337,7 +296,6 @@ Multiple Roots: __Quadratic Formula:__ For quadratic equations $ax^2+bx+c=0$, use the quadratic formula: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ - :::{#exr-solvevar1} ## Finding Roots @@ -351,10 +309,8 @@ Solve for x: 3. $f(x)=e^{-x}-10 = 0$ - ::: - ## Sets __Interior Point__: The point $\bf x$ is an interior point of the set $S$ if $\bf x$ is in $S$ and if there is some $\epsilon$-ball around $\bf x$ that contains only points in $S$. The __interior__ of $S$ is the collection of all interior points in $S$. The interior can also be defined as the union of all open sets in $S$. @@ -385,13 +341,11 @@ __Complement__: The complement of set $S$ is everything outside of $S$. - If the set $S$ is circular, the complement of $S$ is everything outside of the circle. - Example: The complement of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 > 4 \}$. - __Empty__: The empty (or null) set is a unique set that has no elements, denoted by \{\} or $\emptyset$. - The empty set is an example of a set that is open and closed, or a "clopen" set. - Examples: The set of squares with 5 sides; the set of countries south of the South Pole. - ## Answers to Examples and Exercises {.unnumbered} Answer to Example @exm-operators: @@ -412,7 +366,6 @@ Answer to Exercise @exr-operators1: 3. $2^3(7)(11)(2)$ = 1232 - Answer to Example @exm-functions: 1. one-to-one @@ -433,7 +386,6 @@ Answer to Example @exm-log: 3. $3\log_4(x) + 5\log_4(y)$ - Answer to Exercise @exr-log1: 1. 3 @@ -448,7 +400,6 @@ Answer to Example @exm-solvevar: 2. $x = \ln{y}$ - Answer to Exercise @exr-solvevar1: 1. $\frac{-2}{3}$ @@ -457,5 +408,3 @@ Answer to Exercise @exr-solvevar1: 3. x = - $\ln10$ - - diff --git a/03_limits.qmd b/03_limits.qmd index 13a4c4d..41e8b69 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -9,7 +9,6 @@ library(glue) library(patchwork) ``` - Solving limits, i.e. finding out the value of functions as its input moves closer to some value, is important for the social scientist's mathematical toolkit for two related tasks. The first is for the study of calculus, which will be in turn useful to show where certain functions are maximized or minimized. The second is for the study of statistical inference, which is the study of inferring things about things you cannot see by using things you can see. ## Example: The Central Limit Theorem {.unnumbered} @@ -31,7 +30,6 @@ That is, the limit of the distribution of the lefthand side is the distribution The sign of a limit is the arrow "$\rightarrow$". Although we have not yet covered probability (in Section @probability-theory) so we have not described what distributions and random variables are, it is worth foreshadowing the Central Limit Theorem. The Central Limit Theorem is powerful because it gives us a _guarantee_ of what would happen if $n \rightarrow \infty$, which in this case means we collected more data. - ## Example: The Law of Large Numbers {.unnumbered} A finding that perhaps rivals the Central Limit Theorem is the Law of Large Numbers: @@ -44,11 +42,9 @@ For any draw of identically distributed independent variables with mean $\mu$, t $$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ - A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". ::: - Intuitively, the more data, the more accurate is your guess. For example, the Figure @fig-llnsim shows how the sample average from many coin tosses converges to the true value : 0.5. ```{r} @@ -71,8 +67,6 @@ ggplot(tibble(n = 1:n, estimate = means), aes(x = n, y = estimate)) + y = "Estimate of the Probability of Heads after n trials") ``` - - ## Sequences We need a couple of steps until we get to limit theorems in probability. First we will introduce a "sequence", then we will think about the limit of a sequence, then we will think about the limit of a _function_. @@ -94,7 +88,6 @@ How do these sequences behave? ::: - We find the sequence by simply "plugging in" the integers into each $n$. The important thing is to get a sense of how these numbers are going to change. Example 1's numbers seem to come closer and closer to 2, but will it ever surpass 2? Example 2's numbers are also increasing each time, but will it hit a limit? What is the pattern in Example 3? Graphing helps you make this point more clearly. See the sequence of $n = 1, ...20$ for each of the three examples in Figure @fig-seqabc. ```{r} @@ -114,19 +107,14 @@ gA + gB + gC + plot_layout(nrow = 1) ``` - - - ## The Limit of a Sequence - The notion of "converging to a limit" is the behavior of the points in Example @exm-seqbehav. In some sense, that's the counterfactual we want to know. What happens as $n\rightarrow \infty$? 1. Sequences like 1 above that converge to a limit. 2. Sequences like 2 above that increase without bound. 3. Sequences like 3 above that neither converge nor increase without bound --- alternating over the number line. - :::{#def-limseq} The sequence $\{y_n\}$ has the limit $L$, which we write as $$\lim\limits_{n \to \infty} y_n =L$$, if for any $\epsilon>0$ there is an integer $N$ (which depends on $\epsilon$) with the property that $|y_n -L|<\epsilon$ for each $n>N$. $\{y_n\}$ is said to converge to $L$. If the above does not hold, then $\{y_n\}$ diverges. ::: @@ -137,7 +125,6 @@ We can also express the behavior of a sequence as bounded or not: 2. Monotonically Increasing: $y_{n+1}>y_n$ for all $n$ 3. Monotonically Decreasing: $y_{n+1} 0$$ - ::: - - - We will relegate the proofs to small excerpts. - - - ### Derivatives of natural exponential function ($e$) {.unnumbered} To repeat the main rule in Theorem @thm-derivexplog, the intuition is that @@ -287,7 +249,6 @@ To repeat the main rule in Theorem @thm-derivexplog, the intuition is that 3. Same thing no matter how many derivatives there are in front: $\frac{d^n}{dx^n} \alpha e^x = \alpha e^x$ 4. Chain Rule: When the exponent is a function of $x$, remember to take derivative of that function and add to product. $\frac{d}{dx}e^{g(x)}= e^{g(x)} g^\prime(x)$ - ```{r} #| label: fig-derivexponent #| echo: false @@ -319,8 +280,6 @@ Find the derivative for the following. ::: - - ### Derivatives of $\log$ {.unnumbered} The natural log is the mirror image of the natural exponent and has mirroring properties, again, to repeat the theorem, @@ -330,8 +289,6 @@ The natural log is the mirror image of the natural exponent and has mirroring p 3. Chain rule again: $\frac{d}{dx} \log u(x) = \frac{u'(x)}{u(x)}\quad$ 4. For any positive base $b$, $\frac{d}{dx} b^x = (\log b)\left(b^x\right)$. - - ```{r} #| label: fig-derivlog #| echo: false @@ -397,9 +354,6 @@ Then in the special case where $a = e$, $$(e^x)^\prime = (e^x)$$ - - - ## Partial Derivatives What happens when there's more than variable that is changing? @@ -414,7 +368,6 @@ $$\frac{\partial f}{\partial x_i} (x_1,\ldots,x_n) = \lim\limits_{h\to 0} \frac{ Only the $i$th variable changes --- the others are treated as constants. - We can take higher-order partial derivatives, like we did with functions of a single variable, except now the higher-order partials can be with respect to multiple variables. :::{#exm-partialtypes} @@ -431,7 +384,6 @@ Suppose $f(x,y)=x^2+y^2$. Then \end{align*} ::: - :::{#exr-partialderivs} Let $f(x,y)=x^3 y^4 +e^x -\log y$. What are the following partial derivaitves? @@ -445,10 +397,6 @@ Let $f(x,y)=x^3 y^4 +e^x -\log y$. What are the following partial derivaitves? ::: - - - - ## Taylor Series Approximation {#taylorapprox} A common form of approximation used in statistics involves derivatives. A Taylor series is a way to represent common functions as infinite series (a sum of infinite elements) of the function's derivatives at some point $a$. @@ -458,7 +406,6 @@ For example, Taylor series are very helpful in representing nonlinear (read: dif Specifically, a Taylor series of a real or complex function $f(x)$ that is infinitely differentiable in the neighborhood of point $a$ is: - \begin{align*} f(x) &= f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots\\ &= \sum_{n=0}^\infty \frac{f^{(n)} (a)}{n!} (x-a)^n @@ -469,7 +416,6 @@ __Taylor Approximation__: We can often approximate the curvature of a function $ $$f(x) = f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + R_2$$ - $R_2$ is the remainder (R for remainder, 2 for the fact that we took two derivatives) and often treated as negligible, giving us: @@ -477,10 +423,6 @@ $$f(x) \approx f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2} (x-a)^2$$ The more derivatives that are added, the smaller the remainder $R$ and the more accurate the approximation. Proofs involving limits guarantee that the remainder converges to 0 as the order of derivation increases. - - - - ## The Indefinite Integration So far, we've been interested in finding the derivative $f=F'$ of a function $F$. However, sometimes we're interested in exactly the reverse: finding the function $F$ for which $f$ is its derivative. We refer to $F$ as the antiderivative of $f$. @@ -493,12 +435,10 @@ $$F^\prime = f.$$ ::: - Another way to describe is through the inverse formula. Let $DF$ be the derivative of $F$. And let $DF(x)$ be the derivative of $F$ evaluated at $x$. Then the antiderivative is denoted by $D^{-1}$ (i.e., the inverse derivative). If $DF=f$, then $F=D^{-1}f$. This definition bolsters the main takeaway about integrals and derivatives: They are inverses of each other. - :::{#exr-antiderivatives} ## Antiderivative Find the antiderivative of the following: @@ -509,8 +449,6 @@ Find the antiderivative of the following: ::: - - We know from derivatives how to manipulate $F$ to get $f$. But how do you express the procedure to manipulate $f$ to get $F$? For that, we need a new symbol, which we will call indefinite integration. :::{#def-integral} @@ -523,31 +461,25 @@ and is equal to the antiderivative of $f$. ::: - :::{#exm-integralc} Draw the function $f(x)$ and its indefinite integral, $\int\limits f(x) dx$ - $$f(x) = (x^2-4)$$ - ::: - :::{#sol-integralc} The Indefinite Integral of the function $f(x) = (x^2-4)$ can, for example, be $F(x) = \frac{1}{3}x^3 - 4x.$ But it can also be $F(x) = \frac{1}{3}x^3 - 4x + 1$, because the constant 1 disappears when taking the derivative. ::: Some of these functions are plotted in the bottom panel of Figure @fig-integralc as dotted lines. - ```{r} #| label: fig-integralc #| echo: false #| fig-cap: The Many Indefinite Integrals of a Function range1 <- tibble(x = c(-4, 4)) - fx <- ggplot(range1, aes(x = x)) + stat_function(fun = function(x) x^2 - 4, size = 0.5) + labs(y = expression(f(x))) @@ -568,7 +500,6 @@ Notice from these examples that while there is only a single derivative for any Some common rules of integrals follow by virtue of being the inverse of a derivative. - 1. Constants are allowed to slip out: $\int a f(x)dx = a\int f(x)dx$ 1. Integration of the sum is sum of integrations: $\int [f(x)+g(x)]dx=\int f(x)dx + \int g(x)dx$ 1. Reverse Power-rule: $\int x^n dx = \frac{1}{n+1} x^{n+1} + c$ @@ -578,7 +509,6 @@ Some common rules of integrals follow by virtue of being the inverse of a deriva 1. More generally: $\int [f(x)]^n f'(x)dx = \frac{1}{n+1}[f(x)]^{n+1}+c$ 1. Remember the derivative of a log of a function: $\int \frac{f^\prime(x)}{f(x)}dx=\log f(x) + c$ - :::{#exm-common-integration} ## Common Integration Simplify the following indefinite integrals: @@ -587,11 +517,8 @@ Simplify the following indefinite integrals: * $\int (2x+1)dx$ * $\int e^x e^{e^x} dx$ - ::: - - ## The Definite Integral: The Area under the Curve If there is a indefinite integral, there _must_ be a definite integral. Indeed there is, but the notion of definite integrals comes from a different objective: finding the are a under a function. We will find, perhaps remarkably, that the formula we find to get the sum turns out to be expressible by the anti-derivative. @@ -623,7 +550,6 @@ g2 <- fx + geom_col(data = d2, aes(x, f), width = 0.1, fill = "gray", alpha = g1 + g2 + plot_layout(nrow = 1) ``` - One way to calculate the area would be to divide the interval $a\le x\le b$ into $n$ subintervals of length $\Delta x$ and then approximate the region with a series of rectangles, where the base of each rectangle is $\Delta x$ and the height is $f(x)$ at the midpoint of that interval. $A(R)$ would then be approximated by the area of the union of the rectangles, which is given by $$S(f,\Delta x)=\sum\limits_{i=1}^n f(x_i)\Delta x$$ and is called a __Riemann sum__. As we decrease the size of the subintervals $\Delta x$, making the rectangles "thinner," we would expect our approximation of the area of the region to become closer to the true area. This allows us to express the area as a limit of a series: @@ -676,26 +602,19 @@ Let $f(x) = 3x^2$. ::: - - - :::{#exr-defintexp} What is the value of $\int\limits_{-2}^2 e^x e^{e^x} dx$? ::: - - ### Common Rules for Definite Integrals {.unnumbered} The area-interpretation of the definite integral provides some rules for simplification. - 1. There is no area below a point: $$\int\limits_a^a f(x)dx=0$$ 2. Reversing the limits changes the sign of the integral: $$\int\limits_a^b f(x)dx=-\int\limits_b^a f(x)dx$$ 3. Sums can be separated into their own integrals: $$\int\limits_a^b [\alpha f(x)+\beta g(x)]dx = \alpha \int\limits_a^b f(x)dx + \beta \int\limits_a^b g(x)dx$$ 4. Areas can be combined as long as limits are linked: $$\int\limits_a^b f(x) dx +\int\limits_b^c f(x)dx = \int\limits_a^c f(x)dx$$ - :::{#exr-defintshort} ## Definite integral shortcuts Simplify the following definite intergrals. @@ -704,11 +623,8 @@ Simplify the following definite intergrals. 2. $\int\limits_0^4 (2x+1)dx=$ 3. $\int\limits_{-2}^0 e^x e^{e^x} dx + \int\limits_0^2 e^x e^{e^x} dx =$ - ::: - - ## Integration by Substitution From the second fundamental theorem of calculus, we now that a quick way to get a definite integral is to first find the indefinite integral, and then just plug in the bounds. @@ -719,8 +635,6 @@ Suppose we want to find the indefinite integral $$\int g(x)dx$$ but $g(x)$ is co Why does an introduction of yet another function end of simplifying things? Let's refer to the antiderivative of $f$ as $F$. Then the chain rule tells us that $$\frac{d}{dx} F[u(x)]=f[u(x)]u'(x)$$. So, $F[u(x)]$ is the antiderivative of $g$. We can then write $$\int g(x) dx= \int f[u(x)]u'(x)dx = \int \frac{d}{dx} F[u(x)]dx = F[u(x)]+c$$ - - To summarize, the procedure to determine the indefinite integral $\int g(x)dx$ by the method of substitution: 1. Identify some part of $g(x)$ that might be simplified by substituting in a single variable $u$ (which will then be a function of $x$). @@ -728,27 +642,19 @@ To summarize, the procedure to determine the indefinite integral $\int g(x)dx$ b 3. Solve the indefinite integral. 4. Substitute back in for $x$ - Substitution can also be used to calculate a definite integral. Using the same procedure as above, $$\int\limits_a^b g(x)dx=\int\limits_c^d f(u)du = F(d)-F(c)$$ where $c=u(a)$ and $d=u(b)$. - - :::{#exm-intsub1} ## Integration by Substitution I Solve the indefinite integral $$\int x^2 \sqrt{x+1}dx.$$ - ::: - - - For the above problem, we could have also used the substitution $u=\sqrt{x+1}$. Then $x=u^2-1$ and $dx=2u du$. Substituting these in, we get $$\int x^2\sqrt{x+1}dx=\int (u^2-1)^2 u 2u du$$ which when expanded is again a polynomial and gives the same result as above. - Another case in which integration by substitution is is useful is with a fraction. :::{#exm-intsub2} @@ -756,24 +662,17 @@ Another case in which integration by substitution is is useful is with a fractio Simplify $$\int\limits_0^1 \frac{5e^{2x}}{(1+e^{2x})^{1/3}}dx.$$ ::: - - - - ## Integration by Parts - Another useful integration technique is __integration by parts__, which is related to the Product Rule of differentiation. The product rule states that $$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$$ Integrating this and rearranging, we get $$\int u\frac{dv}{dx}dx= u v - \int v \frac{du}{dx}dx$$ or $$\int u(x) v'(x)dx=u(x)v(x) - \int v(x)u'(x)dx$$ More easily remembered with the mnemonic "Ultraviolet Voodoo": $$\int u dv = u v - \int v du$$ where $du=u'(x)dx$ and $dv=v'(x)dx$. - For definite integrals, this is simply $$\int\limits_a^b u\frac{dv}{dx}dx = \left. u v \right|_a^b - \int\limits_a^b v \frac{du}{dx}dx$$ Our goal here is to find expressions for $u$ and $dv$ that, when substituted into the above equation, yield an expression that's more easily evaluated. - :::{#exm-intbyparts} ## Integration by Parts @@ -793,7 +692,6 @@ Let $u=x$ and $\frac{dv}{dx} = e^{ax}$. Then $du=dx$ and $v=(1/a)e^{ax}$. Subst \end{eqnarray} ::: - :::{#exr-intparts-adv} ## Integration by Parts II @@ -801,21 +699,17 @@ Let $u=x$ and $\frac{dv}{dx} = e^{ax}$. Then $du=dx$ and $v=(1/a)e^{ax}$. Subst 1. Integrate $$\int x^n e^{ax} dx$$ - 2. Integrate $$\int x^3 e^{-x^2} dx$$ - ::: - ## Answers to Examples and Exercises {.unnumbered} Exercise @exr-introderivatives :::{#sol-introderivatives} - 1. $f^\prime(x)= 0$ 2. $f^\prime(x)= 1$ 3. $f^\prime(x)= 2x^3$ @@ -827,11 +721,8 @@ Exercise @exr-introderivatives 9. $f\prime(x) = 6x + \frac{2}{3}x^{\frac{-2}{3}}$ 10. $f\prime(x)= \frac{-4x}{x^4 - 2x^2 + 1}$ - - ::: - Example @exm-tothesix :::{#sol-tothesix} @@ -843,36 +734,29 @@ For convenience, define $f(z) = z^6$ and $z = g(x) = 3x^2+5x-7$. Then, $y=f[g(x \end{align*} ::: - Example @exm-exmderivexp :::{#sol-exmderivexp} - 1. Let $u(x)=-3x$. Then $u^\prime(x)=-3$ and $f^\prime(x)=-3e^{-3x}$. 2. Let $u(x)=x^2$. Then $u^\prime(x)=2x$ and $f^\prime(x)=2xe^{x^2}$. - ::: Example @exm-exmderivlog - :::{#sol-exmderivlog} - 1. Let $u(x)=x^2+9$. Then $u^\prime(x)=2x$ and $$\frac{dy}{dx}= \frac{u^\prime(x)}{u(x)} = \frac{2x}{(x^2+9)}$$ 2. Let $u(x)=\log x$. Then $u^\prime(x)=1/x$ and $\frac{dy}{dx} = \frac{1}{(x\log x)}$. 3. Use the generalized power rule. $$\frac{dy}{dx} = \frac{(2 \log x)}{x}$$ 4. We know that $\log e^x=x$ and that $dx/dx=1$, but we can double check. Let $u(x)=e^x$. Then $u^\prime(x)=e^x$ and $\frac{dy}{dx} = \frac{u^\prime(x)}{u(x)} = \frac{e^x}{e^x} = 1.$ - ::: Example @exm-defintmon - :::{#sol-defintmon} What is $F(x)$? From the power rule, recognize $\frac{d}{dx}x^3 = 3x^2$ so @@ -899,7 +783,6 @@ We can easily integrate this, since it is just a polynomial. Doing so and subst ::: - Example @exm-intsub2 :::{#sol-intsub2} @@ -936,7 +819,6 @@ Notice that we now have an integral similar to the previous one, but with $x^{n- For a given $n$, we would repeat the integration by parts procedure until the integrand was directly integratable --- e.g., when the integral became $\int e^{ax}dx$. - 2. $$\int x^3 e^{-x^2} dx$$ We could, as before, choose $u=x^3$ and $dv=e^{-x^2}dx$. But we can't then find $v$ --- i.e., integrating $e^{-x^2}dx$ isn't possible. Instead, notice that $$\frac{d}{dx}e^{-x^2} = -2xe^{-x^2},$$ which can be factored out of the original integrand $$\int x^3 e^{-x^2} dx = \int x^2 (xe^{-x^2})dx.$$ diff --git a/05_optimization.qmd b/05_optimization.qmd index ab4cd02..d4ddd49 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -1,6 +1,5 @@ # Optimization {#optim} - ```{r} #| include: false library(ggplot2) @@ -12,13 +11,11 @@ To optimize, we use derivatives and calculus. Optimization is to find the maximu Optimization also comes up in Economics, Formal Theory, and Political Economy all the time. A go-to model of human behavior is that they optimize a certain utility function. Humans are not pure utility maximizers, of course, but nuanced models of optimization -- for example, adding constraints and adding uncertainty -- will prove to be quite useful. - ## Example: Meltzer-Richard {.unnumbered} A standard backdrop in comparative political economy, the Meltzer-Richard (1981) model states that redistribution of wealth should be higher in societies where the median income is much smaller than the average income. More to the point, typically income distributions wher ethe median is very different from the average is one of high inequality. In other words, the Meltzer-Richard model says that highly unequal economies will have more re-distribution of wealth. Why is that the case? Here is a simplified example that is not the exact model by Meltzer and Richard^[Allan H. Meltzer and Scott F. Richard. ["A Rational Theory of the Size of Government"](https://www.jstor.org/stable/1830813). _Journal of Political Economy_ 89:5 (1981), p. 914-927], but adapted from Persson and Tabellini^[Adapted from Torsten Persson and Guido Tabellini, _Political Economics: Explaining Economic Policy_. MIT Press. ] - We will set the following things about our model human and model democracy. - Individuals are indexed by $i$, and the total population is normalized to unity ("1") without loss of generality. @@ -44,17 +41,14 @@ Income varies by person (In the next section we will cover probability, by then - $E(y)$ is the average income of the society. - $\text{med}(y)$ is the __median income__ of the society. - What will happen in this economy? What will the tax rate be set too? How much public goods will be provided? - We've skipped ahead of some formal theory results of demoracy, but hopefully these are conceptually intuitive. First, if a democracy is competitive, there is no slack in the government's goods, and all tax revenue becomes a public good. So we can go ahead and set the constraint: $$g = \sum_{i} \tau y_i P(y_i) = \tau E(y)$$ We can do this trick because of the "normalizes to unity" setting, but this is a general property of the average. - Now given this constraint we can re-write an individual's welfare as \begin{align*} @@ -74,25 +68,16 @@ $$g_i^\star = {U_g}^{-1}\left(\frac{y_i}{E(y)}\right)$$ Now recall that because we assumed concavity, $U_g$ is a negative sloping function whose value is positive. It can be shown that the inverse of such a function is also decreasing. Thus an individual's preferred level of government is determined by a single continuum, the person's income divided by the average income, and the function is __decreasing__ in $y_i$. This is consistent with our intuition that richer people prefer less redistribution. - That was the amount for any given person. The government has to set one value of $g$, however. So what will that be? Now we will use another result, the median voter theorem. This says that under certain general electoral conditions (single-peaked preferences, two parties, majority rule), the policy winner will be that preferred by the median person in the population. Because the only thing that determines a person's preferred level of government is $y_i / E(y)$, we can presume that the median voter, whose income is $\text{med}(y)$ will prevail in their preferred choice of government. Therefore, we wil see - $$g^\star = {U_g}^{-1}\left(\frac{\text{med}(y)}{E(y)}\right)$$ What does this say about the level of redistribution we observe in an economy? The higher the average income is than the median income, which often (but not always) means _more_ inequality, there should be _more_ redistribution. - - - - - ## Maxima and Minima The first derivative, $f'(x)$, quantifies the slope of a function. Therefore, it can be used to check whether the function $f(x)$ at the point $x$ is increasing or decreasing at $x$. - - 1. __Increasing:__ $f'(x)>0$ 1. __Decreasing:__ $f'(x)<0$ 1. __Neither increasing nor decreasing__: $f'(x)=0$ @@ -100,7 +85,6 @@ The first derivative, $f'(x)$, quantifies the slope of a function. Therefore, it So for example, $f(x) = x^2 + 2$ and $f^\prime(x) = 2x$ - ```{r} #| echo: false #| fig-cap: Maxima and Minima @@ -118,7 +102,6 @@ fprimex <- fx0 + stat_function(fun = function(x) 2*x, linewidth = 0.5, linetype fx + fprimex + plot_layout(nrow = 1) ``` - :::{#exr-maxminplot} ## Plotting a maximum and minimum @@ -126,7 +109,6 @@ Plot $f(x)=x^3+ x^2 + 2$, plot its derivative, and identify where the derivative ::: - ```{r} #| echo: false #| eval: false @@ -145,7 +127,6 @@ fprimex <- fx0 + stat_function(fun = function(x) 3*x^2, size = 0.5, linetype = " fx + fprimex + plot_annotation(expression(f(x)==x^3+2)) ``` - The second derivative $f''(x)$ identifies whether the function $f(x)$ at the point $x$ is 1. Concave down: $f''(x)<0$ @@ -174,24 +155,18 @@ __Global Maxima and Minima__ Sometimes no global max or min exists --- e.g., $f( 2. __Globally concave up or concave down functions.__ If $f''(x)$ is never zero, then there is at most one critical point. That critical point is a global maximum if $f''<0$ and a global minimum if $f''>0$. 3. __Functions over closed and bounded intervals__ must have both a global maximum and a global minimum. - - :::{#exm-maxmindraw} ## Maxima and Minima by drawing Find any critical points and identify whether they are a max, min, or saddle point: - 1. $f(x)=x^2+2$ 1. $f(x)=x^3+2$ 1. $f(x)=|x^2-1|$, $x\in [-2,2]$ - ::: - - ## Concavity of a Function Concavity helps identify the curvature of a function, $f(x)$, in 2 dimensional space. @@ -204,7 +179,6 @@ A function $f$ is strictly concave over the set S \underline{if} $\forall x_1,x_ ::: - ```{r} #| echo: false range1 <- tibble(x = c(-4, 4)) @@ -218,7 +192,6 @@ fx2 <- ggplot(range1, aes(x = x)) + fx1 + fx2 + plot_layout(nrow = 1) ``` - :::{#def-convex} ## Convex Function Convex: A function f is strictly convex over the set S \underline{if} $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) < af(x_1) + (1-a)f(x_2)$$ @@ -226,7 +199,6 @@ Convex: A function f is strictly convex over the set S \underline{if} $\forall x Any line connecting two points on a convex function will lie above the function. ::: - \begin{comment} \parbox{2in}{\includegraphics[scale=.4]{Convex.pdf}} \end{comment} @@ -257,7 +229,6 @@ No matter what two points you select, the \textit{highest} valued point will alw \parbox{1.8in}{\includegraphics[scale=.4]{Quasiconvex.pdf}} \end{comment} - **Second Derivative Test of Concavity**: The second derivative can be used to understand concavity. If @@ -266,7 +237,6 @@ f''(x) < 0 & \Rightarrow & \text{Concave}\\ f''(x) > 0 & \Rightarrow & \text{Convex} \end{array}$$ - ### Quadratic Forms {.unnumbered} Quadratic forms is shorthand for a way to summarize a function. This is important for finding concavity because @@ -310,20 +280,14 @@ For example, the Quadratic on $\mathbf{R}^2$: &= a_{11}x_1^2 + a_{12}x_1x_2 + a_{22}x_2^2 \end{align*} - - - ### Definiteness of Quadratic Forms {.unnumbered} - When the function $f(\mathbf{x})$ has more than two inputs, determining whether it has a maxima and minima (remember, functions may have many inputs but they have only one output) is a bit more tedious. Definiteness helps identify the curvature of a function, $Q(\textbf{x})$, in n dimensional space. __Definiteness__: By definition, a quadratic form always takes on the value of zero when $x = 0$, $Q(\textbf{x})=0$ at $\textbf{x}=0$. The definiteness of the matrix $\textbf{A}$ is determined by whether the quadratic form $Q(\textbf{x})=\textbf{x}^\top\textbf{A}\textbf{x}$ is greater than zero, less than zero, or sometimes both over all $\mathbf{x}\ne 0$. - - ## FOC and SOC We can see from a graphical representation that if a point is a local maxima or minima, it must meet certain conditions regarding its derivative. These are so commonly used that we refer these to "First Order Conditions" (FOCs) and "Second Order Conditions" (SOCs) in the economic tradition. @@ -354,10 +318,6 @@ $\mathbf{x}^*$ is a critical point if and only if $\nabla f(\mathbf{x}^*)=0$. If ::: - - - - :::{#exm-gradcp} Example: Given a function $f(\mathbf{x})=(x_1-1)^2+x_2^2+1$, find the (1) Gradient and (2) Critical point of $f(\mathbf{x})$. @@ -372,7 +332,6 @@ Gradient &= \begin{pmatrix} 2(x_1-1)\\ 2x_2 \end{pmatrix} \end{align*} - Critical Point $\mathbf{x}^* =$ \begin{align*} @@ -382,15 +341,10 @@ Critical Point $\mathbf{x}^* =$ &\Rightarrow x_2^* = 0\\ \end{align*} - So $$\mathbf{x}^* = (1,0)$$ - ::: - - - ### Second Order Conditions {.unnumbered} When we found a critical point for a function of one variable, we used the second derivative as a indicator of the curvature at the point in order to determine whether the point was a min, max, or saddle (second derivative test of concavity). For functions of $n$ variables, we use _second order partial derivatives_ as an indicator of curvature. @@ -411,14 +365,10 @@ $$\mathbf{H(x)}=\begin{pmatrix} \frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}& ::: - - - Note that the hessian will be a symmetric matrix because $\frac{\partial f(\mathbf{x})}{\partial x_1\partial x_2} = \frac{\partial f(\mathbf{x})}{\partial x_2\partial x_1}$. Also note that given that $f(\mathbf{x})$ is of quadratic form, each element of the hessian will be a constant. - These definitions will be employed when we determine the __Second Order Conditions__ of a function: Given a function $f(\mathbf{x})$ and a point $\mathbf{x}^*$ such that $\nabla f(\mathbf{x}^*)=0$, @@ -429,8 +379,6 @@ Given a function $f(\mathbf{x})$ and a point $\mathbf{x}^*$ such that $\nabla f( 4. Hessian is Negative Semidefinite $\forall \mathbf{x}\in B(\mathbf{x}^*,\epsilon)$} $\quad \Longrightarrow \quad$ Local Max 5. Hessian is Indefinite $\quad \Longrightarrow \quad$ Saddle Point - - :::{#exm-maxmin2d} ## Max and min with two dimensions @@ -440,7 +388,6 @@ saddle point? ::: - :::{#sol-maxmin2d} The Hessian is @@ -451,7 +398,6 @@ The Leading principal minors of the Hessian are $M_1=2; M_2=4$. Now we consider Maxima, Minima, or Saddle Point? Since the Hessian is positive definite and the gradient equals 0, $x^\star = (1,0)$ is a strict local minimum. - Note: Alternate check of definiteness. Is $\mathbf{H(x^*)} \geq \leq 0 \quad \forall \quad \mathbf{x}\ne 0$ @@ -465,8 +411,6 @@ For any $\mathbf{x}\ne 0$, $2(x_1^2+x_2^2)>0$, so the Hessian is positive defin ::: - - ### Definiteness and Concavity {.unnumbered} Although definiteness helps us to understand the curvature of an n-dimensional function, it does not necessarily tell us whether the function is globally concave or convex. @@ -476,10 +420,8 @@ We need to know whether a function is globally concave or convex to determine wh 1. Hessian is Positive Semidefinite $\forall \mathbf{x}$} $\quad \Longrightarrow \quad$ Globally Convex 2. Hessian is Negative Semidefinite $\forall \mathbf{x}$} $\quad \Longrightarrow \quad$ Globally Concave - Notice that the definiteness conditions must be satisfied over the entire domain. - ## Global Maxima and Minima __Global Max/Min Conditions__: Given a function $f(\mathbf{x})$ and a point $\mathbf{x}^*$ such that $\nabla f(\mathbf{x}^*)=0$, @@ -496,7 +438,6 @@ not enough to guarantee $\mathbf{x}^*$ is a local max. However, showing that $\mathbf{H(x)}$ is negative semidefinite for all $\mathbf{x}$ guarantees that $x^*$ is a global max. (The same goes for positive semidefinite and minima.)\\ - Example: Take $f_1(x)=x^4$ and $f_2(x)=-x^4$. Both have $x=0$ as a critical point. Unfortunately, $f''_1(0)=0$ and $f''_2(0)=0$, so we can't tell whether $x=0$ is a min or max for either. However, @@ -505,7 +446,6 @@ and $f''_2(x)\le 0$ --- i.e., $f_1(x)$ is globally convex and $f_2(x)$ is globally concave. So $x=0$ is a global min of $f_1(x)$ and a global max of $f_2(x)$. - :::{#exr-maxmin} Given $f(\mathbf{x})=x_1^3-x_2^3+9x_1x_2$, find any maxima or minima. @@ -575,10 +515,8 @@ Given $f(\mathbf{x})=x_1^3-x_2^3+9x_1x_2$, find any maxima or minima. ## Constrained Optimization - We have already looked at optimizing a function in one or more dimensions over the whole domain of the function. Often, however, we want to find the maximum or minimum of a function over some restricted part of its domain. - ex: Maximizing utility subject to a budget constraint ![A typical Utility Function with a Budget Constraint](images/constraint.png) @@ -601,13 +539,10 @@ $$\max_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ $$\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ This tells us to maximize/minimize our function, $f(x_1,x_2)$, with respect to the choice variables, $x_1,x_2$, subject to the constraint. - Example: $$\max_{x_1,x_2} f(x_1, x_2) = -(x_1^2 + 2x_2^2) \text{ s.t. }x_1 + x_2 = 4$$ It is easy to see that the \textit{unconstrained} maximum occurs at $(x_1, x_2) = (0,0)$, but that does not satisfy the constraint. How should we proceed? - - ### Equality Constraints {.unnumbered} Equality constraints are the easiest to deal with because we know that the maximum or minimum has to lie on the (intersection of the) constraint(s). @@ -631,7 +566,6 @@ $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) - \sum_{ $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) + \sum_{i=1}^k\lambda_i(r_i - c_i(x_1,\dots, x_n))$$ Here we add the lagrangian term _and_ we subtract the constraining function from the constraint constant. - **Using the Lagrangian to Find the Critical Points**: To find the critical points, we take the partial derivatives of lagrangian function, $L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k)$, with respect to each of its variables (all choice variables $\mathbf{x}$ _and_ all lagrangian multipliers $\mathbf{\lambda}$). At a critical point, each of these partial derivatives must be equal to zero, so we obtain a system of $n + k$ equations in $n + k$ unknowns: \begin{align*} @@ -674,12 +608,10 @@ $$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ 4. Therefore, the only critical point is $x_1^* = \frac{8}{3}$ and $x_2^* = \frac{4}{3}$ 5. This gives $f(\frac{8}{3}, \frac{4}{3}) = -\frac{96}{9}$, which is less than the unconstrained optimum $f(0,0) = 0$ - ::: Notice that when we take the partial derivative of L with respect to the Lagrangian multiplier and set it equal to 0, we return exactly our constraint! This is why signs matter. - ## Inequality Constraints Inequality constraints define the boundary of a region over which we seek to optimize the function. This makes inequality constraints more challenging because we do not know if the maximum/minimum lies along one of the constraints (the constraint binds) or in the interior of the region. @@ -702,7 +634,6 @@ $$L(x_1,x_2,\lambda_1,s_1) = f(x_1,x_2) - \lambda_1 ( c(x_1,x_2) + s_1^2 - a_1)$ More generally, in n dimensions: $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k, s_1, \dots, s_k) = f(x_1, \dots, x_n) - \sum_{i = 1}^k \lambda_i(c_i(x_1,\dots, x_n) + s_i^2 - a_i)$$ - **Finding the Critical Points**: To find the critical points, we take the partial derivatives of the lagrangian function, $L(x_1,\dots,x_n,\lambda_1,\dots,\lambda_k,s_1,\dots,s_k)$, with respect to each of its variables (all choice variables $x$, all lagrangian multipliers $\lambda$, and all slack variables $s$). At a critical point, _each_ of these partial derivatives must be equal to zero, so we obtain a system of $n + 2k$ equations in $n + 2k$ unknowns: \begin{align*} @@ -723,7 +654,6 @@ $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k, s_1, \dots, s_k) = f(x_1, \do 2. $\lambda_i \neq 0$ and $s_i = 0$: This implies that there is no slack in the constraint and _the constraint does bind_. 3. $\lambda_i = 0$ and $s_i = 0$: In this case, there is no slack but the _constraint binds trivially_, without changing the optimum. - Example: Find the critical points for the following constrained optimization: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4$$ @@ -761,8 +691,6 @@ This shows that there are two critical points: $(0,0)$ and $(\frac{8}{3},\frac{4 5. Find maximum: Looking at the values of $f(x_1,x_2)$ at the critical points, we see that $f(x_1,x_2)$ is maximized at $x_1^* = 0$ and $x_2^*=0$. - - :::{#exr-critical-points-constrained-optimization} Example: Find the critical points for the following constrained optimization: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } @@ -773,8 +701,6 @@ x_2 \ge 0 \end{array}$$ ::: - - 1. Rewrite with the slack variables: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. \begin{array}{l} @@ -818,12 +744,8 @@ This shows that there are four critical points: $(0,0)$, $(4,0)$, $(0,4)$, and $ 4. Find maximum: Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem. - - - ## Kuhn-Tucker Conditions - As you can see, this can be a pain. When dealing explicitly with _non-negativity constraints_, this process is simplified by using the Kuhn-Tucker method. Because the problem of maximizing a function subject to inequality and non-negativity constraints arises frequently in economics, the **Kuhn-Tucker conditions** provides a method that often makes it easier to both calculate the critical points and identify points that are (local) maxima. @@ -842,7 +764,6 @@ $$L(x_1,x_2,\lambda_2) = f(x_1,x_2) - \lambda_1(c(x_1,x_2) - a_1)$$ More generally, in n dimensions: $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) - \sum_{i=1}^k\lambda_i(c_i(x_1,\dots, x_n) - a_i)$$ - **Kuhn-Tucker and Complementary Slackness Conditions**: To find the critical points, we first calculate the Kuhn-Tucker conditions by taking the partial derivatives of the lagrangian function, $L(x_1,\dots,x_n,\lambda_1,\dots,\lambda_k)$, with respect to each of its variables (all choice variable s$x$ and all lagrangian multipliers $\lambda$) and we calculate the _complementary slackness conditions_ by multiplying each partial derivative by its respective variable _and_ include non-negativity conditions for all variables (choice variables $x$ and lagrangian multipliers $\lambda$). **Kuhn-Tucker Conditions** @@ -880,9 +801,6 @@ In a two-dimensional set-up, this means we must check the following cases: 1. $x_1 \neq 0, x_2 = 0$ Border Solution 1. $x_1 \neq 0, x_2 \neq 0$ Interior Solution - - - :::{#exm-k-t-2} ## Kuhn-Tucker with two variables Solve the following optimization problem with inequality constraints @@ -899,7 +817,6 @@ $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)$$ ::: - 1. Write the Lagrangian: $$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ @@ -943,13 +860,10 @@ $x_1 \neq 0, x_2 \neq 0$ & $-\frac{16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\ 4. Find Maximum: Three of the critical points violate the requirement that $\lambda \geq 0$, so the point $(0,0,0)$ is the maximum. - - :::{#exr-ktlogs} ## Kuhn-Tucker with logs Solve the constrained optimization problem, - $$\max_{x_1,x_2} f(x) = \frac{1}{3}\log (x_1 + 1) + \frac{2}{3}\log (x_2 + 1) \text{ s.t. } \begin{array}{l} x_1 + 2x_2 \leq 4\\ @@ -958,8 +872,6 @@ x_1 + 2x_2 \leq 4\\ \end{array}$$ ::: - - 1. Write the Lagrangian: $$L(x_1, x_2, \lambda) = \frac{1}{3}\log(x_1+1) + \frac{2}{3}\log(x_2+1) - \lambda(x_1 + 2x_2 - 4)$$ @@ -1001,14 +913,11 @@ $x_1 \neq 0, x_2 \neq 0$ & & $\frac{4}{3}$ & $\frac{4}{3}$ & $\log\frac{7}{3}$\ Three of the critical points violate the constraints, so the point $(\frac{4}{3},\frac{4}{3})$ is the maximum.\\ - - ## Applications of Quadratic Forms __Curvature and The Taylor Polynomial as a Quadratic Form__: The Hessian is used in a Taylor polynomial approximation to $f(\mathbf{x})$ and provides information about the curvature of $f({\bf x})$ at $\mathbf{x}$ --- e.g., which tells us whether a critical point $\mathbf{x}^*$ is a min, max, or saddle point. - 1. The second order Taylor polynomial about the critical point ${\bf x}^*$ is $$f({\bf x}^*+\bf h)=f({\bf x}^*)+\nabla f({\bf x}^*) \bf h +\frac{1}{2} \bf h^\top {\bf H(x^*)} \bf h + R(\bf h)$$ diff --git a/06_probability.qmd b/06_probability.qmd index c0e8496..e3f0e75 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -14,7 +14,6 @@ options(knitr.graphics.auto_pdf = TRUE) # use pdf for images Probability in high school is usually really about combinatorics: the probability of event _A_ is the number of ways in which _A_ can occur divided by the number of all other possibilities. This is a very simplified version of probability, which we can call the "counting definition of probability", essentially because each possible event to count is often equally likely and discrete. But it is still good to review the underlying rules here. - __Fundamental Theorem of Counting__: If an object has $j$ different characteristics that are independent of each other, and each characteristic $i$ has $n_i$ ways of being expressed, then there are $\prod_{i = 1}^j n_i$ possible unique objects. ```{example} @@ -31,12 +30,8 @@ Suppose we are given a stack of cards. Cards can be either red or black and can 4. Number of Outcomes $=$ - ``` - - - We often need to count the number of ways to choose a subset from some set of possibilities. The number of outcomes depends on two characteristics of the process: does the order matter and is replacement allowed? It is useful to think of any problem concretely, e.g. through a __sampling table__: If there are $n$ objects which are numbered 1 to $n$ and we select $k < n$ of them, how many different outcomes are possible? @@ -45,14 +40,12 @@ If the order in which a given object is selected matters, selecting 4 numbered o If replacement is allowed, there are always the same $n$ objects to select from. However, if replacement is not allowed, there is always one less option than the previous round when making a selection. For example, if replacement is not allowed and I am selecting 3 elements from the following set {1, 2, 3, 4, 5, 6}, I will have 6 options at first, 5 options as I make my second selection, and 4 options as I make my third. - 1. So if ___order matters___ AND we are sampling ___with replacement___, the number of different outcomes is $n^k$. 2. If ___order matters___ AND we are sampling ___without replacement___, the number of different outcomes is $n(n-1)(n-2)...(n-k+1)=\frac{n!}{(n-k)!}$. 3. If ___order doesn't matter___ AND we are sampling ___without replacement___, the number of different outcomes is $\binom{n}{k} = \frac{n!}{(n-k)!k!}$. - Expression $\binom{n}{k}$ is read as "n choose k" and denotes $\frac{n!}{(n-k)!k!}$. Also, note that $0! = 1$. ```{example} @@ -67,8 +60,6 @@ There are five balls numbered from 1 through 5 in a jar. Three balls are chosen. 3. Unordered, without replacement $=$ - - ``` ```{exercise} @@ -81,13 +72,10 @@ Four cards are selected from a deck of 52 cards. Once a card has been drawn, it ``` - - ## Sets {#setoper} Probability is about quantifying the uncertainty of events. _Sets_ (set theory) are the mathematical way we choose to formalize those events. Events are not inherently numerical: the onset of war or the stock market crashing is not inherently a number. Sets can define such events, and we wrap math around so that we have a transparent language to communicate about those events. _Measure theory_ might sound mysterious or hard, but it is also just a mathematical way to quantify things like length, volume, and mass. Probability can be thought of as a particular application of measure theory where we want to quantify the measure of a set. - __Set__ : A set is any well defined collection of elements. If $x$ is an element of $S$, $x \in S$. __Sample Space (S)__: A set or collection of all possible outcomes from some process. Outcomes in the set can be discrete elements (countable) or points along a continuous interval (uncountable). @@ -101,14 +89,12 @@ __Event__: Any collection of possible outcomes of an experiment. Any subset of t __Empty Set__: a set with no elements. $S = \{\}$. It is denoted by the symbol $\emptyset$. - Set operations: 1. __Union__: The union of two sets $A$ and $B$, $A \cup B$, is the set containing all of the elements in $A$ or $B$. $$A_1 \cup A_2 \cup \cdots \cup A_n = \bigcup_{i=1}^n A_i$$ 2. __Intersection__: The intersection of sets $A$ and $B$, $A \cap B$, is the set containing all of the elements in both $A$ and $B$. $$A_1 \cap A_2 \cap \cdots \cap A_n = \bigcap_{i=1}^n A_i$$ 3. __Complement__: If set $A$ is a subset of $S$, then the complement of $A$, denoted $A^C$, is the set containing all of the elements in $S$ that are not in $A$. - Properties of set operations: * __Commutative__: $A \cup B = B \cup A$; $A \cap B = B \cap A$ @@ -117,7 +103,6 @@ Properties of set operations: * __de Morgan's laws__: $(A \cup B)^C = A^C \cap B^C$; $(A \cap B)^C = A^C \cup B^C$ * __Disjointness__: Sets are disjoint when they do not intersect, such that $A \cap B = \emptyset$. A collection of sets is pairwise disjoint (**mutually exclusive**) if, for all $i \neq j$, $A_i \cap A_j = \emptyset$. A collection of sets form a partition of set $S$ if they are pairwise disjoint and they cover set $S$, such that $\bigcup_{i = 1}^k A_i = S$. - ```{example} #| name: Sets #| label: sets @@ -131,9 +116,6 @@ Write out the following sets: 3. $B^c$ 4. $A \cap (B \cup C)$ - - - ``` @@ -141,8 +123,6 @@ Write out the following sets: #| name: Sets #| label: sets1 - - Suppose you had a pair of four-sided dice. You sum the results from a single toss. What is the set of possible outcomes (i.e. the sample space)? @@ -157,10 +137,6 @@ Consider subsets A {2, 8} and B {2,3,7} of the sample space you found. What is ``` - - - - ## Probability {#probdef} ```{r} @@ -175,7 +151,6 @@ knitr::include_graphics('images/probability.png') Many things in the world are uncertain. In everyday speech, we say that we are _uncertain_ about the outcome of random events. Probability is a formal model of uncertainty which provides a measure of uncertainty governed by a particular set of rules (Figure \@ref(fig:prob-image)). A different model of uncertainty would, of course, have a set of rules different from anything we discuss here. Our focus on probability is justified because it has proven to be a particularly useful model of uncertainty. - __Probability Distribution Function__: a mapping of each event in the sample space $S$ to the real numbers that satisfy the following three axioms (also called Kolmogorov's Axioms). Formally, @@ -192,25 +167,19 @@ The axioms of probability make sure that the separate events add up in terms of ```{definition} #| name: Axioms of Probability - 1. For any event $A$, $P(A)\ge 0$. 2. $P(S)=1$ 3. The Countable Additivity Axiom: For any sequence of _disjoint_ (mutually exclusive) events $A_1,A_2,\ldots$ (of which there may be infinitely many), $$P\left( \bigcup\limits_{i=1}^k A_i\right)=\sum\limits_{i=1}^k P(A_i)$$ - The last axiom is an extension of a union to infinite sets. When there are only two events in the space, it boils down to: \begin{align*} P(A_1 \cup A_2) = P(A_1) + P(A_2) \quad\text{for disjoint } A_1, A_2 \end{align*} - ``` - - - ### Probability Operations {.unnumbered} Using these three axioms, we can define all of the common rules of probability. @@ -240,17 +209,14 @@ Then, 6. Let $A=\{ 1,2,3,4,5 \}\subset S$. Then $P(A)=5/6 x) = 1 - P(X \le x)$. ```{example} @@ -527,13 +456,10 @@ For a fair die with its value as $Y$, What are the following? ``` - - ### Continuous Random Variables {.unnumbered} We also have a similar definition for _continuous_ random variables. - ```{definition} #| name: Continuous Random Variable @@ -541,7 +467,6 @@ $X$ is a continuous random variable if there exists a nonnegative function $f(x) ``` - ```{definition} #| name: Probability Density Function @@ -552,13 +477,11 @@ $$\int\limits_{-\infty}^\infty f(x)dx=1$$ Note also that $P(X = x)=0$ --- i.e., the probability of any point $y$ is zero. ``` - \begin{comment} \item[] \parbox[t]{4.5in}{Example: $f(y)=1, \quad 0\le y \le1$}\parbox{1.5in}{\hfill \epsffile{contpdf.eps}} \end{comment} - For both discrete and continuous random variables, we have a unifying concept of another measure: the cumulative distribution: ```{definition} @@ -572,16 +495,12 @@ $$P(a\le x\le b)=\int\limits_a^b f(x)dx$$ The PDF and CDF are linked by the integral: The CDF of the integral of the PDF: $$f(x) = F'(x)=\frac{dF(x)}{dx}$$ - ```{example} For $f(y)=1, \quad 0 0$ * Continuous: $f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)},\quad f_X(x) > 0$ - ```{exercise} #| name: Discrete Outcomes Suppose we are interested in the outcomes of flipping a coin and rolling a 6-sided die at the same time. The sample space for this process contains 12 elements: $$\{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}$$ We can define two random variables $X$ and $Y$ such that $X = 1$ if heads and $X = 0$ if tails, while $Y$ equals the number on the die. We can then make statements about the joint distribution of $X$ and $Y$. What are the following? - 1. $P(X=x)$ 1. $P(Y=y)$ 1. $P(X=x, Y=y)$ @@ -634,11 +549,6 @@ We can then make statements about the joint distribution of $X$ and $Y$. What ar ``` - - - - - ## Expectation We often want to summarize some characteristics of the distribution of a random variable. The most important summary is the expectation (or expected value, or mean), in which the possible values of a random variable are weighted by their probabilities. @@ -650,19 +560,15 @@ In words, it is the weighted average of all possible values of $Y$, weighted by ``` - ```{example} #| name: Expectation of a Discrete Random Variable #| label: expectdiscrete - What is the expectation of a fair, six-sided die? - ``` - __Expectation of a Continuous Random Variable__: The expected value of a continuous random variable is similar in concept to that of the discrete random variable, except that instead of summing using @@ -670,20 +576,14 @@ probabilities as weights, we integrate using the density to weight. Hence, the expected value of the continuous variable $Y$ is defined by $$E(Y)=\int\limits_{y} y f(y) dy$$ - ```{example} #| name: Expectation of a Continuous Random Variable #| label: expectconti - Find $E(Y)$ for $f(y)=\frac{1}{1.5}, \quad 0 2)$ - ``` - \begin{comment} \parbox{1.5in}{\hfill \epsffile{unifpdf.eps}} \end{comment} @@ -947,12 +814,10 @@ A random variable $Y$ is normally distributed with mean $E(Y)=\mu$ and variance $$f(y)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(y-\mu)^2}{2\sigma^2}}$$ - ``` See Figure \@ref(fig:normaldens) are various Normal Distributions with the same $\mu = 1$ and two versions of the variance. - ```{r} #| label: normaldens #| echo: false @@ -972,19 +837,14 @@ fx ``` - - ## Summarizing Observed Events (Data) - So far, we've talked about distributions in a theoretical sense, looking at different properties of random variables. We don't observe random variables; we observe realizations of the random variable. These realizations of events are roughly equivalent to what we mean by "data". - __Sample mean__: This is the most common measure of central tendency, calculated by summing across the observations and dividing by the number of observations. $$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i$$ The sample mean is an _estimate_ of the expected value of a distribution. - \begin{framed} Example: \begin{center} @@ -1004,7 +864,6 @@ Y & 1 & 2 & 1 & 2 & 2 & 1 & 2 & 0 & 2 & 0\\ \end{enumerate} \end{framed} - __Dispersion__: We also typically want to know how spread out the data are relative to the center of the observed distribution. There are several ways to measure dispersion. __Sample variance__: The sample variance is the sum of the squared deviations from the sample mean, divided by the number of observations minus 1. @@ -1015,7 +874,6 @@ Again, this is an _estimate_ of the variance of a random variable; we divide by __Standard deviation__: The sample standard deviation is the square root of the sample variance. $$ \hat{SD}(X) = \sqrt{\hat{\text{Var}}(X)} = \sqrt{\frac{1}{n-1}\sum_{i = 1}^n (x_i - \bar{x})^2}$$ - \begin{framed} Example: Using table above, calculate: \begin{enumerate} @@ -1024,27 +882,19 @@ Example: Using table above, calculate: \end{enumerate} \end{framed} - __Covariance and Correlation__: Both of these quantities measure the degree to which two variables vary together, and are estimates of the covariance and correlation of two random variables as defined above. - 1. **Sample covariance**: $\hat{\text{Cov}}(X,Y) = \frac{1}{n-1}\sum_{i = 1}^n(x_i - \bar{x})(y_i - \bar{y})$ 2. **Sample correlation**: $\hat{\text{Corr}} = \frac{\hat{\text{Cov}}(X,Y)}{\sqrt{\hat{\text{Var}}(X)\hat{\text{Var}}(Y)}}$ - ```{example} Example: Using the above table, calculate the sample versions of: 1. $\text{Cov}(X,Y)$ 2. $\text{Corr}(X, Y)$ - ``` - - - - ## Asymptotic Theory In theoretical and applied research, asymptotic arguments are often made. In this section we briefly introduce some of this material. @@ -1074,7 +924,6 @@ $$\bar{X}_n \approx \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).$$ The standa Intuitively, this result also justifies that whenever a lot of small, independent processes somehow combine together to form the realized observations, practitioners often feel comfortable assuming Normality. - ```{theorem} #| label: lln #| name: Law of Large Numbers (LLN) @@ -1082,14 +931,11 @@ For any draw of independent random variables with the same mean $\mu$, the sampl $$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ - A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". ``` - as $n\to \infty$. In other words, $P( \lim_{n\to\infty}\bar{X}_n = \mu) = 1$. This is an important motivation for the widespread use of the sample mean, as well as the intuition link between averages and expected values. - More precisely this version of the LLN is called the _weak_ law of large numbers because it leaves open the possibility that $|\bar{X}_n - \mu | > \varepsilon$ occurs many times. The _strong_ law of large numbers states that, under a few more conditions, the probability that the limit of the sample average is the true mean is 1 (and other possibilities occur with probability 0), but the difference is rarely consequential in practice. The Strong Law of Large Numbers holds so long as the expected value exists; no other assumptions are needed. However, the rate of convergence will differ greatly depending on the distribution underlying the observed data. When extreme observations occur often (i.e. kurtosis is large), the rate of convergence is much slower. Cf. The distribution of financial returns. @@ -1098,10 +944,8 @@ The Strong Law of Large Numbers holds so long as the expected value exists; no o Some of you may encounter "big-OH''-notation. If $f, g$ are two functions, we say that $f = \mathcal{O}(g)$ if there exists some constant, $c$, such that $f(n) \leq c \times g(n)$ for large enough $n$. This notation is useful for simplifying complex problems in game theory, computer science, and statistics. - Example. - What is $\mathcal{O}( 5\exp(0.5 n) + n^2 + n / 2)$? Answer: $\exp(n)$. Why? Because, for large $n$, $$ \frac{ 5\exp(0.5 n) + n^2 + n / 2 }{ \exp(n)} \leq \frac{ c \exp(n) }{ \exp(n)} = c. @@ -1124,13 +968,10 @@ Answer to Example \@ref(exm:counting): 3. $\binom{5}{3} = \frac{5!}{(5-3)!3!} = \frac{5 \times 4}{2 \times 1} = 10$ - Answer to Exercise \@ref(exr:counting1): 1. $\binom{52}{4} = \frac{52!}{(52-4)!4!} = 270725$ - - Answer to Example \@ref(exm:sets): 1. {1, 2, 3, 4, 5, 6} @@ -1145,10 +986,8 @@ Sample Space: {2, 3, 4, 5, 6, 7, 8} 1. {3, 4, 5, 6, 7} 2. {4, 5, 6} - Answer to Example \@ref(exm:prob): - 1. ${1, 2, 3, 4, 5, 6}$ 2. $\frac{1}{6}$ @@ -1163,7 +1002,6 @@ Answer to Example \@ref(exm:prob): 7. $A\cup B=\{1, 2, 3, 4, 6\}$, $A\cap B=\{2\}$, $\frac{5}{6}$ - Answer to Exercise \@ref(exr:prob1): 1. $P(X = 5) = \frac{4}{16}$, $P(X = 3) = \frac{2}{16}$, $P(X = 6) = \frac{3}{16}$ @@ -1187,7 +1025,6 @@ Answer to Example \@ref(exm:condprobexm2): $P(1|Odd) = \frac{P(1 \cap Odd)}{P(Odd)} = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{3}$ - Answer to Example \@ref(exm:bayesrule): We are given that @@ -1199,12 +1036,10 @@ $$P(D|S) = \frac{.4 \times .5}{.4 \times .5 + .6 \times .9 } = .27$$ Answer to Exercise \@ref(exr:condprobexr): - We are given that $$P(M) = .02, P(C|M) = .95, P(C^c|M^c) = .97$$ - $$P(M|C) = \frac{P(C|M)P(M)}{P(C)}$$ $$= \frac{P(C|M)P(M)}{P(C|M)P(M) + P(C|M^c)P(M^c)}$$ @@ -1212,23 +1047,18 @@ $$= \frac{P(C|M)P(M)}{P(C|M)P(M) + P(C|M^c)P(M^c)}$$ $$= \frac{P(C|M)P(M)}{P(C|M)P(M) + [1-P(C^c|M^c)]P(M^c)}$$ $$ = \frac{.95 \times .02}{.95 \times .02 + .03 \times .98} = .38$$ - Answer to Example \@ref(exm:expectdiscrete): $E(Y)=7/2$ We would never expect the result of a rolled die to be $7/2$, but that would be the average over a large number of rolls of the die. - Answer to Example \@ref(exm:expectconti) 0.75 - - Answer to Example \@ref(exm:var): - $E(X) = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 2 \times \frac{3}{8} + 3 \times \frac{1}{8} = \frac{3}{2}$ Since there is a 1 to 1 mapping from $X$ to $X^2:$ $E(X^2) = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 4 \times \frac{3}{8} + 9 \times \frac{1}{8} = \frac{24}{8} = 3$ @@ -1239,8 +1069,6 @@ Since there is a 1 to 1 mapping from $X$ to $X^2:$ $E(X^2) = 0 \times \frac{1}{8 &= \frac{3}{4} \end{align*} - - Answer to Exercise \@ref(exr:expvar): 1. $E(X) = -2(\frac{1}{5}) + -1(\frac{1}{6}) + 0(\frac{1}{5}) + 1(\frac{1}{15}) + 2(\frac{11}{30}) = \frac{7}{30}$ @@ -1255,8 +1083,6 @@ Answer to Exercise \@ref(exr:expvar): &= \frac{5}{2} - \frac{7}{30}^2 \approx 2.45 \end{align*} - - Answer to Exercise \@ref(exr:expvar2): 1. expectation = $\frac{6}{5}$, variance = $\frac{6}{25}$ diff --git a/07_linear-algebra.qmd b/07_linear-algebra.qmd index 7ac4d62..b91a370 100644 --- a/07_linear-algebra.qmd +++ b/07_linear-algebra.qmd @@ -13,9 +13,6 @@ $\bullet$ Rank $\bullet$ The Inverse of a Matrix $\bullet$ Inverse of Larger Matrices - - - ## Working with Vectors {#vector-def} __Vector__: A vector in $n$-space is an ordered list of $n$ numbers. These numbers can be represented as either a row vector or a column vector: $$ {\bf v} \begin{pmatrix} v_1 & v_2 & \dots & v_n\end{pmatrix} , {\bf v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$$ @@ -50,7 +47,6 @@ Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{p ``` - ```{exercise} #| name: Vector Algebra #| label: vectors1 @@ -66,19 +62,15 @@ Let $u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}$, $v = \begin{pmatrix} 9&-3&2&8 \ 4. $w \cdot v$ - ``` - ## Linear Independence {#linearindependence} __Linear combinations__: The vector ${\bf u}$ is a linear combination of the vectors ${\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k$ if $${\bf u} = c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k$$ - For example, $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of the following three vectors: $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$. This is because $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ = $(2)\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$ $+ (-1)\begin{pmatrix} 2 & 3& 4\end{pmatrix}$ + $3\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$ - __Linear independence__: A set of vectors ${\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k$ is linearly independent if the only solution to the equation $$c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k = 0$$ is $c_1 = c_2 = \cdots = c_k = 0$. If another solution exists, the set of vectors is linearly dependent. @@ -87,7 +79,6 @@ A set $S$ of vectors is linearly dependent if and only if at least one of the ve Linear independence is only defined for sets of vectors with the same number of elements; any linearly independent set of vectors in $n$-space contains at most $n$ vectors. - Since $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$, these 4 vectors constitute a linearly dependent set. ```{example} @@ -99,12 +90,9 @@ Are the following sets of vectors linearly independent? 1. $\begin{pmatrix}2 & 3 & 1 \end{pmatrix}$ and $\begin{pmatrix}4 & 6 & 1 \end{pmatrix}$ 2. $\begin{pmatrix}1 & 0 & 0 \end{pmatrix}$, $\begin{pmatrix}0 & 5 & 0 \end{pmatrix}$, and $\begin{pmatrix}10 & 10 & 0 \end{pmatrix}$ - ``` - - ```{exercise} #| name: Linear Independence #| label: linearindep1 @@ -115,13 +103,8 @@ Are the following sets of vectors linearly independent? 2. $${\bf v}_1 = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} -4 \\ 6 \\ 5 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} -2 \\ 8 \\ 6 \end{pmatrix} $$ - - ``` - - - ## Basics of Matrix Algebra {#matrixbasics} __Matrix__: A matrix is an array of real numbers arranged in $m$ rows by $n$ columns. The dimensionality of the matrix is defined as the number of rows by the number of columns, $m \times n$. @@ -158,7 +141,6 @@ $${\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad ``` - __Scalar Multiplication__: Given the scalar $s$, the scalar multiplication of $s {\bf A}$ is $$ s {\bf A}= s \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ @@ -176,7 +158,6 @@ __Scalar Multiplication__: Given the scalar $s$, the scalar multiplication of $ ```{example} #| label: scalarmulti - $s=2, \qquad {\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ $s {\bf A} =$ @@ -185,26 +166,19 @@ $s {\bf A} =$ ``` - __Matrix Multiplication__: If ${\bf A}$ is an $m\times k$ matrix and $\bf B$ is a $k\times n$ matrix, then their product $\bf C = A B$ is the $m\times n$ matrix where $$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ik}b_{kj}$$ ```{example} #| label: matrixmulti - 1. $\begin{pmatrix} a&b\\c&d\\e&f \end{pmatrix} \begin{pmatrix} A&B\\C&D \end{pmatrix} =$ 2. $\begin{pmatrix} 1&2&-1\\3&1&4 \end{pmatrix} \begin{pmatrix} -2&5\\4&-3\\2&1\end{pmatrix} =$ - - ``` - - - Note that the number of columns of the first matrix must equal the number of rows of the second matrix, in which case they are __conformable for multiplication__. The sizes of the matrices (including the resulting product) must be $$(m\times k)(k\times n)=(m\times n)$$ Also note that if __AB__ exists, __BA__ exists only if $\dim({\bf A}) = m \times n$ and $\dim({\bf B}) = n \times m$. @@ -231,12 +205,10 @@ __Transpose__: The transpose of the $m\times n$ matrix $\bf A$ is the $n\times m For example, - ${\bf A}=\begin{pmatrix} 4&-2&3\\0&5&-1\end{pmatrix}, \qquad {\bf A}^T=\begin{pmatrix} 4&0\\-2&5\\3&-1 \end{pmatrix}$ ${\bf B}=\begin{pmatrix} 2\\-1\\3 \end{pmatrix}, \qquad {\bf B}^T=\begin{pmatrix} 2&-1&3\end{pmatrix}$ - The following rules apply for transposed matrices: \begin{enumerate} \item $({\bf A+B})^T = {\bf A}^T+{\bf B}^T$ @@ -250,14 +222,10 @@ Example of $({\bf AB})^T = {\bf B}^T{\bf A}^T$: $$ ({\bf AB})^T = \left[ \begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix} \begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix} \right]^T = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}$$ $$ {\bf B}^T{\bf A}^T= \begin{pmatrix} 0&2&3\\1&2&-1 \end{pmatrix} \begin{pmatrix} 1&2\\3&-1\\2&3 \end{pmatrix} = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}$$ - - - ```{exercise} #| name: Matrix Multiplication #| label: matrixmulti1 - Let $$A = \begin{pmatrix} 2&0&-1&1\\1&2&0&1 \end{pmatrix}$$ $$B = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix} $$ @@ -272,17 +240,11 @@ Calculate the following: 3. $$(BC)^T$$ 4. $$BC^T$$ - - ``` - - - ## Systems of Linear Equations - __Linear Equation__: $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$ $a_i$ are parameters or coefficients. $x_i$ are variables or unknowns. @@ -290,12 +252,8 @@ $a_i$ are parameters or coefficients. $x_i$ are variables or unknowns. Linear because only one variable per term and degree is at most 1. - - We are often interested in solving linear systems like - - $$\begin{matrix} x & - & 3y & = & -3\\ 2x & + & y & = & 8 @@ -316,10 +274,8 @@ $$\begin{matrix} A __solution__ to a linear system of $m$ equations in $n$ unknowns is a set of $n$ numbers $x_1, x_2, \cdots, x_n$ that satisfy each of the $m$ equations. - Example: $x=3$ and $y=2$ is the solution to the above $2\times 2$ linear system. If you graph the two lines, you will find that they intersect at $(3,2)$. - Does a linear system have one, no, or multiple solutions? For a system of 2 equations with 2 unknowns (i.e., two lines): _ __One solution:__ The lines intersect at exactly one point. @@ -328,9 +284,6 @@ __No solution:__ The lines are parallel. __Infinite solutions:__ The lines coincide. - - - Methods to solve linear systems: 1. Substitution @@ -341,7 +294,6 @@ Methods to solve linear systems: #| name: Linear Equations #| label: lineareq - Provide a system of 2 equations with 2 unknowns that has 1. one solution @@ -350,12 +302,8 @@ Provide a system of 2 equations with 2 unknowns that has 3. infinite solutions - - - ``` - ## Systems of Equations as Matrices Matrices provide an easy and efficient way to represent linear systems such as @@ -380,7 +328,6 @@ The unknown quantities are represented by the vector ${\bf x}=\begin{pmatrix} x_ The right hand side of the linear system is represented by the vector ${\bf b}=\begin{pmatrix} b_1\\b_2\\\vdots\\b_m \end{pmatrix}$. - __Augmented Matrix__: When we append $\bf b$ to the coefficient matrix $\bf A$, we get the augmented matrix $\widehat{\bf A}=[\bf A | b]$ $$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} & | & b_1\\ @@ -389,27 +336,21 @@ __Augmented Matrix__: When we append $\bf b$ to the coefficient matrix $\bf A$, a_{m1} & a_{m2} & \cdots & a_{mn} & | & b_m \end{pmatrix}$$ - ```{exercise} #| name: Augmented Matrix #| label: augmatrix - Create an augmented matrix that represent the following system of equations: $$2x_1 -7x_2 + 9x_3 -4x_4 = 8$$ $$41x_2 + 9x_3 -5x_6 = 11$$ $$x_1 -15x_2 -11x_5 = 9$$ - - ``` - ## Finding Solutions to Augmented Matrices and Systems of Equations - __Row Echelon Form__: Our goal is to translate our augmented matrix or system of equations into row echelon form. This will provide us with the values of the vector __x__ which solve the system. We use the row operations to change coefficients in the lower triangle of the augmented matrix to 0. An augmented matrix of the form $$\begin{pmatrix} @@ -432,12 +373,10 @@ $$\begin{pmatrix} 0 & 0 & 0 & 0 & \fbox{$1$} & | & b^*_m \end{pmatrix}$$ - __Gaussian and Gauss-Jordan elimination__: We can conduct elementary row operations to get our augmented matrix into row echelon or reduced row echelon form. The methods of transforming a matrix or system into row echelon and reduced row echelon form are referred to as Gaussian elimination and Gauss-Jordan elimination, respectively. __Elementary Row Operations__: To do Gaussian and Gauss-Jordan elimination, we use three basic operations to transform the augmented matrix into another augmented matrix that represents an equivalent linear system -- equivalent in the sense that the same values of $x_j$ solve both the original and transformed matrix/system: - __Interchanging Rows__: Suppose we have the augmented matrix $${\widehat{\bf A}}=\begin{pmatrix} a_{11} & a_{12} & | & b_1\\ a_{21} & a_{22} & | & b_2 @@ -466,7 +405,6 @@ __Adding (subtracting) Rows__: If we add (subtract) the first row of matrix $\w ```{example} #| label: solvesys - Solve the following system of equations by using elementary row operations: $\begin{matrix} @@ -475,20 +413,15 @@ $\begin{matrix} \end{matrix}$ - ``` - - ```{exercise} #| name: Solving Systems of Equations #| label: solvesys1 - Put the following system of equations into augmented matrix form. Then, using Gaussian or Gauss-Jordan elimination, solve the system of equations by putting the matrix into row echelon or reduced row echelon form. - $$ 1. \begin{cases} x + y + 2z = 2\\ @@ -505,14 +438,8 @@ $$ \end{cases} $$ - - ``` - - - - ## Rank --- and Whether a System Has One, Infinite, or No Solutions To determine how many solutions exist, we can use information about (1) the number of equations $m$, (2) the number of unknowns $n$, and (3) the __rank__ of the matrix representing the linear system. @@ -533,12 +460,10 @@ $\begin{pmatrix} 1 & 2 & 3 \\ Rank = 2 - ```{exercise} #| name: Rank of Matrices #| label: rank - Find the rank of each matrix below: (Hint: transform the matrices into row echelon form. Remember that the number of nonzero rows of a matrix in row echelon form is the rank of that matrix) @@ -547,7 +472,6 @@ Find the rank of each matrix below: 2 & 1 & 3 \\ 1 & 2 & 3 \end{pmatrix}$ - \bigskip 2.$\begin{pmatrix} 1 & 3 & 3 & -3 & 3\\ @@ -556,24 +480,19 @@ Find the rank of each matrix below: 1 & 3 & 0 & 3 & -2 \end{pmatrix}$ - ``` - - Answer to Exercise \@ref(exr:rank): 1. rank is 2 2. rank is 3 - ## The Inverse of a Matrix __Identity Matrix__: The $n\times n$ identity matrix ${\bf I}_n$ is the matrix whose diagonal elements are 1 and all off-diagonal elements are 0. Examples: $$ {\bf I}_2=\begin{pmatrix} 1&0\\0&1 \end{pmatrix}, \qquad {\bf I}_3=\begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}$$ - __Inverse Matrix__: An $n\times n$ matrix ${\bf A}$ is __nonsingular__ or __invertible__ if there exists an $n\times n$ matrix ${\bf A}^{-1}$ such that $${\bf A} {\bf A}^{-1} = {\bf A}^{-1} {\bf A} = {\bf I}_n$$ where ${\bf A}^{-1}$ is the inverse of ${\bf A}$. If there is no such ${\bf A}^{-1}$, then ${\bf A}$ is singular or not invertible. Example: Let @@ -610,7 +529,6 @@ To summarize: To calculate the inverse of ${\bf A}$ b. If ${\bf C}\ne{\bf I}_n$, then $\bf C$ has a row of zeros. This means ${\bf A}$ is singular and ${\bf A}^{-1}$ does not exist. - ```{example} #| label: inverse @@ -618,30 +536,18 @@ Find the inverse of the following matricies: 1. ${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$ - - ``` - - - - ```{exercise} #| name: Finding the inverse of matrices #| label: inverse1 - Find the inverse of the following matrix: 1. ${\bf A}=\begin{pmatrix} 1&0&4\\0&2&0\\0&0&1 \end{pmatrix}$ - - ``` - - - ## Linear Systems and Inverses Let's return to the matrix representation of a linear system @@ -659,8 +565,6 @@ If $\bf{A}$ is an $n\times n$ matrix,then $\bf{Ax}=\bf{b}$ is a system of $n$ eq Hence, given $\bf{A}$ and $\bf{b}$ and given that $\bf{A}$ is nonsingular, then $\bf{x} = \bf{A}^{-1} \bf{b}$ is a unique solution to this system. - - ```{exercise} #| label: invlinsys #| name: Solve linear system using inverses @@ -672,7 +576,6 @@ Use the inverse matrix to solve the following linear system: 2x - y &= -10 \end{align*} - ___Hint: the linear system above can be written in the matrix form___ $\textbf{A}\textbf{z} = \textbf{b}$ @@ -683,10 +586,6 @@ $$\textbf{z} = \begin{pmatrix} x\\y \end{pmatrix},$$ and $$\textbf{b} = \begin{pmatrix} 5\\-10 \end{pmatrix}$$ - - - - ``` ## Determinants @@ -725,7 +624,6 @@ Let's extend this now to any $n\times n$ matrix. Let's define ${\bf A}_{ij}$ as Then for any $n\times n$ matrix ${\bf A}$ $$|{\bf A}|= a_{11}M_{11} - a_{12}M_{12} + \cdots + (-1)^{n+1} a_{1n} M_{1n}$$ - For example, in figuring out whether the following matrix has an inverse? $${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$$ @@ -737,16 +635,12 @@ For example, in figuring out whether the following matrix has an inverse? \end{eqnarray} 2. Since $|{\bf A}|\ne 0$, we conclude that ${\bf A}$ has an inverse. - - ```{exercise} #| name: Determinants and Inverses #| label: determinants - Determine whether the following matrices are nonsingular: - $$1. \begin{pmatrix} 1 & 0 & 1\\ 2 & 1 & 2\\ @@ -763,7 +657,6 @@ $$2. \begin{pmatrix} ``` - ## Getting Inverse of a Matrix using its Determinant Thus far, we have a number of algorithms to @@ -773,7 +666,6 @@ Thus far, we have a number of algorithms to but these remain just that --- algorithms. At this point, we have no way of telling how the solutions $x_j$ change as the parameters $a_{ij}$ and $b_i$ change, except by changing the values and "rerunning" the algorithms. - With determinants, we can provide an explicit formula for the inverse and therefore provide an explicit formula for the solution of an $n\times n$ linear system. @@ -790,7 +682,6 @@ $$\frac{1}{\det({\bf A})} \begin{pmatrix} For example, Let's calculate the inverse of matrix A from Exercise \@ref(exr:invlinsys) using the determinant formula. - Recall, $$A = \begin{pmatrix} @@ -815,12 +706,10 @@ $$ \begin{pmatrix} \frac{2}{5} & \frac{3}{5}\\ \end{pmatrix}$$ - ```{exercise} #| name: Calculate Inverse using Determinant Formula #| label: calcinverse - Caculate the inverse of A $$A = \begin{pmatrix} @@ -828,17 +717,12 @@ $$A = \begin{pmatrix} -7 & 2\\ \end{pmatrix}$$ - ``` - - ## Answers to Examples and Exercises {.unnumbered} - - Answer to Example \@ref(exm:vectors): 1. $\begin{pmatrix} -1 &-3&-3 \end{pmatrix}$ @@ -870,8 +754,6 @@ Answer to Example \@ref(exm:scalarmulti): $s {\bf A} = \begin{pmatrix} 2 & 4 & 6 \\ 8 & 10 & 12 \end{pmatrix}$ - - Answer to Example \@ref(exm:matrixmulti): 1. $\begin{pmatrix} aA+bC&aB+bD\\cA+dC&cB+dD\\eA+fC&eB+fD \end{pmatrix}$ @@ -890,7 +772,6 @@ Answer to Exercise \@ref(exr:matrixmulti1): 4. $BC^T = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix}\begin{pmatrix} 3&0\\2&4\\-1&6 \end{pmatrix} =\begin{pmatrix}20 & -22 \\ 5 & 4 \\ -3 &2 \\6 & 0\end{pmatrix}$ - Answer to Exercise \@ref(exr:lineareq): There are many answers to this. Some possible simple ones are as follows: @@ -910,8 +791,6 @@ There are many answers to this. Some possible simple ones are as follows: 2x & - & 2y & = & 0 \end{matrix}$$ - - Answer to Exercise \@ref(exr:augmatrix): $\begin{pmatrix} @@ -922,7 +801,6 @@ $\begin{pmatrix} Answer to Example \@ref(exm:solvesys): - $$\begin{matrix} x & - & 3y & = & -3\\ 2x & + & y & = & 8 @@ -949,8 +827,6 @@ Answer to Exercise \@ref(exr:solvesys1): 2. x = -17, y = -3, z = -35 - - Answer to Exercise \@ref(exr:rank): 1. rank is 2 @@ -1001,7 +877,6 @@ ${\bf A}^{-1} = \left(\begin{array}{ccc} 5/4 &0 &-1/4 \end{array} \right)$ - Answer to Exercise \@ref(exr:inverse1): 1. ${\bf A}^{-1}=\begin{pmatrix} 1&0&-4\\0&\frac{1}{2}&0\\0&0&1 \end{pmatrix}$ @@ -1035,4 +910,3 @@ $\begin{pmatrix} \frac{7}{41} & \frac{3}{41}\\ \end{pmatrix}$ - diff --git a/11_data-handling_counting.qmd b/11_data-handling_counting.qmd index 05cc610..66d4c3b 100644 --- a/11_data-handling_counting.qmd +++ b/11_data-handling_counting.qmd @@ -1,24 +1,19 @@ # (PART) Programming {.unnumbered} - # Orientation and Reading in Data^[Module originally written by Shiro Kuriwaki] {#dataimport} - - Welcome to the first in-class session for programming. Up till this point, you should have already: * Completed the R Visualization and Programming primers (under "The Basics") on your own at , * Made an account at RStudio Cloud and join the Math Prefresher 2019 Space, and * Successfully signed up for the University wi-fi: (Access Harvard Secure with your HarvardKey. Try to get a HarvardKey as soon as possible.) - ## Motivation: Data and You {.unnumbered} The modal social science project starts by importing existing datasets. Datasets come in all shapes and sizes. As you search for new data you may encounter dozens of file extensions -- csv, xlsx, dta, sav, por, Rdata, Rds, txt, xml, json, shp ... the list continues. Although these files can often be cumbersome, its a good to be able to find a way to encounter any file that your research may call for. Reviewing data import will allow us to get on the same page on how computer systems work. - ### Where are we? Where are we headed? {.unnumbered} Today we'll cover: @@ -38,7 +33,6 @@ Today we'll cover: * How would you figure out what variables are in the data? size of the data? * How would you read in a `csv` file, a `dta` file, a `sav` file? - ```{r} #| include: false #| message: false @@ -47,7 +41,6 @@ library(dplyr) library(fs) ``` - ## Orienting 1. We will be using a cloud version of RStudio at . You should join the Math Prefresher Space 2019 from the link that was emailed to you. Each day, click on the project with the day's date on it. @@ -62,12 +55,10 @@ library(fs) 3. via the GUI, you the analyst needs to sends instructions, or __commands__, to the R application. The verb for this is "run" or "execute" the command. Computer programs ask users to provide instructions in very specific formats. While a English-speaking human can understand a sentence with a few typos in it by filling in the blanks, the same typo or misplaced character would halt a computer program. Each program has its own requirements for how commands should be typed; after all, each of these is its own language. We refer to the way a program needs its commands to be formatted as its __syntax__. - 4. Theoretically, one could do all their work by typing in commands into the Console. But that would be a lot of work, because you'd have to give instructions each time you start your data analysis. Moreover, you'll have no record of what you did. That's why you need a __script__. This is a type of __code__. It can be referred to as a __source__ because that is the source of your commands. Source is also used as a verb; "source the script" just means execute it. RStudio doesn't start out with a script, so you can make one from "File > New" or the New file icon. ![Opening New Script (as opposed to the Console)](images/11_2_rstudio-script.png) - 4. You can also open scripts that are in folders in your computer. A script is a type of File. Find your Files in the bottom-right "Files" pane. To load a dataset, you need to specify where that file is. Computer files (data, documents, programs) are organized hiearchically, like a branching tree. Folders can contain files, and also other folders. The GUI toolbar makes this lineaer and hiearchical relationship apparent. When we turn to locate the file in our commands, we need another set of syntax. Importantly, denote the hierarchy of a folder by the `/` (slash) symbol. `data/input/2018-08` indicates the `2018-08` folder, which is included in the `input` folder, which is in turn included in the `data` folder. @@ -84,15 +75,12 @@ library(fs) * `.Rmd`: Rmarkdown code (text + code) * `.do`: Stata code (script) - ![Opening an Existing Script from Files](images/11_3_rstudio-files.png) 5. In R, there are two main types of scripts. A classic `.R` file and a `.Rmd` file (for Rmarkdown). A .R file is just lines and lines of R code that is meant to be inserted right into the Console. A .Rmd tries to weave code and English together, to make it easier for users to create reports that interact with data and intersperse R code with explanation. For example, we built this book in Rmds. The Rmarkdown facilitates is the use of __code chunks__, which are used here. These start and end with three back-ticks. In the beginning, we can add options in curly braces (`{}`). Specifying `r` in the beginning tells to render it as R code. Options like `echo = TRUE` switch between showing the code that was executed or not; `eval = TRUE` switch between evaluating the code. More about Rmarkdown in Section \@ref(nonwysiwyg). For example, this code chunk would evaluate `1 + 1` and show its output when compiled, but not display the code that was executed. - - ![A code chunk in Rmarkdown (before rendering)](images/11_4_codechunk.png) \newpage @@ -127,7 +115,6 @@ Object oriented programming makes languages flexible and powerful: It is helpful to think in terms of object manipulation at a high level while programming in R, particularly at the beginning of tackling a new problem. Think about what objects you want to manipulate, what types they are, and how they fit together. Once you have the logic of your solution ready then you can write it in R. - ## The Computer and You: Giving Instructions We'll do the Peanut Butter and Jelly Exercise in class as an introduction to programming for those who are new.^[This Exercise is taken from Harvard's Introductory Undergraduate Class, CS50 (), and many other writeups.] @@ -145,11 +132,8 @@ n_jelly <- 9 # write instructions in R here - ``` - - ## Base-R vs. tidyverse One last thing before we jump into data. Many things in R and other open source packages have competing standards. A lecture on a technique inevitably biases one standard over another. Right now among R users in this area, there are two families of functions: base-R and tidyverse. R instructors thus face a dilemma about which to teach primarily.^[See for example this community discussion: https://community.rstudio.com/t/base-r-and-the-tidyverse/2965/17] @@ -175,10 +159,8 @@ The following side-by-side comparison of commands for a particular function comp | Create a dataframe | `tibble(x = vec1, y = vec2)`| `data.frame(x = vec1, y = vec2)` | | Turn a dataframe into a tidyverse dataframe | `tbl_df(df)` | | - Remember that tidyverse applies to _dataframes_ only, not vectors. For subsetting vectors, use the base-R functions with the square brackets. - ### Read data {.unnumbered} Some non-tidyverse functions are not quite "base-R" but have similar relationships to tidyverse. For these, we recommend using the _tidyverse_ functions as a general rule due to their common format, simplicity, and scalability. @@ -192,7 +174,6 @@ Some non-tidyverse functions are not quite "base-R" but have similar relationshi | Return matching strings | `str_subset()` | `grep(., value = TRUE)` | | Merge `data1` and `data2` on variables `x1` and `x2` | `left_join(data1, data2, by = c("x1", "x2"))` | `merge(data1, data2, by.x = "x1", by.y = "x2", all.x = TRUE)` | - ### Visualization {.unnumbered} Plotting by ggplot2 (from your tutorials) is also a tidyverse family. @@ -204,9 +185,6 @@ Plotting by ggplot2 (from your tutorials) is also a tidyverse family. | Make a histogram | `ggplot(data, aes(x, y)) + geom_histogram()` | `hist(data$x, data$y)` | | Make a barplot | See Section \@ref(dataviz) | See Section \@ref(dataviz)| - - - ## A is for Athens For our first dataset, let's try reading in a dataset on the Ancient Greek world. Political Theorists and Political Historians study the domestic systems, international wars, cultures and writing of this era to understand the first instance of democracy, the rise and overturning of tyranny, and the legacies of political institutions. @@ -223,11 +201,8 @@ dir_ls("data/input") A typical file format is Microsoft Excel. Although this is not usually the best format for R because of its highly formatted structure as opposed to plain text (more on this in Section \ref@(sec:wysiwyg)), recent packages have made this fairly easy. - - ### Reading in Data - In Rstudio, a good way to start is to use the GUI and the Import tool. Once you click a file, an option to "Import Dataset" comes up. RStudio picks the right function for you, and you can copy that code, but it's important to eventually be able to write that code yourself. For the first time using an outside package, you first need to install it. @@ -261,18 +236,14 @@ ober <- read_excel("data/input/ober_2018.xlsx") Review: what does the `/` mean? Why do we need the `data` term first? Does the argument need to be in quotes? - - ### Inspecting For almost any dataset, you usually want to do a couple of standard checks first to understand what you loaded. - ```{r} ober ``` - ```{r} dim(ober) ``` @@ -299,10 +270,8 @@ These `tidyverse` commands from the `dplyr` package are newer and not built-in, * These verbs roughly correspond to the same commands in SQL, another important language in data science. * The `%>%` symbol is a pipe. It takes the thing on the left side and pipes it down to the function on the right side. We could have done `count(cen10, race)` as `cen10 %>% count(race)`. That means take `cen10` and pass it on to the function `count`, which will count observations by race and return a collapsed dataset with the categories in its own variable and their respective counts in `n`. - ### Extra: A sneak peak at Ober's data - Although this is a bit beyond our current stage, it's hard to resist the temptation to see what you can do with data like this. For example, you can map it.^[In mid-2018, changes in Google's services made it no longer possible to render maps on the fly. Therefore, the map is not currently rendered automatically (but can be rendered once the user registers their API). Instead, you now need to register with Google. See the [change](https://github.com/dkahle/ggmap/blob/e55c0b22b0d16a010b4b45dd2fce800ff0ef19b8/NEWS#L6-L12) to the pacakge ggmap.] Using the `ggmap` package @@ -328,7 +297,6 @@ ggmap(greece) I chose the specifications for arguments `zoom` and `maptype` by looking at the webpage and Googling some examples. - Ober's data has the latitude and longitude of each polis. Because the map of Greece has the same coordinates, we can add the polei on the same map. ```{r} @@ -346,9 +314,6 @@ gg_ober + ``` ![](images/ober_ggmap_polis.png) - - - ## Exercises {.unnumbered} ### 1 {.unnumbered} @@ -359,7 +324,6 @@ What is the Fame value of Delphoi? # Enter here ``` - ### 2 {.unnumbered} Find the polis with the top 10 Fame values. @@ -368,8 +332,6 @@ Find the polis with the top 10 Fame values. # Enter here ``` - - ### 3 {.unnumbered} Make a scatterplot with the number of colonies on the x-axis and Fame on the y-axis. @@ -378,11 +340,9 @@ Make a scatterplot with the number of colonies on the x-axis and Fame on the y-a # Enter here ``` - ### 4 {.unnumbered} Find the correct function to read the following datasets (available in your rstudio.cloud session) into your R window. - * `data/input/acs2015_1percent.csv`: A one percent sample of the American Community Survey * `data/input/gapminder_wide.tab`: Country-level wealth and health from Gapminder^[Formatted and taken from ] * `data/input/gapminder_wide.Rds`: A Rds version of the Gapminder (What is a Rds file? What's the difference?) @@ -391,12 +351,10 @@ Find the correct function to read the following datasets (available in your rstu Our Recommendations: Look at the packages `haven` and `readr` - ```{r} # Enter here, perhaps making a chunk for each file. ``` - ### 5 {.unnumbered} Read Ober's codebook and find a variable that you think is interesting. Check the distribution of that variable in your data, get a couple of statistics, and summarize it in English. @@ -404,7 +362,6 @@ Read Ober's codebook and find a variable that you think is interesting. Check th # Enter here ``` - ### 6 {.unnumbered} This is day 1 and we covered a lot of material. Some of you might have found this completely new; others not so. Please click through this survey before you leave so we can adjust accordingly on the next few days. diff --git a/12_matricies-manipulation.qmd b/12_matricies-manipulation.qmd index ac3c56b..802fafa 100644 --- a/12_matricies-manipulation.qmd +++ b/12_matricies-manipulation.qmd @@ -16,7 +16,6 @@ library(ggplot2) Nunn and Wantchekon use a variety of statistical tools to make their case (adding controls, ordered logit, instrumental variables, falsification tests, causal mechanisms), many of which will be covered in future courses. In this module we will only touch on their first set of analysis that use Ordinary Least Squares (OLS). OLS is likely the most common application of linear algebra in the social sciences. We will cover some linear algebra, matrix manipulation, and vector manipulation from this data. - ### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -25,33 +24,25 @@ Up till now, you should have covered: * Data Import * Statistical Summaries. - Today we'll cover * Matrices & Dataframes in R * Manipulating variables * And other `R` tips - - - ## Read Data - ```{r} library(haven) nunn_full <- read_dta("data/input/Nunn_Wantchekon_AER_2011.dta") ``` - Nunn and Wantchekon's main dataset has more than 20,000 observations. Each observation is a respondent from the Afrobarometer survey. ```{r} head(nunn_full) colnames(nunn_full) ``` - - First, let's consider a small subset of this dataset. ```{r} #| include: false @@ -70,15 +61,12 @@ nunn <- read_dta("data/input/Nunn_Wantchekon_sample.dta") nunn ``` - - ## data.frame vs. matricies This is a `data.frame` object. ```{r} class(nunn) ``` - But it can be also consider a matrix in the linear algebra sense. What are the dimensions of this matrix? ```{r} nrow(nunn) @@ -90,13 +78,10 @@ nrow(nunn) X <- as.matrix(nunn) ``` - What is the difference between a `data.frame` and a matrix? A `data.frame` can have columns that are of different types, whereas --- in a matrix --- all columns must be of the same type (usually either "numeric" or "character"). - You can think of data frames maybe as matrices-plus, because a column can take on characters as well as numbers. As we just saw, this is often useful for real data analyses. - Another way to think about data frames is that it is a type of list. Try the `str()` code below and notice how it is organized in slots. Each slot is a vector. They can be vectors of numbers or characters. ```{r} #| eval: false @@ -104,9 +89,6 @@ Another way to think about data frames is that it is a type of list. Try the `st str(cen10) ``` - - - ## Handling matricies in `R` You can easily transpose a matrix @@ -115,7 +97,6 @@ X t(X) ``` - What are the values of all rows in the first column? ```{r} X[, 1] @@ -145,8 +126,6 @@ Why does it give the same output as the following? X[which(X[, "trust_neighbors"] == 0), "export_area"] ``` - - Some more manipulation ```{r} X + X @@ -168,12 +147,10 @@ cbind(X, 1:10) cbind(X, 1) ``` - ```{r} colnames(X) ``` - ## Variable Transformations `exports` is the total number of slaves that were taken from the individual's ethnic group between Africa's four slave trades between 1400-1900. @@ -187,7 +164,6 @@ Question for you: why add the 1? Verify that this is the same as `X[, "ln_exports"]` - ## Linear Combinations In Table 1 we see "OLS Estimates". These are estimates of OLS coefficients and standard errors. You do not need to know what these are for now, but it doesn't hurt to getting used to seeing them. @@ -202,7 +178,6 @@ Take the first number in Table 1, which is -0.00068. Now, multiply this by `expo -0.00068 * X[, "exports"] ``` - Now, just one more step. Make a new matrix with just exports and the value 1 ```{r} X2 <- cbind(1, X[, "exports"]) @@ -218,13 +193,11 @@ colnames(X2) colnames(X2) <- c("intercept", "exports") ``` - What are the dimensions of the matrix `X2`? ```{r} dim(X2) ``` - Now consider a new matrix, called `B`. ```{r} @@ -236,7 +209,6 @@ What are the dimensions of `B`? dim(B) ``` - What is the product of `X2` and `B`? From the dimensions, can you tell if it will be conformable? ```{r} X2 %*% B @@ -244,7 +216,6 @@ X2 %*% B What is this multiplication doing in terms of equations? - ```{r} #| echo: false #| eval: false @@ -255,10 +226,8 @@ lm_1_1 <- lm(as.formula(form), nunn_full) summary(lm_1_1) ``` - ## Matrix Basics - Let's take a look at Matrices in the context of R ```{r} @@ -275,7 +244,6 @@ nrow(cen10) ncol(cen10) ``` - What variables does this dataset hold? What kind of information does it have? ```{r} @@ -294,7 +262,6 @@ head(cen10$race) We can look at a unique set of variable values by calling the unique function - ```{r} #| message: false unique(cen10$state) @@ -314,17 +281,14 @@ A cross-tab can be considered a matrix: table(cen10$race, cen10$sex) ``` - ```{r} cross_tab <- table(cen10$race, cen10$sex) dim(cross_tab) cross_tab[6, 2] ``` - But a subset of your data -- individual values-- can be considered a matrix too. - ```{r} #| warning: false # First 20 rows of the entire data @@ -383,7 +347,6 @@ s3 <- cen10[cen10$state == "New York" | cen10$state == "California", ] s4 <- cen10 %>% filter(state == "New York" | state == "California") all_equal(s3, s4) - # all individuals in any of the following states: California, Ohio, Nevada, Michigan s5 <- cen10[cen10$state %in% c("California", "Ohio", "Nevada", "Michigan"), ] s6 <- cen10 %>% filter(state %in% c("California", "Ohio", "Nevada", "Michigan")) @@ -397,7 +360,6 @@ all_equal(s7, s8) ## Checkpoint {.unnumbered} - ### 1 {.unnumbered} Get the subset of cen10 for non-white individuals (Hint: look at the set of values for the race variable by using the unique function) ```{r} @@ -410,15 +372,12 @@ Get the subset of cen10 for females over the age of 40 # Enter here ``` - ### 3 {.unnumbered} Get all the serial numbers for black, male individuals who don't live in Ohio or Nevada. ```{r} # Enter here ``` - - ## Exercises {.unnumbered} ### 1 {.unnumbered} @@ -437,10 +396,8 @@ Use R to write code that will create the matrix $A$, and then consecutively mul ## Enter yourself ``` - Note that R notation of matrices is different from the math notation. Simply trying `X^n` where `X` is a matrix will only take the power of each element to `n`. Instead, this problem asks you to perform matrix multiplication. - ### 2 {.unnumbered} Let's apply what we learned about subsetting or filtering/selecting. Use the `nunn_full` dataset you have already loaded @@ -466,13 +423,10 @@ c) Lastly, show all values of `"trust_neighbors"` and `"age"` for observations ( Find a way to generate a vector of "column averages" of the matrix `X` from the Nunn and Wantchekon data in one line of code. Each entry in the vector should contain the sample average of the values in the column. So a 100 by 4 matrix should generate a length-4 matrix. - - ### 4 {.unnumbered} Similarly, generate a vector of "column medians". - ### 5 {.unnumbered} Consider the regression that was run to generate Table 1: diff --git a/13_functions_obj_loops.qmd b/13_functions_obj_loops.qmd index 62c53f4..9ebe77b 100644 --- a/13_functions_obj_loops.qmd +++ b/13_functions_obj_loops.qmd @@ -1,6 +1,5 @@ # Objects, Functions, Loops {#robjloops} - ### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -10,15 +9,12 @@ Up till now, you should have covered: * Statistical Summaries * Visualization - Today we'll cover * Objects * Functions * Loops - - ## What is an object? Now that we have covered some hands-on ways to use graphics, let's go into some fundamentals of the R language. @@ -34,37 +30,31 @@ library(haven) library(ggplot2) ``` - ```{r} cen10 <- read_csv("data/input/usc2010_001percent.csv", col_types = cols()) ``` - Objects are abstract symbols in which you store data. Here we will create an object from `copy`, and assign `cen10` to it. ```{r} copy <- cen10 ``` - This looks the same as the original dataset: ```{r} copy ``` - What happens if you do this next? ```{r} copy <- "" ``` - It got reassigned: ```{r} copy ``` - ### lists Lists are one of the most generic and flexible type of object. You can make an empty list by the function `list()` @@ -88,7 +78,6 @@ my_list[[1]] <- c(1, 2, 3, 4, 5) my_list ``` - You can even make nested lists. Let's say we want the 1st slot of the list to be another list of three elements. ```{r} @@ -99,8 +88,6 @@ my_list[[1]][[3]] <- "subitem 1 in slot 3 of my_list" my_list ``` - - ## Making your own objects We've covered one type of object, which is a list. You saw it was quite flexible. How many types of objects are there? @@ -124,13 +111,11 @@ What is type (class) of object is `cen10`? class(cen10) ``` - What about this text? ```{r} class("some random text") ``` - To change or create the class of any object, you can _assign_ it. To do this, assign the name of your class to character to an object's `class()`. We can start from a simple list. For example, say we wanted to store data about pokemon. Because there is no pre-made package for this, we decide to make our own class. @@ -142,7 +127,6 @@ pikachu <- list(name = "Pikachu", color = "Yellow") ``` - and we can give it any class name we want. ```{r} class(pikachu) <- "Pokemon" @@ -151,7 +135,6 @@ pikachu$type ``` - ### Seeing R through objects Most of the R objects that you will see as you advance are their own objects. For example, here's a linear regression object (which you will learn more about in Gov 2000): ```{r} @@ -159,7 +142,6 @@ ols <- lm(mpg ~ wt + vs + gear + carb, mtcars) class(ols) ``` - Anything can be an object! Even graphs (in `ggplot`) can be assigned, re-assigned, and edited. ```{r} @@ -179,15 +161,11 @@ gg_tab <- ggplot(data = grp_race_ordered) + gg_tab ``` - You can change the orientation ```{r} gg_tab<- gg_tab + coord_flip() ``` - - - ### Parsing an object by `str()s` It can be hard to understand an `R` object because it's contents are unknown. The function `str`, short for structure, is a quick way to look into the innards of an object @@ -201,7 +179,6 @@ Same for the object we just made str(pikachu) ``` - What does a `ggplot` object look like? Very complicated, but at least you can see it: ```{r} @@ -210,9 +187,6 @@ What does a `ggplot` object look like? Very complicated, but at least you can se str(gg_tab) ``` - - - ## Types of variables In the social science we often analyze variables. As you saw in the tutorial, different types of variables require different care. @@ -233,7 +207,6 @@ mean(cen10$race == "American Indian or Alaska Native") Hint: you can use the function `mean()` to calcualte the sample mean. The sample proportion is the mean of a sequence of number, where your event of interest is a 1 (or `TRUE`) and others are 0 (or `FALSE`). - ### numeric vectors A sequence of numbers. @@ -270,9 +243,6 @@ my_name2 <- "shiro" my_name == my_name2 ``` - - - ## What is a function? Most of what we do in R is executing a function. `read_csv()`, `nrow()`, `ggplot()` .. pretty much anything with a parentheses is a function. And even things like `<-` and `[` are functions as well. @@ -289,7 +259,6 @@ You'll see below that the most basic functions are quite complicated internally. You'll notice that functions contain other functions. _wrapper_ functions are functions that "wrap around" existing functions. This sounds redundant, but it's an important feature of programming. If you find yourself repeating a command more than two times, you should make your own function, rather than writing the same type of code. - ### Write your own function It's worth remembering the basic structure of a function. You create a new function, call it `my_fun` by this: ```{r} @@ -310,7 +279,6 @@ count_men <- function(data) { } ``` - Then all we need to do is feed this function a dataset ```{r} count_men(cen10) @@ -322,7 +290,6 @@ The point of a function is that you can use it again and again without typing up count_men(cen10[cen10$state == "California",]) ``` - Let's go one step further. What if we want to know the proportion of non-whites in a state, just by entering the name of the state? There's multiple ways to do it, but it could look something like this ```{r} @@ -344,10 +311,8 @@ nw_in_state(cen10, "Massachusetts") ``` - ## Checkpoint {.unnumbered} - ### 1 {.unnumbered} Try making your own function, `average_age_in_state`, that will give you the average age of people in a given state. @@ -356,9 +321,6 @@ Try making your own function, `average_age_in_state`, that will give you the ave ``` - - - ### 2 {.unnumbered} Try making your own function, `asians_in_state`, that will give you the number of `Chinese`, `Japanese`, and `Other Asian or Pacific Islander` people in a given state. @@ -374,7 +336,6 @@ Try making your own function, 'top_10_oldest_cities', that will give you the nam # Enter on your own ``` - ## What is a package? You can think of a package as a suite of functions that other people have already built for you to make your life easier. @@ -382,7 +343,6 @@ You can think of a package as a suite of functions that other people have alread help(package = "ggplot2") ``` - To use a package, you need to do two things: (1) install it, and then (2) load it. Installing is a one-time thing @@ -396,7 +356,6 @@ But you need to load each time you start a R instance. So always keep these com library(ggplot2) ``` - In `rstudio.cloud`, we already installed a set of packages for you. But when you start your own R instance, you need to have installed the package at some point. ## Conditionals @@ -454,7 +413,6 @@ for (fruit in fruits) { Here `for()` and `in` must be part of any for loop. The right hand side `fruits` must be a thing that exists. Finally the `left-hand` side object is "Pick your favor name." It is analogous to how we can index a sum with any letter. $\sum_{i=1}^{10}i$ and `sum_{j = 1}^{10}j` are in fact the same thing. - ```{r} for (i in 1:length(fruits)) { print(paste("I love", fruits[i])) @@ -516,7 +474,6 @@ for (state in states_of_interest) { ``` ## Exercises {.unnumbered} - ### Exercise 1: Write your own function {.unnumbered} Write your own function that makes some task of data analysis simpler. Ideally, it would be a function that helps you do either of the previous tasks in fewer lines of code. You can use the three lines of code that was provided in exercise 1 to wrap that into another function too! ```{r} @@ -524,11 +481,8 @@ Write your own function that makes some task of data analysis simpler. Ideally, ``` - - ### Exercise 2: Using Loops {.unnumbered} - Using a loop, create a crosstab of sex and race for each state in the set "states_of_interest" ```{r} states_of_interest <- c("California", "Massachusetts", "New Hampshire", "Washington") @@ -536,8 +490,6 @@ states_of_interest <- c("California", "Massachusetts", "New Hampshire", "Washing ``` - - ### Exercise 3: Storing information derived within loops in a global dataframe {.unnumbered} Recall the following nested loop @@ -555,9 +507,7 @@ for (state in states_of_interest) { Instead of printing the percentage of each race in each state, create a dataframe, and store all that information in that dataframe. (Hint: look at how I stored information about male percentage in each state of interest in a vector.) - ```{r} - ``` diff --git a/14_visualization.qmd b/14_visualization.qmd index e489c99..b62fcb1 100644 --- a/14_visualization.qmd +++ b/14_visualization.qmd @@ -1,6 +1,5 @@ # Visualization^[Module originally written by Shiro Kuriwaki] {#dataviz} - ```{r} #| include: false #| message: false @@ -12,22 +11,16 @@ library(forcats) library(scales) ``` - ### Motivation: The Law of the Census {.unnumbered} In this module, let's visualize some cross-sectional stats with an actual Census. Then, we'll do an example on time trends with Supreme Court ideal points. - Why care about the Census? The Census is one of the fundamental acts of a government. See the Law Review article by [Persily (2011)](http://cardozolawreview.com/Joomla1.5/content/32-3/Persily.32-3.pdf), "The Law of the Census."^[[Persily, Nathaniel. 2011. "The Law of the Census: How to Count, What to Count, Whom to Count, and Where to Count Them.”](http://cardozolawreview.com/Joomla1.5/content/32-3/Persily.32-3.pdf). _Cardozo Law Review_ 32(3): 755–91.] The Census is government's primary tool for apportionment (allocating seats to districts), appropriations (allocating federal funding), and tracking demographic change. See for example [Hochschild and Powell (2008)](https://dash.harvard.edu/bitstream/handle/1/3153295/hoschschild_racialreorganization.pdf?sequence=2) on how the categorizations of race in the Census during 1850-1930.^[[Hochschild, Jennifer L., and Brenna Marea Powell. 2008. "Racial Reorganization and the United States Census 1850–1930: Mulattoes, Half-Breeds, Mixed Parentage, Hindoos, and the Mexican Race."](https://dash.harvard.edu/bitstream/handle/1/3153295/hoschschild_racialreorganization.pdf?sequence=2). _Studies in American Political Development_ 22(1): 59–96.] Notice also that both of these pieces are not inherently "quantitative" --- the Persily article is a Law Review and the Hochschild and Powell article is on American Historical Development --- but data analysis would be certainly relevant. - Time series data is a common form of data in social science data, and there is growing methodological work on making causal inferences with time series.^[[Blackwell, Matthew, and Adam Glynn. 2018. "How to Make Causal Inferences with Time-Series Cross-Sectional Data under Selection on Observables."](https://doi.org/10.1017/S0003055418000357) _American Political Science Review_] We will use the the ideological estimates of the Supreme court. - - ### Where are we? Where are we headed? {.unnumbered} - Up till now, you should have covered: * The R Visualization and Programming primers at @@ -36,7 +29,6 @@ Up till now, you should have covered: * What does `:` mean in R? What about `==`? `,`?, `!=` , `&`, `|`, `%in% ` * What does `%>%` do? - Today we'll cover: * Visualization @@ -52,14 +44,10 @@ Today we'll cover: ![By Randy Schutt - Own work, CC BY-SA 3.0, [Wikimedia.]( https://commons.wikimedia.org/w/index.php?curid=29585342)](images/Martin-Quinn_Wikipedia.png) - - - ## Read data First, the census. Read in a subset of the 2010 Census that we looked at earlier. This time, it is in Rds form. - ```{r} #| message: false cen10 <- readRDS("data/input/usc2010_001percent.Rds") @@ -67,7 +55,6 @@ cen10 <- readRDS("data/input/usc2010_001percent.Rds") The data comes from IPUMS^[[Ruggles, Steven, Katie Genadek, Ronald Goeken, Josiah Grover, and Matthew Sobek. 2015. Integrated Public Use Microdata Series: Version 6.0 dataset](http://doi.org/10.18128/D010.V6.0)], a great source to extract and analyze Census and Census-conducted survey (ACS, CPS) data. - ## Counting How many people are in your sample? @@ -75,7 +62,6 @@ How many people are in your sample? nrow(cen10) ``` - This and all subsequent tasks involve manipulating and summarizing data, sometimes called "wrangling". As per last time, there are both "base-R" and "tidyverse" approaches. We have already seen several functions from the tidyverse: @@ -87,7 +73,6 @@ We have already seen several functions from the tidyverse: In this visualization section, we'll make use of the pair of functions `group_by()` and `summarize()`. - ## Tabulating Summarizing data is the key part of communication; good data viz gets the point across.^[[Kastellec, Jonathan P., and Eduardo L. Leoni. 2007. "Using Graphs Instead of Tables in Political Science."](http://www.princeton.edu/~jkastell/Tables2Graphs/graphs.pdf). _Perspectives on Politics_ 5 (4): 755–71.] Summaries of data come in two forms: tables and figures. @@ -99,14 +84,11 @@ In base-R Use the `table` function, that provides how many rows exist for an uni table(cen10$race) ``` - With tidyverse, a quick convenience function is `count`, with the variable to count on included. ```{r} count(cen10, race) ``` - - We can check out the arguments of `count` and see that there is a `sort` option. What does this do? ```{r} count(cen10, race, sort = TRUE) @@ -124,24 +106,19 @@ If you are new to tidyverse, what would you _think_ each row did? Reading the fu where `n()` is a function that counts rows. - ## base R graphics and ggplot Two prevalent ways of making graphing are referred to as "base-R" and "ggplot". - ### base R "Base-R" graphics are graphics that are made with R's default graphics commands. First, let's assign our tabulation to an object, then put it in the `barplot()` function. - ```{r} barplot(table(cen10$race)) ``` - - ### ggplot A popular alternative a `ggplot` graphics, that you were introduced to in the tutorial. `gg` stands for grammar of graphics by Hadley Wickham, and it has a new semantics of explaining graphics in R. Again, first let's set up the data. @@ -160,8 +137,6 @@ We will now plot this grouped set of numbers. Recall that the `ggplot()` functio 2. Then enter the `aes`, or aesthetics. This defines which variable in the data the plotting functions should take for pre-set dimensions in graphics. The dimensions `x` and `y` are the most important. We will assign `race` and `count` to them, respectively, 3. After you close `ggplot()` .. add __layers__ by the plus sign. A `geom` is a layer of graphical representation, for example `geom_histogram` renders a histogram, `geom_point` renders a scatter plot. For a barplot, we can use `geom_col()` - - What is the right geometry layer to make a barplot? Turns out: ```{r} @@ -169,12 +144,10 @@ What is the right geometry layer to make a barplot? Turns out: ggplot(data = grp_race, aes(x = race, y = n)) + geom_col() ``` - ## Improving your graphics Adjusting your graphics to make the point clear is an important skill. Here is a base-R example of showing the same numbers but with a different design, in a way that aims to maximize the "data-to-ink ratio". - ```{r} #| fig.fullwidth: true par(oma = c(1, 11, 1, 1)) @@ -203,10 +176,8 @@ ggplot(data = grp_race_ordered, aes(x = race, y = n)) + caption = "Source: 2010 U.S. Census sample") ``` - The data ink ratio was popularized by Ed Tufte (originally a political economy scholar who has recently become well known for his data visualization work). See Tufte (2001), _The Visual Display of Quantitative Information_ and his website . For a R and ggplot focused example using social science examples, check out Healy (2018), _Data Visualization: A Practical Introduction_ with a draft at ^[Healy, Kieran. forthcoming. _Data Visualization: A Practical Introduction_. Princeton University Press]. There are a growing number of excellent books on data visualization. - ## Cross-tabs Visualizations and Tables each have their strengths. A rule of thumb is that more than a dozen numbers on a table is too much to digest, but less than a dozen is too few for a figure to be worth it. Let's look at a table first. @@ -225,12 +196,10 @@ Another function to make a cross-tab is the `xtabs` command, which uses formula xtabs(~ state + race, cen10) ``` - What if we care about proportions within states, rather than counts? Say we'd like to compare the racial composition of a small state (like Delaware) and a large state (like California). In fact, most tasks of inference is about the unobserved population, not the observed data --- and proportions are estimates of a quantity in the population. One way to transform a table of counts to a table of proportions is the function `prop.table`. Be careful what you want to take proportions of -- this is set by the `margin` argument. In R, the first margin (`margin = 1`) is _rows_ and the second (`margin = 2`) is _columns_. - ```{r} ptab_race_state <- prop.table(xtab_race_state, margin = 2) ``` @@ -254,7 +223,6 @@ Can you tell from the code what `grp_race_state` will look like? grp_race_state ``` - Now, we want to tell `ggplot2` something like the following: I want bars by state, where heights indicate racial groups. Each bar should be colored by the race. With some googling, you will get something like this: ```{r} @@ -270,7 +238,6 @@ ggplot(data = grp_race_state, aes(x = state, y = n, fill = race)) + theme_minimal() ``` - ## Line graphs Line graphs are useful for plotting time trends. @@ -279,7 +246,6 @@ The Census does not track individuals over time. So let's take up another exampl This data is adapted from the estimates of Martin and Quinn on their website .^[This exercise inspired from Princeton's R Camp Assignment.] - ```{r} #| message: false justice <- read_csv("data/input/justices_court-median.csv") @@ -291,11 +257,8 @@ What does the data look like? How do you think it is organized? What does each r justice ``` - - As you might have guessed, these data can be shown in a time trend from the range of the `term` variable. As there are only nine justices at any given time and justices have life tenure, there times on the court are staggered. With a common measure of "preference", we can plot time trends of these justices ideal points on the same y-axis scale. - ```{r} ggplot(justice, aes(x = term, y = idealpt)) + geom_line() @@ -310,13 +273,8 @@ If you got the right aesthetic, this seems to "work" off the shelf. But take a m Now, this graphic already indicates a lot, but let's improve the graphics so people can actually read it. This is left for a Exercise. - - As social scientists, we should also not forget to ask ourselves whether these numerical measures are fit for what we care about, or actually succeeds in measuring what we'd like to measure. The estimation of these "ideal points" is a subfield of political methodology beyond this prefresher. For more reading, skim through the original paper by Martin and Quinn (2002).^[[Martin, Andrew D. and Kevin M. Quinn. 2002. "Dynamic Ideal Point Estimation via Markov Chain Monte Carlo for the U.S. Supreme Court, 1953-1999"](http://mqscores.lsa.umich.edu/media/pa02.pdf). _Political Analysis._ 10(2): 134-153.] Also for a methodological discussion on the difficulty of measuring time series of preferences, check out Bailey (2013).^[[Bailey, Michael A. 2013. "Is Today’s Court the Most Conservative in Sixty Years? Challenges and Opportunities in Measuring Judicial Preferences." ](https://michaelbailey.georgetown.domains/wp-content/uploads/2018/05/JOP_proofs_June2013.pdf). _Journal of Politics_ 75(3): 821-834] - - - ## Exercises {.unnumbered} In the time remaining, try the following exercises. Order doesn't matter. @@ -324,13 +282,11 @@ In the time remaining, try the following exercises. Order doesn't matter. Make a well-labelled figure that plots the proportion of the state's population (as per the census) that is 65 years or older. Each state should be visualized as a point, rather than a bar, and there should be 51 points, ordered by their value. All labels should be readable. - ```{r} # Enter yourself ``` - ### 2: The swing justice {.unnumbered} Using the `justices_court-median.csv` dataset and building off of the plot that was given, make an improved plot by implementing as many of the following changes (which hopefully improves the graph): @@ -347,26 +303,21 @@ Using the `justices_court-median.csv` dataset and building off of the plot that * Extend the x-axis label to about 2020 so the text labels of justices are to the right of the trend-lines. * Add a caption to your text describing the data briefly, as well as any features relevant for the reader (such as the median line and the trimming of the y-axis) - ```{r} # Enter yourself ``` - ### 3: Don't sort by the alphabet {.unnumbered} The Figure we made that shows racial composition by state has one notable shortcoming: it orders the states alphabetically, which is not particularly useful if you want see an overall pattern, without having particular states in mind. - Find a way to modify the figures so that the states are ordered by the _proportion_ of White residents in the sample. - ```{r} # Enter yourself ``` - ### 4 What to show and how to show it {.unnumbered} As a student of politics our goal is not necessarily to make pretty pictures, but rather make pictures that tell us something about politics, government, or society. If you could augment either the census dataset or the justices dataset in some way, what would be an substantively significant thing to show as a graphic? diff --git a/15_project-dempeace.qmd b/15_project-dempeace.qmd index 1b37249..29c53c5 100644 --- a/15_project-dempeace.qmd +++ b/15_project-dempeace.qmd @@ -20,10 +20,6 @@ Up till now, you should have covered: Today you will work on your own, but feel free to ask a fellow classmate nearby or the instructor. The objective for this session is to get more experience using R, but in the process (a) test a prominent theory in the political science literature and (b) explore related ideas of interest to you. - - - - ## Setting up ```{r} @@ -35,8 +31,6 @@ library(readr) library(ggplot2) ``` - - ## Create a project directory First start a directory for this project. This can be done manually or through RStudio's Project feature(`File > New Project...`) @@ -45,9 +39,6 @@ Directories is the computer science / programming name for folders. While advice Chapter 4 of Gentzkow and Shapiro's memo, [Code and Data for the Social Scientist](https://web.stanford.edu/~gentzkow/research/CodeAndData.pdf)] provides a good template. - - - ## Data Sources Most projects you do will start with downloading data from elsewhere. For this task, you'll probably want to track down and download the following: @@ -65,7 +56,6 @@ polity <- read_csv("data/input/sample_polity.csv") mid <- read_csv("data/input/sample_mid.csv") ``` - What does `polity` look like? ```{r} unique(polity$country) @@ -76,21 +66,17 @@ ggplot(polity, aes(x = year, y = polity2)) + head(polity) ``` - MID is a dataset that captures a `dispute` for a given country and year. ```{r} mid ``` - ## Loops - Notice that in the `mid` data, we have a start of a dispute vs. an end of a dispute.In order to combine this into the `polity` data, we want a way to give each of the interval years a row. There are many ways to do this, but one is a loop. We go through one row at a time, and then for each we make a new dataset. that has `year` as a sequence of each year. A lengthy loop like this is typically slow, and you'd want to recast the task so you can do things with functions. But, a loop is a good place to start. - ```{r} mid_year_by_year <- data_frame(ccode = numeric(), year = numeric(), @@ -106,7 +92,6 @@ for(i in 1:nrow(mid)) { head(mid_year_by_year) ``` - ## Merging We want to combine these two datasets by merging. Base-R has a function called `merge`. `dplyr` has several types of `joins` (the same thing). Those names are based on SQL syntax. @@ -121,8 +106,6 @@ p_m <- left_join(polity, head(p_m) ``` - - Replace `dispute` = `NA` rows with a zero. ```{r} @@ -149,13 +132,10 @@ Often, files we need are saved in the `.xls` or `xlsx` format. It is possible to `readxl`/`readr`/`haven` packages() is constantly expanding to capture more file types. In day 1, we used the package `readxl`, using the `read_excel()` function. - ```{r} - ``` - ### Task 2: Data Merging {.unnumbered} We will use data to test a version of the Democratic Peace Thesis (DPS). Democracies are said to go to war less because the leaders who wage wars are accountable to voters who have to bear the costs of war. Are democracies less likely to engage in militarized interstate disputes? @@ -167,10 +147,8 @@ To start, let's download and merge some data. ```{r} - ``` - ### Task 3: Tabulations and Visualization {.unnumbered} 1. Calculate the mean Polity2 score by year. Plot the result. Use graphical indicators of your choosing to show where key events fall in this timeline (such as 1914, 1929, 1939, 1989, 2008). Speculate on why the behavior from 1800 to 1920 seems to be qualitatively different than behavior afterwards. @@ -180,10 +158,5 @@ To start, let's download and merge some data. ```{r} - - ``` - - - diff --git a/16_simulation.qmd b/16_simulation.qmd index 7926e21..a9a4886 100644 --- a/16_simulation.qmd +++ b/16_simulation.qmd @@ -1,6 +1,5 @@ # Simulation^[Module originally written by Connor Jerzak and Shiro Kuriwaki] {#simulation} - ```{r} #| include: false library(dplyr) @@ -21,8 +20,6 @@ Statistical methods also incorporate simulation: * Bagging: a method for improving machine learning predictions by re-sampling observations, storing the estimate across many re-samples, and averaging these estimates to form the final estimate. A variance reduction technique. * Statistical reasoning: if you are trying to understand a quantitative problem, a wonderful first-step to understand the problem better is to simulate it! The analytical solution is often very hard (or impossible), but the simulation is often much easier :-) - - ### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -33,10 +30,8 @@ Up till now, you should have covered: * Functions, objects, loops * Joining real data - In this module, we will start to work with generating data within R, from thin air, as it were. Doing simulation also strengthens your understanding of Probability (Section \@ref{probability}). - ### Check your Understanding {.unnumbered} * What does the `sample()` function do? @@ -44,7 +39,6 @@ In this module, we will start to work with generating data within R, from thin a * What is a `seed`? * What is a Monte Carlo? - Check if you have an idea of how you might code the following tasks: * Simulate 100 rolls of a die @@ -54,8 +48,6 @@ Check if you have an idea of how you might code the following tasks: We're going to learn about this today! - - ## Pick a sample, any sample ## The `sample()` function @@ -85,7 +77,6 @@ It follows then that you cannot sample without replacement a sample that is larg sample(x = 1:10, size = 100, replace = FALSE) ``` - So far, every element in `x` has had an equal probability of being chosen. In some application, we want a sampling scheme where some elements are more likely to be chosen than others. The argument `prob` handles this. For example, this simulates 20 fair coin tosses (each outcome is equally likely to happen) @@ -93,7 +84,6 @@ For example, this simulates 20 fair coin tosses (each outcome is equally likely sample(c("Head", "Tail"), size = 20, prob = c(0.5, 0.5), replace = TRUE) ``` - But this simulates 20 biased coin tosses, where say the probability of Tails is 4 times more likely than the number of Heads ```{r} sample(c("Head", "Tail"), size = 20, prob = c(0.2, 0.8), replace = TRUE) @@ -125,7 +115,6 @@ As a side-note, these functions have very practical uses for any type of data an * Inspecting your dataset: using `head()` all the same time and looking over the first few rows might lead you to ignore any issues that end up in the bottom for whatever reason. * Testing your analysis with a small sample: If running analyses on a dataset takes more than a handful of seconds, change your dataset upstream to a fraction of the size so the rest of the code runs in less than a second. Once verifying your analysis code runs, then re-do it with your full dataset (by simply removing the `sample_n` / `sample_frac` line of code in the beginning). While three seconds may not sound like much, they accumulate and eat up time. - ## Random numbers from specific distributions ### `rbinom()` {.unnumbered} @@ -146,12 +135,10 @@ rbinom(n = 10, size = 100, prob = 0.5) The intuition to emphasize here is that one can generate potentially infinite amounts (size `n`) of noise that is a essentially random - ```{r} runif(n = 5) ``` - ### `rnorm()` {.unnumbered} `rnorm` is also a continuous distribution, but draws from a Normal distribution -- perhaps the most important distribution in statistics. It runs the same way as `runif` @@ -171,13 +158,10 @@ hist(from_runif) hist(from_rnorm) ``` - ## r, p, and d Each distribution can do more than generate random numbers (the prefix `r`). We can compute the cumulative probability by the function `pbinom()`, `punif()`, and `pnorm()`. Also the density -- the value of the PDF -- by `dbinom()`, `dunif()` and `dnorm()`. - - ## `set.seed()` `R` doesn't have the ability to generate truly random numbers! Random numbers are actually very hard to generate. (Think: flipping a coin --> can be perfectly predicted if I know wind speed, the angle the coin is flipped, etc.). Some people use random noise in the atmosphere or random behavior in quantum systems to generate "truly" (?) random numbers. Conversely, R uses deterministic algorithms which take as an input a "seed" and which then perform a series of operations to generate a sequence of random-seeming numbers (that is, numbers whose sequence is sufficiently hard to predict). @@ -198,30 +182,24 @@ set.seed(02138) runif(n = 10) ``` - - ## Exercises {.unnumbered} ### Census Sampling {.unnumbered} What can we learn from surveys of populations, and how wrong do we get if our sampling is biased?^[This example is inspired from [Meng, Xiao-Li (2018). Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election. _Annals of Applied Statistics_ 12:2, 685–726. doi:10.1214/18-AOAS1161SF.](https://statistics.fas.harvard.edu/files/statistics-2/files/statistical_paradises_and_paradoxes.pdf)] Suppose we want to estimate the proportion of U.S. residents who are non-white (`race != "White"`). In reality, we do not have any population dataset to utilize and so we _only see the sample survey_. Here, however, to understand how sampling works, let's conveniently use the Census extract in some cases and pretend we didn't in others. - (a) First, load `usc2010_001percent.csv` into your R session. After loading the `library(tidyverse)`, browse it. Although this is only a 0.01 percent extract, treat this as your population for pedagogical purposes. What is the population proportion of non-White residents? ```{r} ``` - - (b) Setting a seed to `1669482`, sample 100 respondents from this sample. What is the proportion of non-White residents in this _particular_ sample? By how many percentage points are you off from (what we labelled as) the true proportion? ```{r} ``` - (c) Now imagine what you did above was one survey. What would we get if we did 20 surveys? To simulate this, write a loop that does the same exercise 20 times, each time computing a sample proportion. Use the same seed at the top, but be careful to position the `set.seed` function such that it generates the same sequence of 20 samples, rather than 20 of the same sample. @@ -232,7 +210,6 @@ Try doing this with a `for` loop and storing your sample proportions in a new le ``` - (d) Now, to make things more real, let's introduce some response bias. The goal here is not to correct response bias but to induce it and see how it affects our estimates. Suppose that non-White residents are 10 percent less likely to respond to enter your survey than White respondents. This is plausible if you think that the Census is from 2010 but you are polling in 2018, and racial minorities are more geographically mobile than Whites. Repeat the same exercise in (c) by modeling this behavior. You can do this by creating a variable, e.g. `propensity`, that is 0.9 for non-Whites and 1 otherwise. Then, you can refer to it in the propensity argument. @@ -241,25 +218,18 @@ You can do this by creating a variable, e.g. `propensity`, that is 0.9 for non-W ``` - (e) Finally, we want to see if more data ("Big Data") will improve our estimates. Using the same unequal response rates framework as (d), repeat the same exercise but instead of each poll collecting 100 responses, we collect 10,000. - ```{r} ``` - (f) Optional - visualize your 2 pairs of 20 estimates, with a bar showing the "correct" population average. ```{r} ``` - - - - ### Conditional Proportions {.unnumbered} This example is not on simulation, but is meant to reinforce some of the probability discussion from math lecture. @@ -274,31 +244,20 @@ In addition to some standard demographic questions, we will focus on one called ``` - (b) Using the dataset after the procedure in (a), find the proportion of _poll respondents_ (those who are in the sample) who support Donald Trump. ```{r} ``` - (c) Among those who supported Donald Trump, what proportion of them has a Bachelor's degree or higher (i.e. have a Bachelor's, Graduate, or other Professional Degree)? - (d) Among those who did not support Donald Trump (i.e. including supporters of Hilary Clinton, another candidate, or those who refused to answer the question), what proportion of them has a Bachelor's degree or higher? - (e) Express the numbers in the previous parts as probabilities of specified events. Define your own symbols: For example, we can let $T$ be the event that a randomly selected respondent in the poll supports Donald Trump, then the proportion in part (b) is the probability $P(T).$ - - (f) Suppose we randomly sampled a person who participated in the survey and found that he/she had a Bachelor's degree or higher. Given this evidence, what is the probability that the same person supports Donald Trump? Use Bayes Rule and show your work -- that is, do not use data or R to compute the quantity directly. Then, verify this is the case via R. - - - - - ### The Birthday problem {.unnumbered} Write code that will answer the well-known birthday problem via simulation.^[This exercise draws from Imai (2017)] @@ -307,37 +266,31 @@ The problem is fairly simple: Suppose $k$ people gather together in a room. What To simplify reality a bit, assume that (1) there are no leap years, and so there are always 365 days in a year, and (2) a given individual's birthday is randomly assigned and independent from each other. - _Step 1_: Set `k` to a concrete number. Pick a number from 1 to 365 randomly, `k` times to simulate birthdays (would this be with replacement or without?). ```{r} # Your code ``` - _Step 2_: Write a line (or two) of code that gives a `TRUE` or `FALSE` statement of whether or not at least two people share the same birth date. ```{r} # Your code ``` - _Step 3_: The above steps will generate a `TRUE` or `FALSE` answer for your event of interest, but only for one realization of an event in the sample space. In order to estimate the _probability_ of your event happening, we need a "stochastic", as opposed to "deterministic", method. To do this, write a loop that does Steps 1 and 2 repeatedly for many times, call that number of times `sims`. For each of `sims` iteration, your code should give you a `TRUE` or `FALSE` answer. Code up a way to store these estimates. ```{r} # Your code ``` - _Step 4_: Finally, generalize the function further by letting `k` be a user-defined number. You have now created a _Monte Carlo simulation_! ```{r} # Your code ``` - _Step 5_: Generate a table or plot that shows how the probability of sharing a birthday changes by `k` (fixing `sims` at a large number like `1000`). Also generate a similar plot that shows how the probability of sharing a birthday changes by `sims` (fixing `k` at some arbitrary number like `10`). ```{r} # Your code ``` - _Extra credit_: Give an "analytical" answer to this problem, that is an answer through deriving the mathematical expressions of the probability. ```{r} # Your equations diff --git a/17_non-wysiwyg.qmd b/17_non-wysiwyg.qmd index 32f9227..581e11a 100644 --- a/17_non-wysiwyg.qmd +++ b/17_non-wysiwyg.qmd @@ -1,6 +1,5 @@ # LaTeX and markdown^[Module originally written by Shiro Kuriwaki] {#nonwysiwyg} - ### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: @@ -23,8 +22,6 @@ and the next chapter (you can read it without reading this LaTeX chapter) covers command-line are a basic set of tools that you may have to use from time to time. It also clarifies what more complicated programs are doing. Markdown is an example of compiling a plain text file. LaTeX is a typesetting program and git is a version control program -- both are useful for non-quantitative work as well. - - ### Check your understanding {.unnumbered} Check if you have an idea of how you might code the following tasks: @@ -39,14 +36,10 @@ Check if you have an idea of how you might code the following tasks: * What is a `.bib` file? * Say you came across a interesting journal article. How would you want to maintain this reference so that you can refer to its citation in all your subsequent papers? - - - ## Motivation Statistical programming is a fast-moving field. The beta version of `R` was released in 2000, `ggplot2` was released on 2005, and `RStudio` started around 2010. Of course, some programming technologies are quite "old": (`C` in 1969, `C++` around 1989, `TeX` in 1978, `Linux` in 1991, Mac OS in 1984). But it is easy to feel you are falling behind in the recent developments of programming. Today we will do a **brief** and rough overview of some fundamental and new tools other than `R`, with the general aim of having you break out of your comfort zone so you won't be shut out from learning these tools in the future. - ## Markdown Markdown is the text we have been using throughout this course! At its core markdown is just plain text. Plain text does not have any formatting embedded in it. Instead, the formatting is coded up as text. Markdown is _not_ a WYSIWYG (What you see is what you get) text editor like Microsoft Word or Google Docs. This will mean that you need to explicitly code for `bold{text}` rather than hitting Command+B and making your text look __bold__ on your own computer. @@ -90,7 +83,6 @@ Some benefits of using Quarto include: * [dozen of output types](https://quarto.org/docs/reference/) * [the ability to make websites interacting only with Quarto](https://quarto.org/docs/websites/) - ### A note on plain-text editors Multiple software exist where you can edit plain-text (roughly speaking, text that is not WYSIWYG). @@ -127,8 +119,6 @@ Hello World \end{document} ``` - - ### compile your first LaTeX document locally LaTeX is a very stable system, and few changes to it have been made since the 1990s. The main benefit: better control over how your papers will look; better methods for writing equations or making tables; overall pleasing aesthetic. @@ -145,7 +135,6 @@ Hello World 3. Open this in your "LaTeX" editor. This can be `TeXMaker`, `Aqumacs`, etc.. 4. Go through the click/dropdown interface and click compile. - ### main LaTeX commands LaTeX can cover most of your typesetting needs, to clean equations and intricate diagrams. @@ -168,7 +157,6 @@ $$\int x^2 dx$$ ``` these compile math symbols: $\displaystyle \int x^2 dx.$^[Enclosing with `$$` instead of `$$` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.] - The `align` environment is useful to align your multi-line math, for example. ``` \begin{align} @@ -191,14 +179,12 @@ For some LaTeX commands you might need to load a separate package that someone e ``` where `package` is the name of the package and `options` are options specific to the package. - ### Further Guides {.unnumbered} For a more comprehensive listing of LaTeX commands, Mayya Komisarchik has a great tutorial set of folders: There is a version of LaTeX called Beamer, which is a popular way of making a slideshow. Slides in markdown is also a competitor. The language of Beamer is the same as LaTeX but has some special functions for slides. - ## BibTeX BibTeX is a reference system for bibliographical tests. We have a `.bib` file separately on our computer. This is also a plain text file, but it encodes bibliographical resources with special syntax so that a program can rearrange parts accordingly for different citation systems. @@ -231,14 +217,12 @@ as part of your text, then when the `.tex` file is compiled the PDF shows someth in whatever citation style (APSA, APA, Chicago) you pre-specified! - Also at the end of your paper you will have a bibliography with entries ordered and formatted in the appropriate citation. ![](images/biblatex_bibliography.png) This is a much less frustrating way of keeping track of your references -- no need to hand-edit formatting the bibliography to conform to citation rules (which biblatex already knows) and no need to update your bibliography as you add and drop references (biblatex will only show entries that are used in the main text). - ### stocking up on your .bib files You should keep your own `.bib` file that has all your bibliographical resources. Storing entries is cheap (does not take much memory), so it is fine to keep all your references in one place (but you'll want to make a new one for collaborative projects where multiple people will compile a `.tex` file). @@ -259,7 +243,6 @@ Each student will have slightly different substantive interests, so we won't imp * Italics and boldface * A figure with a caption and in-text reference to it. - Depending on your subfield or interests, try to implement some of the following: * A bibliographical reference drawing from a separate `.bib` file @@ -269,9 +252,6 @@ Depending on your subfield or interests, try to implement some of the following: * Different page margins * Different line spacing - - - ## Concluding the Prefresher {.unnumbered} Math may not be the perfect tool for every aspiring political scientist, but hopefully it was useful background to have at the least: @@ -279,21 +259,16 @@ Math may not be the perfect tool for every aspiring political scientist, but hop - - But we should be aware that too much slant towards math and programming can miss the point: - - Keep on learning, trying new techniques to improve your work, and learn from others! - ### Your Feedback Matters {.unnumbered} _Please tell us how we can improve the Prefresher_: The Prefresher is a work in progress, with material mainly driven by graduate students. Please tell us how we should change (or not change) each of its elements: diff --git a/18_text.qmd b/18_text.qmd index 7e6a671..5c1e1eb 100644 --- a/18_text.qmd +++ b/18_text.qmd @@ -1,7 +1,5 @@ # Text^[Module originally written by Connor Jerzak] {#rtext} - - ```{r} #| include: false library(dplyr) @@ -58,7 +56,6 @@ Today, we will learn more about using text data. Our objectives are: * To learn how to use regular expressions; * To learn about other tools for representing + analyzing text in ```R```. - ## Reading and writing text in R * To read in a text file, use readLines @@ -77,7 +74,6 @@ write.table(my_string_vector, "~/mydata.txt", sep="\t") paste and sprintf are useful commands in text processing, such as for automatically naming files or automatically performing a series of command over a subset of your data. Table making also will often need these commands. - Paste concatenates vectors together. ````{R} #use collapse for inputs of length > 1 @@ -98,7 +94,6 @@ sprintf("Coefficient for %s: %.3f (%.2f)", "Gender", 1.52324, 0.03143) #%.2f is replaced by a floating point digit with 2 decimal places ```` - ## Regular expressions A regular expression is a special text string for describing a search pattern. @@ -110,7 +105,6 @@ Use cases: 2. WEB SCRAPING. E.g. Parse html code in order to extract research information from an online table. 3. CLEANING DATA. E.g. After loading in a dataset, we might need to remove mistakes from the dataset, orsubset the data using regular expression tools. - Example in ```R```. Extract the tweet mentioning Indonesia. ```{r} s1 <- "If only Bradley's arm was longer. RT" @@ -144,7 +138,6 @@ regexpr(pattern = "was", text = my_string)[2] Seems simple? The problem: the patterns can get pretty complex! - ### Character classes Some types of symbols are stand in for some more complex thing, rather than taken literally. @@ -215,7 +208,6 @@ grepl(my_string, pattern = "^(?!presidential1).*\\.png", perl = TRUE) ``` - * Indicates which file names don't start with `presidential1` but do end in `.png` * `^` indicates that the pattern should start at the beginning of the string. * `?!` indicates negative lookahead -- we're looking for any pattern NOT following presidential1 which meets the subsequent conditions. (see below) @@ -238,7 +230,6 @@ Representing text numerically. 1. Document term matrix. The document term matrix (DTM) is a common method for representing text. The DTM is a matrix. Each row of this matrix corresponds to a document; each column corresponds to a word. It is often useful to look at summary statistics such as the percentage of speaches in which a Democratic lawmaker used the word "inequality" compared to a Republican; the DTM would be very helpful for this and other tasks. - ```{R} doc1 <- "Rage---Goddess, sing the rage of Peleus’ son Achilles, murderous, doomed, that cost the Achaeans countless losses, @@ -284,7 +275,6 @@ Stemming is the process of reducing inflected/derived words to their word stem o 2. tm -- Useful for converting text into a numerical representation (forming DTMs). 3. stringr -- Useful for string parsing. - ## Exercises {.unnumbered} ### 1 {.unnumbered} @@ -314,8 +304,6 @@ grep('\'', c("To dare is to lose one's footing momentarily.", "To not dare is to lose oneself."), value = TRUE) ``` - - ### 4 {.unnumbered} Write code to automatically extract the file names that DO end start with presidential and DO end in .pdf @@ -339,7 +327,6 @@ Using the same string as in the above, write code to automatically extract the # Your code here ``` - ### 6 {.unnumbered} Combine these two strings into a single string separated by a "-". Desired output: "The carbonyl group in aldehydes and ketones is an oxygen analog of the carbon–carbon double bond." @@ -350,7 +337,6 @@ string1 <- "The carbonyl group in aldehydes and ketones string2 <- "–carbon double bond." ``` - ### 7 {.unnumbered} Challenge problem! Download this webpage @@ -362,7 +348,6 @@ Challenge problem! Download this webpage * - GitHub is the GUI to git. Making an account there is free. Making an account will allow you to be a part of the collaborative programming community. It will also allow you to "fork" other people's "repositories". "Forking" is making your own copy of the project that forks off from the master project at a point in time. A "repository" is simply the name of your main project directory. "cloning" someone else's repository is similar to forking -- it gives you your own copy. @@ -149,7 +137,6 @@ Repositories become a drop-down menu. It also provides a visual difference interface, which shows the changes you are making to files before you "push" them. It can't do everything, but it provides a way to become familiar with GitHub without the (potentially) intimidating aspects of diving full-on into the command line. - ### is git worth it? While git is a powerful tool, you may choose to not use it for everything because diff --git a/21_solutions-warmup.qmd b/21_solutions-warmup.qmd index c42f21a..7e5c2bf 100644 --- a/21_solutions-warmup.qmd +++ b/21_solutions-warmup.qmd @@ -2,8 +2,6 @@ # Solutions to Warmup Questions {.unnumbered} - - ## Linear Algebra {.unnumbered} ### Vectors {.unnumbered} @@ -14,11 +12,8 @@ Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pm 2. $cv = \begin{pmatrix}8\\10\\12\end{pmatrix}$ 3. $u \cdot v = 1(4) + 2(5) + 3(6) = 32$ - - If you are having trouble with these problems, please review Section \@ref(vector-def) "Working with Vectors" in Chapter \@ref(linearalgebra). - Are the following sets of vectors linearly independent? 1. $u = \begin{pmatrix} 1\\ 2\end{pmatrix}$, $v = \begin{pmatrix} 2\\4\end{pmatrix}$ @@ -26,13 +21,10 @@ Are the following sets of vectors linearly independent? $\leadsto$ No: $$2u = \begin{pmatrix} 2\\ 4\end{pmatrix}, v = \begin{pmatrix} 2\\ 4\end{pmatrix}$$ so infinitely many linear combinations of $u$ and $v$ that amount to 0 exist. - 2. $u = \begin{pmatrix} 1\\ 2\\ 5 \end{pmatrix}$, $v = \begin{pmatrix} 3\\ 7\\ 9 \end{pmatrix}$ $\leadsto$ Yes: we cannot find linear combination of these two vectors that would amount to zero. - - 3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ $\leadsto$ No: After playing around with some numbers, we can find that @@ -43,10 +35,8 @@ $$-2a + 3b - c = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ i.e., a linear combination of these three vectors that would amount to zero exists. - If you are having trouble with these problems, please review Section \@ref(linearindependence). - ### Matrices {.unnumbered} $${\bf A}=\begin{pmatrix} @@ -56,7 +46,6 @@ $${\bf A}=\begin{pmatrix} 4 & 1 & 5 \end{pmatrix}$$ - What is the dimensionality of matrix ${\bf A}$? 4 $\times$ 3 What is the element $a_{23}$ of ${\bf A}$? 3 @@ -77,7 +66,6 @@ $$\mathbf{A} + \mathbf{B} = \begin{pmatrix} 9 & 2 & 14 \end{pmatrix}$$ - Given that $${\bf C}=\begin{pmatrix} @@ -88,8 +76,6 @@ $${\bf C}=\begin{pmatrix} $$\mathbf{A} + \mathbf{C} = \text{No solution, matrices non-conformable}$$ - - Given that $$c = 2$$ @@ -103,8 +89,6 @@ $$c\textbf{A} = \begin{pmatrix} If you are having trouble with these problems, please review Section \@ref(matrixbasics). - - ## Operations {.unnumbered} ### Summation {.unnumbered} @@ -117,17 +101,14 @@ Simplify the following 3. $\sum\limits_{i= 1}^4 (3k + i + 2) = 3\sum\limits_{i= 1}^4k + \sum\limits_{i= 1}^4i + \sum\limits_{i= 1}^42 = 12k + 10 + 8 = 12k + 18$ - ### Products {.unnumbered} 1. $\prod\limits_{i= 1}^3 i = 1\cdot 2\cdot 3 = 6$ 2. $\prod\limits_{k=1}^3(3k + 2) = (3 + 2)\cdot (6 + 2)\cdot (9 + 2) = 440$ - To review this material, please see Section \@ref(sum-notation). - ### Logs and exponents {.unnumbered} Simplify the following @@ -145,8 +126,6 @@ Simplify the following To review this material, please see Section \@ref(logexponents) - - ## Limits {.unnumbered} Find the limit of the following. @@ -157,7 +136,6 @@ Find the limit of the following. To review this material please see Section \@ref(limitsfun) - ## Calculus {.unnumbered} For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\frac{d}{dx}f(x)$ @@ -190,7 +168,5 @@ If you are stuck, please try sketching out a picture of each of the functions. 3. If we roll two fair dice, what is the probability that their sum would be 11? $\leadsto \frac{1}{18}$ 4. If we roll two fair dice, what is the probability that their sum would be 12? $\leadsto \frac{1}{36}$. There are two independent dice, so $6^2 = 36$ options in total. While the previous question had two possibilities for a sum of 11 (5,6 and 6,5), there is only one possibility out of 36 for a sum of 12 (6,6). - For a review, please see Sections \@ref(setoper) - \@ref(probdef) - diff --git a/23_solution_programming.qmd b/23_solution_programming.qmd index a17fb69..c29d9a5 100644 --- a/23_solution_programming.qmd +++ b/23_solution_programming.qmd @@ -10,18 +10,14 @@ library(forcats) library(scales) ``` - - ## Chapter \@ref(dataviz): Visualization - ### 1 State Proportions {.unnumbered} ```{r} cen10 <- readRDS("data/input/usc2010_001percent.Rds") ``` - Group by state, noting that the mean of a set of logicals is a mean of 1s (`TRUE`) and 0s (`FALSE`). ```{r} @@ -32,7 +28,6 @@ grp_st <- cen10 %>% mutate(state = as_factor(state)) ``` - Plot points ```{r} ggplot(grp_st, aes(x = state, y = prop)) + @@ -46,13 +41,8 @@ ggplot(grp_st, aes(x = state, y = prop)) + ) ``` - - - - ### 2 Swing Justice {.unnumbered} - ```{r} #| message: false justices <- read_csv("data/input/justices_court-median.csv") @@ -91,20 +81,16 @@ ggplot(df_indicator, aes(x = term, y = idealpt, group = justice_id)) + theme_bw() ``` - ## Chapter \@ref(robjloops): Objects and Loops - ```{r} #| message: false cen10 <- read_csv("data/input/usc2010_001percent.csv") sample_acs <- read_csv("data/input/acs2015_1percent.csv") ``` - ### Checkpoint #3 {.unnumbered} - ```{r} cen10 %>% group_by(state) %>% @@ -113,11 +99,8 @@ cen10 %>% slice(1:10) ``` - - ### Exercise 2 {.unnumbered} - ```{r} states_of_interest <- c("California", "Massachusetts", "New Hampshire", "Washington") @@ -130,11 +113,8 @@ for (state_i in states_of_interest) { } ``` - - ### Exercise 3 {.unnumbered} - ```{r} race_d <- c() state_d <- c() @@ -156,7 +136,6 @@ for (state in states_of_interest) { } ``` - ## Chapter \@ref(dempeace): Demoratic Peace Project ### Task 1: Data Input and Standardization {.unnumbered} @@ -168,8 +147,6 @@ polity <- read_excel("data/input/p4v2017.xls") ``` - - ### Task 2: Data Merging {.unnumbered} ```{r} @@ -190,10 +167,8 @@ merged_mid_polity <- left_join(polity, by = c("ccode", "year")) ``` - ### Task 3: Tabulations and Visualization {.unnumbered} - ```{r} #| eval: false @@ -209,7 +184,6 @@ mean_polity_by_year_mid_ordered <- arrange(mean_polity_by_year_mid, year) mean_polity_no_mid <- mean_polity_by_year_mid_ordered %>% filter(dispute == 0) mean_polity_yes_mid <- mean_polity_by_year_mid_ordered %>% filter(dispute == 1) - answer <- ggplot(data = mean_polity_by_year_ordered, aes(x = year, y = mean_polity)) + geom_line() + labs(y = "Mean Polity Score", @@ -218,8 +192,6 @@ answer <- ggplot(data = mean_polity_by_year_ordered, aes(x = year, y = mean_poli answer + geom_line(data =mean_polity_no_mid, aes(x = year, y = mean_polity_mid), col = "indianred") + geom_line(data =mean_polity_yes_mid, aes(x = year, y = mean_polity_mid), col = "dodgerblue") - - ``` ## Chapter \@ref(simulation): Simulation @@ -234,14 +206,12 @@ pop <- read_csv("data/input/usc2010_001percent.csv") mean(pop$race != "White") ``` - ```{r} set.seed(1669482) samp <- sample_n(pop, 100) mean(samp$race != "White") ``` - ```{r} #| eval: false ests <- c() @@ -252,12 +222,9 @@ for (i in 1:20) { ests[i] <- mean(samp$race != "White") } - mean(ests) ``` - - ```{r} pop_with_prop <- mutate(pop, propensity = ifelse(race != "White", 0.9, 1)) ``` @@ -275,7 +242,6 @@ for (i in 1:20) { mean(ests) ``` - ```{r} #| eval: false ests <- c() @@ -289,4 +255,3 @@ for (i in 1:20) { mean(ests) ``` - diff --git a/index.qmd b/index.qmd index 0cef19e..66c26d6 100644 --- a/index.qmd +++ b/index.qmd @@ -10,7 +10,6 @@ This booklet serves as the text for the Prefresher, available as a [webpage](htt For information about the role of the prefresher (or "math camp") as a introduction to graduate school, you may also be interested in ["The Math Prefresher and The Collective Future of Political Science Graduate Training"](https://gking.harvard.edu/prefresher), in _PS: Political Science & Politics_, by Gary King, Shiro Kuriwaki, and Yon Soo Park. - ### Authors and Contributors {.unnumbered} - Past Authors and Instructors: Curt Signorino 1996-1997; Ken Scheve 1997-1998; Eric Dickson 1998-2000; Orit Kedar 1999; James Fowler 2000-2001; Kosuke Imai 2001-2002; Jacob Kline 2002; Dan Epstein 2003; Ben Ansell 2003-2004; Ryan Moore 2004-2005; Mike Kellermann 2005-2006; Ellie Powell 2006-2007; Jen Katkin 2007-2008; Patrick Lam 2008-2009; Viridiana Rios 2009-2010; Jennifer Pan 2010-2011; Konstantin Kashin 2011-2012; Soledad Prillaman 2013; Stephen Pettigrew 2013-2014; Anton Strezhnev 2014-2015; Mayya Komisarchik 2015-2016; Connor Jerzak 2016-2017; Shiro Kuriwaki 2017-2018; Yon Soo Park 2018; Meg Schwenzfeier 2019; Shannon Parker 2019; Laura Royden 2020-2021; Hunter Rendleman 2020-2021; Christopher T. Kenny 2022; Jialu Li 2022; Sooahn Shin 2023; María Ballesteros 2023. @@ -19,7 +18,6 @@ For information about the role of the prefresher (or "math camp") as a introduct - Shiro Kurikwaki ([kuriwaki](https://github.com/kuriwaki)) 2018-2023. - Contributors: Thanks to Juan Dodyk ([juandodyk](https://github.com/juandodyk)), Hunter Rendleman ([hrendleman](https://github.com/hrendleman)), and Tyler Simko ([tylersimko](https://github.com/tylersimko)) for contributing to the booklet for corrections and improvements as students. - ### Contributing {.unnumbered} We transitioned the booklet into a bookdown [github repository](https://github.com/IQSS/prefresher) in 2018. As we update this version, we appreciate any bug reports or fixes appreciated. From 31030bced43a91f804f1c99201f7353309015edd Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 21:19:29 -0400 Subject: [PATCH 10/34] explicit latex --- 05_optimization.qmd | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/05_optimization.qmd b/05_optimization.qmd index d4ddd49..30522cc 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -702,14 +702,18 @@ x_2 \ge 0 ::: 1. Rewrite with the slack variables: + +```{=latex} $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. \begin{array}{l} x_1 + x_2 \le 4 - s_1^2\\ -x_1 \le 0 - s_2^2\\ -x_2 \le 0 - s_3^2 \end{array}}$$ +``` 2. Write the Lagrangian: + $$L(x_1, x_2, \lambda_1, \lambda_2, \lambda_3, s_1, s_2, s_3) = -(x_1^2 + 2x_2^2) - \lambda_1(x_1 + x_2 + s_1^2 - 4) - \lambda_2(-x_1 + s_2^2) - \lambda_3(-x_2 + s_3^2)$$ 3. Take the partial derivatives and set equal to zero: From 1dedd8cca5ec19b3b39c71bd2cb06287b88a8248 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 21:22:56 -0400 Subject: [PATCH 11/34] use quarto visual mode to auto clean --- 05_optimization.qmd | 752 ++++++++++++++++++++------------------------ 1 file changed, 343 insertions(+), 409 deletions(-) diff --git a/05_optimization.qmd b/05_optimization.qmd index 30522cc..22db493 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -7,58 +7,60 @@ library(tibble) library(patchwork) ``` -To optimize, we use derivatives and calculus. Optimization is to find the maximum or minimum of a functon, and to find what value of an input gives that extremum. This has obvious uses in engineering. Many tools in the statistical toolkit use optimization. One of the most common ways of estimating a model is through "Maximum Likelihood Estimation", done via optimizing a function (the likelihood). +To optimize, we use derivatives and calculus. Optimization is to find the maximum or minimum of a functon, and to find what value of an input gives that extremum. This has obvious uses in engineering. Many tools in the statistical toolkit use optimization. One of the most common ways of estimating a model is through "Maximum Likelihood Estimation", done via optimizing a function (the likelihood). Optimization also comes up in Economics, Formal Theory, and Political Economy all the time. A go-to model of human behavior is that they optimize a certain utility function. Humans are not pure utility maximizers, of course, but nuanced models of optimization -- for example, adding constraints and adding uncertainty -- will prove to be quite useful. ## Example: Meltzer-Richard {.unnumbered} -A standard backdrop in comparative political economy, the Meltzer-Richard (1981) model states that redistribution of wealth should be higher in societies where the median income is much smaller than the average income. More to the point, typically income distributions wher ethe median is very different from the average is one of high inequality. In other words, the Meltzer-Richard model says that highly unequal economies will have more re-distribution of wealth. Why is that the case? Here is a simplified example that is not the exact model by Meltzer and Richard^[Allan H. Meltzer and Scott F. Richard. ["A Rational Theory of the Size of Government"](https://www.jstor.org/stable/1830813). _Journal of Political Economy_ -89:5 (1981), p. 914-927], but adapted from Persson and Tabellini^[Adapted from Torsten Persson and Guido Tabellini, _Political Economics: Explaining Economic Policy_. MIT Press. ] +A standard backdrop in comparative political economy, the Meltzer-Richard (1981) model states that redistribution of wealth should be higher in societies where the median income is much smaller than the average income. More to the point, typically income distributions wher ethe median is very different from the average is one of high inequality. In other words, the Meltzer-Richard model says that highly unequal economies will have more re-distribution of wealth. Why is that the case? Here is a simplified example that is not the exact model by Meltzer and Richard[^05_optimization-1], but adapted from Persson and Tabellini[^05_optimization-2] + +[^05_optimization-1]: Allan H. Meltzer and Scott F. Richard. ["A Rational Theory of the Size of Government"](https://www.jstor.org/stable/1830813). *Journal of Political Economy* 89:5 (1981), p. 914-927 + +[^05_optimization-2]: Adapted from Torsten Persson and Guido Tabellini, *Political Economics: Explaining Economic Policy*. MIT Press. We will set the following things about our model human and model democracy. -- Individuals are indexed by $i$, and the total population is normalized to unity ("1") without loss of generality. -- $U(\cdot)$, u for "utility", is a function that is concave and increasing, and expresses the utility gained from public goods. This tells us that its first derivative is _positive_, and its second derivative is __negative__. -- $y_i$ is the income of person $i$ -- $W_i$, w for "welfare", is the welfare of person $i$ -- $c_i$, c for "consumption", is the consumption utility of person $i$ +- Individuals are indexed by $i$, and the total population is normalized to unity ("1") without loss of generality. +- $U(\cdot)$, u for "utility", is a function that is concave and increasing, and expresses the utility gained from public goods. This tells us that its first derivative is *positive*, and its second derivative is **negative**. +- $y_i$ is the income of person $i$ +- $W_i$, w for "welfare", is the welfare of person $i$ +- $c_i$, c for "consumption", is the consumption utility of person $i$ Also, the government is democratically elected and sets the following redistribution output: -- $\tau$, t for "tax", is a flat tax rate between 0 and 1 that is applied to everyone's income. -- $g$, "g" for "goods", is the amount of public goods that the government provides. +- $\tau$, t for "tax", is a flat tax rate between 0 and 1 that is applied to everyone's income. +- $g$, "g" for "goods", is the amount of public goods that the government provides. -Suppose an individual's welfare is given by: -$$W_i = c_i + U(g)$$ +Suppose an individual's welfare is given by: $$W_i = c_i + U(g)$$ -The consumption good is the person's post-tax income. +The consumption good is the person's post-tax income. $$c_i = (1 - \tau) y_i$$ Income varies by person (In the next section we will cover probability, by then we will know that we can express this by saying that $y$ is a random variable with the cumulative distribution function $F$, i.e. $y \sim F$.). Every distribution has a mean and an median. -- $E(y)$ is the average income of the society. -- $\text{med}(y)$ is the __median income__ of the society. +- $E(y)$ is the average income of the society. +- $\text{med}(y)$ is the **median income** of the society. -What will happen in this economy? What will the tax rate be set too? How much public goods will be provided? +What will happen in this economy? What will the tax rate be set too? How much public goods will be provided? We've skipped ahead of some formal theory results of demoracy, but hopefully these are conceptually intuitive. First, if a democracy is competitive, there is no slack in the government's goods, and all tax revenue becomes a public good. So we can go ahead and set the constraint: $$g = \sum_{i} \tau y_i P(y_i) = \tau E(y)$$ -We can do this trick because of the "normalizes to unity" setting, but this is a general property of the average. +We can do this trick because of the "normalizes to unity" setting, but this is a general property of the average. Now given this constraint we can re-write an individual's welfare as +```{=tex} \begin{align*} W_i &= \left(1 - \frac{g}{E(y)}\right)y_i + U(g)\\ &= \left(E(y) - g\right) \frac{1}{E(y)} y_i + U(g)\\ &= \left(E(y) - g\right) \frac{y_i}{E(y)} + U(g)\\ \end{align*} - -When is the individual's welfare maximized, __as a function of the public good__? -\begin{align*} +``` +When is the individual's welfare maximized, **as a function of the public good**? \begin{align*} \frac{d}{dg}W_i &= - \frac{y_i}{E(y)} + \frac{d}{dg}U(g)\\ \end{align*} @@ -66,24 +68,23 @@ $\frac{d}{dg}W_i = 0$ when $\frac{d}{dg}U(g) = \frac{y_i}{E(y)}$, and so after e $$g_i^\star = {U_g}^{-1}\left(\frac{y_i}{E(y)}\right)$$ -Now recall that because we assumed concavity, $U_g$ is a negative sloping function whose value is positive. It can be shown that the inverse of such a function is also decreasing. Thus an individual's preferred level of government is determined by a single continuum, the person's income divided by the average income, and the function is __decreasing__ in $y_i$. This is consistent with our intuition that richer people prefer less redistribution. +Now recall that because we assumed concavity, $U_g$ is a negative sloping function whose value is positive. It can be shown that the inverse of such a function is also decreasing. Thus an individual's preferred level of government is determined by a single continuum, the person's income divided by the average income, and the function is **decreasing** in $y_i$. This is consistent with our intuition that richer people prefer less redistribution. That was the amount for any given person. The government has to set one value of $g$, however. So what will that be? Now we will use another result, the median voter theorem. This says that under certain general electoral conditions (single-peaked preferences, two parties, majority rule), the policy winner will be that preferred by the median person in the population. Because the only thing that determines a person's preferred level of government is $y_i / E(y)$, we can presume that the median voter, whose income is $\text{med}(y)$ will prevail in their preferred choice of government. Therefore, we wil see $$g^\star = {U_g}^{-1}\left(\frac{\text{med}(y)}{E(y)}\right)$$ -What does this say about the level of redistribution we observe in an economy? The higher the average income is than the median income, which often (but not always) means _more_ inequality, there should be _more_ redistribution. +What does this say about the level of redistribution we observe in an economy? The higher the average income is than the median income, which often (but not always) means *more* inequality, there should be *more* redistribution. ## Maxima and Minima The first derivative, $f'(x)$, quantifies the slope of a function. Therefore, it can be used to check whether the function $f(x)$ at the point $x$ is increasing or decreasing at $x$. -1. __Increasing:__ $f'(x)>0$ -1. __Decreasing:__ $f'(x)<0$ -1. __Neither increasing nor decreasing__: $f'(x)=0$ - i.e. a maximum, minimum, or saddle point +1. **Increasing:** $f'(x)>0$ +2. **Decreasing:** $f'(x)<0$ +3. **Neither increasing nor decreasing**: $f'(x)=0$ i.e. a maximum, minimum, or saddle point -So for example, $f(x) = x^2 + 2$ and $f^\prime(x) = 2x$ +So for example, $f(x) = x^2 + 2$ and $f^\prime(x) = 2x$ ```{r} #| echo: false @@ -102,11 +103,10 @@ fprimex <- fx0 + stat_function(fun = function(x) 2*x, linewidth = 0.5, linetype fx + fprimex + plot_layout(nrow = 1) ``` -:::{#exr-maxminplot} +::: {#exr-maxminplot} +## Plotting a maximum and minimum -## Plotting a maximum and minimum Plot $f(x)=x^3+ x^2 + 2$, plot its derivative, and identify where the derivative is zero. Is there a maximum or minimum? - ::: ```{r} @@ -129,54 +129,49 @@ fx + fprimex + plot_annotation(expression(f(x)==x^3+2)) The second derivative $f''(x)$ identifies whether the function $f(x)$ at the point $x$ is -1. Concave down: $f''(x)<0$ -1. Concave up: $f''(x)>0$ +1. Concave down: $f''(x)<0$ +2. Concave up: $f''(x)>0$ -__Maximum (Minimum)__: $x_0$ is a __local maximum (minimum)__ if $f(x_0)>f(x)$ ($f(x_0)f(x)$ ($f(x_0)f(x)$ ($f(x_0)f(x)$ ($f(x_0)0$ -1. Need more info: $f'(x)=0$ and $f''(x)=0$ - +1. Local Maximum: $f'(x)=0$ and $f''(x)<0$ +2. Local Minimum: $f'(x)=0$ and $f''(x)>0$ +3. Need more info: $f'(x)=0$ and $f''(x)=0$ -__Global Maxima and Minima__ Sometimes no global max or min exists --- e.g., $f(x)$ not bounded above or below. However, there are three situations where we can fairly easily identify global max or min. +**Global Maxima and Minima** Sometimes no global max or min exists --- e.g., $f(x)$ not bounded above or below. However, there are three situations where we can fairly easily identify global max or min. -1. __Functions with only one critical point.__ If $x_0$ is a local max or min of $f$ and it is the only critical point, then it is the global max or min. -2. __Globally concave up or concave down functions.__ If $f''(x)$ is never zero, then there is at most one critical point. That critical point is a global maximum if $f''<0$ and a global minimum if $f''>0$. -3. __Functions over closed and bounded intervals__ must have both a global maximum and a global minimum. +1. **Functions with only one critical point.** If $x_0$ is a local max or min of $f$ and it is the only critical point, then it is the global max or min. +2. **Globally concave up or concave down functions.** If $f''(x)$ is never zero, then there is at most one critical point. That critical point is a global maximum if $f''<0$ and a global minimum if $f''>0$. +3. **Functions over closed and bounded intervals** must have both a global maximum and a global minimum. -:::{#exm-maxmindraw} - -## Maxima and Minima by drawing +::: {#exm-maxmindraw} +## Maxima and Minima by drawing Find any critical points and identify whether they are a max, min, or saddle point: -1. $f(x)=x^2+2$ -1. $f(x)=x^3+2$ -1. $f(x)=|x^2-1|$, $x\in [-2,2]$ - +1. $f(x)=x^2+2$ +2. $f(x)=x^3+2$ +3. $f(x)=|x^2-1|$, $x\in [-2,2]$ ::: ## Concavity of a Function Concavity helps identify the curvature of a function, $f(x)$, in 2 dimensional space. -:::{#def-concave} - -## Concave Function -A function $f$ is strictly concave over the set S \underline{if} $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) > af(x_1) + (1-a)f(x_2)$$ -\textit{Any} line connecting two points on a concave function will lie \textit{below} the function. +::: {#def-concave} +## Concave Function +A function $f$ is strictly concave over the set S \underline{if} $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) > af(x_1) + (1-a)f(x_2)$$ \textit{Any} line connecting two points on a concave function will lie \textit{below} the function. ::: ```{r} @@ -192,47 +187,48 @@ fx2 <- ggplot(range1, aes(x = x)) + fx1 + fx2 + plot_layout(nrow = 1) ``` -:::{#def-convex} -## Convex Function +::: {#def-convex} +## Convex Function + Convex: A function f is strictly convex over the set S \underline{if} $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) < af(x_1) + (1-a)f(x_2)$$ - Any line connecting two points on a convex function will lie above the function. +Any line connecting two points on a convex function will lie above the function. ::: +```{=tex} \begin{comment} \parbox{2in}{\includegraphics[scale=.4]{Convex.pdf}} \end{comment} - +``` Sometimes, concavity and convexity are strict of a requirement. For most purposes of getting solutions, what we call quasi-concavity is enough. -:::{#def-quasiconcave} -## Quasiconcave Function -A function f is quasiconcave over the set S if $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) \ge \min(f(x_1),f(x_2))$$ +::: {#def-quasiconcave} +## Quasiconcave Function - No matter what two points you select, the \textit{lowest} valued point will always be an end point. +A function f is quasiconcave over the set S if $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) \ge \min(f(x_1),f(x_2))$$ +No matter what two points you select, the \textit{lowest} valued point will always be an end point. ::: +```{=tex} \begin{comment} \parbox{2in}{\includegraphics[scale=.4]{Quasiconcave.pdf}} \end{comment} +``` +::: {#def-quasiconvex} +## Quasiconvex -:::{#def-quasiconvex} - -## Quasiconvex -A function f is quasiconvex over the set $S$ if $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) \le \max(f(x_1),f(x_2))$$ -No matter what two points you select, the \textit{highest} valued point will always be an end point. - +A function f is quasiconvex over the set $S$ if $\forall x_1,x_2 \in S$ and $\forall a \in (0,1)$, $$f(ax_1 + (1-a)x_2) \le \max(f(x_1),f(x_2))$$ No matter what two points you select, the \textit{highest} valued point will always be an end point. ::: +```{=tex} \begin{comment} \parbox{1.8in}{\includegraphics[scale=.4]{Quasiconvex.pdf}} \end{comment} +``` +**Second Derivative Test of Concavity**: The second derivative can be used to understand concavity. -**Second Derivative Test of Concavity**: The second derivative can be used to understand concavity. - -If -$$\begin{array}{lll} +If $$\begin{array}{lll} f''(x) < 0 & \Rightarrow & \text{Concave}\\ f''(x) > 0 & \Rightarrow & \text{Convex} \end{array}$$ @@ -241,28 +237,26 @@ f''(x) > 0 & \Rightarrow & \text{Convex} Quadratic forms is shorthand for a way to summarize a function. This is important for finding concavity because -1. Approximates local curvature around a point --- e.g., used to -identify max vs min vs saddle point. -2. They are simple to express even in $n$ dimensions: -3. Have a matrix representation. +1. Approximates local curvature around a point --- e.g., used to identify max vs min vs saddle point. +2. They are simple to express even in $n$ dimensions: +3. Have a matrix representation. -__Quadratic Form__: A polynomial where each term is a monomial -of degree 2 in any number of variables: +**Quadratic Form**: A polynomial where each term is a monomial of degree 2 in any number of variables: +```{=tex} \begin{align*} \text{One variable: }& Q(x_1) = a_{11}x_1^2\\ \text{Two variables: }& Q(x_1,x_2) = a_{11}x_1^2 + a_{12}x_1x_2 + a_{22}x_2^2\\ \text{N variables: }& Q(x_1,\cdots,x_n)=\sum\limits_{i\le j} a_{ij}x_i x_j \end{align*} - +``` which can be written in matrix terms: One variable $$Q(\mathbf{x}) = x_1^\top a_{11} x_1$$ -N variables: -\begin{align*} +N variables: \begin{align*} Q(\mathbf{x}) &=\begin{pmatrix} x_1 & x_2 & \cdots & x_n \end{pmatrix}\begin{pmatrix} a_{11}&\frac{1}{2}a_{12}&\cdots&\frac{1}{2}a_{1n}\\ \frac{1}{2}a_{12}&a_{22}&\cdots&\frac{1}{2}a_{2n}\\ @@ -273,8 +267,7 @@ a_{11}&\frac{1}{2}a_{12}&\cdots&\frac{1}{2}a_{1n}\\ &= \mathbf{x}^\top\mathbf{Ax} \end{align*} -For example, the Quadratic on $\mathbf{R}^2$: -\begin{align*} +For example, the Quadratic on $\mathbf{R}^2$: \begin{align*} Q(x_1,x_2)&=\begin{pmatrix} x_1& x_2 \end{pmatrix} \begin{pmatrix} a_{11}&\frac{1}{2} a_{12}\\ \frac{1}{2}a_{12}&a_{22}\end{pmatrix} \begin{pmatrix} x_1\\x_2 \end{pmatrix} \\ &= a_{11}x_1^2 + a_{12}x_1x_2 + a_{22}x_2^2 @@ -282,11 +275,9 @@ For example, the Quadratic on $\mathbf{R}^2$: ### Definiteness of Quadratic Forms {.unnumbered} -When the function $f(\mathbf{x})$ has more than two inputs, determining whether it has a maxima and minima (remember, functions may have many inputs but they have only one output) is a bit more tedious. Definiteness helps identify the curvature of a function, $Q(\textbf{x})$, in n dimensional space. +When the function $f(\mathbf{x})$ has more than two inputs, determining whether it has a maxima and minima (remember, functions may have many inputs but they have only one output) is a bit more tedious. Definiteness helps identify the curvature of a function, $Q(\textbf{x})$, in n dimensional space. -__Definiteness__: By definition, a quadratic form always takes on the value of zero when $x = 0$, $Q(\textbf{x})=0$ at $\textbf{x}=0$. The definiteness of the matrix $\textbf{A}$ is determined by whether the -quadratic form $Q(\textbf{x})=\textbf{x}^\top\textbf{A}\textbf{x}$ is greater than zero, less than -zero, or sometimes both over all $\mathbf{x}\ne 0$. +**Definiteness**: By definition, a quadratic form always takes on the value of zero when $x = 0$, $Q(\textbf{x})=0$ at $\textbf{x}=0$. The definiteness of the matrix $\textbf{A}$ is determined by whether the quadratic form $Q(\textbf{x})=\textbf{x}^\top\textbf{A}\textbf{x}$ is greater than zero, less than zero, or sometimes both over all $\mathbf{x}\ne 0$. ## FOC and SOC @@ -294,67 +285,62 @@ We can see from a graphical representation that if a point is a local maxima or ### First Order Conditions {.unnumbered} -When we examined functions of one variable $x$, we found critical points by taking the first derivative, setting it to zero, and solving for $x$. For functions of $n$ variables, the critical points are found in much the same way, except now we set the partial derivatives equal to zero. Note: We will only consider critical points on the interior of a function's domain. +When we examined functions of one variable $x$, we found critical points by taking the first derivative, setting it to zero, and solving for $x$. For functions of $n$ variables, the critical points are found in much the same way, except now we set the partial derivatives equal to zero. Note: We will only consider critical points on the interior of a function's domain. -In a derivative, we only took the derivative with respect to one variable at a time. When we take the derivative separately with respect to all variables in the elements of $\mathbf{x}$ and then express the result as a vector, we use the term Gradient and Hessian. +In a derivative, we only took the derivative with respect to one variable at a time. When we take the derivative separately with respect to all variables in the elements of $\mathbf{x}$ and then express the result as a vector, we use the term Gradient and Hessian. -:::{#def-gradient} -## Gradient +::: {#def-gradient} +## Gradient -Given a function $f(\textbf{x})$ in $n$ variables, the gradient $\nabla f(\mathbf{x})$ (the greek letter nabla ) is a column vector, where the $i$th element is the partial derivative of $f(\textbf{x})$ with respect to $x_i$: +Given a function $f(\textbf{x})$ in $n$ variables, the gradient $\nabla f(\mathbf{x})$ (the greek letter nabla ) is a column vector, where the $i$th element is the partial derivative of $f(\textbf{x})$ with respect to $x_i$: $$\nabla f(\mathbf{x}) = \begin{pmatrix} \frac{\partial f(\mathbf{x})}{\partial x_1}\\ \frac{\partial f(\mathbf{x})}{\partial x_2}\\ \vdots \\ \frac{\partial f(\mathbf{x})}{\partial x_n} \end{pmatrix}$$ - ::: Before we know whether a point is a maxima or minima, if it meets the FOC it is a "Critical Point". -:::{#def-criticalpoint} -## Critical Point +::: {#def-criticalpoint} +## Critical Point $\mathbf{x}^*$ is a critical point if and only if $\nabla f(\mathbf{x}^*)=0$. If the partial derivative of f(x) with respect to $x^*$ is 0, then $\mathbf{x}^*$ is a critical point. To solve for $\mathbf{x}^*$, find the gradient, set each element equal to 0, and solve the system of equations. $$\mathbf{x}^* = \begin{pmatrix} x_1^*\\x_2^*\\ \vdots \\ x_n^*\end{pmatrix}$$ - ::: -:::{#exm-gradcp} - +::: {#exm-gradcp} Example: Given a function $f(\mathbf{x})=(x_1-1)^2+x_2^2+1$, find the (1) Gradient and (2) Critical point of $f(\mathbf{x})$. - ::: -:::{#sol-gradcp} -Gradient +::: {#sol-gradcp} +Gradient +```{=tex} \begin{align*} \nabla f(\mathbf{x}) &= \begin{pmatrix}\frac{\partial f(\mathbf{x})}{\partial x_1}\\ \frac{\partial f(\mathbf{x})}{\partial x_2} \end{pmatrix}\\ &= \begin{pmatrix} 2(x_1-1)\\ 2x_2 \end{pmatrix} \end{align*} - +``` Critical Point $\mathbf{x}^* =$ - + +```{=tex} \begin{align*} &\frac{\partial f(\mathbf{x})}{\partial x_1} = 2(x_1-1) = 0\\ &\Rightarrow x_1^* = 1\\ &\frac{\partial f(\mathbf{x})}{\partial x_2} = 2x_2 = 0\\ &\Rightarrow x_2^* = 0\\ \end{align*} - +``` So $$\mathbf{x}^* = (1,0)$$ - ::: ### Second Order Conditions {.unnumbered} -When we found a critical point for a function of one variable, we used the second derivative as a indicator of the curvature at the point in order to determine whether the point was a min, max, or saddle (second derivative test of concavity). For functions of $n$ variables, we use _second order partial derivatives_ as an indicator of curvature. -:::{#def-hessian} +When we found a critical point for a function of one variable, we used the second derivative as a indicator of the curvature at the point in order to determine whether the point was a min, max, or saddle (second derivative test of concavity). For functions of $n$ variables, we use *second order partial derivatives* as an indicator of curvature. -## Hessian +::: {#def-hessian} +## Hessian -Given a function $f(\mathbf{x})$ in $n$ variables, the hessian $\mathbf{H(x)}$ is -an $n\times n$ matrix, where the $(i,j)$th element is the second order -partial derivative of $f(\mathbf{x})$ with respect to $x_i$ and $x_j$: +Given a function $f(\mathbf{x})$ in $n$ variables, the hessian $\mathbf{H(x)}$ is an $n\times n$ matrix, where the $(i,j)$th element is the second order partial derivative of $f(\mathbf{x})$ with respect to $x_i$ and $x_j$: $$\mathbf{H(x)}=\begin{pmatrix} \frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}&\frac{\partial^2f(\mathbf{x})}{\partial x_1 \partial x_2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_2^2}& @@ -362,212 +348,186 @@ $$\mathbf{H(x)}=\begin{pmatrix} \frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}& \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f(\mathbf{x})}{\partial x_n \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_n \partial x_2}& \cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_n^2}\end{pmatrix}$$ - ::: Note that the hessian will be a symmetric matrix because $\frac{\partial f(\mathbf{x})}{\partial x_1\partial x_2} = \frac{\partial f(\mathbf{x})}{\partial x_2\partial x_1}$. Also note that given that $f(\mathbf{x})$ is of quadratic form, each element of the hessian will be a constant. -These definitions will be employed when we determine the __Second Order Conditions__ of a function: +These definitions will be employed when we determine the **Second Order Conditions** of a function: Given a function $f(\mathbf{x})$ and a point $\mathbf{x}^*$ such that $\nabla f(\mathbf{x}^*)=0$, -1. Hessian is Positive Definite $\quad \Longrightarrow \quad$ Strict Local Min -2. Hessian is Positive Semidefinite $\forall \mathbf{x}\in B(\mathbf{x}^*,\epsilon)$} $\quad \Longrightarrow \quad$ Local Min -3. Hessian is Negative Definite $\quad \Longrightarrow \quad$ Strict Local Max -4. Hessian is Negative Semidefinite $\forall \mathbf{x}\in B(\mathbf{x}^*,\epsilon)$} $\quad \Longrightarrow \quad$ Local Max -5. Hessian is Indefinite $\quad \Longrightarrow \quad$ Saddle Point - -:::{#exm-maxmin2d} -## Max and min with two dimensions +1. Hessian is Positive Definite $\quad \Longrightarrow \quad$ Strict Local Min +2. Hessian is Positive Semidefinite $\forall \mathbf{x}\in B(\mathbf{x}^*,\epsilon)$} $\quad \Longrightarrow \quad$ Local Min +3. Hessian is Negative Definite $\quad \Longrightarrow \quad$ Strict Local Max +4. Hessian is Negative Semidefinite $\forall \mathbf{x}\in B(\mathbf{x}^*,\epsilon)$} $\quad \Longrightarrow \quad$ Local Max +5. Hessian is Indefinite $\quad \Longrightarrow \quad$ Saddle Point -We found that the only critical point of -$f(\mathbf{x})=(x_1-1)^2+x_2^2+1$ is at $\mathbf{x}^*=(1,0)$. Is it a min, max, or -saddle point? +::: {#exm-maxmin2d} +## Max and min with two dimensions +We found that the only critical point of $f(\mathbf{x})=(x_1-1)^2+x_2^2+1$ is at $\mathbf{x}^*=(1,0)$. Is it a min, max, or saddle point? ::: -:::{#sol-maxmin2d} - -The Hessian is -\begin{align*} +::: {#sol-maxmin2d} +The Hessian is \begin{align*} \mathbf{H(x)} &= \begin{pmatrix} 2&0\\0&2 \end{pmatrix} -\end{align*} -The Leading principal minors of the Hessian are $M_1=2; M_2=4$. Now we consider Definiteness. Since both leading principal minors are positive, the Hessian is positive definite. +\end{align*} The Leading principal minors of the Hessian are $M_1=2; M_2=4$. Now we consider Definiteness. Since both leading principal minors are positive, the Hessian is positive definite. Maxima, Minima, or Saddle Point? Since the Hessian is positive definite and the gradient equals 0, $x^\star = (1,0)$ is a strict local minimum. Note: Alternate check of definiteness. Is $\mathbf{H(x^*)} \geq \leq 0 \quad \forall \quad \mathbf{x}\ne 0$ - - + +```{=tex} \begin{align*} \mathbf{x}^\top H(\mathbf{x}^*) \mathbf{x} &= \begin{pmatrix} x_1 & x_2 \end{pmatrix}\\ &= \begin{pmatrix} 2&0\\0&2 \end{pmatrix}\\ \begin{pmatrix} x_1\\x_2\end{pmatrix} &= 2x_1^2+2x_2^2 \end{align*} - -For any $\mathbf{x}\ne 0$, $2(x_1^2+x_2^2)>0$, so the Hessian is positive definite and $\mathbf{x}^*$ is a strict local minimum. - +``` +For any $\mathbf{x}\ne 0$, $2(x_1^2+x_2^2)>0$, so the Hessian is positive definite and $\mathbf{x}^*$ is a strict local minimum. ::: ### Definiteness and Concavity {.unnumbered} Although definiteness helps us to understand the curvature of an n-dimensional function, it does not necessarily tell us whether the function is globally concave or convex. -We need to know whether a function is globally concave or convex to determine whether a critical point is a global min or max. We can use the definiteness of the Hessian to determine whether a function is globally concave or convex: +We need to know whether a function is globally concave or convex to determine whether a critical point is a global min or max. We can use the definiteness of the Hessian to determine whether a function is globally concave or convex: -1. Hessian is Positive Semidefinite $\forall \mathbf{x}$} $\quad \Longrightarrow \quad$ Globally Convex -2. Hessian is Negative Semidefinite $\forall \mathbf{x}$} $\quad \Longrightarrow \quad$ Globally Concave +1. Hessian is Positive Semidefinite $\forall \mathbf{x}$} $\quad \Longrightarrow \quad$ Globally Convex +2. Hessian is Negative Semidefinite $\forall \mathbf{x}$} $\quad \Longrightarrow \quad$ Globally Concave Notice that the definiteness conditions must be satisfied over the entire domain. ## Global Maxima and Minima -__Global Max/Min Conditions__: Given a function $f(\mathbf{x})$ and a point $\mathbf{x}^*$ such that $\nabla f(\mathbf{x}^*)=0$, +**Global Max/Min Conditions**: Given a function $f(\mathbf{x})$ and a point $\mathbf{x}^*$ such that $\nabla f(\mathbf{x}^*)=0$, +```{=tex} \begin{enumerate} \item \parbox[t]{2in}{$f(\mathbf{x})$ Globally Convex} $\quad \Longrightarrow \quad$ Global Min \item \parbox[t]{2in}{$f(\mathbf{x})$ Globally Concave} $\quad \Longrightarrow \quad$ Global Max \end{enumerate} +``` +Note that showing that $\mathbf{H(x^*)}$ is negative semidefinite is not enough to guarantee $\mathbf{x}^*$ is a local max. However, showing that $\mathbf{H(x)}$ is negative semidefinite for all $\mathbf{x}$ guarantees that $x^*$ is a global max. (The same goes for positive semidefinite and minima.)\\ -Note that showing that $\mathbf{H(x^*)}$ is negative semidefinite is -not enough to guarantee $\mathbf{x}^*$ is a local max. However, showing that -$\mathbf{H(x)}$ is negative semidefinite for all $\mathbf{x}$ guarantees that $x^*$ -is a global max. (The same goes for positive semidefinite and minima.)\\ - -Example: Take $f_1(x)=x^4$ and $f_2(x)=-x^4$. Both have $x=0$ as -a critical point. Unfortunately, $f''_1(0)=0$ and $f''_2(0)=0$, so we -can't tell whether $x=0$ is a min or max for either. However, -$f''_1(x)=12x^2$ and $f''_2(x)=-12x^2$. For all $x$, $f''_1(x)\ge 0$ -and $f''_2(x)\le 0$ --- i.e., $f_1(x)$ is globally convex and $f_2(x)$ -is globally concave. So $x=0$ is a global min of $f_1(x)$ and a -global max of $f_2(x)$. - -:::{#exr-maxmin} +Example: Take $f_1(x)=x^4$ and $f_2(x)=-x^4$. Both have $x=0$ as a critical point. Unfortunately, $f''_1(0)=0$ and $f''_2(0)=0$, so we can't tell whether $x=0$ is a min or max for either. However, $f''_1(x)=12x^2$ and $f''_2(x)=-12x^2$. For all $x$, $f''_1(x)\ge 0$ and $f''_2(x)\le 0$ --- i.e., $f_1(x)$ is globally convex and $f_2(x)$ is globally concave. So $x=0$ is a global min of $f_1(x)$ and a global max of $f_2(x)$. +::: {#exr-maxmin} Given $f(\mathbf{x})=x_1^3-x_2^3+9x_1x_2$, find any maxima or minima. - ::: ```{=latex} \begin{enumerate} \item First order conditions. - \begin{enumerate} - \item Gradient $\nabla f(\mathbf{x}) = $ - $$\begin{pmatrix} \frac{\partial f}{\partial x_1} \\ - \frac{\partial f}{\partial x_2}\end{pmatrix} = - \begin{pmatrix} 3x_1^2+9x_2 \\ -3x_2^2+9x_1 \end{pmatrix}$$ - \item Critical Points $\mathbf{x^*} =$\\ - Set the gradient equal to zero and solve for $x_1$ and - $x_2$.We have two equations and two unknowns. Solving for $x_1$ - and $x_2$, we get two critical points: $\mathbf{x}_1^*=(0,0)$ and - $\mathbf{x}_2^*=(3,-3)$. - $$3x_1^2 + 9x_2 = 0 \quad \Rightarrow \quad 9x_2 = - -3x_1^2 \quad \Rightarrow \quad x_2 = -\frac{1}{3}x_1^2$$ - $$-3x_2^2 + 9x_1 = 0 \quad \Rightarrow \quad - -3(-\frac{1} {3}x_1^2)^2 + 9x_1 = 0$$ - $$\Rightarrow \quad -\frac{1}{3}x_1^4 - + 9x_1 = 0 \quad \Rightarrow \quad x_1^3 = 27x_1 \quad - \Rightarrow \quad x_1 = 3$$ - $$3(3)^2 + 9x_2 = 0 \quad \Rightarrow \quad x_2 = -3$$ - \end{enumerate} + \begin{enumerate} + \item Gradient $\nabla f(\mathbf{x}) = $ + $$\begin{pmatrix} \frac{\partial f}{\partial x_1} \\ + \frac{\partial f}{\partial x_2}\end{pmatrix} = + \begin{pmatrix} 3x_1^2+9x_2 \\ -3x_2^2+9x_1 \end{pmatrix}$$ + \item Critical Points $\mathbf{x^*} =$\\ + Set the gradient equal to zero and solve for $x_1$ and + $x_2$.We have two equations and two unknowns. Solving for $x_1$ + and $x_2$, we get two critical points: $\mathbf{x}_1^*=(0,0)$ and + $\mathbf{x}_2^*=(3,-3)$. + $$3x_1^2 + 9x_2 = 0 \quad \Rightarrow \quad 9x_2 = + -3x_1^2 \quad \Rightarrow \quad x_2 = -\frac{1}{3}x_1^2$$ + $$-3x_2^2 + 9x_1 = 0 \quad \Rightarrow \quad + -3(-\frac{1} {3}x_1^2)^2 + 9x_1 = 0$$ + $$\Rightarrow \quad -\frac{1}{3}x_1^4 + + 9x_1 = 0 \quad \Rightarrow \quad x_1^3 = 27x_1 \quad + \Rightarrow \quad x_1 = 3$$ + $$3(3)^2 + 9x_2 = 0 \quad \Rightarrow \quad x_2 = -3$$ + \end{enumerate} \item Second order conditions. - \begin{enumerate} - \item Hessian $\mathbf{H(x)} = $ - $$\begin{pmatrix} 6x_1&9\\9&-6x_2 \end{pmatrix}$$ - \item Hessian $\mathbf{H(x_1^*)} = $ - $$\begin{pmatrix} 0&9\\9&0\end{pmatrix}$$ - \item Leading principal minors of $\mathbf{H(x_1^*)} = $ - $$M_1=0; M_2=-81$$\\ - \item Definiteness of $\mathbf{H(x_1^*)}$?\\ - $\mathbf{H(x_1^*)}$ is indefinite\\ - \item Maxima, Minima, or Saddle Point for $\mathbf{x_1^*}$?\\ - Since $\mathbf{H(x_1^*)}$ is indefinite, $\mathbf{x}_1^*=(0,0)$ - is a saddle point.\\ - \item Hessian $\mathbf{H(x_2^*)} = $ - $$\begin{pmatrix} 18&9\\9&18\end{pmatrix}$$ - \item Leading principal minors of $\mathbf{H(x_2^*)} = $ - $$M_1=18; M_2=243$$\\ - \item Definiteness of $\mathbf{H(x_2^*)}$?\\ - $\mathbf{H(x_2^*)}$ is positive definite\\ - \item Maxima, Minima, or Saddle Point for $\mathbf{x}_2^*$?\\ - Since $\mathbf{H(x_2^*)}$ is positive definite, $\mathbf{x}_1^*=(3,-3)$ is a strict local minimum\\ - \end{enumerate} - + \begin{enumerate} + \item Hessian $\mathbf{H(x)} = $ + $$\begin{pmatrix} 6x_1&9\\9&-6x_2 \end{pmatrix}$$ + \item Hessian $\mathbf{H(x_1^*)} = $ + $$\begin{pmatrix} 0&9\\9&0\end{pmatrix}$$ + \item Leading principal minors of $\mathbf{H(x_1^*)} = $ + $$M_1=0; M_2=-81$$\\ + \item Definiteness of $\mathbf{H(x_1^*)}$?\\ + $\mathbf{H(x_1^*)}$ is indefinite\\ + \item Maxima, Minima, or Saddle Point for $\mathbf{x_1^*}$?\\ + Since $\mathbf{H(x_1^*)}$ is indefinite, $\mathbf{x}_1^*=(0,0)$ + is a saddle point.\\ + \item Hessian $\mathbf{H(x_2^*)} = $ + $$\begin{pmatrix} 18&9\\9&18\end{pmatrix}$$ + \item Leading principal minors of $\mathbf{H(x_2^*)} = $ + $$M_1=18; M_2=243$$\\ + \item Definiteness of $\mathbf{H(x_2^*)}$?\\ + $\mathbf{H(x_2^*)}$ is positive definite\\ + \item Maxima, Minima, or Saddle Point for $\mathbf{x}_2^*$?\\ + Since $\mathbf{H(x_2^*)}$ is positive definite, $\mathbf{x}_1^*=(3,-3)$ is a strict local minimum\\ + \end{enumerate} + \item Global concavity/convexity. - \begin{enumerate} - \item Is f(x) globally concave/convex?\\ - No. In evaluating the Hessians for $\mathbf{x}_1^*$ - and $\mathbf{x}_2^*$ we saw that the Hessian is not positive - semidefinite at x $=$ (0,0).\\ - \item Are any $\mathbf{x^*}$ global minima or maxima?\\ - No. Since the function is not globally concave/convex, - we can't infer that $\mathbf{x}_2^*=(3,-3)$ is a global minimum. - In fact, if we set $x_1=0$, the $f(\mathbf{x})=-x_2^3$, which will - go to $-\infty$ as $x_2\to \infty$.\\ - \end{enumerate} + \begin{enumerate} + \item Is f(x) globally concave/convex?\\ + No. In evaluating the Hessians for $\mathbf{x}_1^*$ + and $\mathbf{x}_2^*$ we saw that the Hessian is not positive + semidefinite at x $=$ (0,0).\\ + \item Are any $\mathbf{x^*}$ global minima or maxima?\\ + No. Since the function is not globally concave/convex, + we can't infer that $\mathbf{x}_2^*=(3,-3)$ is a global minimum. + In fact, if we set $x_1=0$, the $f(\mathbf{x})=-x_2^3$, which will + go to $-\infty$ as $x_2\to \infty$.\\ + \end{enumerate} \end{enumerate} ``` - ## Constrained Optimization -We have already looked at optimizing a function in one or more dimensions over the whole domain of the function. Often, however, we want to find the maximum or minimum of a function over some restricted part of its domain. +We have already looked at optimizing a function in one or more dimensions over the whole domain of the function. Often, however, we want to find the maximum or minimum of a function over some restricted part of its domain. ex: Maximizing utility subject to a budget constraint ![A typical Utility Function with a Budget Constraint](images/constraint.png) -__Types of Constraints__: For a function $f(x_1, \dots, x_n)$, there are two types of constraints that can be imposed: +**Types of Constraints**: For a function $f(x_1, \dots, x_n)$, there are two types of constraints that can be imposed: + +```{=tex} \begin{enumerate} \item \textbf{Equality constraints:} constraints of the form $c(x_1,\dots, x_n) = r$. Budget constraints are the classic example of equality constraints in social science. \item \textbf{Inequality constraints:} constraints of the form $c(x_1, \dots, x_n) \leq r$. These might arise from non-negativity constraints or other threshold effects. \end{enumerate} - -In any constrained optimization problem, the constrained maximum will always be less than or equal to the unconstrained maximum. If the constrained maximum is less than the unconstrained maximum, then the constraint is binding. Essentially, this means that you can treat your constraint as an equality constraint rather than an inequality constraint. +``` +In any constrained optimization problem, the constrained maximum will always be less than or equal to the unconstrained maximum. If the constrained maximum is less than the unconstrained maximum, then the constraint is binding. Essentially, this means that you can treat your constraint as an equality constraint rather than an inequality constraint. For example, the budget constraint binds when you spend your entire budget. This generally happens because we believe that utility is strictly increasing in consumption, i.e. you always want more so you spend everything you have. Any number of constraints can be placed on an optimization problem. When working with multiple constraints, always make sure that the set of constraints are not pathological; it must be possible for all of the constraints to be satisfied simultaneously. -\textbf{Set-up for Constrained Optimization:} -$$\max_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ -$$\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ -This tells us to maximize/minimize our function, $f(x_1,x_2)$, with respect to the choice variables, $x_1,x_2$, subject to the constraint. +\textbf{Set-up for Constrained Optimization:} $$\max_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ $$\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ This tells us to maximize/minimize our function, $f(x_1,x_2)$, with respect to the choice variables, $x_1,x_2$, subject to the constraint. -Example: -$$\max_{x_1,x_2} f(x_1, x_2) = -(x_1^2 + 2x_2^2) \text{ s.t. }x_1 + x_2 = 4$$ -It is easy to see that the \textit{unconstrained} maximum occurs at $(x_1, x_2) = (0,0)$, but that does not satisfy the constraint. How should we proceed? +Example: $$\max_{x_1,x_2} f(x_1, x_2) = -(x_1^2 + 2x_2^2) \text{ s.t. }x_1 + x_2 = 4$$ It is easy to see that the \textit{unconstrained} maximum occurs at $(x_1, x_2) = (0,0)$, but that does not satisfy the constraint. How should we proceed? ### Equality Constraints {.unnumbered} Equality constraints are the easiest to deal with because we know that the maximum or minimum has to lie on the (intersection of the) constraint(s). -The trick is to change the problem from a constrained optimization problem in $n$ variables to an unconstrained optimization problem in $n + k$ variables, adding _one_ variable for _each_ equality constraint. We do this using a lagrangian multiplier. +The trick is to change the problem from a constrained optimization problem in $n$ variables to an unconstrained optimization problem in $n + k$ variables, adding *one* variable for *each* equality constraint. We do this using a lagrangian multiplier. -__Lagrangian function__: The Lagrangian function allows us to combine the function we want to optimize and the constraint function into a single function. Once we have this single function, we can proceed as if this were an _unconstrained_ optimization problem. +**Lagrangian function**: The Lagrangian function allows us to combine the function we want to optimize and the constraint function into a single function. Once we have this single function, we can proceed as if this were an *unconstrained* optimization problem. For each constraint, we must include a **Lagrange multiplier** ($\lambda_i$) as an additional variable in the analysis. These terms are the link between the constraint and the Lagrangian function. -Given a _two dimensional_ set-up: -$$\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) = a$$ +Given a *two dimensional* set-up: $$\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) = a$$ -We define the Lagrangian function $L(x_1,x_2,\lambda_1)$ as follows: -$$L(x_1,x_2,\lambda_1) = f(x_1,x_2) - \lambda_1 (c(x_1,x_2) - a)$$ +We define the Lagrangian function $L(x_1,x_2,\lambda_1)$ as follows: $$L(x_1,x_2,\lambda_1) = f(x_1,x_2) - \lambda_1 (c(x_1,x_2) - a)$$ -More generally, in _n dimensions_: -$$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) - \sum_{i=1}^k\lambda_i(c_i(x_1,\dots, x_n) - r_i)$$ +More generally, in *n dimensions*: $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) - \sum_{i=1}^k\lambda_i(c_i(x_1,\dots, x_n) - r_i)$$ -**Getting the sign right:** Note that above we subtract the lagrangian term _and_ we subtract the constraint constant from the constraint function. Occasionally, you may see the following alternative form of the Lagrangian, which is _equivalent_: -$$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) + \sum_{i=1}^k\lambda_i(r_i - c_i(x_1,\dots, x_n))$$ -Here we add the lagrangian term _and_ we subtract the constraining function from the constraint constant. +**Getting the sign right:** Note that above we subtract the lagrangian term *and* we subtract the constraint constant from the constraint function. Occasionally, you may see the following alternative form of the Lagrangian, which is *equivalent*: $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) + \sum_{i=1}^k\lambda_i(r_i - c_i(x_1,\dots, x_n))$$ Here we add the lagrangian term *and* we subtract the constraining function from the constraint constant. -**Using the Lagrangian to Find the Critical Points**: To find the critical points, we take the partial derivatives of lagrangian function, $L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k)$, with respect to each of its variables (all choice variables $\mathbf{x}$ _and_ all lagrangian multipliers $\mathbf{\lambda}$). At a critical point, each of these partial derivatives must be equal to zero, so we obtain a system of $n + k$ equations in $n + k$ unknowns: +**Using the Lagrangian to Find the Critical Points**: To find the critical points, we take the partial derivatives of lagrangian function, $L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k)$, with respect to each of its variables (all choice variables $\mathbf{x}$ *and* all lagrangian multipliers $\mathbf{\lambda}$). At a critical point, each of these partial derivatives must be equal to zero, so we obtain a system of $n + k$ equations in $n + k$ unknowns: +```{=tex} \begin{align*} \frac{\partial L}{\partial x_1} &= \frac{\partial f}{\partial x_1} - \sum_{i = 1}^k\lambda_i\frac{\partial c_i}{\partial x_1} = 0\\ \vdots &= \vdots \nonumber \\ @@ -576,66 +536,58 @@ Here we add the lagrangian term _and_ we subtract the constraining function from \vdots &= \vdots \nonumber \\ \frac{\partial L}{\partial \lambda_k} &= c_k(x_i, \dots, x_n) - r_k = 0 \end{align*} - +``` We can then solve this system of equations, because there are $n+k$ equations and $n+k$ unknowns, to calculate the critical point $(x_1^*,\dots,x_n^*,\lambda_1^*,\dots,\lambda_k^*)$. -**Second-order Conditions and Unconstrained Optimization:** There may be more than one critical point, i.e. we need to verify that the critical point we find is a maximum/minimum. Similar to unconstrained optimization, we can do this by checking the second-order conditions. +**Second-order Conditions and Unconstrained Optimization:** There may be more than one critical point, i.e. we need to verify that the critical point we find is a maximum/minimum. Similar to unconstrained optimization, we can do this by checking the second-order conditions. -:::{#exm-constropt} -## Constrained optimization with two goods and a budget constraint +::: {#exm-constropt} +## Constrained optimization with two goods and a budget constraint -Find the constrained optimization of -$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 = 4$$ - +Find the constrained optimization of $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 = 4$$ ::: -:::{#sol-constropt} - -1. Begin by writing the Lagrangian: -$$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ -2. Take the partial derivatives and set equal to zero: +::: {#sol-constropt} +1. Begin by writing the Lagrangian: $$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ +2. Take the partial derivatives and set equal to zero: +```{=tex} \begin{align*} \frac{\partial L}{\partial x_1} = -2x_1 - \lambda \quad \quad \quad &= 0\\ \frac{\partial L}{\partial x_2} = -4x_2 - \lambda \quad \quad \quad &= 0\\ \frac{\partial L}{\partial \lambda} = -(x_1 + x_2 - 4) \quad & = & 0\\ \end{align*} +``` +3. Solve the system of equations: Using the first two partials, we see that $\lambda = -2x_1$ and $\lambda = -4x_2$ Set these equal to see that $x_1 = 2x_2$. Using the third partial and the above equality, $4 = 2x_2 + x_2$ from which we get $$x_2^* = 4/3, x_1^* = 8/3, \lambda = -16/3$$ -3. Solve the system of equations: Using the first two partials, we see that $\lambda = -2x_1$ and $\lambda = -4x_2$ - Set these equal to see that $x_1 = 2x_2$. - Using the third partial and the above equality, $4 = 2x_2 + x_2$ from which we get $$x_2^* = 4/3, x_1^* = 8/3, \lambda = -16/3$$ - -4. Therefore, the only critical point is $x_1^* = \frac{8}{3}$ and $x_2^* = \frac{4}{3}$ -5. This gives $f(\frac{8}{3}, \frac{4}{3}) = -\frac{96}{9}$, which is less than the unconstrained optimum $f(0,0) = 0$ +4. Therefore, the only critical point is $x_1^* = \frac{8}{3}$ and $x_2^* = \frac{4}{3}$ +5. This gives $f(\frac{8}{3}, \frac{4}{3}) = -\frac{96}{9}$, which is less than the unconstrained optimum $f(0,0) = 0$ ::: Notice that when we take the partial derivative of L with respect to the Lagrangian multiplier and set it equal to 0, we return exactly our constraint! This is why signs matter. ## Inequality Constraints -Inequality constraints define the boundary of a region over which we seek to optimize the function. This makes inequality constraints more challenging because we do not know if the maximum/minimum lies along one of the constraints (the constraint binds) or in the interior of the region. +Inequality constraints define the boundary of a region over which we seek to optimize the function. This makes inequality constraints more challenging because we do not know if the maximum/minimum lies along one of the constraints (the constraint binds) or in the interior of the region. We must introduce more variables in order to turn the problem into an unconstrained optimization. -__Slack:__ For each inequality constraint $c_i(x_1, \dots, x_n) \leq a_i$, we define a slack variable $s_i^2$ for which the expression $c_i(x_1, \dots, x_n) \leq a_i - s_i^2$ would hold with equality. These slack variables capture how close the constraint comes to binding. We use $s^2$ rather than $s$ to ensure that the slack is positive. +**Slack:** For each inequality constraint $c_i(x_1, \dots, x_n) \leq a_i$, we define a slack variable $s_i^2$ for which the expression $c_i(x_1, \dots, x_n) \leq a_i - s_i^2$ would hold with equality. These slack variables capture how close the constraint comes to binding. We use $s^2$ rather than $s$ to ensure that the slack is positive. Slack is just a way to transform our constraints. -Given a two-dimensional set-up and these edited constraints: -$$\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) \le a_1$$ +Given a two-dimensional set-up and these edited constraints: $$\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) \le a_1$$ -Adding in Slack: -$$\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) \le a_1 - s_1^2$$ +Adding in Slack: $$\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) \le a_1 - s_1^2$$ -We define the Lagrangian function $L(x_1,x_2,\lambda_1,s_1)$ as follows: -$$L(x_1,x_2,\lambda_1,s_1) = f(x_1,x_2) - \lambda_1 ( c(x_1,x_2) + s_1^2 - a_1)$$ +We define the Lagrangian function $L(x_1,x_2,\lambda_1,s_1)$ as follows: $$L(x_1,x_2,\lambda_1,s_1) = f(x_1,x_2) - \lambda_1 ( c(x_1,x_2) + s_1^2 - a_1)$$ -More generally, in n dimensions: -$$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k, s_1, \dots, s_k) = f(x_1, \dots, x_n) - \sum_{i = 1}^k \lambda_i(c_i(x_1,\dots, x_n) + s_i^2 - a_i)$$ +More generally, in n dimensions: $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k, s_1, \dots, s_k) = f(x_1, \dots, x_n) - \sum_{i = 1}^k \lambda_i(c_i(x_1,\dots, x_n) + s_i^2 - a_i)$$ -**Finding the Critical Points**: To find the critical points, we take the partial derivatives of the lagrangian function, $L(x_1,\dots,x_n,\lambda_1,\dots,\lambda_k,s_1,\dots,s_k)$, with respect to each of its variables (all choice variables $x$, all lagrangian multipliers $\lambda$, and all slack variables $s$). At a critical point, _each_ of these partial derivatives must be equal to zero, so we obtain a system of $n + 2k$ equations in $n + 2k$ unknowns: +**Finding the Critical Points**: To find the critical points, we take the partial derivatives of the lagrangian function, $L(x_1,\dots,x_n,\lambda_1,\dots,\lambda_k,s_1,\dots,s_k)$, with respect to each of its variables (all choice variables $x$, all lagrangian multipliers $\lambda$, and all slack variables $s$). At a critical point, *each* of these partial derivatives must be equal to zero, so we obtain a system of $n + 2k$ equations in $n + 2k$ unknowns: +```{=tex} \begin{align*} \frac{\partial L}{\partial x_1} &= \frac{\partial f}{\partial x_1} - \sum_{i = 1}^k\lambda_i\frac{\partial c_i}{\partial x_1} = 0\\ \vdots & = \vdots \\ @@ -647,53 +599,52 @@ $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k, s_1, \dots, s_k) = f(x_1, \do \vdots =\vdots \\ \frac{\partial L}{\partial s_k} &= 2s_k\lambda_k = 0 \end{align*} - +``` **Complementary slackness conditions**: The last set of first order conditions of the form $2s_i\lambda_i = 0$ (the partials taken with respect to the slack variables) are known as complementary slackness conditions. These conditions can be satisfied one of three ways: -1. $\lambda_i = 0$ and $s_i \neq 0$: This implies that the slack is positive and thus _the constraint does not bind_. -2. $\lambda_i \neq 0$ and $s_i = 0$: This implies that there is no slack in the constraint and _the constraint does bind_. -3. $\lambda_i = 0$ and $s_i = 0$: In this case, there is no slack but the _constraint binds trivially_, without changing the optimum. +1. $\lambda_i = 0$ and $s_i \neq 0$: This implies that the slack is positive and thus *the constraint does not bind*. +2. $\lambda_i \neq 0$ and $s_i = 0$: This implies that there is no slack in the constraint and *the constraint does bind*. +3. $\lambda_i = 0$ and $s_i = 0$: In this case, there is no slack but the *constraint binds trivially*, without changing the optimum. -Example: Find the critical points for the following constrained optimization: -$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4$$ +Example: Find the critical points for the following constrained optimization: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4$$ -1. Rewrite with the slack variables: -$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4 - s_1^2$$ +1. Rewrite with the slack variables: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4 - s_1^2$$ -2. Write the Lagrangian: -$$L(x_1,x_2,\lambda_1,s_1) = -(x_1^2 + 2x_2^2) - \lambda_1 (x_1 + x_2 + s_1^2 - 4)$$ +2. Write the Lagrangian: $$L(x_1,x_2,\lambda_1,s_1) = -(x_1^2 + 2x_2^2) - \lambda_1 (x_1 + x_2 + s_1^2 - 4)$$ -3. Take the partial derivatives and set equal to 0: +3. Take the partial derivatives and set equal to 0: +```{=tex} \begin{align*} \frac{\partial L}{\partial x_1} = -2x_1 - \lambda_1 &= 0\\ \frac{\partial L}{\partial x_2} = -4x_2 - \lambda_1 &= 0\\ \frac{\partial L}{\partial \lambda_1} = -(x_1 + x_2 + s_1^2 - 4)&= 0\\ \frac{\partial L}{\partial s_1} = -2s_1\lambda_1 &= 0\\ \end{align*} - +``` 4. Consider all ways that the complementary slackness conditions are solved: -\begin{center} -\begin{tabular}{|l|cccc|c|} -\hline -Hypothesis & $s_1$ & $\lambda_1$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ -\hline -$s_1 = 0$ $\lambda_1 = 0$ & \multicolumn{4}{l|}{No solution} & \\ -$s_1 \neq 0$ $\lambda_1 = 0$ & 2 & 0 & 0 & 0 & 0\\ -$s_1 = 0$ $\lambda_1 \neq 0$ & 0 & $\frac{-16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ -$s_1 \neq 0$ $\lambda_1 \neq 0$ & \multicolumn{4}{l|}{No solution} &\\ -\hline -\end{tabular} -\end{center} + + ```{=tex} + \begin{center} + \begin{tabular}{|l|cccc|c|} + \hline + Hypothesis & $s_1$ & $\lambda_1$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ + \hline + $s_1 = 0$ $\lambda_1 = 0$ & \multicolumn{4}{l|}{No solution} & \\ + $s_1 \neq 0$ $\lambda_1 = 0$ & 2 & 0 & 0 & 0 & 0\\ + $s_1 = 0$ $\lambda_1 \neq 0$ & 0 & $\frac{-16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ + $s_1 \neq 0$ $\lambda_1 \neq 0$ & \multicolumn{4}{l|}{No solution} &\\ + \hline + \end{tabular} + \end{center} + ``` This shows that there are two critical points: $(0,0)$ and $(\frac{8}{3},\frac{4}{3})$. -5. Find maximum: -Looking at the values of $f(x_1,x_2)$ at the critical points, we see that $f(x_1,x_2)$ is maximized at $x_1^* = 0$ and $x_2^*=0$. +5. Find maximum: Looking at the values of $f(x_1,x_2)$ at the critical points, we see that $f(x_1,x_2)$ is maximized at $x_1^* = 0$ and $x_2^*=0$. -:::{#exr-critical-points-constrained-optimization} -Example: Find the critical points for the following constrained optimization: -$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } +::: {#exr-critical-points-constrained-optimization} +Example: Find the critical points for the following constrained optimization: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } \begin{array}{l} x_1 + x_2 \le 4\\ x_1 \ge 0\\ @@ -711,81 +662,73 @@ x_1 + x_2 \le 4 - s_1^2\\ -x_2 \le 0 - s_3^2 \end{array}}$$ ``` - -2. Write the Lagrangian: - -$$L(x_1, x_2, \lambda_1, \lambda_2, \lambda_3, s_1, s_2, s_3) = -(x_1^2 + 2x_2^2) - \lambda_1(x_1 + x_2 + s_1^2 - 4) - \lambda_2(-x_1 + s_2^2) - \lambda_3(-x_2 + s_3^2)$$ -3. Take the partial derivatives and set equal to zero: - -$\frac{\partial L}{\partial x_1} = -2x_1 - \lambda_1 + \lambda_2 = 0$\\ -$\frac{\partial L}{\partial x_2} = -4x_2 - \lambda_1 + \lambda_3 = 0$\\ -$\frac{\partial L}{\partial \lambda_1} = -(x_1 + x_2 + s_1^2 - 4) = 0$\\ -$\frac{\partial L}{\partial \lambda_2} = -(-x_1 + s_2^2) = 0$\\ -$\frac{\partial L}{\partial \lambda_3} = -(-x_2 + s_3^2) = 0$\\ -$\frac{\partial L}{\partial s_1} = 2s_1\lambda_1 = 0$\\ -$\frac{\partial L}{\partial s_2} = 2s_2\lambda_2 = 0$\\ -$\frac{\partial L}{\partial s_3} = 2s_3\lambda_3 = 0$ - -3. Consider all ways that the complementary slackness conditions are solved: -\begin{center} -\begin{tabular}{|l|cccccccc|c|} -\hline -Hypothesis & $s_1$ & $s_2$ & $s_3$ & $\lambda_1$ & $\lambda_2$ & $\lambda_3$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ -\hline -$s_1 = s_2 = s_3 = 0$ & \multicolumn{8}{l|}{No solution} & \\ -$s_1 \neq 0, s_2 = s_3 = 0$ & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -$s_2 \neq 0, s_1 = s_3 = 0$ & 0 & 2 & 0 & -8 & 0 & -8 & 4 & 0 & -16\\ -$s_3 \neq 0, s_1 = s_2 = 0$ & 0 & 0 & 2 & -16 & -16 & 0 & 0 & 4 & -32\\ -$s_1 \neq 0, s_2 \neq 0, s_3 = 0$ &\multicolumn{8}{l|}{No solution} & \\ -$s_1 \neq 0, s_3 \neq 0, s_2 = 0$ &\multicolumn{8}{l|}{No solution} & \\ -$s_2 \neq 0, s_3 \neq 0, s_1 = 0$ &0 & $\sqrt{\frac{8}{3}}$ & $\sqrt{\frac{4}{3}}$ & $-\frac{16}{3}$ & 0 & 0 & $\frac{8}{3}$& $\frac{4}{3}$ & $-\frac{32}{3}$\\ -$s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{No solution}& \\ -\hline -\end{tabular} -\end{center} +2. Write the Lagrangian: + +$$L(x_1, x_2, \lambda_1, \lambda_2, \lambda_3, s_1, s_2, s_3) = -(x_1^2 + 2x_2^2) - \lambda_1(x_1 + x_2 + s_1^2 - 4) - \lambda_2(-x_1 + s_2^2) - \lambda_3(-x_2 + s_3^2)$$ 3. Take the partial derivatives and set equal to zero: + +$\frac{\partial L}{\partial x_1} = -2x_1 - \lambda_1 + \lambda_2 = 0$\\ $\frac{\partial L}{\partial x_2} = -4x_2 - \lambda_1 + \lambda_3 = 0$\\ $\frac{\partial L}{\partial \lambda_1} = -(x_1 + x_2 + s_1^2 - 4) = 0$\\ $\frac{\partial L}{\partial \lambda_2} = -(-x_1 + s_2^2) = 0$\\ $\frac{\partial L}{\partial \lambda_3} = -(-x_2 + s_3^2) = 0$\\ $\frac{\partial L}{\partial s_1} = 2s_1\lambda_1 = 0$\\ $\frac{\partial L}{\partial s_2} = 2s_2\lambda_2 = 0$\\ $\frac{\partial L}{\partial s_3} = 2s_3\lambda_3 = 0$ + +3. Consider all ways that the complementary slackness conditions are solved: + + ```{=tex} + \begin{center} + \begin{tabular}{|l|cccccccc|c|} + \hline + Hypothesis & $s_1$ & $s_2$ & $s_3$ & $\lambda_1$ & $\lambda_2$ & $\lambda_3$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ + \hline + $s_1 = s_2 = s_3 = 0$ & \multicolumn{8}{l|}{No solution} & \\ + $s_1 \neq 0, s_2 = s_3 = 0$ & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ + $s_2 \neq 0, s_1 = s_3 = 0$ & 0 & 2 & 0 & -8 & 0 & -8 & 4 & 0 & -16\\ + $s_3 \neq 0, s_1 = s_2 = 0$ & 0 & 0 & 2 & -16 & -16 & 0 & 0 & 4 & -32\\ + $s_1 \neq 0, s_2 \neq 0, s_3 = 0$ &\multicolumn{8}{l|}{No solution} & \\ + $s_1 \neq 0, s_3 \neq 0, s_2 = 0$ &\multicolumn{8}{l|}{No solution} & \\ + $s_2 \neq 0, s_3 \neq 0, s_1 = 0$ &0 & $\sqrt{\frac{8}{3}}$ & $\sqrt{\frac{4}{3}}$ & $-\frac{16}{3}$ & 0 & 0 & $\frac{8}{3}$& $\frac{4}{3}$ & $-\frac{32}{3}$\\ + $s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{No solution}& \\ + \hline + \end{tabular} + \end{center} + ``` This shows that there are four critical points: $(0,0)$, $(4,0)$, $(0,4)$, and $(\frac{8}{3},\frac{4}{3})$ -4. Find maximum: Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem. +4. Find maximum: Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem. ## Kuhn-Tucker Conditions -As you can see, this can be a pain. When dealing explicitly with _non-negativity constraints_, this process is simplified by using the Kuhn-Tucker method. +As you can see, this can be a pain. When dealing explicitly with *non-negativity constraints*, this process is simplified by using the Kuhn-Tucker method. Because the problem of maximizing a function subject to inequality and non-negativity constraints arises frequently in economics, the **Kuhn-Tucker conditions** provides a method that often makes it easier to both calculate the critical points and identify points that are (local) maxima. -Given a _two-dimensional set-up_: -$$\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } +Given a *two-dimensional set-up*: $$\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } \begin{array}{l} c(x_1,x_2) \le a_1\\ x_1 \ge 0 \\ gx_2 \ge 0 \end{array}$$ -We define the Lagrangian function $L(x_1,x_2,\lambda_1)$ the same as if we did not have the non-negativity constraints: -$$L(x_1,x_2,\lambda_2) = f(x_1,x_2) - \lambda_1(c(x_1,x_2) - a_1)$$ +We define the Lagrangian function $L(x_1,x_2,\lambda_1)$ the same as if we did not have the non-negativity constraints: $$L(x_1,x_2,\lambda_2) = f(x_1,x_2) - \lambda_1(c(x_1,x_2) - a_1)$$ -More generally, in n dimensions: -$$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) - \sum_{i=1}^k\lambda_i(c_i(x_1,\dots, x_n) - a_i)$$ +More generally, in n dimensions: $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) - \sum_{i=1}^k\lambda_i(c_i(x_1,\dots, x_n) - a_i)$$ -**Kuhn-Tucker and Complementary Slackness Conditions**: To find the critical points, we first calculate the Kuhn-Tucker conditions by taking the partial derivatives of the lagrangian function, $L(x_1,\dots,x_n,\lambda_1,\dots,\lambda_k)$, with respect to each of its variables (all choice variable s$x$ and all lagrangian multipliers $\lambda$) and we calculate the _complementary slackness conditions_ by multiplying each partial derivative by its respective variable _and_ include non-negativity conditions for all variables (choice variables $x$ and lagrangian multipliers $\lambda$). +**Kuhn-Tucker and Complementary Slackness Conditions**: To find the critical points, we first calculate the Kuhn-Tucker conditions by taking the partial derivatives of the lagrangian function, $L(x_1,\dots,x_n,\lambda_1,\dots,\lambda_k)$, with respect to each of its variables (all choice variable s$x$ and all lagrangian multipliers $\lambda$) and we calculate the *complementary slackness conditions* by multiplying each partial derivative by its respective variable *and* include non-negativity conditions for all variables (choice variables $x$ and lagrangian multipliers $\lambda$). **Kuhn-Tucker Conditions** +```{=tex} \begin{align*} \frac{\partial L}{\partial x_1} \leq 0, & \dots, \frac{\partial L}{\partial x_n} \leq 0\\ \frac{\partial L}{\partial \lambda_1} \geq 0, & \dots, \frac{\partial L}{\partial \lambda_m} \geq 0 \end{align*} - +``` **Complementary Slackness Conditions** +```{=tex} \begin{align*} x_1\frac{\partial L}{\partial x_1} = 0, & \dots, x_n\frac{\partial L}{\partial x_n} = 0\\ \lambda_1\frac{\partial L}{\partial \lambda_1} = 0, & \dots, \lambda_m \frac{\partial L}{\partial \lambda_m} = 0 \end{align*} - -**Non-negativity Conditions** -\begin{eqnarray*} +``` +**Non-negativity Conditions** \begin{eqnarray*} x_1 \geq 0 & \dots & x_n \geq 0\\ \lambda_1 \geq 0 & \dots & \lambda_m \geq 0 \end{eqnarray*} @@ -800,16 +743,17 @@ There are additional assumptions (notably, f(x) is quasi-concave and the constra In a two-dimensional set-up, this means we must check the following cases: -1. $x_1 = 0, x_2 = 0$ Border Solution -1. $x_1 = 0, x_2 \neq 0$ Border Solution -1. $x_1 \neq 0, x_2 = 0$ Border Solution -1. $x_1 \neq 0, x_2 \neq 0$ Interior Solution +1. $x_1 = 0, x_2 = 0$ Border Solution +2. $x_1 = 0, x_2 \neq 0$ Border Solution +3. $x_1 \neq 0, x_2 = 0$ Border Solution +4. $x_1 \neq 0, x_2 \neq 0$ Interior Solution + +::: {#exm-k-t-2} +## Kuhn-Tucker with two variables -:::{#exm-k-t-2} -## Kuhn-Tucker with two variables -Solve the following optimization problem with inequality constraints -$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)$$ +Solve the following optimization problem with inequality constraints $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)$$ +```{=tex} \begin{align*} \text{ s.t. } \begin{cases} @@ -818,100 +762,95 @@ $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)$$ &x_2 *\ge 0 \end{cases} \end{align*} - +``` ::: -1. Write the Lagrangian: -$$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ +1. Write the Lagrangian: $$L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)$$ -2. Find the First Order Conditions: +2. Find the First Order Conditions: -Kuhn-Tucker Conditions - \begin{align*} +Kuhn-Tucker Conditions \begin{align*} \frac{\partial L}{\partial x_1} = -2x_1 - \lambda &\leq 0\\ \frac{\partial L}{\partial x_2} = -4x_2 - \lambda & \leq 0\\ \frac{\partial L}{\partial \lambda} = -(x_1 + x_2 - 4)& \geq 0 \end{align*} -Complementary Slackness Conditions -\begin{align*} +Complementary Slackness Conditions \begin{align*} x_1\frac{\partial L}{\partial x_2} = x_1(-2x_1 - \lambda) &= 0\\ x_2\frac{\partial L}{\partial x_2} = x_2(-4x_2 - \lambda) &= 0\\ \lambda\frac{\partial L}{\partial \lambda} = -\lambda(x_1 + x_2 - 4)&= 0 \end{align*} -Non-negativity Conditions -\begin{align*} +Non-negativity Conditions \begin{align*} x_1 & \geq 0\\ x_2 & \geq 0\\ \lambda & \geq 0 \end{align*} 3. Consider all border and interior cases: -\begin{center} -\begin{tabular}{|l|ccc|c|} -\hline -Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ -\hline -$x_1 = 0, x_2 = 0$ &0 & 0 & 0 & 0\\ -$x_1 = 0, x_2 \neq 0$ &-16 & 0 & 4 & -32\\ -$x_1 \neq 0, x_2 = 0$ &-8 & 4 & 0 & -16\\ -$x_1 \neq 0, x_2 \neq 0$ & $-\frac{16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ -\hline -\end{tabular} -\end{center} -4. Find Maximum: -Three of the critical points violate the requirement that $\lambda \geq 0$, so the point $(0,0,0)$ is the maximum. + ```{=tex} + \begin{center} + \begin{tabular}{|l|ccc|c|} + \hline + Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ + \hline + $x_1 = 0, x_2 = 0$ &0 & 0 & 0 & 0\\ + $x_1 = 0, x_2 \neq 0$ &-16 & 0 & 4 & -32\\ + $x_1 \neq 0, x_2 = 0$ &-8 & 4 & 0 & -16\\ + $x_1 \neq 0, x_2 \neq 0$ & $-\frac{16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ + \hline + \end{tabular} + \end{center} + ``` + +4. Find Maximum: Three of the critical points violate the requirement that $\lambda \geq 0$, so the point $(0,0,0)$ is the maximum. + +::: {#exr-ktlogs} +## Kuhn-Tucker with logs -:::{#exr-ktlogs} -## Kuhn-Tucker with logs Solve the constrained optimization problem, $$\max_{x_1,x_2} f(x) = \frac{1}{3}\log (x_1 + 1) + \frac{2}{3}\log (x_2 + 1) \text{ s.t. } \begin{array}{l} x_1 + 2x_2 \leq 4\\ - x_1 \geq 0\\ - x_2 \geq 0 + x_1 \geq 0\\ + x_2 \geq 0 \end{array}$$ ::: -1. Write the Lagrangian: -$$L(x_1, x_2, \lambda) = \frac{1}{3}\log(x_1+1) + \frac{2}{3}\log(x_2+1) - \lambda(x_1 + 2x_2 - 4)$$ +1. Write the Lagrangian: $$L(x_1, x_2, \lambda) = \frac{1}{3}\log(x_1+1) + \frac{2}{3}\log(x_2+1) - \lambda(x_1 + 2x_2 - 4)$$ -2. Find the First Order Conditions: +2. Find the First Order Conditions: Kuhn-Tucker Conditions -$\frac{\partial L}{\partial x_1} = \frac{1}{3(x_1+1)} - \lambda \leq 0$\\ -$\frac{\partial L}{\partial x_2} = \frac{2}{3(x_2+1)} - \lambda \leq 0$\\ -$\frac{\partial L}{\partial \lambda} = -(x_1 + 2x_2 - 4) \geq 0$\\ - +$\frac{\partial L}{\partial x_1} = \frac{1}{3(x_1+1)} - \lambda \leq 0$\\ $\frac{\partial L}{\partial x_2} = \frac{2}{3(x_2+1)} - \lambda \leq 0$\\ $\frac{\partial L}{\partial \lambda} = -(x_1 + 2x_2 - 4) \geq 0$\\ + Complementary Slackness Conditions -$x_1\frac{\partial L}{\partial x_2} = x_1(\frac{1}{3(x_1+1)} - \lambda) = 0$\\ -$x_2\frac{\partial L}{\partial x_2} = x_2(\frac{2}{3(x_2+1)} - \lambda) = 0$\\ -$\lambda\frac{\partial L}{\partial \lambda} = -\lambda(x_1 + 2x_2 - 4) = 0$\\ +$x_1\frac{\partial L}{\partial x_2} = x_1(\frac{1}{3(x_1+1)} - \lambda) = 0$\\ $x_2\frac{\partial L}{\partial x_2} = x_2(\frac{2}{3(x_2+1)} - \lambda) = 0$\\ $\lambda\frac{\partial L}{\partial \lambda} = -\lambda(x_1 + 2x_2 - 4) = 0$\\ Non-negativity Conditions -$x_1 \geq 0$\\ -$x_2 \geq 0$\\ -$\lambda \geq $0\\ - -3. Consider all border and interior cases: -\begin{center} -\begin{tabular}{|l|ccc|c|} -\hline -Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ -\hline -$x_1 = 0, x_2 = 0$ &\multicolumn{3}{l|}{No Solution}& \\ -$x_1 = 0, x_2 \neq 0$ &\multicolumn{3}{l|}{No Solution}& \\ -$x_1 \neq 0, x_2 = 0$ & \multicolumn{3}{l|}{No Solution}& \\ -$x_1 \neq 0, x_2 \neq 0$ & & $\frac{4}{3}$ & $\frac{4}{3}$ & $\log\frac{7}{3}$\\ -\hline -\end{tabular} -\end{center} +$x_1 \geq 0$\\ $x_2 \geq 0$\\ \$\lambda \\geq \$0\\ + +3. Consider all border and interior cases: + + ```{=tex} + \begin{center} + \begin{tabular}{|l|ccc|c|} + \hline + Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ + \hline + $x_1 = 0, x_2 = 0$ &\multicolumn{3}{l|}{No Solution}& \\ + $x_1 = 0, x_2 \neq 0$ &\multicolumn{3}{l|}{No Solution}& \\ + $x_1 \neq 0, x_2 = 0$ & \multicolumn{3}{l|}{No Solution}& \\ + $x_1 \neq 0, x_2 \neq 0$ & & $\frac{4}{3}$ & $\frac{4}{3}$ & $\log\frac{7}{3}$\\ + \hline + \end{tabular} + \end{center} + ``` 4. Find Maximum: @@ -919,13 +858,8 @@ Three of the critical points violate the constraints, so the point $(\frac{4}{3} ## Applications of Quadratic Forms -__Curvature and The Taylor Polynomial as a Quadratic Form__: -The Hessian is used in a Taylor polynomial approximation to $f(\mathbf{x})$ and provides information about the curvature of $f({\bf x})$ at $\mathbf{x}$ --- e.g., which tells us whether a critical point $\mathbf{x}^*$ is a min, max, or saddle point. +**Curvature and The Taylor Polynomial as a Quadratic Form**: The Hessian is used in a Taylor polynomial approximation to $f(\mathbf{x})$ and provides information about the curvature of $f({\bf x})$ at $\mathbf{x}$ --- e.g., which tells us whether a critical point $\mathbf{x}^*$ is a min, max, or saddle point. -1. The second order Taylor polynomial about the critical point -${\bf x}^*$ is - $$f({\bf x}^*+\bf h)=f({\bf x}^*)+\nabla f({\bf x}^*) \bf h +\frac{1}{2} \bf h^\top {\bf H(x^*)} \bf h + R(\bf h)$$ -2. Since we're looking at a critical point, $\nabla f({\bf x}^*)=0$; -and for small $\bf h$, $R(\bf h)$ is negligible. Rearranging, we get -$$f({\bf x}^*+\bf h)-f({\bf x}^*)\approx \frac{1}{2} \bf h^\top {\bf H(x^*)} \bf h $$ -3. The Righthand side here is a quadratic form and we can determine the definiteness of $\bf H(x^*)$. +1. The second order Taylor polynomial about the critical point ${\bf x}^*$ is $$f({\bf x}^*+\bf h)=f({\bf x}^*)+\nabla f({\bf x}^*) \bf h +\frac{1}{2} \bf h^\top {\bf H(x^*)} \bf h + R(\bf h)$$ +2. Since we're looking at a critical point, $\nabla f({\bf x}^*)=0$; and for small $\bf h$, $R(\bf h)$ is negligible. Rearranging, we get $$f({\bf x}^*+\bf h)-f({\bf x}^*)\approx \frac{1}{2} \bf h^\top {\bf H(x^*)} \bf h $$ +3. The Righthand side here is a quadratic form and we can determine the definiteness of $\bf H(x^*)$. From 55e282f86502432e62636231fccad0f324cf78f4 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 21:25:55 -0400 Subject: [PATCH 12/34] weird visual spacing --- 05_optimization.qmd | 120 ++++++++++++++++++++++---------------------- 1 file changed, 60 insertions(+), 60 deletions(-) diff --git a/05_optimization.qmd b/05_optimization.qmd index 22db493..b43f926 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -624,20 +624,20 @@ Example: Find the critical points for the following constrained optimization: $$ ``` 4. Consider all ways that the complementary slackness conditions are solved: - ```{=tex} - \begin{center} - \begin{tabular}{|l|cccc|c|} - \hline - Hypothesis & $s_1$ & $\lambda_1$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ - \hline - $s_1 = 0$ $\lambda_1 = 0$ & \multicolumn{4}{l|}{No solution} & \\ - $s_1 \neq 0$ $\lambda_1 = 0$ & 2 & 0 & 0 & 0 & 0\\ - $s_1 = 0$ $\lambda_1 \neq 0$ & 0 & $\frac{-16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ - $s_1 \neq 0$ $\lambda_1 \neq 0$ & \multicolumn{4}{l|}{No solution} &\\ - \hline - \end{tabular} - \end{center} - ``` +```{=tex} +\begin{center} +\begin{tabular}{|l|cccc|c|} +\hline +Hypothesis & $s_1$ & $\lambda_1$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ +\hline +$s_1 = 0$ $\lambda_1 = 0$ & \multicolumn{4}{l|}{No solution} & \\ +$s_1 \neq 0$ $\lambda_1 = 0$ & 2 & 0 & 0 & 0 & 0\\ +$s_1 = 0$ $\lambda_1 \neq 0$ & 0 & $\frac{-16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ +$s_1 \neq 0$ $\lambda_1 \neq 0$ & \multicolumn{4}{l|}{No solution} &\\ +\hline +\end{tabular} +\end{center} +``` This shows that there are two critical points: $(0,0)$ and $(\frac{8}{3},\frac{4}{3})$. @@ -670,24 +670,24 @@ $\frac{\partial L}{\partial x_1} = -2x_1 - \lambda_1 + \lambda_2 = 0$\\ $\frac 3. Consider all ways that the complementary slackness conditions are solved: - ```{=tex} - \begin{center} - \begin{tabular}{|l|cccccccc|c|} - \hline - Hypothesis & $s_1$ & $s_2$ & $s_3$ & $\lambda_1$ & $\lambda_2$ & $\lambda_3$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ - \hline - $s_1 = s_2 = s_3 = 0$ & \multicolumn{8}{l|}{No solution} & \\ - $s_1 \neq 0, s_2 = s_3 = 0$ & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ - $s_2 \neq 0, s_1 = s_3 = 0$ & 0 & 2 & 0 & -8 & 0 & -8 & 4 & 0 & -16\\ - $s_3 \neq 0, s_1 = s_2 = 0$ & 0 & 0 & 2 & -16 & -16 & 0 & 0 & 4 & -32\\ - $s_1 \neq 0, s_2 \neq 0, s_3 = 0$ &\multicolumn{8}{l|}{No solution} & \\ - $s_1 \neq 0, s_3 \neq 0, s_2 = 0$ &\multicolumn{8}{l|}{No solution} & \\ - $s_2 \neq 0, s_3 \neq 0, s_1 = 0$ &0 & $\sqrt{\frac{8}{3}}$ & $\sqrt{\frac{4}{3}}$ & $-\frac{16}{3}$ & 0 & 0 & $\frac{8}{3}$& $\frac{4}{3}$ & $-\frac{32}{3}$\\ - $s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{No solution}& \\ - \hline - \end{tabular} - \end{center} - ``` +```{=tex} +\begin{center} +\begin{tabular}{|l|cccccccc|c|} +\hline +Hypothesis & $s_1$ & $s_2$ & $s_3$ & $\lambda_1$ & $\lambda_2$ & $\lambda_3$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ +\hline +$s_1 = s_2 = s_3 = 0$ & \multicolumn{8}{l|}{No solution} & \\ +$s_1 \neq 0, s_2 = s_3 = 0$ & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ +$s_2 \neq 0, s_1 = s_3 = 0$ & 0 & 2 & 0 & -8 & 0 & -8 & 4 & 0 & -16\\ +$s_3 \neq 0, s_1 = s_2 = 0$ & 0 & 0 & 2 & -16 & -16 & 0 & 0 & 4 & -32\\ +$s_1 \neq 0, s_2 \neq 0, s_3 = 0$ &\multicolumn{8}{l|}{No solution} & \\ +$s_1 \neq 0, s_3 \neq 0, s_2 = 0$ &\multicolumn{8}{l|}{No solution} & \\ +$s_2 \neq 0, s_3 \neq 0, s_1 = 0$ &0 & $\sqrt{\frac{8}{3}}$ & $\sqrt{\frac{4}{3}}$ & $-\frac{16}{3}$ & 0 & 0 & $\frac{8}{3}$& $\frac{4}{3}$ & $-\frac{32}{3}$\\ +$s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{No solution}& \\ +\hline +\end{tabular} +\end{center} +``` This shows that there are four critical points: $(0,0)$, $(4,0)$, $(0,4)$, and $(\frac{8}{3},\frac{4}{3})$ @@ -789,20 +789,20 @@ x_2 & \geq 0\\ 3. Consider all border and interior cases: - ```{=tex} - \begin{center} - \begin{tabular}{|l|ccc|c|} - \hline - Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ - \hline - $x_1 = 0, x_2 = 0$ &0 & 0 & 0 & 0\\ - $x_1 = 0, x_2 \neq 0$ &-16 & 0 & 4 & -32\\ - $x_1 \neq 0, x_2 = 0$ &-8 & 4 & 0 & -16\\ - $x_1 \neq 0, x_2 \neq 0$ & $-\frac{16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ - \hline - \end{tabular} - \end{center} - ``` +```{=tex} +\begin{center} +\begin{tabular}{|l|ccc|c|} +\hline +Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ +\hline +$x_1 = 0, x_2 = 0$ &0 & 0 & 0 & 0\\ +$x_1 = 0, x_2 \neq 0$ &-16 & 0 & 4 & -32\\ +$x_1 \neq 0, x_2 = 0$ &-8 & 4 & 0 & -16\\ +$x_1 \neq 0, x_2 \neq 0$ & $-\frac{16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ +\hline +\end{tabular} +\end{center} +``` 4. Find Maximum: Three of the critical points violate the requirement that $\lambda \geq 0$, so the point $(0,0,0)$ is the maximum. @@ -837,20 +837,20 @@ $x_1 \geq 0$\\ $x_2 \geq 0$\\ \$\lambda \\geq \$0\\ 3. Consider all border and interior cases: - ```{=tex} - \begin{center} - \begin{tabular}{|l|ccc|c|} - \hline - Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ - \hline - $x_1 = 0, x_2 = 0$ &\multicolumn{3}{l|}{No Solution}& \\ - $x_1 = 0, x_2 \neq 0$ &\multicolumn{3}{l|}{No Solution}& \\ - $x_1 \neq 0, x_2 = 0$ & \multicolumn{3}{l|}{No Solution}& \\ - $x_1 \neq 0, x_2 \neq 0$ & & $\frac{4}{3}$ & $\frac{4}{3}$ & $\log\frac{7}{3}$\\ - \hline - \end{tabular} - \end{center} - ``` +```{=tex} +\begin{center} +\begin{tabular}{|l|ccc|c|} +\hline +Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ +\hline +$x_1 = 0, x_2 = 0$ &\multicolumn{3}{l|}{No Solution}& \\ +$x_1 = 0, x_2 \neq 0$ &\multicolumn{3}{l|}{No Solution}& \\ +$x_1 \neq 0, x_2 = 0$ & \multicolumn{3}{l|}{No Solution}& \\ +$x_1 \neq 0, x_2 \neq 0$ & & $\frac{4}{3}$ & $\frac{4}{3}$ & $\log\frac{7}{3}$\\ +\hline +\end{tabular} +\end{center} +``` 4. Find Maximum: From dbc3c4b44890a6a70de0b43a0dbcb27920949acb Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 21:29:47 -0400 Subject: [PATCH 13/34] more whitespace trimming --- 01_warmup.qmd | 10 ++- 02_functions.qmd | 83 ++++++++++------------- 03_limits.qmd | 19 +++--- 04_calculus.qmd | 36 ++++------ 05_optimization.qmd | 4 +- 06_probability.qmd | 47 ++++--------- 07_linear-algebra.qmd | 124 ++++++++++++++-------------------- 11_data-handling_counting.qmd | 4 +- 13_functions_obj_loops.qmd | 13 ++-- 16_simulation.qmd | 2 +- 17_non-wysiwyg.qmd | 2 +- 18_text.qmd | 6 +- 21_solutions-warmup.qmd | 11 ++- 13 files changed, 144 insertions(+), 217 deletions(-) diff --git a/01_warmup.qmd b/01_warmup.qmd index b7deff2..fdba91a 100644 --- a/01_warmup.qmd +++ b/01_warmup.qmd @@ -35,30 +35,29 @@ $${\bf A}=\begin{pmatrix} 4 & 1 & 5 \end{pmatrix}$$ - What is the dimensionality of matrix ${\bf A}$? What is the element $a_{23}$ of ${\bf A}$? Given that - + $${\bf B}=\begin{pmatrix} 1 & 2 & 8 \\ 3 & 9 & 11 \\ 4 & 7 & 5 \\ 5 & 1 & 9 \end{pmatrix}$$ - + What is ${\bf A}$ + ${\bf B}$? Given that - + $${\bf C}=\begin{pmatrix} 1 & 2 & 8 \\ 3 & 9 & 11 \\ 4 & 7 & 5 \\ \end{pmatrix}$$ - + What is ${\bf A}$ + ${\bf C}$? Given that @@ -138,7 +137,6 @@ For each of the followng functions $f(x)$, does a maximum and minimum exist in t 3. $f(x) = -(x - 2)^2$ If you are stuck, please try sketching out a picture of each of the functions. - ## Probability {.unnumbered} diff --git a/02_functions.qmd b/02_functions.qmd index ed2d20c..f3dc398 100644 --- a/02_functions.qmd +++ b/02_functions.qmd @@ -59,11 +59,11 @@ __Modulo:__ Tells you the remainder when you divide the first number by the seco ## Operators 1. $\sum\limits_{i=1}^{5} i =$ - + 2. $\prod\limits_{i=1}^{5} i =$ - + 3. $14 \mod 4 =$ - + 4. $4! =$ ::: @@ -71,7 +71,7 @@ __Modulo:__ Tells you the remainder when you divide the first number by the seco :::{#exr-operators1} ## Operators - + Let $x_1 = 4, x_2 = 3, x_3 = 7, x_4 = 11, x_5 = 2$ 1. $\sum\limits_{i=1}^{3} (7)x_i$ @@ -80,8 +80,6 @@ Let $x_1 = 4, x_2 = 3, x_3 = 7, x_4 = 11, x_5 = 2$ 3. $\prod\limits_{i=3}^{5} (2)x_i$ - - ::: ## Introduction to Functions @@ -107,7 +105,7 @@ Examples of mapping notation: Function of one variable: $f:{\bf R}^1\to{\bf R}^1$ - $f(x)=x+1$. For each $x$ in ${\bf R}^1$, $f(x)$ assigns the number $x+1$. - + Function of two variables: $f: {\bf R}^2\to{\bf R}^1$. - $f(x,y)=x^2+y^2$. For each ordered pair $(x,y)$ in ${\bf R}^2$, $f(x,y)$ assigns the number $x^2+y^2$. @@ -135,7 +133,6 @@ For each of the following, state whether they are one-to-one or many-to-one func 1. For $x \in [-3, \infty]$, $f: x \rightarrow x^2$. 2. For $x \in [0, \infty]$, $f: x \rightarrow \sqrt{x}$ - ::: @@ -145,22 +142,20 @@ Some functions are defined only on proper subsets of ${\bf R}^n$. - __Range__: elements of $Y$ assigned by $f(x)$ to elements of $X$, or $$f(X)=\{ y : y=f(x), x\in X\}$$ Most often used when talking about a function $f:{\bf R}^1\to{\bf R}^1$. - __Image__: same as range, but more often used when talking about a function $f:{\bf R}^n\to{\bf R}^1$. - Some General Types of Functions __Monomials__: $f(x)=a x^k$ $a$ is the coefficient. $k$ is the degree. - + Examples: $y=x^2$, $y=-\frac{1}{2}x^3$ - + __Polynomials__: sum of monomials. Examples: $y=-\frac{1}{2}x^3+x^2$, $y=3x+5$ The degree of a polynomial is the highest degree of its monomial terms. Also, it's often a good idea to write polynomials with terms in decreasing degree. - __Exponential Functions__: Example: $y=2^x$ @@ -170,37 +165,37 @@ __Relationship of logarithmic and exponential functions__: $$y=\log_a(x) \iff a^y=x$$ The log function can be thought of as an inverse for exponential functions. $a$ is referred to as the "base" of the logarithm. - + __Common Bases__: The two most common logarithms are base 10 and base $e$. - + 1. Base 10: $\quad y=\log_{10}(x) \iff 10^y=x$. The base 10 logarithm is often simply written as "$\log(x)$" with no base denoted. 2. Base $e$: $\quad y=\log_e(x) \iff e^y=x$. The base $e$ logarithm is referred to as the "natural" logarithm and is written as ``$\ln(x)$". \begin{comment} {\includegraphics[width=1in, angle = 270]{ln.eps}} \, {\includegraphics[width=1in, angle = 270]{exp.eps}} \end{comment} - + __Properties of exponential functions:__ - + - $a^x a^y = a^{x+y}$ - $a^{-x} = 1/a^x$ - $a^x/a^y = a^{x-y}$ - $(a^x)^y = a^{x y}$ - $a^0 = 1$ - + __Properties of logarithmic functions__ (any base): Generally, when statisticians or social scientists write $\log(x)$ they mean $\log_e(x)$. In other words: $\log_e(x) \equiv \ln(x) \equiv \log(x)$ - + $$\log_a(a^x)=x$$ and $$a^{\log_a(x)}=x$$ - + - $\log(x y)=\log(x)+\log(y)$ - $\log(x^y)=y\log(x)$ - $\log(1/x)=\log(x^{-1})=-\log(x)$ - $\log(x/y)=\log(x\cdot y^{-1})=\log(x)+\log(y^{-1})=\log(x)-\log(y)$ - $\log(1)=\log(e^0)=0$ - + __Change of Base Formula__: Use the change of base formula to switch bases as necessary: $$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$$ @@ -213,9 +208,9 @@ You can use logs to go between sum and product notation. This will be particular &=& \log(x_1) + \log(x_2) + \log(x_3) + \cdots + \log(x_n)\\ &=& \sum\limits_{i=1}^n \log (x_i) \end{eqnarray*} - + Therefore, you can see that the log of a product is equal to the sum of the logs. We can write this more generally by adding in a constant, $c$: - + \begin{eqnarray*} \log \bigg(\prod\limits_{i=1}^n c x_i\bigg) &=& \log(cx_1 \cdot cx_2 \cdots cx_n)\\ &=& \log(c^n \cdot x_1 \cdot x_2 \cdots x_n)\\ @@ -232,7 +227,7 @@ Evaluate each of the following logarithms 1. $\log_4(16)$ 2. $\log_2(16)$ - + Simplify the following logarithm. By "simplify", we actually really mean - use as many of the logarithmic properties as you can. 3. $\log_4(x^3y^5)$ @@ -246,13 +241,12 @@ Simplify the following logarithm. By "simplify", we actually really mean - use a Evaluate each of the following logarithms 1. $\log_\frac{3}{2}(\frac{27}{8})$ - + Simplify each of the following logarithms. By "simplify", we actually really mean - use as many of the logarithmic properties as you can. 2. $\log(\frac{x^9y^5}{z^3})$ 3. $\ln{\sqrt{xy}}$ - ::: @@ -269,7 +263,7 @@ What can a graph tell you about a function? - Other questions related to the substance of the problem at hand. ## Solving for Variables and Finding Roots - + Sometimes we're given a function $y=f(x)$ and we want to find how $x$ varies as a function of $y$. Use algebra to move $x$ to the left hand side (LHS) of the equation and so that the right hand side (RHS) is only a function of $y$. :::{#exm-solvevar} @@ -279,10 +273,8 @@ Sometimes we're given a function $y=f(x)$ and we want to find how $x$ varies as Solve for x: 1. $y=3x+2$ - -2. $y=e^x$ - +2. $y=e^x$ ::: @@ -294,20 +286,18 @@ Multiple Roots: $$f(x)=x^2 - 9 \quad\Longrightarrow\quad 0=x^2 - 9 \quad\Longrightarrow\quad 9=x^2 \quad\Longrightarrow\quad \pm \sqrt{9}=\sqrt{x^2} \quad\Longrightarrow\quad \pm 3=x$$ __Quadratic Formula:__ For quadratic equations $ax^2+bx+c=0$, use the quadratic formula: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ - :::{#exr-solvevar1} ## Finding Roots Solve for x: - + 1. $f(x)=3x+2 = 0$ - + 2. $f(x)=x^2+3x-4=0$ 3. $f(x)=e^{-x}-10 = 0$ - ::: @@ -317,17 +307,17 @@ __Interior Point__: The point $\bf x$ is an interior point of the set $S$ if $\b - If the set $S$ is circular, the interior points are everything inside of the circle, but not on the circle's rim. - Example: The interior of the set $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2< 4 \}$ . - + __Boundary Point__: The point $\bf x$ is a boundary point of the set $S$ if every $\epsilon$-ball around $\bf x$ contains both points that are in $S$ and points that are outside $S$. The __boundary__ is the collection of all boundary points. - If the set $S$ is circular, the boundary points are everything on the circle's rim. - Example: The boundary of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 = 4 \}$. - + __Open__: A set $S$ is open if for each point $\bf x$ in $S$, there exists an open $\epsilon$-ball around $\bf x$ completely contained in $S$. - If the set $S$ is circular and open, the points contained within the set get infinitely close to the circle's rim, but do not touch it. - Example: $\{ (x,y) : x^2+y^2<4 \}$ - + __Closed__: A set $S$ is closed if it contains all of its boundary points. - Alternatively: A set is closed if its complement is open. @@ -335,12 +325,11 @@ __Closed__: A set $S$ is closed if it contains all of its boundary points. - Example: $\{ (x,y) : x^2+y^2\le 4 \}$ - Note: a set may be neither open nor closed. Example: $\{ (x,y) : 2 < x^2+y^2\le 4 \}$ - __Complement__: The complement of set $S$ is everything outside of $S$. - If the set $S$ is circular, the complement of $S$ is everything outside of the circle. - Example: The complement of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 > 4 \}$. - + __Empty__: The empty (or null) set is a unique set that has no elements, denoted by \{\} or $\emptyset$. - The empty set is an example of a set that is open and closed, or a "clopen" set. @@ -349,23 +338,23 @@ __Empty__: The empty (or null) set is a unique set that has no elements, denoted ## Answers to Examples and Exercises {.unnumbered} Answer to Example @exm-operators: - + 1. 1 + 2 + 3 + 4 + 5 = 15 - + 2. 1 * 2 * 3 * 4 * 5 = 120 - + 3. 2 - + 4. 4 * 3 * 2 * 1 = 24 - + Answer to Exercise @exr-operators1: - + 1. 7(4 + 3 + 7) = 98 - + 2. 2 + 2 + 2 + 2 + 2 = 10 - + 3. $2^3(7)(11)(2)$ = 1232 - + Answer to Example @exm-functions: 1. one-to-one diff --git a/03_limits.qmd b/03_limits.qmd index 41e8b69..74cdea6 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -39,9 +39,9 @@ A finding that perhaps rivals the Central Limit Theorem is the Law of Large Numb ## (Weak) Law of Large Numbers For any draw of identically distributed independent variables with mean $\mu$, the sample average after $n$ draws, $\bar{X}_n$, converges in probability to the true mean as $n \rightarrow \infty$: - + $$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ - + A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". ::: @@ -74,14 +74,13 @@ We need a couple of steps until we get to limit theorems in probability. First w A **sequence** $$\{x_n\}=\{x_1, x_2, x_3, \ldots, x_n\}$$ is an ordered set of real numbers, where $x_1$ is the first term in the sequence and $y_n$ is the $n$th term. Generally, a sequence is infinite, that is it extends to $n=\infty$. We can also write the sequence as $$\{x_n\}^\infty_{n=1}$$ where the subscript and superscript are read together as "from 1 to infinity." - :::{#exm-seqbehav} ## Sequences How do these sequences behave? - + 1. $\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\}$ 2. $\{B_n\}=\left\{\frac{n^2+1}{n} \right\}$ 3. $\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\}$ @@ -148,7 +147,7 @@ $$\lim_{n\to \infty} \frac{n + 3}{n},$$ :::{#sol-simplfrac} At first glance, $n + 3$ and $n$ both grow to $\infty$, so it looks like we need to divide infinity by infinity. However, we can express this fraction as a sum, then the limits apply separately: - + $$\lim_{n\to \infty} \frac{n + 3}{n} = \lim_{n\to \infty} \left(1 + \frac{3}{n}\right) = \underbrace{\lim_{n\to \infty}1}_{1} + \underbrace{\lim_{n\to \infty}\left(\frac{3}{n}\right)}_{0}$$ so, the limit is actually 1. @@ -170,9 +169,9 @@ Find the following limits of sequences, then explain in English the intuition fo We've now covered functions and just covered limits of sequences, so now is the time to combine the two. A function $f$ is a compact representation of some behavior we care about. Like for sequences, we often want to know if $f(x)$ approaches some number $L$ as its independent variable $x$ moves to some number $c$ (which is usually 0 or $\pm\infty$). If it does, we say that the limit of $f(x)$, as $x$ approaches $c$, is $L$: $\lim\limits_{x \to c} f(x)=L$. Unlike a sequence, $x$ is a continuous number, and we can move in decreasing order as well as increasing. - + For a limit $L$ to exist, the function $f(x)$ must approach $L$ from both the left (increasing) and the right (decreasing). - + :::{#def-limfunc} ## Limit of a function @@ -228,8 +227,6 @@ Solve the following limits of functions ::: - - So there are a few more alternatives about what a limit of a function could be: 1. Right-hand limit: The value approached by $f(x)$ when you move from right to left. @@ -280,7 +277,7 @@ For each function, determine if it is continuous or discontinuous. 2. $f(x) = e^x$ 3. $f(x) = 1 + \frac{1}{x^2}$ 4. $f(x) = \text{floor}(x)$. - + The floor is the smaller of the two integers bounding a number. So $\text{floor}(x = 2.999) = 2$, $\text{floor}(x = 2.0001) = 2$, and $\text{floor}(x = 2) = 2.$ ::: @@ -345,7 +342,7 @@ Example @exm-seqbehav 1. $\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\} = \left\{1, \frac{7}{4}, \frac{17}{9}, \frac{31}{16}, \frac{49}{25}, \ldots\right\} = 2$ 2. $\{B_n\}=\left\{\frac{n^2+1}{n} \right\} = \left\{2, \frac{5}{2}, \frac{10}{3}, \frac{17}{4}..., \right\}$ 3. $\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\} = \left\{0, \frac{1}{2}, -\frac{2}{3}, \frac{3}{4}, -\frac{4}{5}\right\}$ - + ::: Exercise @exr-limseq2 diff --git a/04_calculus.qmd b/04_calculus.qmd index b5e43bd..08716b5 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -39,7 +39,7 @@ Let $f$ be a function whose domain includes an open interval containing the poin $$\frac{d}{dx}f(x) =\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x} = \lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ There are a two main ways to denote a derivative: - + * Leibniz Notation: $\frac{d}{dx}(f(x))$ * Prime or Lagrange Notation: $f'(x)$ @@ -68,7 +68,7 @@ gx <- fx0 + stat_function(fun = function(x) x^3, size = 0.5) + labs(y = expression(g(x)), title = expression(g(x)==x^3)) + expand_limits(y = 0) - + gprimex <- fx0 + stat_function(fun = function(x) 3*(x^2), size = 0.5, linetype = "dashed") + labs(x = expression(x), y = expression(g~plain("'")~(x))) @@ -148,7 +148,6 @@ For each of the following functions, find the first-order derivative $f^\prime(x 1. $f(x) = (x^2 + 1)(x^3 - 1)$ 1. $f(x) = 3x^2 + 2x^{1/3}$ 1. $f(x)=\frac{x^2+1}{x^2-1}$ - ::: @@ -206,10 +205,9 @@ We can read this as: "the derivative of the composite function $y$ is the deriva The chain rule can be thought of as the derivative of the "outside" times the derivative of the "inside", remembering that the derivative of the outside function is evaluated at the value of the inside function. * The chain rule can also be written as $$\frac{dy}{dx}=\frac{dy}{dg(x)} \frac{dg(x)}{dx}$$ This expression does not imply that the $dg(x)$'s cancel out, as in fractions. They are part of the derivative notation and you can't separate them out or cancel them.) - :::{#exm-tothesix} - + ## Composite Exponent Find $f^\prime(x)$ for $f(x) = (3x^2+5x-7)^6$. @@ -393,7 +391,6 @@ Let $f(x,y)=x^3 y^4 +e^x -\log y$. What are the following partial derivaitves? \frac{\partial^2 f}{\partial x^2}(x,y) &=\\ \frac{\partial^2 f}{\partial x \partial y}(x,y) &= \end{align*} - ::: @@ -402,7 +399,6 @@ Let $f(x,y)=x^3 y^4 +e^x -\log y$. What are the following partial derivaitves? A common form of approximation used in statistics involves derivatives. A Taylor series is a way to represent common functions as infinite series (a sum of infinite elements) of the function's derivatives at some point $a$. For example, Taylor series are very helpful in representing nonlinear (read: difficult) functions as linear (read: manageable) functions. One can thus __approximate__ functions by using lower-order, finite series known as __Taylor polynomials__. If $a=0$, the series is called a Maclaurin series. - Specifically, a Taylor series of a real or complex function $f(x)$ that is infinitely differentiable in the neighborhood of point $a$ is: @@ -410,7 +406,7 @@ Specifically, a Taylor series of a real or complex function $f(x)$ that is infi f(x) &= f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots\\ &= \sum_{n=0}^\infty \frac{f^{(n)} (a)}{n!} (x-a)^n \end{align*} - + __Taylor Approximation__: We can often approximate the curvature of a function $f(x)$ at point $a$ using a 2nd order Taylor polynomial around point $a$: $$f(x) = f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 @@ -442,10 +438,9 @@ This definition bolsters the main takeaway about integrals and derivatives: They :::{#exr-antiderivatives} ## Antiderivative Find the antiderivative of the following: - + 1. $f(x) = \frac{1}{x^2}$ 2. $f(x) = 3e^{3x}$ - ::: @@ -465,7 +460,7 @@ and is equal to the antiderivative of $f$. Draw the function $f(x)$ and its indefinite integral, $\int\limits f(x) dx$ $$f(x) = (x^2-4)$$ - + ::: :::{#sol-integralc} @@ -493,7 +488,6 @@ Fx <- ggplot(range1, aes(x = x)) + fx + Fx + plot_layout(ncol = 1) ``` - Notice from these examples that while there is only a single derivative for any function, there are multiple antiderivatives: one for any arbitrary constant $c$. $c$ just shifts the curve up or down on the $y$-axis. If more information is present about the antiderivative --- e.g., that it passes through a particular point --- then we can solve for a specific value of $c$. ### Common Rules of Integration {.unnumbered} @@ -512,7 +506,7 @@ Some common rules of integrals follow by virtue of being the inverse of a deriva :::{#exm-common-integration} ## Common Integration Simplify the following indefinite integrals: - + * $\int 3x^2 dx$ * $\int (2x+1)dx$ * $\int e^x e^{e^x} dx$ @@ -562,7 +556,7 @@ This is how we define the "Definite" Integral: :::{#def-defintegral} ## The Definite Integral (Riemann) If for a given function $f$ the Riemann sum approaches a limit as $\Delta x \to 0$, then that limit is called the Riemann integral of $f$ from $a$ to $b$. We express this with the $\int$, symbol, and write $$\int\limits_a^b f(x) dx= \lim\limits_{\Delta x\to 0} \sum\limits_{i=1}^n f(x_i)\Delta x$$ - + The most straightforward of a definite integral is the definite integral. That is, we read $$\int\limits_a^b f(x) dx$$ as the definite integral of $f$ from $a$ to $b$ and we defined as the area under the "curve" $f(x)$ from point $x=a$ to $x=b$. ::: @@ -622,7 +616,7 @@ Simplify the following definite intergrals. 1. $\int\limits_1^1 3x^2 dx =$ 2. $\int\limits_0^4 (2x+1)dx=$ 3. $\int\limits_{-2}^0 e^x e^{e^x} dx + \int\limits_0^2 e^x e^{e^x} dx =$ - + ::: ## Integration by Substitution @@ -646,7 +640,7 @@ Substitution can also be used to calculate a definite integral. Using the same p where $c=u(a)$ and $d=u(b)$. :::{#exm-intsub1} - + ## Integration by Substitution I Solve the indefinite integral $$\int x^2 \sqrt{x+1}dx.$$ @@ -673,13 +667,12 @@ $$\int\limits_a^b u\frac{dv}{dx}dx = \left. u v \right|_a^b - \int\limits_a^b v Our goal here is to find expressions for $u$ and $dv$ that, when substituted into the above equation, yield an expression that's more easily evaluated. - :::{#exm-intbyparts} ## Integration by Parts Simplify the following integrals. These seemingly obscure forms of integrals come up often when integrating distributions. $$\int x e^{ax} dx$$ - + ::: :::{#sol-intbyparts} @@ -740,7 +733,7 @@ Example @exm-exmderivexp 1. Let $u(x)=-3x$. Then $u^\prime(x)=-3$ and $f^\prime(x)=-3e^{-3x}$. 2. Let $u(x)=x^2$. Then $u^\prime(x)=2x$ and $f^\prime(x)=2xe^{x^2}$. - + ::: Example @exm-exmderivlog @@ -752,14 +745,13 @@ Example @exm-exmderivlog 3. Use the generalized power rule. $$\frac{dy}{dx} = \frac{(2 \log x)}{x}$$ 4. We know that $\log e^x=x$ and that $dx/dx=1$, but we can double check. Let $u(x)=e^x$. Then $u^\prime(x)=e^x$ and $\frac{dy}{dx} = \frac{u^\prime(x)}{u(x)} = \frac{e^x}{e^x} = 1.$ - ::: Example @exm-defintmon :::{#sol-defintmon} What is $F(x)$? From the power rule, recognize $\frac{d}{dx}x^3 = 3x^2$ so - + \begin{align*} F(x) &= x^3\\ \int\limits_1^3 f(x) dx &= F(x = 3) - F(x - 1)\\ @@ -822,7 +814,7 @@ For a given $n$, we would repeat the integration by parts procedure until the in 2. $$\int x^3 e^{-x^2} dx$$ We could, as before, choose $u=x^3$ and $dv=e^{-x^2}dx$. But we can't then find $v$ --- i.e., integrating $e^{-x^2}dx$ isn't possible. Instead, notice that $$\frac{d}{dx}e^{-x^2} = -2xe^{-x^2},$$ which can be factored out of the original integrand $$\int x^3 e^{-x^2} dx = \int x^2 (xe^{-x^2})dx.$$ - + We can then let $u=x^2$ and $dv=x e^{-x^2}dx$. The$du=2x dx$ and $v=-\frac{1}{2}e^{-x^2}$. Substituting these in, we have \begin{eqnarray} \int x^3 e^{-x^2} dx &=& u v - \int v du\nonumber\\ diff --git a/05_optimization.qmd b/05_optimization.qmd index b43f926..959e3d4 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -442,7 +442,7 @@ Given $f(\mathbf{x})=x_1^3-x_2^3+9x_1x_2$, find any maxima or minima. \Rightarrow \quad x_1 = 3$$ $$3(3)^2 + 9x_2 = 0 \quad \Rightarrow \quad x_2 = -3$$ \end{enumerate} - + \item Second order conditions. \begin{enumerate} \item Hessian $\mathbf{H(x)} = $ @@ -465,7 +465,7 @@ Given $f(\mathbf{x})=x_1^3-x_2^3+9x_1x_2$, find any maxima or minima. \item Maxima, Minima, or Saddle Point for $\mathbf{x}_2^*$?\\ Since $\mathbf{H(x_2^*)}$ is positive definite, $\mathbf{x}_1^*=(3,-3)$ is a strict local minimum\\ \end{enumerate} - + \item Global concavity/convexity. \begin{enumerate} \item Is f(x) globally concave/convex?\\ diff --git a/06_probability.qmd b/06_probability.qmd index e3f0e75..109c454 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -2,13 +2,11 @@ Probability and Inferences are mirror images of each other, and both are integral to social science. Probability quantifies uncertainty, which is important because many things in the social world are at first uncertain. Inference is then the study of how to learn about facts you don't observe from facts you do observe. - ```{r} #| include: false library(ggplot2) options(knitr.graphics.auto_pdf = TRUE) # use pdf for images ``` - ## Counting rules @@ -68,8 +66,6 @@ There are five balls numbered from 1 through 5 in a jar. Three balls are chosen. Four cards are selected from a deck of 52 cards. Once a card has been drawn, it is not reshuffled back into the deck. Moreover, we care only about the complete hand that we get (i.e. we care about the set of selected cards, not the sequence in which it was drawn). How many possible outcomes are there? - - ``` ## Sets {#setoper} @@ -115,7 +111,6 @@ Write out the following sets: 2. $C \cap B$ 3. $B^c$ 4. $A \cap (B \cup C)$ - ``` @@ -127,14 +122,11 @@ Suppose you had a pair of four-sided dice. You sum the results from a single tos What is the set of possible outcomes (i.e. the sample space)? - Consider subsets A {2, 8} and B {2,3,7} of the sample space you found. What is 1. $A^c$ 2. $(A \cup B)^c$ - - - + ``` ## Probability {#probdef} @@ -173,7 +165,7 @@ The axioms of probability make sure that the separate events add up in terms of A_i\right)=\sum\limits_{i=1}^k P(A_i)$$ The last axiom is an extension of a union to infinite sets. When there are only two events in the space, it boils down to: - + \begin{align*} P(A_1 \cup A_2) = P(A_1) + P(A_2) \quad\text{for disjoint } A_1, A_2 \end{align*} @@ -219,9 +211,7 @@ Suppose you had a pair of four-sided dice. You sum the results from a single tos 1. What is $P(X = 5)$, $P(X = 3)$, $P(X = 6)$? - 2. What is $P(X=5 \cup X = 3)^C$? - ``` @@ -243,7 +233,6 @@ Assume $A$ and $B$ occur with the following frequencies: $\quad$ --------- --------- --------- $B$ $n_{ab}$ $n_{a^cb}$ $B^C$ $n_{ab^c}$ $n_{(ab)^c}$ - and let $n_{ab}+n_{a^Cb}+n_{ab^C}+n_{(ab)^C}=N$. Then @@ -260,7 +249,7 @@ and let $n_{ab}+n_{a^Cb}+n_{ab^C}+n_{(ab)^C}=N$. Then #| label: condprobexm2 A six-sided die is rolled. What is the probability of a 1, given the outcome is an odd number? - + ``` You could rearrange the fraction to highlight how a joint probability could be expressed as the product of a conditional probability. @@ -313,7 +302,6 @@ In a given town, 40% of the voters are Democrat and 60% are Republican. The pres #| label: condprobexr Assume that 2% of the population of the U.S. are members of some extremist militia group. We develop a survey that positively classifies someone as being a member of a militia group given that they are a member 95% of the time and negatively classifies someone as not being a member of a militia group given that they are not a member 97% of the time. What is the probability that someone positively classified as being a member of a militia group is actually a militia member? - ``` @@ -347,7 +335,6 @@ No. If A and B are mutually exclusive, then they cannot happen simultaneously. I Just because two events are conditionally independent does not mean that they are independent. Actually it is hard to think of real-world things that are "unconditionally" independent. That's why it's always important to ask about a finding: What was it conditioned on? For example, suppose that a graduate school admission decisions are done by only one professor, who picks a group of 50 bright students and flips a coin for each student to generate a class of about 25 students. Then the the probability that two students get accepted are conditionally independent, because they are determined by two separate coin tosses. However, this does not mean that their admittance is not completely independent. Knowing that student $A$ got in gives us information about whether student $B$ got in, if we think that the professor originally picked her pool of 50 students by merit. Perhaps more counter-intuitively: If two events are already independent, then it might seem that no amount of "conditioning" will make them dependent. But this is not always so. For example^[Example taken from Blitzstein and Hwang, Example 2.5.10], suppose I only get a call from two people, Alice and Bob. Let $A$ be the event that Alice calls, and $B$ be the event that Bob calls. Alice and Bob do not communicate, so $P(A \mid B) = P(A).$ But now let $C$ be the event that your phone rings. For conditional independence to hold here, then $P(A \mid C)$ must be equal to $P(A \mid B \cap C).$ But this is not true -- $A \mid C$ may or may not be true, but $P(A \mid B \cap C)$ certainly is true. - ## Random Variables @@ -448,12 +435,11 @@ Note that $P(X > x) = 1 - P(X \le x)$. ```{example} For a fair die with its value as $Y$, What are the following? - + 1. $P(Y\le 1)$ 2. $P(Y\le 3)$ 3. $P(Y\le 6)$ - ``` ### Continuous Random Variables {.unnumbered} @@ -545,8 +531,6 @@ We can then make statements about the joint distribution of $X$ and $Y$. What ar 1. $P(X=x|Y=y)$ 1. Are X and Y independent? - - ``` ## Expectation @@ -565,7 +549,6 @@ In words, it is the weighted average of all possible values of $Y$, weighted by #| label: expectdiscrete What is the expectation of a fair, six-sided die? - ``` @@ -623,10 +606,9 @@ We can also look at other summaries of the distribution, which build on the idea ```{definition} #| name: Variance The Variance of a Random Variable $Y$ is - + $$\text{Var}(Y) = E[(Y - E(Y))^2] = E(Y^2)-[E(Y)]^2$$ - The Standard Deviation is the square root of the variance : $$SD(Y) = \sigma_Y= \sqrt{\text{Var}(Y)}$$ ``` @@ -643,10 +625,8 @@ $$f(x) = \begin{cases} $$ What is $\text{Var}(x)$? - -__Hint:__ First calculate $E(X)$ and $E(X^2)$ - +__Hint:__ First calculate $E(X)$ and $E(X^2)$ ``` @@ -701,7 +681,7 @@ Suppose we have a PMF with the following characteristics: P(X = 1) = \frac{1}{15}\\ P(X = 2) = \frac{11}{30} \end{eqnarray*} - + 1. Calculate the expected value of X Define the random variable $Y = X^2$. @@ -732,7 +712,7 @@ $$f(x) = \begin{cases} #| label: expvar3 1. Find the mean and standard deviation of random variable X. The PDF of this X is as follows: - + $$f(x) = \begin{cases} \frac{1}{4}x \quad 0 \leq x \leq 2\\ \frac{1}{4}(4 - x) \quad 2 \leq x \leq 4\\ @@ -741,7 +721,6 @@ $$f(x) = \begin{cases} $$ 2. Next, calculate $P(X < \mu - \sigma)$ Remember, $\mu$ is the mean and $\sigma$ is the standard deviation - ``` @@ -776,7 +755,6 @@ $$P(Y = y)=\frac{\lambda^y}{y!}e^{-\lambda}, \quad y=0,1,2,\ldots, \quad \lambda The Poisson has the unusual feature that its expectation equals its variance: $E(Y)=\text{Var}(Y)=\lambda$. The Poisson distribution is often used to model rare event counts: counts of the number of events that occur during some unit of time. $\lambda$ is often called the "arrival rate." ``` - ```{example} Border disputes occur between two countries through a Poisson Distribution, at a rate of 2 per month. What is the probability of 0, 2, and less than 5 disputes occurring in a month? @@ -889,7 +867,7 @@ __Covariance and Correlation__: Both of these quantities measure the degree to w ```{example} Example: Using the above table, calculate the sample versions of: - + 1. $\text{Cov}(X,Y)$ 2. $\text{Corr}(X, Y)$ @@ -921,16 +899,16 @@ as $n\to \infty$, where $$\Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-\f This result means that, as $n$ grows, the distribution of the sample mean $\bar X_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n)$ is approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt n}$, i.e., $$\bar{X}_n \approx \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).$$ The standard deviation of $\bar X_n$ (which is roughly a measure of the precision of $\bar X_n$ as an estimator of $\mu$) decreases at the rate $1/\sqrt{n}$, so, for example, to increase its precision by $10$ (i.e., to get one more digit right), one needs to collect $10^2=100$ times more units of data. - + Intuitively, this result also justifies that whenever a lot of small, independent processes somehow combine together to form the realized observations, practitioners often feel comfortable assuming Normality. ```{theorem} #| label: lln #| name: Law of Large Numbers (LLN) For any draw of independent random variables with the same mean $\mu$, the sample average after $n$ draws, $\bar{X}_n = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)$, converges in probability to the expected value of $X$, $\mu$ as $n \rightarrow \infty$: - + $$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ - + A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". ``` @@ -1006,7 +984,6 @@ Answer to Exercise \@ref(exr:prob1): 1. $P(X = 5) = \frac{4}{16}$, $P(X = 3) = \frac{2}{16}$, $P(X = 6) = \frac{3}{16}$ - 2. What is $P(X=5 \cup X = 3)^C = \frac{10}{16}$? Answer to Example \@ref(exm:condprobexm1): diff --git a/07_linear-algebra.qmd b/07_linear-algebra.qmd index b91a370..6370b07 100644 --- a/07_linear-algebra.qmd +++ b/07_linear-algebra.qmd @@ -16,7 +16,7 @@ $\bullet$ Inverse of Larger Matrices ## Working with Vectors {#vector-def} __Vector__: A vector in $n$-space is an ordered list of $n$ numbers. These numbers can be represented as either a row vector or a column vector: $$ {\bf v} \begin{pmatrix} v_1 & v_2 & \dots & v_n\end{pmatrix} , {\bf v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$$ - + We can also think of a vector as defining a point in $n$-dimensional space, usually ${\bf R}^n$; each element of the vector defines the coordinate of the point in a particular direction. __Vector Addition and Subtraction__: If two vectors, ${\bf u}$ and ${\bf v}$, have the same length (i.e. have the same number of elements), they can be added (subtracted) together: @@ -43,8 +43,6 @@ Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{p 2. $a \cdot b$ - - ``` ```{exercise} @@ -52,15 +50,14 @@ Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{p #| label: vectors1 Let $u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}$, $v = \begin{pmatrix} 9&-3&2&8 \end{pmatrix}$, $w = \begin{pmatrix} 1&13& -7&2 &15 \end{pmatrix}$, and $c = 2$. Calculate the following: - + 1. $u-v$ - + 2. $cw$ - + 3. $u \cdot v$ - + 4. $w \cdot v$ - ``` @@ -74,11 +71,11 @@ For example, $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination __Linear independence__: A set of vectors ${\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k$ is linearly independent if the only solution to the equation $$c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k = 0$$ is $c_1 = c_2 = \cdots = c_k = 0$. If another solution exists, the set of vectors is linearly dependent. - + A set $S$ of vectors is linearly dependent if and only if at least one of the vectors in $S$ can be written as a linear combination of the other vectors in $S$. Linear independence is only defined for sets of vectors with the same number of elements; any linearly independent set of vectors in $n$-space contains at most $n$ vectors. - + Since $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$, these 4 vectors constitute a linearly dependent set. ```{example} @@ -86,11 +83,10 @@ A set $S$ of vectors is linearly dependent if and only if at least one of the ve #| label: linearindep Are the following sets of vectors linearly independent? - + 1. $\begin{pmatrix}2 & 3 & 1 \end{pmatrix}$ and $\begin{pmatrix}4 & 6 & 1 \end{pmatrix}$ 2. $\begin{pmatrix}1 & 0 & 0 \end{pmatrix}$, $\begin{pmatrix}0 & 5 & 0 \end{pmatrix}$, and $\begin{pmatrix}10 & 10 & 0 \end{pmatrix}$ - ``` ```{exercise} @@ -98,9 +94,9 @@ Are the following sets of vectors linearly independent? #| label: linearindep1 Are the following sets of vectors linearly independent? - + 1. $${\bf v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} $$ - + 2. $${\bf v}_1 = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} -4 \\ 6 \\ 5 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} -2 \\ 8 \\ 6 \end{pmatrix} $$ ``` @@ -115,12 +111,12 @@ $${\bf A}=\begin{pmatrix} \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$$ - + Note that you can think of vectors as special cases of matrices; a column vector of length $k$ is a $k \times 1$ matrix, while a row vector of the same length is a $1 \times k$ matrix. It's also useful to think of matrices as being made up of a collection of row or column vectors. For example, $$\bf A = \begin{pmatrix} {\bf a}_1 & {\bf a}_2 & \cdots & {\bf a}_m \end{pmatrix}$$ - + __Matrix Addition__: Let $\bf A$ and $\bf B$ be two $m\times n$ matrices. $${\bf A+B}=\begin{pmatrix} a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ @@ -130,15 +126,13 @@ __Matrix Addition__: Let $\bf A$ and $\bf B$ be two $m\times n$ matrices. \end{pmatrix}$$ Note that matrices ${\bf A}$ and ${\bf B}$ must have the same dimensionality, in which case they are __conformable for addition__. - + ```{example} #| label: matrixaddition $${\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & 2 \end{pmatrix}$$ ${\bf A+B}=$ - - ``` __Scalar Multiplication__: Given the scalar $s$, the scalar multiplication of $s {\bf A}$ is @@ -159,10 +153,8 @@ __Scalar Multiplication__: Given the scalar $s$, the scalar multiplication of $ #| label: scalarmulti $s=2, \qquad {\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ - -$s {\bf A} =$ - +$s {\bf A} =$ ``` @@ -174,15 +166,15 @@ __Matrix Multiplication__: If ${\bf A}$ is an $m\times k$ matrix and $\bf B$ is 1. $\begin{pmatrix} a&b\\c&d\\e&f \end{pmatrix} \begin{pmatrix} A&B\\C&D \end{pmatrix} =$ - + 2. $\begin{pmatrix} 1&2&-1\\3&1&4 \end{pmatrix} \begin{pmatrix} -2&5\\4&-3\\2&1\end{pmatrix} =$ ``` Note that the number of columns of the first matrix must equal the number of rows of the second matrix, in which case they are __conformable for multiplication__. The sizes of the matrices (including the resulting product) must be $$(m\times k)(k\times n)=(m\times n)$$ - + Also note that if __AB__ exists, __BA__ exists only if $\dim({\bf A}) = m \times n$ and $\dim({\bf B}) = n \times m$. - + This does not mean that __AB__ = __BA__. __AB__ = __BA__ is true only in special circumstances, like when ${\bf A}$ or ${\bf B}$ is an identity matrix or ${\bf A} = {\bf B}^{-1}$. __Laws of Matrix Algebra__: @@ -193,7 +185,7 @@ __Laws of Matrix Algebra__: \item \parbox[t]{1.5in}{Distributive:} $\bf A(B+C)=AB+AC$\\ \parbox[t]{1.5in}{\quad} $\bf (A+B)C=AC+BC$ \end{enumerate} - + Commutative law for multiplication does not hold -- the order of multiplication matters: $$\bf AB\ne BA$$ @@ -229,26 +221,23 @@ Example of $({\bf AB})^T = {\bf B}^T{\bf A}^T$: Let $$A = \begin{pmatrix} 2&0&-1&1\\1&2&0&1 \end{pmatrix}$$ $$B = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix} $$ - $$C = \begin{pmatrix} 3&2&-1\\0&4&6 \end{pmatrix}$$ Calculate the following: - + 1. $$AB$$ 2. $$BA$$ 3. $$(BC)^T$$ 4. $$BC^T$$ ``` - ## Systems of Linear Equations __Linear Equation__: $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$ $a_i$ are parameters or coefficients. $x_i$ are variables or unknowns. - Linear because only one variable per term and degree is at most 1. @@ -258,7 +247,7 @@ $$\begin{matrix} x & - & 3y & = & -3\\ 2x & + & y & = & 8 \end{matrix}$$ - + \begin{comment} \parbox[t]{1in}{\includegraphics[angle=270, width = 1in]{linsys.eps}} \end{comment} @@ -341,14 +330,13 @@ __Augmented Matrix__: When we append $\bf b$ to the coefficient matrix $\bf A$, #| label: augmatrix Create an augmented matrix that represent the following system of equations: - + $$2x_1 -7x_2 + 9x_3 -4x_4 = 8$$ $$41x_2 + 9x_3 -5x_6 = 11$$ $$x_1 -15x_2 -11x_5 = 9$$ ``` - ## Finding Solutions to Augmented Matrices and Systems of Equations __Row Echelon Form__: Our goal is to translate our augmented matrix or system of equations into row echelon form. This will provide us with the values of the vector __x__ which solve the system. We use the row operations to change coefficients in the lower triangle of the augmented matrix to 0. An augmented matrix of the form @@ -374,7 +362,7 @@ $$\begin{pmatrix} \end{pmatrix}$$ __Gaussian and Gauss-Jordan elimination__: We can conduct elementary row operations to get our augmented matrix into row echelon or reduced row echelon form. The methods of transforming a matrix or system into row echelon and reduced row echelon form are referred to as Gaussian elimination and Gauss-Jordan elimination, respectively. - + __Elementary Row Operations__: To do Gaussian and Gauss-Jordan elimination, we use three basic operations to transform the augmented matrix into another augmented matrix that represents an equivalent linear system -- equivalent in the sense that the same values of $x_j$ solve both the original and transformed matrix/system: __Interchanging Rows__: Suppose we have the augmented matrix @@ -411,11 +399,9 @@ $\begin{matrix} x & - & 3y & = & -3\\ 2x & + & y & = & 8 \end{matrix}$ - ``` - ```{exercise} #| name: Solving Systems of Equations #| label: solvesys1 @@ -443,7 +429,7 @@ $$ ## Rank --- and Whether a System Has One, Infinite, or No Solutions To determine how many solutions exist, we can use information about (1) the number of equations $m$, (2) the number of unknowns $n$, and (3) the __rank__ of the matrix representing the linear system. - + __Rank__: The maximum number of linearly independent row or column vectors in the matrix. This is equivalent to the number of nonzero rows of a matrix in row echelon form. For any matrix __A__, the row rank always equals column rank, and we refer to this number as the rank of __A__. For example @@ -451,13 +437,13 @@ For example $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}$ - + Rank = 3 - + $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 0 \end{pmatrix}$ - + Rank = 2 ```{exercise} @@ -478,7 +464,6 @@ Find the rank of each matrix below: 1 & 3 & 1 & 1 & 3 \\ 1 & 3 & 2 & -1 & -2 \\ 1 & 3 & 0 & 3 & -2 \end{pmatrix}$ - ``` @@ -492,7 +477,7 @@ Answer to Exercise \@ref(exr:rank): __Identity Matrix__: The $n\times n$ identity matrix ${\bf I}_n$ is the matrix whose diagonal elements are 1 and all off-diagonal elements are 0. Examples: $$ {\bf I}_2=\begin{pmatrix} 1&0\\0&1 \end{pmatrix}, \qquad {\bf I}_3=\begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}$$ - + __Inverse Matrix__: An $n\times n$ matrix ${\bf A}$ is __nonsingular__ or __invertible__ if there exists an $n\times n$ matrix ${\bf A}^{-1}$ such that $${\bf A} {\bf A}^{-1} = {\bf A}^{-1} {\bf A} = {\bf I}_n$$ where ${\bf A}^{-1}$ is the inverse of ${\bf A}$. If there is no such ${\bf A}^{-1}$, then ${\bf A}$ is singular or not invertible. Example: Let @@ -514,26 +499,25 @@ __Properties of the Inverse__: - If ${\bf A}$ is nonsingular, then $({\bf A}^T)^{-1}=({\bf A}^{-1})^T$ - __Procedure to Find__ ${\bf A}^{-1}$: We know that if ${\bf B}$ is the inverse of ${\bf A}$, then $${\bf A} {\bf B} = {\bf B} {\bf A} = {\bf I}_n$$ Looking only at the first and last parts of this $${\bf A} {\bf B} = {\bf I}_n$$ Solving for ${\bf B}$ is equivalent to solving for $n$ linear systems, where each column of ${\bf B}$ is solved for the corresponding column in ${\bf I}_n$. We can solve the systems simultaneously by augmenting ${\bf A}$ with ${\bf I}_n$ and performing Gauss-Jordan elimination on ${\bf A}$. If Gauss-Jordan elimination on $[{\bf A} | {\bf I}_n]$ results in $[{\bf I}_n | {\bf B} ]$, then ${\bf B}$ is the inverse of ${\bf A}$. Otherwise, ${\bf A}$ is singular. To summarize: To calculate the inverse of ${\bf A}$ - + 1. Form the augmented matrix $[ {\bf A} | {\bf I}_n]$ 2. Using elementary row operations, transform the augmented matrix to reduced row echelon form. 3. The result of step 2 is an augmented matrix $[ {\bf C} | {\bf B} ]$. - + a. If ${\bf C}={\bf I}_n$, then ${\bf B}={\bf A}^{-1}$. - + b. If ${\bf C}\ne{\bf I}_n$, then $\bf C$ has a row of zeros. This means ${\bf A}$ is singular and ${\bf A}^{-1}$ does not exist. ```{example} #| label: inverse Find the inverse of the following matricies: - + 1. ${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$ ``` @@ -543,7 +527,7 @@ Find the inverse of the following matricies: #| label: inverse1 Find the inverse of the following matrix: - + 1. ${\bf A}=\begin{pmatrix} 1&0&4\\0&2&0\\0&0&1 \end{pmatrix}$ ``` @@ -551,11 +535,11 @@ Find the inverse of the following matrix: ## Linear Systems and Inverses Let's return to the matrix representation of a linear system - + $$\bf{Ax} = \bf{b}$$ If $\bf{A}$ is an $n\times n$ matrix,then $\bf{Ax}=\bf{b}$ is a system of $n$ equations in $n$ unknowns. Suppose $\bf{A}$ is nonsingular. Then $\bf{A}^{-1}$ exists. To solve this system, we can multiply each side by $\bf{A}^{-1}$ and reduce it as follows: - + \begin{eqnarray*} \bf{A}^{-1} (\bf{A} \bf{x}) & = & \bf{A}^{-1} \bf{b} \\ (\bf{A}^{-1} \bf{A})\bf{x} & = & \bf{A}^{-1} \bf{b}\\ @@ -579,7 +563,7 @@ Use the inverse matrix to solve the following linear system: ___Hint: the linear system above can be written in the matrix form___ $\textbf{A}\textbf{z} = \textbf{b}$ - + given $$\textbf{A} = \begin{pmatrix} -3&4\\2&-1 \end{pmatrix},$$ $$\textbf{z} = \begin{pmatrix} x\\y \end{pmatrix},$$ @@ -591,9 +575,8 @@ $$\textbf{b} = \begin{pmatrix} 5\\-10 \end{pmatrix}$$ ## Determinants __Singularity__: Determinants can be used to determine whether a square matrix is nonsingular. - + A square matrix is nonsingular if and only if its determinant is not zero. - Determinant of a $1 \times 1$ matrix, __A__, equals $a_{11}$ @@ -624,7 +607,6 @@ Let's extend this now to any $n\times n$ matrix. Let's define ${\bf A}_{ij}$ as Then for any $n\times n$ matrix ${\bf A}$ $$|{\bf A}|= a_{11}M_{11} - a_{12}M_{12} + \cdots + (-1)^{n+1} a_{1n} M_{1n}$$ - For example, in figuring out whether the following matrix has an inverse? $${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$$ 1. Calculate its determinant. @@ -640,21 +622,19 @@ For example, in figuring out whether the following matrix has an inverse? #| label: determinants Determine whether the following matrices are nonsingular: - + $$1. \begin{pmatrix} 1 & 0 & 1\\ 2 & 1 & 2\\ 1 & 0 & -1 \end{pmatrix}$$ - + $$2. \begin{pmatrix} 2 & 1 & 2\\ 1 & 0 & 1\\ 4 & 1 & 4 \end{pmatrix}$$ - - ``` ## Getting Inverse of a Matrix using its Determinant @@ -678,7 +658,6 @@ $$\frac{1}{\det({\bf A})} \begin{pmatrix} d & -b\\ -c & a\\ \end{pmatrix}$$ - For example, Let's calculate the inverse of matrix A from Exercise \@ref(exr:invlinsys) using the determinant formula. @@ -688,19 +667,19 @@ $$A = \begin{pmatrix} -3 & 4\\ 2 & -1\\ \end{pmatrix}$$ - + $$\det({\bf A}) = (-3)(-1) - (4)(2) = 3 - 8 = -5$$ $$\frac{1}{\det({\bf A})} \begin{pmatrix} -1 & -4\\ -2 & -3\\ \end{pmatrix}$$ - + $$\frac{1}{-5} \begin{pmatrix} -1 & -4\\ -2 & -3\\ \end{pmatrix}$$ - + $$ \begin{pmatrix} \frac{1}{5} & \frac{4}{5}\\ \frac{2}{5} & \frac{3}{5}\\ @@ -717,8 +696,6 @@ $$A = \begin{pmatrix} -7 & 2\\ \end{pmatrix}$$ - - ``` ## Answers to Examples and Exercises {.unnumbered} @@ -744,12 +721,11 @@ Answer to Exercise \@ref(exr:linearindep1): 1. yes 2. no ($-v_1 -v_2 + v_3 = 0$) - Answer to Example \@ref(exm:matrixaddition): ${\bf A+B}=\begin{pmatrix} 2 & 4 & 4 \\ 6 & 6 & 8 \end{pmatrix}$ - + Answer to Example \@ref(exm:scalarmulti): $s {\bf A} = \begin{pmatrix} 2 & 4 & 6 \\ 8 & 10 & 12 \end{pmatrix}$ @@ -763,13 +739,13 @@ Answer to Example \@ref(exm:matrixmulti): \begin{pmatrix} 4&-2\\6&16\end{pmatrix}$ Answer to Exercise \@ref(exr:matrixmulti1): - + 1. $AB = \begin{pmatrix} 4 & 11 & -15 \\ 5 & 7 & -7 \end{pmatrix}$ - + 2. $BA =$ undefined - + 3. $(BC)^T =$ undefined - + 4. $BC^T = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix}\begin{pmatrix} 3&0\\2&4\\-1&6 \end{pmatrix} =\begin{pmatrix}20 & -22 \\ 5 & 4 \\ -3 &2 \\6 & 0\end{pmatrix}$ Answer to Exercise \@ref(exr:lineareq): @@ -815,12 +791,12 @@ $$\begin{matrix} x & - & 3y & = & -3\\ & & y & = & 2\\ \end{matrix}$$ - + $$\begin{matrix} x & = & 3\\ y & = & 2\\ \end{matrix}$$ - + Answer to Exercise \@ref(exr:solvesys1): 1. x = 2, y = 2, z = -1 @@ -840,7 +816,7 @@ $\left(\begin{array}{ccc|ccc} 0&2&3&0&1&0\\ 5&5&1&0&0&1 \end{array} \right)$ - + $\left(\begin{array}{ccc|ccc} 1&1&1 &1 &0&0\\ 0&2&3 &0 &1&0\\ @@ -864,7 +840,7 @@ $\left(\begin{array}{ccc|ccc} 0&1&0&-15/8&1/2&3/8\\ 0&0&1&5/4 &0 &-1/4 \end{array} \right)$ - + $\left(\begin{array}{ccc|ccc} 1&0&0&13/8 &-1/2&-1/8\\ 0&1&0&-15/8&1/2 &3/8\\ diff --git a/11_data-handling_counting.qmd b/11_data-handling_counting.qmd index 66d4c3b..eb269a5 100644 --- a/11_data-handling_counting.qmd +++ b/11_data-handling_counting.qmd @@ -64,13 +64,13 @@ library(fs) To load a dataset, you need to specify where that file is. Computer files (data, documents, programs) are organized hiearchically, like a branching tree. Folders can contain files, and also other folders. The GUI toolbar makes this lineaer and hiearchical relationship apparent. When we turn to locate the file in our commands, we need another set of syntax. Importantly, denote the hierarchy of a folder by the `/` (slash) symbol. `data/input/2018-08` indicates the `2018-08` folder, which is included in the `input` folder, which is in turn included in the `data` folder. Files (but not folders) have "file extensions" which you are probably familiar with already: `.docx`, `.pdf`, and `.pdf`. The file extensions you will see in a stats or quantitative social science class are: - + * `.pdf`: PDF, a convenient format to view documents and slides in, regardless of Mac/Windows. * `.csv`: A comma separated values file * `.xlsx`: Microsoft Excel file * `.dta`: Stata data * `.sav`: SPSS data - + * `.R`: R code (script) * `.Rmd`: Rmarkdown code (text + code) * `.do`: Stata code (script) diff --git a/13_functions_obj_loops.qmd b/13_functions_obj_loops.qmd index 9ebe77b..0355db4 100644 --- a/13_functions_obj_loops.qmd +++ b/13_functions_obj_loops.qmd @@ -104,7 +104,6 @@ __class__ - Book, __object__ - To Kill a Mockingbird __class__ - DataFrame, __object__ - 2010 census data __class__ - Character, __object__ - "Programming is Fun" - What is type (class) of object is `cen10`? ```{r} @@ -264,7 +263,7 @@ It's worth remembering the basic structure of a function. You create a new funct ```{r} #| eval: !expr F my_fun <- function() { - + } ``` @@ -272,9 +271,9 @@ If we wanted to generate a function that computed the number of men in your data ```{r} count_men <- function(data) { - + nmen <- sum(data$sex == "Male") - + return(nmen) } ``` @@ -294,11 +293,11 @@ Let's go one step further. What if we want to know the proportion of non-whites ```{r} nw_in_state <- function(data, state) { - + s.subset <- data[data$state == state,] total.s <- nrow(s.subset) nw.s <- sum(s.subset$race != "White") - + nw.s / total.s } ``` @@ -445,7 +444,7 @@ for( state in states_of_interest){ nmen <- sum(state_data$sex == "Male") n <- nrow(state_data) men_perc <- round(100*(nmen/n), digits=2) - + male_percentages <- c(male_percentages, men_perc) names(male_percentages)[iter] <- state iter <- iter + 1 diff --git a/16_simulation.qmd b/16_simulation.qmd index a9a4886..9b79edf 100644 --- a/16_simulation.qmd +++ b/16_simulation.qmd @@ -15,7 +15,7 @@ An increasing amount of political science contributions now include a simulation Statistical Software, 45(7), 1-47.](http://www.jstatsoft.org/v45/i07/)]) Statistical methods also incorporate simulation: - + * The bootstrap: a statistical method for estimating uncertainty around some parameter by re-sampling observations. * Bagging: a method for improving machine learning predictions by re-sampling observations, storing the estimate across many re-samples, and averaging these estimates to form the final estimate. A variance reduction technique. * Statistical reasoning: if you are trying to understand a quantitative problem, a wonderful first-step to understand the problem better is to simulate it! The analytical solution is often very hard (or impossible), but the simulation is often much easier :-) diff --git a/17_non-wysiwyg.qmd b/17_non-wysiwyg.qmd index 581e11a..6d7c80d 100644 --- a/17_non-wysiwyg.qmd +++ b/17_non-wysiwyg.qmd @@ -210,7 +210,7 @@ The first entry, `nunn2011slave`, is "pick your favorite" -- pick your own name Now, in LaTeX, if you type \textcite{nunn2011slave} argue that current variation in the trust among citizens of African countries has historical roots in the European slave trade in the 1600s. - + as part of your text, then when the `.tex` file is compiled the PDF shows something like ![](images/biblatex_inline.png) diff --git a/18_text.qmd b/18_text.qmd index 5c1e1eb..21dda64 100644 --- a/18_text.qmd +++ b/18_text.qmd @@ -117,7 +117,7 @@ my_string[ grepl(my_string, pattern = "Indonesia")] Key point: Many R commands use regular expressions. See ```?grepl```. Assume that ```x``` is a character vector and that ```pattern``` is the target pattern. In the earlier example, ```x``` could have been something like ```my_string``` and ```pattern``` would have been "```Indonesia```". Here are other key uses: 1. DETECT PATTERNS. ```grepl(pattern, x)``` goes through all the entries of ```x``` and returns a string of TRUE and FALSE values of the same size as ```x```. It will return a ```TRUE``` whenever that string entry has the target pattern, and ```FALSE``` whenever it doesn't. - + 2. REPLACE PATTERNS. ```gsub(pattern, x, replacement)``` goes through all the entries of ```x``` replaces the ```pattern``` with ```replacement```. ```{r} @@ -159,7 +159,7 @@ my_string <- "Do you think that 34% of apples are red?" gsub(my_string, pattern = "[[:digit:]]", replace ="DIGIT") gsub(my_string, pattern = "[[:alpha:]]", replace ="") ``` - + ### Special Characters. Certain characters (such as ```., *, \```) have special meaning in the regular expressions framework (they are used to form conditional patterns as discussed below). Thus, when we want our pattern to explicitly include those characters as characters, we must "escape" them by using \\ or encoding them in \\Q...\\E. @@ -340,7 +340,7 @@ string2 <- "–carbon double bond." ### 7 {.unnumbered} Challenge problem! Download this webpage - + * Read the html file into your R workspace. * Remove all of the htlm tags (you may need Google to help with this one). * Remove all punctuation. diff --git a/21_solutions-warmup.qmd b/21_solutions-warmup.qmd index 7e5c2bf..2f4c609 100644 --- a/21_solutions-warmup.qmd +++ b/21_solutions-warmup.qmd @@ -45,20 +45,20 @@ $${\bf A}=\begin{pmatrix} 2 & 14 & 21 \\ 4 & 1 & 5 \end{pmatrix}$$ - + What is the dimensionality of matrix ${\bf A}$? 4 $\times$ 3 What is the element $a_{23}$ of ${\bf A}$? 3 Given that - + $${\bf B}=\begin{pmatrix} 1 & 2 & 8 \\ 3 & 9 & 11 \\ 4 & 7 & 5 \\ 5 & 1 & 9 \end{pmatrix}$$ - + $$\mathbf{A} + \mathbf{B} = \begin{pmatrix} 8 & 7 & 9 \\ 14 & 18 & 14 \\ @@ -67,13 +67,13 @@ $$\mathbf{A} + \mathbf{B} = \begin{pmatrix} \end{pmatrix}$$ Given that - + $${\bf C}=\begin{pmatrix} 1 & 2 & 8 \\ 3 & 9 & 11 \\ 4 & 7 & 5 \\ \end{pmatrix}$$ - + $$\mathbf{A} + \mathbf{C} = \text{No solution, matrices non-conformable}$$ Given that @@ -158,7 +158,6 @@ For each of the followng functions $f(x)$, does a maximum and minimum exist in t 3. $f(x) = -(x - 2)^2$ $\leadsto$ a maximum $f(x) = 0$ exists at $x = 2$, but not a minimum. If you are stuck, please try sketching out a picture of each of the functions. - ## Probability {.unnumbered} From b78e59d28704c44cb53deceee442301456c3dbd8 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 21:55:05 -0400 Subject: [PATCH 14/34] fix probability --- 06_probability.qmd | 341 +++++++++++++++++++++------------------------ _quarto.yml | 1 + 2 files changed, 163 insertions(+), 179 deletions(-) diff --git a/06_probability.qmd b/06_probability.qmd index 109c454..089fedd 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -14,9 +14,8 @@ Probability in high school is usually really about combinatorics: the probabilit __Fundamental Theorem of Counting__: If an object has $j$ different characteristics that are independent of each other, and each characteristic $i$ has $n_i$ ways of being expressed, then there are $\prod_{i = 1}^j n_i$ possible unique objects. -```{example} -#| name: Counting Possibilities -#| label: countingrules +:::{#exm-countingrules} +## Counting Possibilities Suppose we are given a stack of cards. Cards can be either red or black and can take on any of 13 values. There is only one of each color-number combination. In this case, @@ -28,7 +27,7 @@ Suppose we are given a stack of cards. Cards can be either red or black and can 4. Number of Outcomes $=$ -``` +::: We often need to count the number of ways to choose a subset from some set of possibilities. The number of outcomes depends on two characteristics of the process: does the order matter and is replacement allowed? @@ -46,9 +45,8 @@ If replacement is allowed, there are always the same $n$ objects to select from. Expression $\binom{n}{k}$ is read as "n choose k" and denotes $\frac{n!}{(n-k)!k!}$. Also, note that $0! = 1$. -```{example} -#| name: Counting -#| label: counting +:::{#exm-counting} +## Counting There are five balls numbered from 1 through 5 in a jar. Three balls are chosen. How many possible choices are there? @@ -58,15 +56,14 @@ There are five balls numbered from 1 through 5 in a jar. Three balls are chosen. 3. Unordered, without replacement $=$ -``` +::: -```{exercise} -#| name: Counting -#| label: counting1 +:::{#exr-counting1} +## Counting Four cards are selected from a deck of 52 cards. Once a card has been drawn, it is not reshuffled back into the deck. Moreover, we care only about the complete hand that we get (i.e. we care about the set of selected cards, not the sequence in which it was drawn). How many possible outcomes are there? -``` +::: ## Sets {#setoper} @@ -99,9 +96,8 @@ Properties of set operations: * __de Morgan's laws__: $(A \cup B)^C = A^C \cap B^C$; $(A \cap B)^C = A^C \cup B^C$ * __Disjointness__: Sets are disjoint when they do not intersect, such that $A \cap B = \emptyset$. A collection of sets is pairwise disjoint (**mutually exclusive**) if, for all $i \neq j$, $A_i \cap A_j = \emptyset$. A collection of sets form a partition of set $S$ if they are pairwise disjoint and they cover set $S$, such that $\bigcup_{i = 1}^k A_i = S$. -```{example} -#| name: Sets -#| label: sets +:::{#exm-sets} +## Sets Let set $A$ be {1, 2, 3, 4}, $B$ be {3, 4, 5, 6}, and $C$ be {5, 6, 7, 8}. Sets $A$, $B$, and $C$ are all subsets of the sample space $S$ which is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} @@ -112,11 +108,10 @@ Write out the following sets: 3. $B^c$ 4. $A \cap (B \cup C)$ -``` +::: -```{exercise} -#| name: Sets -#| label: sets1 +:::{#exr-sets1} +## Sets Suppose you had a pair of four-sided dice. You sum the results from a single toss. @@ -127,7 +122,7 @@ Consider subsets A {2, 8} and B {2,3,7} of the sample space you found. What is 1. $A^c$ 2. $(A \cup B)^c$ -``` +::: ## Probability {#probdef} @@ -147,17 +142,17 @@ __Probability Distribution Function__: a mapping of each event in the sample spa Formally, -```{definition} -#| name: Probability +:::{#def-probability} +## Probability Probability is a function that maps events to a real number, obeying the axioms of probability. -``` +::: The axioms of probability make sure that the separate events add up in terms of probability, and -- for standardization purposes -- that they add up to 1. -```{definition} -#| name: Axioms of Probability +:::{#def-axiomsofprobability} +## Axioms of Probability 1. For any event $A$, $P(A)\ge 0$. 2. $P(S)=1$ @@ -170,7 +165,7 @@ The last axiom is an extension of a union to infinite sets. When there are only P(A_1 \cup A_2) = P(A_1) + P(A_2) \quad\text{for disjoint } A_1, A_2 \end{align*} -``` +::: ### Probability Operations {.unnumbered} @@ -185,9 +180,8 @@ B)=P(A)+P(B)-P(A\cap B)$ 6. Boole's Inequality: For any sequence of $n$ events (which need not be disjoint) $A_1,A_2,\ldots,A_n$, then $P\left( \bigcup\limits_{i=1}^n A_i\right) \leq \sum\limits_{i=1}^n P(A_i)$. -```{example} -#| name: Probability -#| label: prob +:::{#exm-prob} +## Probability Assume we have an evenly-balanced, six-sided die. @@ -201,11 +195,10 @@ Then, 6. Let $A=\{ 1,2,3,4,5 \}\subset S$. Then $P(A)=5/6 x) = 1 - P(X \le x)$. -```{example} +:::{#exm-die-pmf} For a fair die with its value as $Y$, What are the following? 1. $P(Y\le 1)$ 2. $P(Y\le 3)$ 3. $P(Y\le 6)$ -``` +::: ### Continuous Random Variables {.unnumbered} We also have a similar definition for _continuous_ random variables. -```{definition} -#| name: Continuous Random Variable +:::{#def-rv-continuous} +## Continuous Random Variable $X$ is a continuous random variable if there exists a nonnegative function $f(x)$ defined for all real $x\in (-\infty,\infty)$, such that for any interval $A$, $P(X\in A)=\int\limits_A f(x)dx$. Examples: age, income, GNP, temperature. -``` +::: -```{definition} -#| name: Probability Density Function +:::{#def-pdf} +## Probability Density Function The function $f$ above is called the probability density function (pdf) of $X$ and must satisfy $$f(x)\ge 0$$ $$\int\limits_{-\infty}^\infty f(x)dx=1$$ Note also that $P(X = x)=0$ --- i.e., the probability of any point $y$ is zero. -``` +::: \begin{comment} \item[] \parbox[t]{4.5in}{Example: $f(y)=1, \quad 0\le y \le1$}\parbox{1.5in}{\hfill @@ -470,22 +459,22 @@ $$\int\limits_{-\infty}^\infty f(x)dx=1$$ For both discrete and continuous random variables, we have a unifying concept of another measure: the cumulative distribution: -```{definition} -#| name: Cumulative Density Function +:::{#def-cdf} +## Cumulative Density Function Because the probability that a continuous random variable will assume any particular value is zero, we can only make statements about the probability of a continuous random variable being within an interval. The cumulative distribution gives the probability that $Y$ lies on the interval $(-\infty,y)$ and is defined as $$F(x)=P(X\le x)=\int\limits_{-\infty}^x f(s)ds$$ Note that $F(x)$ has similar properties with continuous distributions as it does with discrete - non-decreasing, continuous (not just right-continuous), and $\lim\limits_{x \to -\infty} F(x) = 0$ and $\lim\limits_{x \to \infty} F(x) = 1$. -``` +::: We can also make statements about the probability of $Y$ falling in an interval $a\le y\le b$. $$P(a\le x\le b)=\int\limits_a^b f(x)dx$$ The PDF and CDF are linked by the integral: The CDF of the integral of the PDF: $$f(x) = F'(x)=\frac{dF(x)}{dx}$$ -```{example} +:::{#exm-continuous-pdf} For $f(y)=1, \quad 0 0$ * Continuous: $f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)},\quad f_X(x) > 0$ -```{exercise} -#| name: Discrete Outcomes +:::{#exr-joint-distribution} +## Discrete Outcomes Suppose we are interested in the outcomes of flipping a coin and rolling a 6-sided die at the same time. The sample space for this process contains 12 elements: $$\{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}$$ We can define two random variables $X$ and $Y$ such that $X = 1$ if heads and $X = 0$ if tails, while $Y$ equals the number on the die. We can then make statements about the joint distribution of $X$ and $Y$. What are the following? @@ -531,26 +520,25 @@ We can then make statements about the joint distribution of $X$ and $Y$. What ar 1. $P(X=x|Y=y)$ 1. Are X and Y independent? -``` +::: ## Expectation We often want to summarize some characteristics of the distribution of a random variable. The most important summary is the expectation (or expected value, or mean), in which the possible values of a random variable are weighted by their probabilities. -```{definition} -#| name: Expectation of a Discrete Random Variable +:::{#def-expectation} +## Expectation of a Discrete Random Variable The expected value of a discrete random variable $Y$ is $$E(Y)=\sum\limits_{y} y P(Y=y)= \sum\limits_{y} y p(y)$$ In words, it is the weighted average of all possible values of $Y$, weighted by the probability that $y$ occurs. It is not necessarily the number we would expect $Y$ to take on, but the average value of $Y$ after a large number of repetitions of an experiment. -``` +::: -```{example} -#| name: Expectation of a Discrete Random Variable -#| label: expectdiscrete +:::{#exm-expectdiscrete} +## Expectation of a Discrete Random Variable What is the expectation of a fair, six-sided die? -``` +::: __Expectation of a Continuous Random Variable__: The expected value of a continuous random variable is similar in concept to that of @@ -559,13 +547,12 @@ probabilities as weights, we integrate using the density to weight. Hence, the expected value of the continuous variable $Y$ is defined by $$E(Y)=\int\limits_{y} y f(y) dy$$ -```{example} -#| name: Expectation of a Continuous Random Variable -#| label: expectconti +:::{#exm-expectconti} +## Expectation of a Continuous Random Variable Find $E(Y)$ for $f(y)=\frac{1}{1.5}, \quad 00$$ The Poisson has the unusual feature that its expectation equals its variance: $E(Y)=\text{Var}(Y)=\lambda$. The Poisson distribution is often used to model rare event counts: counts of the number of events that occur during some unit of time. $\lambda$ is often called the "arrival rate." -``` +::: -```{example} +:::{#exm-pois} Border disputes occur between two countries through a Poisson Distribution, at a rate of 2 per month. What is the probability of 0, 2, and less than 5 disputes occurring in a month? -``` +::: \begin{comment} \parbox{1.5in}{\hfill \epsffile{poispmf.eps}} @@ -766,12 +749,12 @@ Border disputes occur between two countries through a Poisson Distribution, at a Two _continuous_ distributions used often are: -```{definition} -#| name: Uniform Distribution +:::{#def-unif} +## Uniform Distribution A random variable $Y$ has a continuous uniform distribution on the interval $(\alpha,\beta)$ if its density is given by $$f(y)=\frac{1}{\beta-\alpha}, \quad \alpha\le y\le \beta$$ The mean and variance of $Y$ are $E(Y)=\frac{\alpha+\beta}{2}$ and $\text{Var}(Y)=\frac{(\beta-\alpha)^2}{12}$. -``` +::: -```{example} +:::{#exm-unif} For $Y$ uniformly distributed over $(1,3)$, what are the following probabilities? 1. $P(Y=2)$ @@ -779,25 +762,25 @@ For $Y$ uniformly distributed over $(1,3)$, what are the following probabilities 1. $P(Y \le 2)$ 1. $P(Y > 2)$ -``` +::: \begin{comment} \parbox{1.5in}{\hfill \epsffile{unifpdf.eps}} \end{comment} -```{definition} -#| name: Normal Distribution +:::{#def-normal} +## Normal Distribution A random variable $Y$ is normally distributed with mean $E(Y)=\mu$ and variance $\text{Var}(Y)=\sigma^2$ if its density is $$f(y)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(y-\mu)^2}{2\sigma^2}}$$ -``` +::: -See Figure \@ref(fig:normaldens) are various Normal Distributions with the same $\mu = 1$ and two versions of the variance. +See Figure @fig-normaldens are various Normal Distributions with the same $\mu = 1$ and two versions of the variance. ```{r} -#| label: normaldens +#| label: fig-normaldens #| echo: false #| fig-cap: Normal Distribution Density range <- tibble::tibble(x = c(-5, 5)) @@ -806,8 +789,8 @@ fx0 <- ggplot(range, aes(x = x)) + labs(x = expression(x), y = expression(f(x))) fx <- fx0 + - stat_function(fun = function(x) dnorm(x, mean = 0, sd = 1), size = 0.5) + - stat_function(fun = function(x) dnorm(x, mean = 0, sd = sqrt(2)), size = 1.5) + + stat_function(fun = function(x) dnorm(x, mean = 0, sd = 1), linewidth = 0.5) + + stat_function(fun = function(x) dnorm(x, mean = 0, sd = sqrt(2)), linewidth = 1.5) + expand_limits(y = 0) + labs(caption = "Thick line: variance = 2, Normal line: variance = 1") @@ -865,13 +848,13 @@ __Covariance and Correlation__: Both of these quantities measure the degree to w 1. **Sample covariance**: $\hat{\text{Cov}}(X,Y) = \frac{1}{n-1}\sum_{i = 1}^n(x_i - \bar{x})(y_i - \bar{y})$ 2. **Sample correlation**: $\hat{\text{Corr}} = \frac{\hat{\text{Cov}}(X,Y)}{\sqrt{\hat{\text{Var}}(X)\hat{\text{Var}}(Y)}}$ -```{example} +:::{#exm-sample} Example: Using the above table, calculate the sample versions of: 1. $\text{Cov}(X,Y)$ 2. $\text{Corr}(X, Y)$ -``` +::: ## Asymptotic Theory @@ -885,13 +868,14 @@ Why is this important? We rarely know the true process governing the events we We are now finally ready to revisit, with a bit more precise terms, the two pillars of statistical theory we motivated Section \@ref(limitsfun) with. -```{theorem} -#| label: clt -#| name: Central Limit Theorem (i.i.d. case) +:::{#thm-clt} +## Central Limit Theorem (i.i.d. case) Let $\{X_n\} = \{X_1, X_2, \ldots\}$ be a sequence of i.i.d. random variables with finite mean ($\mu$) and variance ($\sigma^2$). Then, the sample mean $\bar{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n}$ increasingly converges into a Normal distribution. $$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),$$ +::: + Another way to write this as a probability statement is that for all real numbers $a$, $$P\left(\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \le a\right) \rightarrow \Phi(a)$$ @@ -902,15 +886,16 @@ $$\bar{X}_n \approx \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).$$ The standa Intuitively, this result also justifies that whenever a lot of small, independent processes somehow combine together to form the realized observations, practitioners often feel comfortable assuming Normality. -```{theorem} -#| label: lln -#| name: Law of Large Numbers (LLN) +:::{#thm-lln} + +## Law of Large Numbers (LLN) For any draw of independent random variables with the same mean $\mu$, the sample average after $n$ draws, $\bar{X}_n = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)$, converges in probability to the expected value of $X$, $\mu$ as $n \rightarrow \infty$: $$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". -``` + +::: as $n\to \infty$. In other words, $P( \lim_{n\to\infty}\bar{X}_n = \mu) = 1$. This is an important motivation for the widespread use of the sample mean, as well as the intuition link between averages and expected values. @@ -925,9 +910,7 @@ Some of you may encounter "big-OH''-notation. If $f, g$ are two functions, we sa Example. What is $\mathcal{O}( 5\exp(0.5 n) + n^2 + n / 2)$? Answer: $\exp(n)$. Why? Because, for large $n$, -$$ -\frac{ 5\exp(0.5 n) + n^2 + n / 2 }{ \exp(n)} \leq \frac{ c \exp(n) }{ \exp(n)} = c. -$$ +$$\frac{ 5\exp(0.5 n) + n^2 + n / 2 }{ \exp(n)} \leq \frac{ c \exp(n) }{ \exp(n)} = c. $$ whenever $n > 4$ and where $c = 1$. \begin{comment} @@ -938,7 +921,7 @@ numerator / denominator <= (1 * denominator) / denominator ## Answers to Examples and Exercises {.unnumbered} -Answer to Example \@ref(exm:counting): +Answer to Example @exm-counting: 1. $5 \times 5 \times 5 = 125$ @@ -946,25 +929,25 @@ Answer to Example \@ref(exm:counting): 3. $\binom{5}{3} = \frac{5!}{(5-3)!3!} = \frac{5 \times 4}{2 \times 1} = 10$ -Answer to Exercise \@ref(exr:counting1): +Answer to Exercise @exr-counting1: 1. $\binom{52}{4} = \frac{52!}{(52-4)!4!} = 270725$ -Answer to Example \@ref(exm:sets): +Answer to Example @exm-sets: 1. {1, 2, 3, 4, 5, 6} 2. {5, 6} 3. {1, 2, 7, 8, 9, 10} 4. {3, 4} -Answer to Exercise \@ref(exr:sets1): +Answer to Exercise @exr-sets1: Sample Space: {2, 3, 4, 5, 6, 7, 8} 1. {3, 4, 5, 6, 7} 2. {4, 5, 6} -Answer to Example \@ref(exm:prob): +Answer to Example @exm-prob: 1. ${1, 2, 3, 4, 5, 6}$ @@ -980,13 +963,13 @@ Answer to Example \@ref(exm:prob): 7. $A\cup B=\{1, 2, 3, 4, 6\}$, $A\cap B=\{2\}$, $\frac{5}{6}$ -Answer to Exercise \@ref(exr:prob1): +Answer to Exercise @exr-prob1: 1. $P(X = 5) = \frac{4}{16}$, $P(X = 3) = \frac{2}{16}$, $P(X = 6) = \frac{3}{16}$ 2. What is $P(X=5 \cup X = 3)^C = \frac{10}{16}$? -Answer to Example \@ref(exm:condprobexm1): +Answer to Example @exm-condprobexm1: 1. $\frac{n_{ab} + n_{ab^c}}{N}$ @@ -998,11 +981,11 @@ Answer to Example \@ref(exm:condprobexm1): 5. $\frac{\frac{n_{ab}}{N}}{\frac{n_{ab} + n_{ab^c}}{N}} = \frac{n_{ab}}{n_{ab} + n_{ab^c}}$ -Answer to Example \@ref(exm:condprobexm2): +Answer to Example @exm-condprobexm2: $P(1|Odd) = \frac{P(1 \cap Odd)}{P(Odd)} = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{3}$ -Answer to Example \@ref(exm:bayesrule): +Answer to Example @exm-bayesrule: We are given that $$P(D) = .4, P(D^c) = .6, P(S|D) = .5, P(S|D^c) = .9$$ @@ -1011,7 +994,7 @@ Using this, Bayes' Law and the Law of Total Probability, we know: $$P(D|S) = \frac{P(D)P(S|D)}{P(D)P(S|D) + P(D^c)P(S|D^c)}$$ $$P(D|S) = \frac{.4 \times .5}{.4 \times .5 + .6 \times .9 } = .27$$ -Answer to Exercise \@ref(exr:condprobexr): +Answer to Exercise @exr-condprobexr: We are given that @@ -1024,17 +1007,17 @@ $$= \frac{P(C|M)P(M)}{P(C|M)P(M) + P(C|M^c)P(M^c)}$$ $$= \frac{P(C|M)P(M)}{P(C|M)P(M) + [1-P(C^c|M^c)]P(M^c)}$$ $$ = \frac{.95 \times .02}{.95 \times .02 + .03 \times .98} = .38$$ -Answer to Example \@ref(exm:expectdiscrete): +Answer to Example @exm-expectdiscrete: $E(Y)=7/2$ We would never expect the result of a rolled die to be $7/2$, but that would be the average over a large number of rolls of the die. -Answer to Example \@ref(exm:expectconti) +Answer to Example @exm-expectconti 0.75 -Answer to Example \@ref(exm:var): +Answer to Example @exm-var: $E(X) = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 2 \times \frac{3}{8} + 3 \times \frac{1}{8} = \frac{3}{2}$ @@ -1046,7 +1029,7 @@ Since there is a 1 to 1 mapping from $X$ to $X^2:$ $E(X^2) = 0 \times \frac{1}{8 &= \frac{3}{4} \end{align*} -Answer to Exercise \@ref(exr:expvar): +Answer to Exercise @exr-expvar: 1. $E(X) = -2(\frac{1}{5}) + -1(\frac{1}{6}) + 0(\frac{1}{5}) + 1(\frac{1}{15}) + 2(\frac{11}{30}) = \frac{7}{30}$ @@ -1060,11 +1043,11 @@ Answer to Exercise \@ref(exr:expvar): &= \frac{5}{2} - \frac{7}{30}^2 \approx 2.45 \end{align*} -Answer to Exercise \@ref(exr:expvar2): +Answer to Exercise @exr-expvar2: 1. expectation = $\frac{6}{5}$, variance = $\frac{6}{25}$ -Answer to Exercise \@ref(exr:expvar3): +Answer to Exercise @exr-expvar3: 1. mean = 2, standard deviation = $\sqrt(\frac{2}{3})$ diff --git a/_quarto.yml b/_quarto.yml index d3bd31c..11f2d65 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -16,6 +16,7 @@ book: - 03_limits.qmd - 04_calculus.qmd - 05_optimization.qmd + - 06_probability.qmd delete_merged_file: true From 07fc75c30dc26cdc66a1e480bb4a47256bd9f00d Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 22:06:49 -0400 Subject: [PATCH 15/34] fix refs + file 07 --- 01_warmup.qmd | 2 +- 04_calculus.qmd | 2 +- 06_probability.qmd | 6 +- 07_linear-algebra.qmd | 814 ++++++++++++++++++++---------------------- _quarto.yml | 4 +- 5 files changed, 389 insertions(+), 439 deletions(-) diff --git a/01_warmup.qmd b/01_warmup.qmd index fdba91a..be5429f 100644 --- a/01_warmup.qmd +++ b/01_warmup.qmd @@ -86,7 +86,7 @@ Simplify the following 2. $\prod\limits_{k=1}^3(3k + 2)$ -To review this material, please see Section \@ref(sum-notation). +To review this material, please see Section @sec-sum-notation. ### Logs and exponents {.unnumbered} diff --git a/04_calculus.qmd b/04_calculus.qmd index 08716b5..793b892 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -30,7 +30,7 @@ $$E(X) = \large \sum_{x} \quad\left( \underbrace{x}_{\text{value}}\cdot \underbr ## Derivatives {#derivintro} -The derivative of $f$ at $x$ is its rate of change at $x$: how much $f(x)$ changes with a change in $x$. The rate of change is a fraction --- rise over run --- but because not all lines are straight and the rise over run formula will give us different values depending on the range we examine, we need to take a limit (Section \@ref(limits-precalc)). +The derivative of $f$ at $x$ is its rate of change at $x$: how much $f(x)$ changes with a change in $x$. The rate of change is a fraction --- rise over run --- but because not all lines are straight and the rise over run formula will give us different values depending on the range we examine, we need to take a limit (Section @sec-limits-precalc). :::{#def-derivative} ## Derivative diff --git a/06_probability.qmd b/06_probability.qmd index 089fedd..27aef8e 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -136,7 +136,7 @@ knitr::include_graphics('images/probability.png') ### Probability Definitions: Formal and Informal {.unnumbered} -Many things in the world are uncertain. In everyday speech, we say that we are _uncertain_ about the outcome of random events. Probability is a formal model of uncertainty which provides a measure of uncertainty governed by a particular set of rules (Figure \@ref(fig:prob-image)). A different model of uncertainty would, of course, have a set of rules different from anything we discuss here. Our focus on probability is justified because it has proven to be a particularly useful model of uncertainty. +Many things in the world are uncertain. In everyday speech, we say that we are _uncertain_ about the outcome of random events. Probability is a formal model of uncertainty which provides a measure of uncertainty governed by a particular set of rules (Figure @fig-prob-image). A different model of uncertainty would, of course, have a set of rules different from anything we discuss here. Our focus on probability is justified because it has proven to be a particularly useful model of uncertainty. __Probability Distribution Function__: a mapping of each event in the sample space $S$ to the real numbers that satisfy the following three axioms (also called Kolmogorov's Axioms). @@ -343,7 +343,7 @@ A random variable is a measurable function $X$ that maps from the sample space $ ::: -Figure \@ref(fig:rv-image) shows a image of the function. It might seem strange to define a random variable as a function -- which is neither random nor variable. The randomness comes from the realization of an event from the sample space $s$. +Figure @fig-rv-image shows a image of the function. It might seem strange to define a random variable as a function -- which is neither random nor variable. The randomness comes from the realization of an event from the sample space $s$. __Randomness__ means that the outcome of some experiment is not deterministic, i.e. there is some probability ($0 < P(A) < 1$) that the event will occur. @@ -866,7 +866,7 @@ Why is this important? We rarely know the true process governing the events we ### CLT and LLN -We are now finally ready to revisit, with a bit more precise terms, the two pillars of statistical theory we motivated Section \@ref(limitsfun) with. +We are now finally ready to revisit, with a bit more precise terms, the two pillars of statistical theory we motivated Section @sec-limitsfun with. :::{#thm-clt} ## Central Limit Theorem (i.i.d. case) diff --git a/07_linear-algebra.qmd b/07_linear-algebra.qmd index 6370b07..bfcf6ab 100644 --- a/07_linear-algebra.qmd +++ b/07_linear-algebra.qmd @@ -1,41 +1,36 @@ # Linear Algebra {#linearalgebra} Topics: -$\bullet$ Working with Vectors -$\bullet$ Linear Independence -$\bullet$ Basics of Matrix Algebra -$\bullet$ Square Matrices -$\bullet$ Linear Equations -$\bullet$ Systems of Linear Equations -$\bullet$ Systems of Equations as Matrices -$\bullet$ Solving Augmented Matrices and Systems of Equations -$\bullet$ Rank -$\bullet$ The Inverse of a Matrix -$\bullet$ Inverse of Larger Matrices + +- Working with Vectors +- Linear Independence +- Basics of Matrix Algebra +- Square Matrices +- Linear Equations +- Systems of Linear Equations +- Systems of Equations as Matrices +- Solving Augmented Matrices and Systems of Equations +- Rank +- The Inverse of a Matrix +- Inverse of Larger Matrices ## Working with Vectors {#vector-def} -__Vector__: A vector in $n$-space is an ordered list of $n$ numbers. These numbers can be represented as either a row vector or a column vector: - $$ {\bf v} \begin{pmatrix} v_1 & v_2 & \dots & v_n\end{pmatrix} , {\bf v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$$ + +**Vector**: A vector in $n$-space is an ordered list of $n$ numbers. These numbers can be represented as either a row vector or a column vector: $$ {\bf v} \begin{pmatrix} v_1 & v_2 & \dots & v_n\end{pmatrix} , {\bf v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$$ We can also think of a vector as defining a point in $n$-dimensional space, usually ${\bf R}^n$; each element of the vector defines the coordinate of the point in a particular direction. -__Vector Addition and Subtraction__: If two vectors, ${\bf u}$ and ${\bf v}$, have the same length (i.e. have the same number of elements), they can be added (subtracted) together: - $$ {\bf u} + {\bf v} = \begin{pmatrix} u_1 + v_1 & u_2 + v_2 & \cdots & u_k + v_n \end{pmatrix}$$ - $$ {\bf u} - {\bf v} = \begin{pmatrix} u_1 - v_1 & u_2 - v_2 & \cdots & u_k - v_n \end{pmatrix}$$ +**Vector Addition and Subtraction**: If two vectors, ${\bf u}$ and ${\bf v}$, have the same length (i.e. have the same number of elements), they can be added (subtracted) together: $$ {\bf u} + {\bf v} = \begin{pmatrix} u_1 + v_1 & u_2 + v_2 & \cdots & u_k + v_n \end{pmatrix}$$ $$ {\bf u} - {\bf v} = \begin{pmatrix} u_1 - v_1 & u_2 - v_2 & \cdots & u_k - v_n \end{pmatrix}$$ -__Scalar Multiplication__: The product of a scalar $c$ (i.e. a constant) and vector ${\bf v}$ is: \ - $$ c{\bf v} = \begin{pmatrix} cv_1 & cv_2 & \dots & cv_n \end{pmatrix} $$ +**Scalar Multiplication**: The product of a scalar $c$ (i.e. a constant) and vector ${\bf v}$ is:\ +$$ c{\bf v} = \begin{pmatrix} cv_1 & cv_2 & \dots & cv_n \end{pmatrix} $$ -__Vector Inner Product__: The inner product (also called the dot product or scalar product) of two vectors ${\bf u}$ and ${\bf v}$ is again defined if and only if they have the same number of elements - $$ {\bf u} \cdot {\bf v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n = \sum_{i = 1}^n u_iv_i$$ - If ${\bf u} \cdot {\bf v} = 0$, the two vectors are orthogonal (or perpendicular). +**Vector Inner Product**: The inner product (also called the dot product or scalar product) of two vectors ${\bf u}$ and ${\bf v}$ is again defined if and only if they have the same number of elements $$ {\bf u} \cdot {\bf v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n = \sum_{i = 1}^n u_iv_i$$ If ${\bf u} \cdot {\bf v} = 0$, the two vectors are orthogonal (or perpendicular). -__Vector Norm__: The norm of a vector is a measure of its length. There are many different ways to calculate the norm, but the most common is the Euclidean norm (which corresponds to our usual conception of distance in three-dimensional space): - $$ ||{\bf v}|| = \sqrt{{\bf v}\cdot{\bf v}} = \sqrt{ v_1v_1 + v_2v_2 + \cdots + v_nv_n}$$ +**Vector Norm**: The norm of a vector is a measure of its length. There are many different ways to calculate the norm, but the most common is the Euclidean norm (which corresponds to our usual conception of distance in three-dimensional space): $$ ||{\bf v}|| = \sqrt{{\bf v}\cdot{\bf v}} = \sqrt{ v_1v_1 + v_2v_2 + \cdots + v_nv_n}$$ -```{example} -#| name: Vector Algebra -#| label: vectors +:::{#exm-vectors} +## Vector Algebra Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{pmatrix}$. Calculate the following: @@ -43,11 +38,10 @@ Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{p 2. $a \cdot b$ -``` +::: -```{exercise} -#| name: Vector Algebra -#| label: vectors1 +:::{#exr-vectors1} +## Vector Algebra Let $u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}$, $v = \begin{pmatrix} 9&-3&2&8 \end{pmatrix}$, $w = \begin{pmatrix} 1&13& -7&2 &15 \end{pmatrix}$, and $c = 2$. Calculate the following: @@ -59,39 +53,34 @@ Let $u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}$, $v = \begin{pmatrix} 9&-3&2&8 \ 4. $w \cdot v$ -``` +::: ## Linear Independence {#linearindependence} -__Linear combinations__: The vector ${\bf u}$ is a linear combination of the vectors ${\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k$ if - $${\bf u} = c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k$$ +**Linear combinations**: The vector ${\bf u}$ is a linear combination of the vectors ${\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k$ if $${\bf u} = c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k$$ -For example, $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of the following three vectors: $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$. This is because $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ = $(2)\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$ $+ (-1)\begin{pmatrix} 2 & 3& 4\end{pmatrix}$ + $3\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$ +For example, $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of the following three vectors: $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$. This is because $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ = $(2)\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$ $+ (-1)\begin{pmatrix} 2 & 3& 4\end{pmatrix}$ + $3\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$ -__Linear independence__: A set of vectors ${\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k$ is linearly independent if the only solution to the equation - $$c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k = 0$$ - is $c_1 = c_2 = \cdots = c_k = 0$. If another solution exists, the set of vectors is linearly dependent. +**Linear independence**: A set of vectors ${\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k$ is linearly independent if the only solution to the equation $$c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k = 0$$ is $c_1 = c_2 = \cdots = c_k = 0$. If another solution exists, the set of vectors is linearly dependent. A set $S$ of vectors is linearly dependent if and only if at least one of the vectors in $S$ can be written as a linear combination of the other vectors in $S$. - Linear independence is only defined for sets of vectors with the same number of elements; any linearly independent set of vectors in $n$-space contains at most $n$ vectors. +Linear independence is only defined for sets of vectors with the same number of elements; any linearly independent set of vectors in $n$-space contains at most $n$ vectors. - Since $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$, these 4 vectors constitute a linearly dependent set. +Since $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$, these 4 vectors constitute a linearly dependent set. -```{example} -#| name: Linear Independence -#| label: linearindep +:::{#exm-linearindep} +## Linear Independence Are the following sets of vectors linearly independent? 1. $\begin{pmatrix}2 & 3 & 1 \end{pmatrix}$ and $\begin{pmatrix}4 & 6 & 1 \end{pmatrix}$ 2. $\begin{pmatrix}1 & 0 & 0 \end{pmatrix}$, $\begin{pmatrix}0 & 5 & 0 \end{pmatrix}$, and $\begin{pmatrix}10 & 10 & 0 \end{pmatrix}$ -``` +::: -```{exercise} -#| name: Linear Independence -#| label: linearindep1 +:::{#exr-linearindep1} +## Linear Independence Are the following sets of vectors linearly independent? @@ -99,101 +88,94 @@ Are the following sets of vectors linearly independent? 2. $${\bf v}_1 = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} -4 \\ 6 \\ 5 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} -2 \\ 8 \\ 6 \end{pmatrix} $$ -``` +::: ## Basics of Matrix Algebra {#matrixbasics} -__Matrix__: A matrix is an array of real numbers arranged in $m$ rows by $n$ columns. The dimensionality of the matrix is defined as the number of rows by the number of columns, $m \times n$. +**Matrix**: A matrix is an array of real numbers arranged in $m$ rows by $n$ columns. The dimensionality of the matrix is defined as the number of rows by the number of columns, $m \times n$. $${\bf A}=\begin{pmatrix} - a_{11} & a_{12} & \cdots & a_{1n} \\ - a_{21} & a_{22} & \cdots & a_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - a_{m1} & a_{m2} & \cdots & a_{mn} - \end{pmatrix}$$ + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & a_{22} & \cdots & a_{2n} \\ + \vdots & \vdots & \ddots & \vdots \\ + a_{m1} & a_{m2} & \cdots & a_{mn} + \end{pmatrix}$$ Note that you can think of vectors as special cases of matrices; a column vector of length $k$ is a $k \times 1$ matrix, while a row vector of the same length is a $1 \times k$ matrix. -It's also useful to think of matrices as being made up of a collection of row or column vectors. For example, - $$\bf A = \begin{pmatrix} {\bf a}_1 & {\bf a}_2 & \cdots & {\bf a}_m \end{pmatrix}$$ +It's also useful to think of matrices as being made up of a collection of row or column vectors. For example, $$\bf A = \begin{pmatrix} {\bf a}_1 & {\bf a}_2 & \cdots & {\bf a}_m \end{pmatrix}$$ -__Matrix Addition__: Let $\bf A$ and $\bf B$ be two $m\times n$ matrices. - $${\bf A+B}=\begin{pmatrix} - a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ - a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{mn}+b_{mn} - \end{pmatrix}$$ +**Matrix Addition**: Let $\bf A$ and $\bf B$ be two $m\times n$ matrices. $${\bf A+B}=\begin{pmatrix} + a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ + a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ + \vdots & \vdots & \ddots & \vdots \\ + a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{mn}+b_{mn} + \end{pmatrix}$$ -Note that matrices ${\bf A}$ and ${\bf B}$ must have the same dimensionality, in which case they are __conformable for addition__. +Note that matrices ${\bf A}$ and ${\bf B}$ must have the same dimensionality, in which case they are **conformable for addition**. + +:::{#exm-matrixaddition} -```{example} -#| label: matrixaddition $${\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & 2 \end{pmatrix}$$ ${\bf A+B}=$ -``` +::: -__Scalar Multiplication__: Given the scalar $s$, the scalar multiplication of $s {\bf A}$ is - $$ s {\bf A}= s \begin{pmatrix} - a_{11} & a_{12} & \cdots & a_{1n} \\ - a_{21} & a_{22} & \cdots & a_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - a_{m1} & a_{m2} & \cdots & a_{mn} - \end{pmatrix} - = \begin{pmatrix} - s a_{11} & s a_{12} & \cdots & s a_{1n} \\ - s a_{21} & s a_{22} & \cdots & s a_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - s a_{m1} & s a_{m2} & \cdots & s a_{mn} - \end{pmatrix}$$ +**Scalar Multiplication**: Given the scalar $s$, the scalar multiplication of $s {\bf A}$ is $$ s {\bf A}= s \begin{pmatrix} + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & a_{22} & \cdots & a_{2n} \\ + \vdots & \vdots & \ddots & \vdots \\ + a_{m1} & a_{m2} & \cdots & a_{mn} + \end{pmatrix} + = \begin{pmatrix} + s a_{11} & s a_{12} & \cdots & s a_{1n} \\ + s a_{21} & s a_{22} & \cdots & s a_{2n} \\ + \vdots & \vdots & \ddots & \vdots \\ + s a_{m1} & s a_{m2} & \cdots & s a_{mn} + \end{pmatrix}$$ -```{example} -#| label: scalarmulti +:::{#exm-scalarmulti} $s=2, \qquad {\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ $s {\bf A} =$ -``` +::: -__Matrix Multiplication__: If ${\bf A}$ is an $m\times k$ matrix and $\bf B$ is a $k\times n$ matrix, then their product $\bf C = A B$ is the $m\times n$ matrix where - $$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ik}b_{kj}$$ +**Matrix Multiplication**: If ${\bf A}$ is an $m\times k$ matrix and $\bf B$ is a $k\times n$ matrix, then their product $\bf C = A B$ is the $m\times n$ matrix where $$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ik}b_{kj}$$ -```{example} -#| label: matrixmulti +:::{#exm-matrixmulti} 1. $\begin{pmatrix} a&b\\c&d\\e&f \end{pmatrix} \begin{pmatrix} A&B\\C&D \end{pmatrix} =$ 2. $\begin{pmatrix} 1&2&-1\\3&1&4 \end{pmatrix} \begin{pmatrix} -2&5\\4&-3\\2&1\end{pmatrix} =$ -``` +::: + +Note that the number of columns of the first matrix must equal the number of rows of the second matrix, in which case they are **conformable for multiplication**. The sizes of the matrices (including the resulting product) must be $$(m\times k)(k\times n)=(m\times n)$$ -Note that the number of columns of the first matrix must equal the number of rows of the second matrix, in which case they are __conformable for multiplication__. The sizes of the matrices (including the resulting product) must be $$(m\times k)(k\times n)=(m\times n)$$ +Also note that if **AB** exists, **BA** exists only if $\dim({\bf A}) = m \times n$ and $\dim({\bf B}) = n \times m$. -Also note that if __AB__ exists, __BA__ exists only if $\dim({\bf A}) = m \times n$ and $\dim({\bf B}) = n \times m$. +This does not mean that **AB** = **BA**. **AB** = **BA** is true only in special circumstances, like when ${\bf A}$ or ${\bf B}$ is an identity matrix or ${\bf A} = {\bf B}^{-1}$. -This does not mean that __AB__ = __BA__. __AB__ = __BA__ is true only in special circumstances, like when ${\bf A}$ or ${\bf B}$ is an identity matrix or ${\bf A} = {\bf B}^{-1}$. +**Laws of Matrix Algebra**: -__Laws of Matrix Algebra__: +```{=tex} \begin{enumerate} - \item \parbox[t]{1.5in}{Associative:} $\bf (A+B)+C = A+(B+C)$\\ - \parbox[t]{1.5in}{\quad} $\bf (AB)C = A(BC)$ - \item \parbox[t]{1.5in}{Commutative:} $\bf A+B=B+A$ - \item \parbox[t]{1.5in}{Distributive:} $\bf A(B+C)=AB+AC$\\ - \parbox[t]{1.5in}{\quad} $\bf (A+B)C=AC+BC$ + \item \parbox[t]{1.5in}{Associative:} $\bf (A+B)+C = A+(B+C)$\\ + \parbox[t]{1.5in}{\quad} $\bf (AB)C = A(BC)$ + \item \parbox[t]{1.5in}{Commutative:} $\bf A+B=B+A$ + \item \parbox[t]{1.5in}{Distributive:} $\bf A(B+C)=AB+AC$\\ + \parbox[t]{1.5in}{\quad} $\bf (A+B)C=AC+BC$ \end{enumerate} +``` +Commutative law for multiplication does not hold -- the order of multiplication matters: $$\bf AB\ne BA$$ -Commutative law for multiplication does not hold -- the order of multiplication matters: - $$\bf AB\ne BA$$ - -For example, - $${\bf A}=\begin{pmatrix} 1&2\\-1&3\end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 2&1\\0&1\end{pmatrix}$$ - $${\bf AB}=\begin{pmatrix} 2&3\\-2&2\end{pmatrix}, \qquad {\bf BA}=\begin{pmatrix} 1&7\\-1&3\end{pmatrix}$$ +For example, $${\bf A}=\begin{pmatrix} 1&2\\-1&3\end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 2&1\\0&1\end{pmatrix}$$ $${\bf AB}=\begin{pmatrix} 2&3\\-2&2\end{pmatrix}, \qquad {\bf BA}=\begin{pmatrix} 1&7\\-1&3\end{pmatrix}$$ -__Transpose__: The transpose of the $m\times n$ matrix $\bf A$ is the $n\times m$ matrix ${\bf A}^T$ (also written ${\bf A}'$) obtained by interchanging the rows and columns of $\bf A$. +**Transpose**: The transpose of the $m\times n$ matrix $\bf A$ is the $n\times m$ matrix ${\bf A}^T$ (also written ${\bf A}'$) obtained by interchanging the rows and columns of $\bf A$. For example, @@ -202,21 +184,19 @@ ${\bf A}=\begin{pmatrix} 4&-2&3\\0&5&-1\end{pmatrix}, \qquad {\bf A}^T=\begin{pm ${\bf B}=\begin{pmatrix} 2\\-1\\3 \end{pmatrix}, \qquad {\bf B}^T=\begin{pmatrix} 2&-1&3\end{pmatrix}$ The following rules apply for transposed matrices: - \begin{enumerate} - \item $({\bf A+B})^T = {\bf A}^T+{\bf B}^T$ - \item $({\bf A}^T)^T={\bf A}$ - \item $(s{\bf A})^T = s{\bf A}^T$ - \item $({\bf AB})^T = {\bf B}^T{\bf A}^T$; and by induction $({\bf ABC})^T = {\bf C}^T{\bf B}^T{\bf A}^T$ -\end{enumerate} -Example of $({\bf AB})^T = {\bf B}^T{\bf A}^T$: - $${\bf A}=\begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix}$$ - $$ ({\bf AB})^T = \left[ \begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix} \begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix} \right]^T = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}$$ - $$ {\bf B}^T{\bf A}^T= \begin{pmatrix} 0&2&3\\1&2&-1 \end{pmatrix} \begin{pmatrix} 1&2\\3&-1\\2&3 \end{pmatrix} = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}$$ +```{=tex} +\begin{enumerate} + \item $({\bf A+B})^T = {\bf A}^T+{\bf B}^T$ + \item $({\bf A}^T)^T={\bf A}$ + \item $(s{\bf A})^T = s{\bf A}^T$ + \item $({\bf AB})^T = {\bf B}^T{\bf A}^T$; and by induction $({\bf ABC})^T = {\bf C}^T{\bf B}^T{\bf A}^T$ +\end{enumerate} +``` +Example of $({\bf AB})^T = {\bf B}^T{\bf A}^T$: $${\bf A}=\begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix}$$ $$ ({\bf AB})^T = \left[ \begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix} \begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix} \right]^T = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}$$ $$ {\bf B}^T{\bf A}^T= \begin{pmatrix} 0&2&3\\1&2&-1 \end{pmatrix} \begin{pmatrix} 1&2\\3&-1\\2&3 \end{pmatrix} = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}$$ -```{exercise} -#| name: Matrix Multiplication -#| label: matrixmulti1 +:::{#exr-matrixmulti1} +## Matrix Multiplication Let $$A = \begin{pmatrix} 2&0&-1&1\\1&2&0&1 \end{pmatrix}$$ @@ -231,57 +211,56 @@ Calculate the following: 3. $$(BC)^T$$ 4. $$BC^T$$ -``` +::: ## Systems of Linear Equations -__Linear Equation__: $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$ +**Linear Equation**: $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$ -$a_i$ are parameters or coefficients. $x_i$ are variables or unknowns. +$a_i$ are parameters or coefficients. $x_i$ are variables or unknowns. Linear because only one variable per term and degree is at most 1. We are often interested in solving linear systems like $$\begin{matrix} - x & - & 3y & = & -3\\ - 2x & + & y & = & 8 - \end{matrix}$$ + x & - & 3y & = & -3\\ + 2x & + & y & = & 8 + \end{matrix}$$ +```{=tex} \begin{comment} - \parbox[t]{1in}{\includegraphics[angle=270, width = 1in]{linsys.eps}} + \parbox[t]{1in}{\includegraphics[angle=270, width = 1in]{linsys.eps}} \end{comment} - +``` More generally, we might have a system of $m$ equations in $n$ unknowns $$\begin{matrix} - a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ - a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ - \vdots & & & & \vdots & & & \vdots & \\ - a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m - \end{matrix}$$ + a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ + a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ + \vdots & & & & \vdots & & & \vdots & \\ + a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m + \end{matrix}$$ -A __solution__ to a linear system of $m$ equations in $n$ unknowns is a set of $n$ numbers $x_1, x_2, \cdots, x_n$ that satisfy each of the $m$ equations. +A **solution** to a linear system of $m$ equations in $n$ unknowns is a set of $n$ numbers $x_1, x_2, \cdots, x_n$ that satisfy each of the $m$ equations. -Example: $x=3$ and $y=2$ is the solution to the above $2\times 2$ linear system. If you graph the two lines, you will find that they intersect at $(3,2)$. +Example: $x=3$ and $y=2$ is the solution to the above $2\times 2$ linear system. If you graph the two lines, you will find that they intersect at $(3,2)$. -Does a linear system have one, no, or multiple solutions? For a system of 2 equations with 2 unknowns (i.e., two lines): -_ -__One solution:__ The lines intersect at exactly one point. +Does a linear system have one, no, or multiple solutions? For a system of 2 equations with 2 unknowns (i.e., two lines): \_\ +**One solution:** The lines intersect at exactly one point. -__No solution:__ The lines are parallel. +**No solution:** The lines are parallel. -__Infinite solutions:__ The lines coincide. +**Infinite solutions:** The lines coincide. Methods to solve linear systems: -1. Substitution +1. Substitution 2. Elimination of variables 3. Matrix methods -```{exercise} -#| name: Linear Equations -#| label: lineareq +:::{#exr-lineareq} +## Linear Equations Provide a system of 2 equations with 2 unknowns that has @@ -291,43 +270,39 @@ Provide a system of 2 equations with 2 unknowns that has 3. infinite solutions -``` +::: ## Systems of Equations as Matrices -Matrices provide an easy and efficient way to represent linear systems such as - $$\begin{matrix} - a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ - a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ - \vdots & & & & \vdots & & & \vdots & \\ - a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m - \end{matrix}$$ +Matrices provide an easy and efficient way to represent linear systems such as $$\begin{matrix} + a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ + a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ + \vdots & & & & \vdots & & & \vdots & \\ + a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m + \end{matrix}$$ as $${\bf A x = b}$$ where - The $m \times n$ \textbf{coefficient matrix} ${\bf A}$ is an array of $m n$ real numbers arranged in $m$ rows by $n$ columns: - $${\bf A}=\begin{pmatrix} - a_{11} & a_{12} & \cdots & a_{1n} \\ - a_{21} & a_{22} & \cdots & a_{2n} \\ - \vdots & & \ddots & \vdots \\ - a_{m1} & a_{m2} & \cdots & a_{mn} - \end{pmatrix}$$ +The $m \times n$ \textbf{coefficient matrix} ${\bf A}$ is an array of $m n$ real numbers arranged in $m$ rows by $n$ columns: $${\bf A}=\begin{pmatrix} + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & a_{22} & \cdots & a_{2n} \\ + \vdots & & \ddots & \vdots \\ + a_{m1} & a_{m2} & \cdots & a_{mn} + \end{pmatrix}$$ The unknown quantities are represented by the vector ${\bf x}=\begin{pmatrix} x_1\\x_2\\\vdots\\x_n \end{pmatrix}$. The right hand side of the linear system is represented by the vector ${\bf b}=\begin{pmatrix} b_1\\b_2\\\vdots\\b_m \end{pmatrix}$. -__Augmented Matrix__: When we append $\bf b$ to the coefficient matrix $\bf A$, we get the augmented matrix $\widehat{\bf A}=[\bf A | b]$ - $$\begin{pmatrix} - a_{11} & a_{12} & \cdots & a_{1n} & | & b_1\\ - a_{21} & a_{22} & \cdots & a_{2n} & | & b_2\\ - \vdots & & \ddots & \vdots & | & \vdots\\ - a_{m1} & a_{m2} & \cdots & a_{mn} & | & b_m - \end{pmatrix}$$ +**Augmented Matrix**: When we append $\bf b$ to the coefficient matrix $\bf A$, we get the augmented matrix $\widehat{\bf A}=[\bf A | b]$ $$\begin{pmatrix} + a_{11} & a_{12} & \cdots & a_{1n} & | & b_1\\ + a_{21} & a_{22} & \cdots & a_{2n} & | & b_2\\ + \vdots & & \ddots & \vdots & | & \vdots\\ + a_{m1} & a_{m2} & \cdots & a_{mn} & | & b_m + \end{pmatrix}$$ -```{exercise} -#| name: Augmented Matrix -#| label: augmatrix +:::{#exr-augmatrix} +## Augmented Matrix Create an augmented matrix that represent the following system of equations: @@ -335,63 +310,54 @@ Create an augmented matrix that represent the following system of equations: $$41x_2 + 9x_3 -5x_6 = 11$$ $$x_1 -15x_2 -11x_5 = 9$$ -``` +::: ## Finding Solutions to Augmented Matrices and Systems of Equations -__Row Echelon Form__: Our goal is to translate our augmented matrix or system of equations into row echelon form. This will provide us with the values of the vector __x__ which solve the system. We use the row operations to change coefficients in the lower triangle of the augmented matrix to 0. An augmented matrix of the form +**Row Echelon Form**: Our goal is to translate our augmented matrix or system of equations into row echelon form. This will provide us with the values of the vector **x** which solve the system. We use the row operations to change coefficients in the lower triangle of the augmented matrix to 0. An augmented matrix of the form $$\begin{pmatrix} - \fbox{$a'_{11}$}& a'_{12} & a'_{13}& \cdots & a'_{1n} & | & b'_1\\ - 0 & \fbox{$a'_{22}$} & a'_{23}& \cdots & a'_{2n} & | & b'_2\\ - 0 & 0 & \fbox{$a'_{33}$}& \cdots & a'_{3n} & | & b'_3\\ - 0 & 0 &0 & \ddots & \vdots & | & \vdots \\ - 0 & 0 &0 &0 & \fbox{$a'_{mn}$} & | & b'_m - \end{pmatrix}$$ + \fbox{$a'_{11}$}& a'_{12} & a'_{13}& \cdots & a'_{1n} & | & b'_1\\ + 0 & \fbox{$a'_{22}$} & a'_{23}& \cdots & a'_{2n} & | & b'_2\\ + 0 & 0 & \fbox{$a'_{33}$}& \cdots & a'_{3n} & | & b'_3\\ + 0 & 0 &0 & \ddots & \vdots & | & \vdots \\ + 0 & 0 &0 &0 & \fbox{$a'_{mn}$} & | & b'_m + \end{pmatrix}$$ is said to be in row echelon form --- each row has more leading zeros than the row preceding it. -__Reduced Row Echelon Form__: We can go one step further and put the matrix into reduced row echelon form. Reduced row echelon form makes the value of __x__ which solves the system very obvious. For a system of $m$ equations in $m$ unknowns, with no all-zero rows, the reduced row echelon form would be +**Reduced Row Echelon Form**: We can go one step further and put the matrix into reduced row echelon form. Reduced row echelon form makes the value of **x** which solves the system very obvious. For a system of $m$ equations in $m$ unknowns, with no all-zero rows, the reduced row echelon form would be $$\begin{pmatrix} - \fbox{$1$} & 0 & 0 & 0 & 0 & | & b^*_1\\ - 0 & \fbox{$1$} & 0 & 0 & 0 & | & b^*_2\\ - 0 & 0 & \fbox{$1$} & 0 & 0 & | & b^*_3\\ - 0 & 0 & 0 &\ddots & 0 & | &\vdots\\ - 0 & 0 & 0 & 0 & \fbox{$1$} & | & b^*_m - \end{pmatrix}$$ - -__Gaussian and Gauss-Jordan elimination__: We can conduct elementary row operations to get our augmented matrix into row echelon or reduced row echelon form. The methods of transforming a matrix or system into row echelon and reduced row echelon form are referred to as Gaussian elimination and Gauss-Jordan elimination, respectively. - -__Elementary Row Operations__: To do Gaussian and Gauss-Jordan elimination, we use three basic operations to transform the augmented matrix into another augmented matrix that represents an equivalent linear system -- equivalent in the sense that the same values of $x_j$ solve both the original and transformed matrix/system: - -__Interchanging Rows__: Suppose we have the augmented matrix - $${\widehat{\bf A}}=\begin{pmatrix} a_{11} & a_{12} & | & b_1\\ - a_{21} & a_{22} & | & b_2 - \end{pmatrix}$$ - If we interchange the two rows, we get the augmented matrix - $$\begin{pmatrix} - a_{21} & a_{22} & | & b_2\\ - a_{11} & a_{12} & | & b_1 - \end{pmatrix}$$ - which represents a linear system equivalent to that represented by matrix $\widehat{\bf A}$. - -__Multiplying by a Constant__: If we multiply the second row of matrix $\widehat{\bf A}$ by a constant $c$, we get the augmented matrix - $$\begin{pmatrix} - a_{11} & a_{12} & | & b_1\\ - c a_{21} & c a_{22} & | & c b_2 - \end{pmatrix}$$ - which represents a linear system equivalent to that represented by matrix $\widehat{\bf A}$. - -__Adding (subtracting) Rows__: If we add (subtract) the first row of matrix $\widehat{\bf A}$ to the second, we obtain the augmented matrix - $$\begin{pmatrix} - a_{11} & a_{12} & | & b_1\\ - a_{11}+a_{21} & a_{12}+a_{22} & | & b_1+b_2 - \end{pmatrix}$$ - which represents a linear system equivalent to that represented by matrix $\widehat{\bf A}$. - -```{example} -#| label: solvesys + \fbox{$1$} & 0 & 0 & 0 & 0 & | & b^*_1\\ + 0 & \fbox{$1$} & 0 & 0 & 0 & | & b^*_2\\ + 0 & 0 & \fbox{$1$} & 0 & 0 & | & b^*_3\\ + 0 & 0 & 0 &\ddots & 0 & | &\vdots\\ + 0 & 0 & 0 & 0 & \fbox{$1$} & | & b^*_m + \end{pmatrix}$$ + +**Gaussian and Gauss-Jordan elimination**: We can conduct elementary row operations to get our augmented matrix into row echelon or reduced row echelon form. The methods of transforming a matrix or system into row echelon and reduced row echelon form are referred to as Gaussian elimination and Gauss-Jordan elimination, respectively. + +**Elementary Row Operations**: To do Gaussian and Gauss-Jordan elimination, we use three basic operations to transform the augmented matrix into another augmented matrix that represents an equivalent linear system -- equivalent in the sense that the same values of $x_j$ solve both the original and transformed matrix/system: + +**Interchanging Rows**: Suppose we have the augmented matrix $${\widehat{\bf A}}=\begin{pmatrix} a_{11} & a_{12} & | & b_1\\ + a_{21} & a_{22} & | & b_2 + \end{pmatrix}$$ If we interchange the two rows, we get the augmented matrix $$\begin{pmatrix} + a_{21} & a_{22} & | & b_2\\ + a_{11} & a_{12} & | & b_1 + \end{pmatrix}$$ which represents a linear system equivalent to that represented by matrix $\widehat{\bf A}$. + +**Multiplying by a Constant**: If we multiply the second row of matrix $\widehat{\bf A}$ by a constant $c$, we get the augmented matrix $$\begin{pmatrix} + a_{11} & a_{12} & | & b_1\\ + c a_{21} & c a_{22} & | & c b_2 + \end{pmatrix}$$ which represents a linear system equivalent to that represented by matrix $\widehat{\bf A}$. + +**Adding (subtracting) Rows**: If we add (subtract) the first row of matrix $\widehat{\bf A}$ to the second, we obtain the augmented matrix $$\begin{pmatrix} + a_{11} & a_{12} & | & b_1\\ + a_{11}+a_{21} & a_{12}+a_{22} & | & b_1+b_2 + \end{pmatrix}$$ which represents a linear system equivalent to that represented by matrix $\widehat{\bf A}$. + +:::{#exm-solvesys} Solve the following system of equations by using elementary row operations: @@ -400,11 +366,10 @@ $\begin{matrix} 2x & + & y & = & 8 \end{matrix}$ -``` +::: -```{exercise} -#| name: Solving Systems of Equations -#| label: solvesys1 +:::{#exr-solvesys1} +## Solving Systems of Equations Put the following system of equations into augmented matrix form. Then, using Gaussian or Gauss-Jordan elimination, solve the system of equations by putting the matrix into row echelon or reduced row echelon form. @@ -424,31 +389,30 @@ $$ \end{cases} $$ -``` +::: ## Rank --- and Whether a System Has One, Infinite, or No Solutions -To determine how many solutions exist, we can use information about (1) the number of equations $m$, (2) the number of unknowns $n$, and (3) the __rank__ of the matrix representing the linear system. +To determine how many solutions exist, we can use information about (1) the number of equations $m$, (2) the number of unknowns $n$, and (3) the **rank** of the matrix representing the linear system. -__Rank__: The maximum number of linearly independent row or column vectors in the matrix. This is equivalent to the number of nonzero rows of a matrix in row echelon form. For any matrix __A__, the row rank always equals column rank, and we refer to this number as the rank of __A__. +**Rank**: The maximum number of linearly independent row or column vectors in the matrix. This is equivalent to the number of nonzero rows of a matrix in row echelon form. For any matrix **A**, the row rank always equals column rank, and we refer to this number as the rank of **A**. For example $\begin{pmatrix} 1 & 2 & 3 \\ - 0 & 4 & 5 \\ - 0 & 0 & 6 \end{pmatrix}$ + 0 & 4 & 5 \\ + 0 & 0 & 6 \end{pmatrix}$ Rank = 3 -$\begin{pmatrix} 1 & 2 & 3 \\ +$\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 0 \end{pmatrix}$ Rank = 2 -```{exercise} -#| name: Rank of Matrices -#| label: rank +:::{#exr-rank} +## Rank of Matrices Find the rank of each matrix below: @@ -465,93 +429,90 @@ Find the rank of each matrix below: 1 & 3 & 2 & -1 & -2 \\ 1 & 3 & 0 & 3 & -2 \end{pmatrix}$ -``` +::: -Answer to Exercise \@ref(exr:rank): +Answer to Exercise @exr-rank: -1. rank is 2 +1. rank is 2 -2. rank is 3 +2. rank is 3 ## The Inverse of a Matrix -__Identity Matrix__: The $n\times n$ identity matrix ${\bf I}_n$ is the matrix whose diagonal elements are 1 and all off-diagonal elements are 0. Examples: - $$ {\bf I}_2=\begin{pmatrix} 1&0\\0&1 \end{pmatrix}, \qquad {\bf I}_3=\begin{pmatrix} 1&0&0\\ 0&1&0\\ - 0&0&1 \end{pmatrix}$$ -__Inverse Matrix__: An $n\times n$ matrix ${\bf A}$ is __nonsingular__ or __invertible__ if there exists an $n\times n$ matrix ${\bf A}^{-1}$ such that $${\bf A} {\bf A}^{-1} = {\bf A}^{-1} {\bf A} = {\bf I}_n$$ where ${\bf A}^{-1}$ is the inverse of ${\bf A}$. If there is no such ${\bf A}^{-1}$, then ${\bf A}$ is singular or not invertible. +**Identity Matrix**: The $n\times n$ identity matrix ${\bf I}_n$ is the matrix whose diagonal elements are 1 and all off-diagonal elements are 0. Examples: $$ {\bf I}_2=\begin{pmatrix} 1&0\\0&1 \end{pmatrix}, \qquad {\bf I}_3=\begin{pmatrix} 1&0&0\\ 0&1&0\\ + 0&0&1 \end{pmatrix}$$ -Example: Let - $${\bf A} = \begin{pmatrix} 2&3\\2&2 \end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} -1&\frac{3}{2}\\ 1&-1 - \end{pmatrix}$$ - Since $${\bf A} {\bf B} = {\bf B} {\bf A} = {\bf I}_n$$ we conclude that ${\bf B}$ is the inverse, ${\bf A}^{-1}$, of ${\bf A}$ and that ${\bf A}$ is nonsingular. +**Inverse Matrix**: An $n\times n$ matrix ${\bf A}$ is **nonsingular** or **invertible** if there exists an $n\times n$ matrix ${\bf A}^{-1}$ such that $${\bf A} {\bf A}^{-1} = {\bf A}^{-1} {\bf A} = {\bf I}_n$$ where ${\bf A}^{-1}$ is the inverse of ${\bf A}$. If there is no such ${\bf A}^{-1}$, then ${\bf A}$ is singular or not invertible. + +Example: Let $${\bf A} = \begin{pmatrix} 2&3\\2&2 \end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} -1&\frac{3}{2}\\ 1&-1 + \end{pmatrix}$$ Since $${\bf A} {\bf B} = {\bf B} {\bf A} = {\bf I}_n$$ we conclude that ${\bf B}$ is the inverse, ${\bf A}^{-1}$, of ${\bf A}$ and that ${\bf A}$ is nonsingular. -__Properties of the Inverse__: +**Properties of the Inverse**: -- If the inverse exists, it is unique. +- If the inverse exists, it is unique. -- If ${\bf A}$ is nonsingular, then ${\bf A}^{-1}$ is nonsingular. +- If ${\bf A}$ is nonsingular, then ${\bf A}^{-1}$ is nonsingular. -- $({\bf A}^{-1})^{-1} = {\bf A}$ +- $({\bf A}^{-1})^{-1} = {\bf A}$ -- If ${\bf A}$ and ${\bf B}$ are nonsingular, then ${\bf A}{\bf B}$ is nonsingular +- If ${\bf A}$ and ${\bf B}$ are nonsingular, then ${\bf A}{\bf B}$ is nonsingular -- $({\bf A}{\bf B})^{-1} = {\bf B}^{-1}{\bf A}^{-1}$ +- $({\bf A}{\bf B})^{-1} = {\bf B}^{-1}{\bf A}^{-1}$ -- If ${\bf A}$ is nonsingular, then $({\bf A}^T)^{-1}=({\bf A}^{-1})^T$ +- If ${\bf A}$ is nonsingular, then $({\bf A}^T)^{-1}=({\bf A}^{-1})^T$ -__Procedure to Find__ ${\bf A}^{-1}$: We know that if ${\bf B}$ is the inverse of ${\bf A}$, then $${\bf A} {\bf B} = {\bf B} {\bf A} = {\bf I}_n$$ Looking only at the first and last parts of this $${\bf A} {\bf B} = {\bf I}_n$$ Solving for ${\bf B}$ is equivalent to solving for $n$ linear systems, where each column of ${\bf B}$ is solved for the corresponding column in ${\bf I}_n$. We can solve the systems simultaneously by augmenting ${\bf A}$ with ${\bf I}_n$ and performing Gauss-Jordan elimination on ${\bf A}$. If Gauss-Jordan elimination on $[{\bf A} | {\bf I}_n]$ results in $[{\bf I}_n | {\bf B} ]$, then ${\bf B}$ is the inverse of ${\bf A}$. Otherwise, ${\bf A}$ is singular. +**Procedure to Find** ${\bf A}^{-1}$: We know that if ${\bf B}$ is the inverse of ${\bf A}$, then $${\bf A} {\bf B} = {\bf B} {\bf A} = {\bf I}_n$$ Looking only at the first and last parts of this $${\bf A} {\bf B} = {\bf I}_n$$ Solving for ${\bf B}$ is equivalent to solving for $n$ linear systems, where each column of ${\bf B}$ is solved for the corresponding column in ${\bf I}_n$. We can solve the systems simultaneously by augmenting ${\bf A}$ with ${\bf I}_n$ and performing Gauss-Jordan elimination on ${\bf A}$. If Gauss-Jordan elimination on $[{\bf A} | {\bf I}_n]$ results in $[{\bf I}_n | {\bf B} ]$, then ${\bf B}$ is the inverse of ${\bf A}$. Otherwise, ${\bf A}$ is singular. -To summarize: To calculate the inverse of ${\bf A}$ +To summarize: To calculate the inverse of ${\bf A}$ -1. Form the augmented matrix $[ {\bf A} | {\bf I}_n]$ +1. Form the augmented matrix $[ {\bf A} | {\bf I}_n]$ -2. Using elementary row operations, transform the augmented matrix to reduced row echelon form. +2. Using elementary row operations, transform the augmented matrix to reduced row echelon form. -3. The result of step 2 is an augmented matrix $[ {\bf C} | {\bf B} ]$. +3. The result of step 2 is an augmented matrix $[ {\bf C} | {\bf B} ]$. - a. If ${\bf C}={\bf I}_n$, then ${\bf B}={\bf A}^{-1}$. + a. If ${\bf C}={\bf I}_n$, then ${\bf B}={\bf A}^{-1}$. - b. If ${\bf C}\ne{\bf I}_n$, then $\bf C$ has a row of zeros. This means ${\bf A}$ is singular and ${\bf A}^{-1}$ does not exist. + b. If ${\bf C}\ne{\bf I}_n$, then $\bf C$ has a row of zeros. This means ${\bf A}$ is singular and ${\bf A}^{-1}$ does not exist. -```{example} -#| label: inverse +:::{#exm-inverse} Find the inverse of the following matricies: 1. ${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$ -``` +::: -```{exercise} -#| name: Finding the inverse of matrices -#| label: inverse1 +:::{#exr-inverse1} +## Finding the inverse of matrices Find the inverse of the following matrix: 1. ${\bf A}=\begin{pmatrix} 1&0&4\\0&2&0\\0&0&1 \end{pmatrix}$ -``` +::: ## Linear Systems and Inverses -Let's return to the matrix representation of a linear system +Let's return to the matrix representation of a linear system $$\bf{Ax} = \bf{b}$$ -If $\bf{A}$ is an $n\times n$ matrix,then $\bf{Ax}=\bf{b}$ is a system of $n$ equations in $n$ unknowns. Suppose $\bf{A}$ is nonsingular. Then $\bf{A}^{-1}$ exists. To solve this system, we can multiply each side by $\bf{A}^{-1}$ and reduce it as follows: +If $\bf{A}$ is an $n\times n$ matrix,then $\bf{Ax}=\bf{b}$ is a system of $n$ equations in $n$ unknowns. Suppose $\bf{A}$ is nonsingular. Then $\bf{A}^{-1}$ exists. To solve this system, we can multiply each side by $\bf{A}^{-1}$ and reduce it as follows: +```{=tex} \begin{eqnarray*} \bf{A}^{-1} (\bf{A} \bf{x}) & = & \bf{A}^{-1} \bf{b} \\ (\bf{A}^{-1} \bf{A})\bf{x} & = & \bf{A}^{-1} \bf{b}\\ \bf{I}_n \bf{x} & = & \bf{A}^{-1} \bf{b}\\ \bf{x} & = & \bf{A}^{-1} \bf{b} \end{eqnarray*} - +``` Hence, given $\bf{A}$ and $\bf{b}$ and given that $\bf{A}$ is nonsingular, then $\bf{x} = \bf{A}^{-1} \bf{b}$ is a unique solution to this system. -```{exercise} -#| label: invlinsys -#| name: Solve linear system using inverses +:::{#exr-invlinsys} + +## Solve linear system using inverses Use the inverse matrix to solve the following linear system: @@ -570,56 +531,49 @@ $$\textbf{z} = \begin{pmatrix} x\\y \end{pmatrix},$$ and $$\textbf{b} = \begin{pmatrix} 5\\-10 \end{pmatrix}$$ -``` +::: ## Determinants -__Singularity__: Determinants can be used to determine whether a square matrix is nonsingular. +**Singularity**: Determinants can be used to determine whether a square matrix is nonsingular. - A square matrix is nonsingular if and only if its determinant is not zero. +A square matrix is nonsingular if and only if its determinant is not zero. -Determinant of a $1 \times 1$ matrix, __A__, equals $a_{11}$ +Determinant of a $1 \times 1$ matrix, **A**, equals $a_{11}$ -Determinant of a $2 \times 2$ matrix, __A__, - $\begin{vmatrix} a_{11}&a_{12}\\ - a_{21}&a_{22} \end{vmatrix}$: +Determinant of a $2 \times 2$ matrix, **A**, $\begin{vmatrix} a_{11}&a_{12}\\ + a_{21}&a_{22} \end{vmatrix}$: +```{=tex} \begin{eqnarray*} \det({\bf A}) &=& |{\bf A}|\\ - &=& a_{11}|a_{22}| - a_{12}|a_{21}|\\ - &=& a_{11}a_{22} - a_{12}a_{21} + &=& a_{11}|a_{22}| - a_{12}|a_{21}|\\ + &=& a_{11}a_{22} - a_{12}a_{21} \end{eqnarray*} - +``` We can extend the second to last equation above to get the definition of the determinant of a $3 \times 3$ matrix: +```{=tex} \begin{eqnarray*} - \begin{vmatrix} a_{11}&a_{12}&a_{13}\\ a_{21} & a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{vmatrix} - &=& - a_{11} \begin{vmatrix} a_{22}&a_{23}\\ a_{32}&a_{33} \end{vmatrix} - - a_{12} \begin{vmatrix} a_{21}&a_{23}\\ a_{31}&a_{33} \end{vmatrix} - + a_{13} \begin{vmatrix} a_{21}&a_{22}\\ a_{31}&a_{32} - \end{vmatrix}\\ - &=& a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) + \begin{vmatrix} a_{11}&a_{12}&a_{13}\\ a_{21} & a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{vmatrix} + &=& + a_{11} \begin{vmatrix} a_{22}&a_{23}\\ a_{32}&a_{33} \end{vmatrix} + - a_{12} \begin{vmatrix} a_{21}&a_{23}\\ a_{31}&a_{33} \end{vmatrix} + + a_{13} \begin{vmatrix} a_{21}&a_{22}\\ a_{31}&a_{32} + \end{vmatrix}\\ + &=& a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \end{eqnarray*} +``` +Let's extend this now to any $n\times n$ matrix. Let's define ${\bf A}_{ij}$ as the $(n-1)\times (n-1)$ submatrix of ${\bf A}$ obtained by deleting row $i$ and column $j$. Let the $(i,j)$th **minor** of ${\bf A}$ be the determinant of ${\bf A}_{ij}$: $$M_{ij}=|{\bf A}_{ij}|$$ Then for any $n\times n$ matrix ${\bf A}$ $$|{\bf A}|= a_{11}M_{11} - a_{12}M_{12} + \cdots + (-1)^{n+1} a_{1n} M_{1n}$$ + +For example, in figuring out whether the following matrix has an inverse? $${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$$ 1. Calculate its determinant. \begin{eqnarray} + &=& 1(2-15) - 1(0-15) + 1(0-10) \nonumber\\ + &=& -13+15-10 \nonumber\\ + &=& -8\nonumber +\end{eqnarray} 2. Since $|{\bf A}|\ne 0$, we conclude that ${\bf A}$ has an inverse. -Let's extend this now to any $n\times n$ matrix. Let's define ${\bf A}_{ij}$ as the $(n-1)\times (n-1)$ submatrix of ${\bf A}$ obtained by deleting row $i$ and column $j$. Let the $(i,j)$th __minor__ of ${\bf A}$ be the determinant of ${\bf A}_{ij}$: - $$M_{ij}=|{\bf A}_{ij}|$$ - Then for any $n\times n$ matrix ${\bf A}$ - $$|{\bf A}|= a_{11}M_{11} - a_{12}M_{12} + \cdots + (-1)^{n+1} a_{1n} M_{1n}$$ - -For example, in figuring out whether the following matrix has an inverse? - $${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$$ -1. Calculate its determinant. -\begin{eqnarray} - &=& 1(2-15) - 1(0-15) + 1(0-10) \nonumber\\ - &=& -13+15-10 \nonumber\\ - &=& -8\nonumber -\end{eqnarray} -2. Since $|{\bf A}|\ne 0$, we conclude that ${\bf A}$ has an inverse. - -```{exercise} -#| name: Determinants and Inverses -#| label: determinants +:::{#exr-determinants} +## Determinants and Inverses Determine whether the following matrices are nonsingular: @@ -635,59 +589,56 @@ $$2. \begin{pmatrix} 4 & 1 & 4 \end{pmatrix}$$ -``` +::: ## Getting Inverse of a Matrix using its Determinant Thus far, we have a number of algorithms to -1. Find the solution of a linear system, -2. Find the inverse of a matrix +1. Find the solution of a linear system, +2. Find the inverse of a matrix -but these remain just that --- algorithms. At this point, we have no way of telling how the solutions $x_j$ change as the parameters $a_{ij}$ and $b_i$ change, except by changing the values and "rerunning" the algorithms. +but these remain just that --- algorithms. At this point, we have no way of telling how the solutions $x_j$ change as the parameters $a_{ij}$ and $b_i$ change, except by changing the values and "rerunning" the algorithms. -With determinants, we can provide an explicit formula for the inverse and -therefore provide an explicit formula for the solution of an $n\times n$ linear system. +With determinants, we can provide an explicit formula for the inverse and therefore provide an explicit formula for the solution of an $n\times n$ linear system. Hence, we can examine how changes in the parameters and $b_i$ affect the solutions $x_j$. -__Determinant Formula for the Inverse of a $2 \times 2$__: +**Determinant Formula for the Inverse of a** $2 \times 2$: -The determinant of a $2 \times 2$ matrix __A__ $\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}$ is defined as: -$$\frac{1}{\det({\bf A})} \begin{pmatrix} - d & -b\\ - -c & a\\ - \end{pmatrix}$$ +The determinant of a $2 \times 2$ matrix **A** $\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}$ is defined as: $$\frac{1}{\det({\bf A})} \begin{pmatrix} + d & -b\\ + -c & a\\ + \end{pmatrix}$$ -For example, Let's calculate the inverse of matrix A from Exercise \@ref(exr:invlinsys) using the determinant formula. +For example, Let's calculate the inverse of matrix A from Exercise @exr-invlinsys using the determinant formula. -Recall, +Recall, $$A = \begin{pmatrix} - -3 & 4\\ - 2 & -1\\ - \end{pmatrix}$$ + -3 & 4\\ + 2 & -1\\ + \end{pmatrix}$$ $$\det({\bf A}) = (-3)(-1) - (4)(2) = 3 - 8 = -5$$ $$\frac{1}{\det({\bf A})} \begin{pmatrix} - -1 & -4\\ - -2 & -3\\ - \end{pmatrix}$$ + -1 & -4\\ + -2 & -3\\ + \end{pmatrix}$$ $$\frac{1}{-5} \begin{pmatrix} - -1 & -4\\ - -2 & -3\\ - \end{pmatrix}$$ + -1 & -4\\ + -2 & -3\\ + \end{pmatrix}$$ $$ \begin{pmatrix} - \frac{1}{5} & \frac{4}{5}\\ - \frac{2}{5} & \frac{3}{5}\\ - \end{pmatrix}$$ + \frac{1}{5} & \frac{4}{5}\\ + \frac{2}{5} & \frac{3}{5}\\ + \end{pmatrix}$$ -```{exercise} -#| name: Calculate Inverse using Determinant Formula -#| label: calcinverse +:::{#exr-calcinverse} +## Calculate Inverse using Determinant Formula Caculate the inverse of A @@ -696,193 +647,192 @@ $$A = \begin{pmatrix} -7 & 2\\ \end{pmatrix}$$ -``` +::: ## Answers to Examples and Exercises {.unnumbered} -Answer to Example \@ref(exm:vectors): +Answer to Example @exm-vectors: - 1. $\begin{pmatrix} -1 &-3&-3 \end{pmatrix}$ - 2. 6 + 4 + 10 = 20 +1. $\begin{pmatrix} -1 &-3&-3 \end{pmatrix}$ +2. 6 + 4 + 10 = 20 -Answer to Exercise \@ref(exr:vectors1): +Answer to Exercise @exr-vectors1: - 1. $\begin{pmatrix} -2 &4&-7&-5 \end{pmatrix}$ - 2. $\begin{pmatrix} 2 &26&-14&4&30 \end{pmatrix}$ - 3. 63 -3 -10 + 24 = 74 - 4. undefined +1. $\begin{pmatrix} -2 &4&-7&-5 \end{pmatrix}$ +2. $\begin{pmatrix} 2 &26&-14&4&30 \end{pmatrix}$ +3. 63 -3 -10 + 24 = 74 +4. undefined -Answer to Example \@ref(exm:linearindep): +Answer to Example @exm-linearindep: - 1. yes - 2. no +1. yes +2. no -Answer to Exercise \@ref(exr:linearindep1): +Answer to Exercise @exr-linearindep1: - 1. yes - 2. no ($-v_1 -v_2 + v_3 = 0$) +1. yes +2. no ($-v_1 -v_2 + v_3 = 0$) -Answer to Example \@ref(exm:matrixaddition): +Answer to Example @exm-matrixaddition: ${\bf A+B}=\begin{pmatrix} 2 & 4 & 4 \\ 6 & 6 & 8 \end{pmatrix}$ -Answer to Example \@ref(exm:scalarmulti): +Answer to Example @exm-scalarmulti: $s {\bf A} = \begin{pmatrix} 2 & 4 & 6 \\ 8 & 10 & 12 \end{pmatrix}$ -Answer to Example \@ref(exm:matrixmulti): +Answer to Example @exm-matrixmulti: -1. $\begin{pmatrix} aA+bC&aB+bD\\cA+dC&cB+dD\\eA+fC&eB+fD \end{pmatrix}$ +1. $\begin{pmatrix} aA+bC&aB+bD\\cA+dC&cB+dD\\eA+fC&eB+fD \end{pmatrix}$ -2. $\begin{pmatrix} 1(-2)+2(4)-1(2)&1(5)+2(-3)-1(1)\\ - 3(-2)+1(4)+4(2)&3(5)+1(-3)+4(1)\end{pmatrix} = - \begin{pmatrix} 4&-2\\6&16\end{pmatrix}$ +2. $\begin{pmatrix} 1(-2)+2(4)-1(2)&1(5)+2(-3)-1(1)\\ + 3(-2)+1(4)+4(2)&3(5)+1(-3)+4(1)\end{pmatrix} = + \begin{pmatrix} 4&-2\\6&16\end{pmatrix}$ -Answer to Exercise \@ref(exr:matrixmulti1): +Answer to Exercise @exr-matrixmulti1: -1. $AB = \begin{pmatrix} 4 & 11 & -15 \\ 5 & 7 & -7 \end{pmatrix}$ +1. $AB = \begin{pmatrix} 4 & 11 & -15 \\ 5 & 7 & -7 \end{pmatrix}$ -2. $BA =$ undefined +2. $BA =$ undefined -3. $(BC)^T =$ undefined +3. $(BC)^T =$ undefined -4. $BC^T = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix}\begin{pmatrix} 3&0\\2&4\\-1&6 \end{pmatrix} =\begin{pmatrix}20 & -22 \\ 5 & 4 \\ -3 &2 \\6 & 0\end{pmatrix}$ +4. $BC^T = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix}\begin{pmatrix} 3&0\\2&4\\-1&6 \end{pmatrix} =\begin{pmatrix}20 & -22 \\ 5 & 4 \\ -3 &2 \\6 & 0\end{pmatrix}$ -Answer to Exercise \@ref(exr:lineareq): +Answer to Exercise @exr-lineareq: There are many answers to this. Some possible simple ones are as follows: -1. One solution: $$\begin{matrix} - -x & + & y & = & 0\\ - x & + & y & = & 2 - \end{matrix}$$ +1. One solution: $$\begin{matrix} + -x & + & y & = & 0\\ + x & + & y & = & 2 + \end{matrix}$$ 2. No solution: $$\begin{matrix} - -x & + & y & = & 0\\ - x & - & y & = & 2 - \end{matrix}$$ + -x & + & y & = & 0\\ + x & - & y & = & 2 + \end{matrix}$$ -3. Infinite solutions: $$\begin{matrix} - -x & + & y & = & 0\\ - 2x & - & 2y & = & 0 - \end{matrix}$$ +3. Infinite solutions: $$\begin{matrix} + -x & + & y & = & 0\\ + 2x & - & 2y & = & 0 + \end{matrix}$$ -Answer to Exercise \@ref(exr:augmatrix): +Answer to Exercise @exr-augmatrix: $\begin{pmatrix} - 2 & -7 & 9 & -4 & 0 & 0| & 8\\ - 0 & 41 & 9 & 0 & 0 & 5 | & 11\\ - 1 & -15 & 0 & 0 & -11 & 0 | & 9 - \end{pmatrix}$ + 2 & -7 & 9 & -4 & 0 & 0| & 8\\ + 0 & 41 & 9 & 0 & 0 & 5 | & 11\\ + 1 & -15 & 0 & 0 & -11 & 0 | & 9 + \end{pmatrix}$ -Answer to Example \@ref(exm:solvesys): +Answer to Example @exm-solvesys: $$\begin{matrix} - x & - & 3y & = & -3\\ - 2x & + & y & = & 8 - \end{matrix}$$ + x & - & 3y & = & -3\\ + 2x & + & y & = & 8 + \end{matrix}$$ $$\begin{matrix} - x & - & 3y & = & -3\\ - & & 7y & = & 14\\ - \end{matrix}$$ + x & - & 3y & = & -3\\ + & & 7y & = & 14\\ + \end{matrix}$$ $$\begin{matrix} - x & - & 3y & = & -3\\ - & & y & = & 2\\ - \end{matrix}$$ + x & - & 3y & = & -3\\ + & & y & = & 2\\ + \end{matrix}$$ $$\begin{matrix} - x & = & 3\\ - y & = & 2\\ - \end{matrix}$$ + x & = & 3\\ + y & = & 2\\ + \end{matrix}$$ -Answer to Exercise \@ref(exr:solvesys1): +Answer to Exercise @exr-solvesys1: -1. x = 2, y = 2, z = -1 +1. x = 2, y = 2, z = -1 -2. x = -17, y = -3, z = -35 +2. x = -17, y = -3, z = -35 -Answer to Exercise \@ref(exr:rank): +Answer to Exercise @exr-rank: -1. rank is 2 +1. rank is 2 -2. rank is 3 +2. rank is 3 -Answer to Example \@ref(exm:inverse): +Answer to Example @exm-inverse: $\left(\begin{array}{ccc|ccc} - 1&1&1&1&0&0\\ - 0&2&3&0&1&0\\ - 5&5&1&0&0&1 + 1&1&1&1&0&0\\ + 0&2&3&0&1&0\\ + 5&5&1&0&0&1 \end{array} \right)$ $\left(\begin{array}{ccc|ccc} - 1&1&1 &1 &0&0\\ - 0&2&3 &0 &1&0\\ - 0&0&-4&-5&0&1 + 1&1&1 &1 &0&0\\ + 0&2&3 &0 &1&0\\ + 0&0&-4&-5&0&1 \end{array} \right)$ $\left(\begin{array}{ccc|ccc} - 1&1&1&1 &0&0\\ - 0&2&3&0 &1&0\\ - 0&0&1&5/4&0&-1/4 + 1&1&1&1 &0&0\\ + 0&2&3&0 &1&0\\ + 0&0&1&5/4&0&-1/4 \end{array} \right)$ $\left(\begin{array}{ccc|ccc} - 1&1&0&-1/4 &0&1/4\\ - 0&2&0&-15/4&1&3/4\\ - 0&0&1&5/4 &0&-1/4 + 1&1&0&-1/4 &0&1/4\\ + 0&2&0&-15/4&1&3/4\\ + 0&0&1&5/4 &0&-1/4 \end{array} \right)$ $\left(\begin{array}{ccc|ccc} - 1&1&0&-1/4 &0 &1/4\\ - 0&1&0&-15/8&1/2&3/8\\ - 0&0&1&5/4 &0 &-1/4 + 1&1&0&-1/4 &0 &1/4\\ + 0&1&0&-15/8&1/2&3/8\\ + 0&0&1&5/4 &0 &-1/4 \end{array} \right)$ $\left(\begin{array}{ccc|ccc} - 1&0&0&13/8 &-1/2&-1/8\\ - 0&1&0&-15/8&1/2 &3/8\\ - 0&0&1&5/4 &0 &-1/4 + 1&0&0&13/8 &-1/2&-1/8\\ + 0&1&0&-15/8&1/2 &3/8\\ + 0&0&1&5/4 &0 &-1/4 \end{array} \right)$ ${\bf A}^{-1} = \left(\begin{array}{ccc} - 13/8 &-1/2&-1/8\\ - -15/8&1/2 &3/8\\ - 5/4 &0 &-1/4 + 13/8 &-1/2&-1/8\\ + -15/8&1/2 &3/8\\ + 5/4 &0 &-1/4 \end{array} \right)$ -Answer to Exercise \@ref(exr:inverse1): +Answer to Exercise @exr-inverse1: -1. ${\bf A}^{-1}=\begin{pmatrix} 1&0&-4\\0&\frac{1}{2}&0\\0&0&1 \end{pmatrix}$ +1. ${\bf A}^{-1}=\begin{pmatrix} 1&0&-4\\0&\frac{1}{2}&0\\0&0&1 \end{pmatrix}$ -Answer to Exercise \@ref(exr:invlinsys): +Answer to Exercise @exr-invlinsys: $\textbf{z} = \bf{A}^{-1} \bf{b} = \begin{pmatrix} - 1/5 &4/5\\ - 2/5&3/5 + 1/5 &4/5\\ + 2/5&3/5 \end{pmatrix} \begin{pmatrix} - 5 \\ - -10 + 5 \\ + -10 \end{pmatrix}= \begin{pmatrix} - -7 \\ - -4 + -7 \\ + -4 \end{pmatrix} = \begin{pmatrix} - x \\ - y + x \\ + y \end{pmatrix}$ -Answer to Exercise \@ref(exr:determinants): +Answer to Exercise @exr-determinants: -1. nonsingular +1. nonsingular -2. singular +2. singular -Answer to Exercise \@ref(exr:calcinverse): +Answer to Exercise @exr-calcinverse: $\begin{pmatrix} - \frac{2}{41} & \frac{-5}{41}\\ - \frac{7}{41} & \frac{3}{41}\\ - \end{pmatrix}$ - + \frac{2}{41} & \frac{-5}{41}\\ + \frac{7}{41} & \frac{3}{41}\\ + \end{pmatrix}$ diff --git a/_quarto.yml b/_quarto.yml index 11f2d65..e7342c2 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -17,8 +17,8 @@ book: - 04_calculus.qmd - 05_optimization.qmd - 06_probability.qmd - - + - 07_linear-algebra.qmd + delete_merged_file: true language: ui: From 3351b29cd7bee8212c584045648d4d24c54d6a17 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 22:10:09 -0400 Subject: [PATCH 16/34] use visual mode as styling --- 01_warmup.qmd | 99 +++--- 02_functions.qmd | 350 ++++++++++----------- 03_limits.qmd | 211 ++++++------- 04_calculus.qmd | 552 ++++++++++++++++---------------- 06_probability.qmd | 714 ++++++++++++++++++++---------------------- 07_linear-algebra.qmd | 184 +++++------ 6 files changed, 981 insertions(+), 1129 deletions(-) diff --git a/01_warmup.qmd b/01_warmup.qmd index be5429f..2fb6a29 100644 --- a/01_warmup.qmd +++ b/01_warmup.qmd @@ -1,8 +1,8 @@ # Pre-Prefresher Exercises {.unnumbered} -Before our first meeting, please try solving these questions. They are a sample of the very beginning of each math section. We have provided links to the parts of the book you can read if the concepts are new to you. +Before our first meeting, please try solving these questions. They are a sample of the very beginning of each math section. We have provided links to the parts of the book you can read if the concepts are new to you. -The goal of this "pre"-prefresher assignment is not to intimidate you but to set common expectations so you can make the most out of the actual Prefresher. Even if you do not understand some or all of these questions after skimming through the linked sections, your effort will pay off and you will be better prepared for the math prefresher. We are also open to adjusting these expectations based on feedback (this class is for _you_), so please do not hesitate to write to the instructors for feedback. +The goal of this "pre"-prefresher assignment is not to intimidate you but to set common expectations so you can make the most out of the actual Prefresher. Even if you do not understand some or all of these questions after skimming through the linked sections, your effort will pay off and you will be better prepared for the math prefresher. We are also open to adjusting these expectations based on feedback (this class is for *you*), so please do not hesitate to write to the instructors for feedback. ## Linear Algebra {.unnumbered} @@ -10,63 +10,63 @@ The goal of this "pre"-prefresher assignment is not to intimidate you but to set Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pmatrix} 4\\5\\6 \end{pmatrix}$, and the scalar $c = 2$. Calculate the following: -1. $u + v$ -2. $cv$ -3. $u \cdot v$ +1. $u + v$ +2. $cv$ +3. $u \cdot v$ -If you are having trouble with these problems, please review Section @sec-vector-def "Working with Vectors" in Chapter @sec-linearalgebra. +If you are having trouble with these problems, please review Section @sec-vector-def "Working with Vectors" in Chapter @sec-linearalgebra. Are the following sets of vectors linearly independent? -1. $u = \begin{pmatrix} 1\\ 2\end{pmatrix}$, $v = \begin{pmatrix} 2\\4\end{pmatrix}$ +1. $u = \begin{pmatrix} 1\\ 2\end{pmatrix}$, $v = \begin{pmatrix} 2\\4\end{pmatrix}$ -2. $u = \begin{pmatrix} 1\\ 2\\ 5 \end{pmatrix}$, $v = \begin{pmatrix} 3\\ 7\\ 9 \end{pmatrix}$ +2. $u = \begin{pmatrix} 1\\ 2\\ 5 \end{pmatrix}$, $v = \begin{pmatrix} 3\\ 7\\ 9 \end{pmatrix}$ -3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ (this requires some guesswork) +3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ (this requires some guesswork) -If you are having trouble with these problems, please review Section @sec-linearindependence. +If you are having trouble with these problems, please review Section @sec-linearindependence. ### Matrices {.unnumbered} $${\bf A}=\begin{pmatrix} - 7 & 5 & 1 \\ - 11 & 9 & 3 \\ - 2 & 14 & 21 \\ - 4 & 1 & 5 - \end{pmatrix}$$ + 7 & 5 & 1 \\ + 11 & 9 & 3 \\ + 2 & 14 & 21 \\ + 4 & 1 & 5 + \end{pmatrix}$$ What is the dimensionality of matrix ${\bf A}$? What is the element $a_{23}$ of ${\bf A}$? -Given that +Given that $${\bf B}=\begin{pmatrix} - 1 & 2 & 8 \\ - 3 & 9 & 11 \\ - 4 & 7 & 5 \\ - 5 & 1 & 9 - \end{pmatrix}$$ + 1 & 2 & 8 \\ + 3 & 9 & 11 \\ + 4 & 7 & 5 \\ + 5 & 1 & 9 + \end{pmatrix}$$ What is ${\bf A}$ + ${\bf B}$? -Given that +Given that $${\bf C}=\begin{pmatrix} - 1 & 2 & 8 \\ - 3 & 9 & 11 \\ - 4 & 7 & 5 \\ - \end{pmatrix}$$ + 1 & 2 & 8 \\ + 3 & 9 & 11 \\ + 4 & 7 & 5 \\ + \end{pmatrix}$$ What is ${\bf A}$ + ${\bf C}$? -Given that +Given that $$c = 2$$ What is $c$${\bf A}$? -If you are having trouble with these problems, please review Section @sec-matrixbasics. +If you are having trouble with these problems, please review Section @sec-matrixbasics. ## Operations {.unnumbered} @@ -74,17 +74,17 @@ If you are having trouble with these problems, please review Section @sec-matrix Simplify the following -1. $$\sum\limits_{i = 1}^3 i$$ +1. $$\sum\limits_{i = 1}^3 i$$ -2. $\sum\limits_{k = 1}^3(3k + 2)$ +2. $\sum\limits_{k = 1}^3(3k + 2)$ -3. $\sum\limits_{i= 1}^4 (3k + i + 2)$ +3. $\sum\limits_{i= 1}^4 (3k + i + 2)$ ### Products {.unnumbered} -1. $\prod\limits_{i= 1}^3 i$ +1. $\prod\limits_{i= 1}^3 i$ -2. $\prod\limits_{k=1}^3(3k + 2)$ +2. $\prod\limits_{k=1}^3(3k + 2)$ To review this material, please see Section @sec-sum-notation. @@ -109,9 +109,9 @@ To review this material, please see Section @sec-logexponents Find the limit of the following. -1. $\lim\limits_{x \to 2} (x - 1)$ -2. $\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)}$ -3. $\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2}$ +1. $\lim\limits_{x \to 2} (x - 1)$ +2. $\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)}$ +3. $\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2}$ To review this material please see Section @sec-limitsfun @@ -119,12 +119,12 @@ To review this material please see Section @sec-limitsfun For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\frac{d}{dx}f(x)$ -1. $f(x)=c$ -2. $f(x)=x$ -3. $f(x)=x^2$ -4. $f(x)=x^3$ -5. $f(x)=3x^2+2x^{1/3}$ -6. $f(x)=(x^3)(2x^4)$ +1. $f(x)=c$ +2. $f(x)=x$ +3. $f(x)=x^2$ +4. $f(x)=x^3$ +5. $f(x)=3x^2+2x^{1/3}$ +6. $f(x)=(x^3)(2x^4)$ For a review, please see Section @sec-derivintro - @sec-derivpoly @@ -132,18 +132,17 @@ For a review, please see Section @sec-derivintro - @sec-derivpoly For each of the followng functions $f(x)$, does a maximum and minimum exist in the domain $x \in \mathbf{R}$? If so, for what are those values and for which values of $x$? -1. $f(x) = x$ -2. $f(x) = x^2$ -3. $f(x) = -(x - 2)^2$ +1. $f(x) = x$ +2. $f(x) = x^2$ +3. $f(x) = -(x - 2)^2$ If you are stuck, please try sketching out a picture of each of the functions. ## Probability {.unnumbered} -1. If there are 12 cards, numbered 1 to 12, and 4 cards are chosen, how many distinct possible choices are there? (unordered, without replacement) -2. Let $A = \{1,3,5,7,8\}$ and $B = \{2,4,7,8,12,13\}$. What is $A \cup B$? What is $A \cap B$? If $A$ is a subset of the Sample Space $S = \{1,2,3,4,5,6,7,8,9,10\}$, what is the complement $A^C$? -3. If we roll two fair dice, what is the probability that their sum would be 11? -4. If we roll two fair dice, what is the probability that their sum would be 12? +1. If there are 12 cards, numbered 1 to 12, and 4 cards are chosen, how many distinct possible choices are there? (unordered, without replacement) +2. Let $A = \{1,3,5,7,8\}$ and $B = \{2,4,7,8,12,13\}$. What is $A \cup B$? What is $A \cap B$? If $A$ is a subset of the Sample Space $S = \{1,2,3,4,5,6,7,8,9,10\}$, what is the complement $A^C$? +3. If we roll two fair dice, what is the probability that their sum would be 11? +4. If we roll two fair dice, what is the probability that their sum would be 12? For a review, please see Sections @sec-setoper - @sec-probdef. - diff --git a/02_functions.qmd b/02_functions.qmd index f3dc398..10f5503 100644 --- a/02_functions.qmd +++ b/02_functions.qmd @@ -2,398 +2,368 @@ title: Functions and Operations --- -__Topics__ - Dimensionality; - Interval Notation for ${\bf R}^1$; - Neighborhoods: Intervals, Disks, and Balls; Introduction to Functions; - Domain and Range; - Some General Types of Functions; - $\log$, $\ln$, and $\exp$; - Other Useful Functions; - Graphing Functions; - Solving for Variables; - Finding Roots; - Limit of a Function; - Continuity; Sets, Sets, and More Sets. +**Topics** Dimensionality; Interval Notation for ${\bf R}^1$; Neighborhoods: Intervals, Disks, and Balls; Introduction to Functions; Domain and Range; Some General Types of Functions; $\log$, $\ln$, and $\exp$; Other Useful Functions; Graphing Functions; Solving for Variables; Finding Roots; Limit of a Function; Continuity; Sets, Sets, and More Sets. ## Summation Operators $\sum$ and $\prod$ {#sum-notation} -Addition (+), Subtraction (-), multiplication and division are basic operations of arithmetic -- combining numbers. In statistics and calculus, we want to add a _sequence_ of numbers that can be expressed as a pattern without needing to write down all its components. For example, how would we express the sum of all numbers from 1 to 100 without writing a hundred numbers? +Addition (+), Subtraction (-), multiplication and division are basic operations of arithmetic -- combining numbers. In statistics and calculus, we want to add a *sequence* of numbers that can be expressed as a pattern without needing to write down all its components. For example, how would we express the sum of all numbers from 1 to 100 without writing a hundred numbers? -For this we use the summation operator $\sum$ and the product operator $\prod$. +For this we use the summation operator $\sum$ and the product operator $\prod$. -__Summation:__ +**Summation:** $$\sum\limits_{i=1}^{100} x_i = x_1+x_2+x_3+\cdots+x_{100}$$ -The bottom of the $\sum$ symbol indicates an index (here, $i$), and its start value $1$. At the top is where the index ends. The notion of "addition" is part of the $\sum$ symbol. The content to the right of the summation is the meat of what we add. While you can pick your favorite index, start, and end values, the content must also have the index. +The bottom of the $\sum$ symbol indicates an index (here, $i$), and its start value $1$. At the top is where the index ends. The notion of "addition" is part of the $\sum$ symbol. The content to the right of the summation is the meat of what we add. While you can pick your favorite index, start, and end values, the content must also have the index. -- $\sum\limits_{i=1}^n c x_i = c \sum\limits_{i=1}^n x_i$ -- $\sum\limits_{i=1}^n (x_i + y_i) = \sum\limits_{i=1}^n x_i + \sum\limits_{i=1}^n y_i$ -- $\sum\limits_{i=1}^n c = n c$ +- $\sum\limits_{i=1}^n c x_i = c \sum\limits_{i=1}^n x_i$ +- $\sum\limits_{i=1}^n (x_i + y_i) = \sum\limits_{i=1}^n x_i + \sum\limits_{i=1}^n y_i$ +- $\sum\limits_{i=1}^n c = n c$ -__Product:__ +**Product:** $$\prod\limits_{i=1}^n x_i = x_1 x_2 x_3 \cdots x_n$$ Properties: -- $\prod\limits_{i=1}^n c x_i = c^n \prod\limits_{i=1}^n x_i$ -- $\prod\limits_{i=k}^n c x_i = c^{n-k+1} \prod\limits_{i=k}^n x_i$ -- $\prod\limits_{i=1}^n (x_i + y_i) =$ a total mess -- $\prod\limits_{i=1}^n c = c^n$ +- $\prod\limits_{i=1}^n c x_i = c^n \prod\limits_{i=1}^n x_i$ +- $\prod\limits_{i=k}^n c x_i = c^{n-k+1} \prod\limits_{i=k}^n x_i$ +- $\prod\limits_{i=1}^n (x_i + y_i) =$ a total mess +- $\prod\limits_{i=1}^n c = c^n$ Other Useful Functions -__Factorials!:__ +**Factorials!:** $$x! = x\cdot (x-1) \cdot (x-2) \cdots (1)$$ -__Modulo:__ Tells you the remainder when you divide the first number by the second. +**Modulo:** Tells you the remainder when you divide the first number by the second. -- $17\mod3 = 2$ -- $100 \ \% \ 30 = 10$ - -:::{#exm-operators} +- $17\mod3 = 2$ +- $100 \ \% \ 30 = 10$ +::: {#exm-operators} ## Operators -1. $\sum\limits_{i=1}^{5} i =$ - -2. $\prod\limits_{i=1}^{5} i =$ +1. $\sum\limits_{i=1}^{5} i =$ -3. $14 \mod 4 =$ +2. $\prod\limits_{i=1}^{5} i =$ -4. $4! =$ +3. $14 \mod 4 =$ +4. $4! =$ ::: -:::{#exr-operators1} - +::: {#exr-operators1} ## Operators Let $x_1 = 4, x_2 = 3, x_3 = 7, x_4 = 11, x_5 = 2$ -1. $\sum\limits_{i=1}^{3} (7)x_i$ +1. $\sum\limits_{i=1}^{3} (7)x_i$ -2. $\sum\limits_{i=1}^{5} 2$ - -3. $\prod\limits_{i=3}^{5} (2)x_i$ +2. $\sum\limits_{i=1}^{5} 2$ +3. $\prod\limits_{i=3}^{5} (2)x_i$ ::: ## Introduction to Functions -A __function__ (in ${\bf R}^1$) is a mapping, or transformation, that relates members of one set to members of another set. For instance, if you have two sets: set $A$ and set $B$, a function from $A$ to $B$ maps every value $a$ in set $A$ such that $f(a) \in B$. Functions can be "many-to-one", where many values or combinations of values from set $A$ produce a single output in set $B$, or they can be "one-to-one", where each value in set $A$ corresponds to a single value in set $B$. A function by definition has a single function value for each element of its domain. This means, there cannot be "one-to-many" mapping. +A **function** (in ${\bf R}^1$) is a mapping, or transformation, that relates members of one set to members of another set. For instance, if you have two sets: set $A$ and set $B$, a function from $A$ to $B$ maps every value $a$ in set $A$ such that $f(a) \in B$. Functions can be "many-to-one", where many values or combinations of values from set $A$ produce a single output in set $B$, or they can be "one-to-one", where each value in set $A$ corresponds to a single value in set $B$. A function by definition has a single function value for each element of its domain. This means, there cannot be "one-to-many" mapping. -__Dimensionality__: ${\bf R}^1$ is the set of all real numbers extending from $-\infty$ to $+\infty$ --- i.e., the real number line. ${\bf R}^n$ is an $n$-dimensional space, where each of the $n$ axes extends from $-\infty$ to $+\infty$. +**Dimensionality**: ${\bf R}^1$ is the set of all real numbers extending from $-\infty$ to $+\infty$ --- i.e., the real number line. ${\bf R}^n$ is an $n$-dimensional space, where each of the $n$ axes extends from $-\infty$ to $+\infty$. -- ${\bf R}^1$ is a one dimensional line. -- ${\bf R}^2$ is a two dimensional plane. -- ${\bf R}^3$ is a three dimensional space. +- ${\bf R}^1$ is a one dimensional line. +- ${\bf R}^2$ is a two dimensional plane. +- ${\bf R}^3$ is a three dimensional space. Points in ${\bf R}^n$ are ordered $n$-tuples (just means an combination of $n$ elements where order matters), where each element of the $n$-tuple represents the coordinate along that dimension. For example: -- ${\bf R}^1$: (3) -- ${\bf R}^2$: (-15, 5) -- ${\bf R}^3$: (86, 4, 0) +- ${\bf R}^1$: (3) +- ${\bf R}^2$: (-15, 5) +- ${\bf R}^3$: (86, 4, 0) -Examples of mapping notation: +Examples of mapping notation: Function of one variable: $f:{\bf R}^1\to{\bf R}^1$ -- $f(x)=x+1$. For each $x$ in ${\bf R}^1$, $f(x)$ assigns the number $x+1$. +- $f(x)=x+1$. For each $x$ in ${\bf R}^1$, $f(x)$ assigns the number $x+1$. -Function of two variables: $f: {\bf R}^2\to{\bf R}^1$. +Function of two variables: $f: {\bf R}^2\to{\bf R}^1$. -- $f(x,y)=x^2+y^2$. For each ordered pair $(x,y)$ in ${\bf R}^2$, $f(x,y)$ assigns the number $x^2+y^2$. +- $f(x,y)=x^2+y^2$. For each ordered pair $(x,y)$ in ${\bf R}^2$, $f(x,y)$ assigns the number $x^2+y^2$. We often use variable $x$ as input and another $y$ as output, e.g. $y=x+1$ -:::{#exm-functions} - +::: {#exm-functions} ## Functions -For each of the following, state whether they are one-to-one or many-to-one functions. - -1. For $x \in [0,\infty]$, $f : x \rightarrow x^2$ (this could also be written as $f(x) = x^2$). +For each of the following, state whether they are one-to-one or many-to-one functions. -2. For $x \in [-\infty, \infty]$, $f: x \rightarrow x^2$. +1. For $x \in [0,\infty]$, $f : x \rightarrow x^2$ (this could also be written as $f(x) = x^2$). +2. For $x \in [-\infty, \infty]$, $f: x \rightarrow x^2$. ::: -:::{#exr-functions1} - +::: {#exr-functions1} ## Functions -For each of the following, state whether they are one-to-one or many-to-one functions. - -1. For $x \in [-3, \infty]$, $f: x \rightarrow x^2$. +For each of the following, state whether they are one-to-one or many-to-one functions. -2. For $x \in [0, \infty]$, $f: x \rightarrow \sqrt{x}$ +1. For $x \in [-3, \infty]$, $f: x \rightarrow x^2$. +2. For $x \in [0, \infty]$, $f: x \rightarrow \sqrt{x}$ ::: Some functions are defined only on proper subsets of ${\bf R}^n$. -- __Domain__: the set of numbers in $X$ at which $f(x)$ is defined. -- __Range__: elements of $Y$ assigned by $f(x)$ to elements of $X$, or $$f(X)=\{ y : y=f(x), x\in X\}$$ - Most often used when talking about a function $f:{\bf R}^1\to{\bf R}^1$. -- __Image__: same as range, but more often used when talking about a function $f:{\bf R}^n\to{\bf R}^1$. +- **Domain**: the set of numbers in $X$ at which $f(x)$ is defined. +- **Range**: elements of $Y$ assigned by $f(x)$ to elements of $X$, or $$f(X)=\{ y : y=f(x), x\in X\}$$ Most often used when talking about a function $f:{\bf R}^1\to{\bf R}^1$. +- **Image**: same as range, but more often used when talking about a function $f:{\bf R}^n\to{\bf R}^1$. Some General Types of Functions -__Monomials__: $f(x)=a x^k$ +**Monomials**: $f(x)=a x^k$ -$a$ is the coefficient. $k$ is the degree. +$a$ is the coefficient. $k$ is the degree. Examples: $y=x^2$, $y=-\frac{1}{2}x^3$ -__Polynomials__: sum of monomials. +**Polynomials**: sum of monomials. Examples: $y=-\frac{1}{2}x^3+x^2$, $y=3x+5$ -The degree of a polynomial is the highest degree of its monomial terms. Also, it's often a good idea to write polynomials with terms in decreasing degree. +The degree of a polynomial is the highest degree of its monomial terms. Also, it's often a good idea to write polynomials with terms in decreasing degree. -__Exponential Functions__: Example: $y=2^x$ +**Exponential Functions**: Example: $y=2^x$ -## $\log$ and $\exp$ {#logexponents} +## $\log$ and $\exp$ {#logexponents} -__Relationship of logarithmic and exponential functions__: - $$y=\log_a(x) \iff a^y=x$$ +**Relationship of logarithmic and exponential functions**: $$y=\log_a(x) \iff a^y=x$$ The log function can be thought of as an inverse for exponential functions. $a$ is referred to as the "base" of the logarithm. -__Common Bases__: The two most common logarithms are base 10 and base $e$. +**Common Bases**: The two most common logarithms are base 10 and base $e$. -1. Base 10: $\quad y=\log_{10}(x) \iff 10^y=x$. The base 10 logarithm is often simply written as "$\log(x)$" with no base denoted. -2. Base $e$: $\quad y=\log_e(x) \iff e^y=x$. The base $e$ logarithm is referred to as the "natural" logarithm and is written as ``$\ln(x)$". +1. Base 10: $\quad y=\log_{10}(x) \iff 10^y=x$. The base 10 logarithm is often simply written as "$\log(x)$" with no base denoted. +2. Base $e$: $\quad y=\log_e(x) \iff e^y=x$. The base $e$ logarithm is referred to as the "natural" logarithm and is written as \`\`$\ln(x)$". +```{=tex} \begin{comment} - {\includegraphics[width=1in, angle = 270]{ln.eps}} \, {\includegraphics[width=1in, angle = 270]{exp.eps}} - \end{comment} - -__Properties of exponential functions:__ + {\includegraphics[width=1in, angle = 270]{ln.eps}} \, {\includegraphics[width=1in, angle = 270]{exp.eps}} + \end{comment} +``` +**Properties of exponential functions:** -- $a^x a^y = a^{x+y}$ -- $a^{-x} = 1/a^x$ -- $a^x/a^y = a^{x-y}$ -- $(a^x)^y = a^{x y}$ -- $a^0 = 1$ +- $a^x a^y = a^{x+y}$ +- $a^{-x} = 1/a^x$ +- $a^x/a^y = a^{x-y}$ +- $(a^x)^y = a^{x y}$ +- $a^0 = 1$ -__Properties of logarithmic functions__ (any base): +**Properties of logarithmic functions** (any base): Generally, when statisticians or social scientists write $\log(x)$ they mean $\log_e(x)$. In other words: $\log_e(x) \equiv \ln(x) \equiv \log(x)$ -$$\log_a(a^x)=x$$ and -$$a^{\log_a(x)}=x$$ +$$\log_a(a^x)=x$$ and $$a^{\log_a(x)}=x$$ -- $\log(x y)=\log(x)+\log(y)$ -- $\log(x^y)=y\log(x)$ -- $\log(1/x)=\log(x^{-1})=-\log(x)$ -- $\log(x/y)=\log(x\cdot y^{-1})=\log(x)+\log(y^{-1})=\log(x)-\log(y)$ -- $\log(1)=\log(e^0)=0$ +- $\log(x y)=\log(x)+\log(y)$ +- $\log(x^y)=y\log(x)$ +- $\log(1/x)=\log(x^{-1})=-\log(x)$ +- $\log(x/y)=\log(x\cdot y^{-1})=\log(x)+\log(y^{-1})=\log(x)-\log(y)$ +- $\log(1)=\log(e^0)=0$ -__Change of Base Formula__: Use the change of base formula to switch bases as necessary: -$$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$$ +**Change of Base Formula**: Use the change of base formula to switch bases as necessary: $$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$$ Example: $$\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$$ You can use logs to go between sum and product notation. This will be particularly important when you're learning maximum likelihood estimation. +```{=tex} \begin{eqnarray*} - \log \bigg(\prod\limits_{i=1}^n x_i \bigg) &=& \log(x_1 \cdot x_2 \cdot x_3 \cdots \cdot x_n)\\ - &=& \log(x_1) + \log(x_2) + \log(x_3) + \cdots + \log(x_n)\\ - &=& \sum\limits_{i=1}^n \log (x_i) + \log \bigg(\prod\limits_{i=1}^n x_i \bigg) &=& \log(x_1 \cdot x_2 \cdot x_3 \cdots \cdot x_n)\\ + &=& \log(x_1) + \log(x_2) + \log(x_3) + \cdots + \log(x_n)\\ + &=& \sum\limits_{i=1}^n \log (x_i) \end{eqnarray*} - +``` Therefore, you can see that the log of a product is equal to the sum of the logs. We can write this more generally by adding in a constant, $c$: +```{=tex} \begin{eqnarray*} - \log \bigg(\prod\limits_{i=1}^n c x_i\bigg) &=& \log(cx_1 \cdot cx_2 \cdots cx_n)\\ - &=& \log(c^n \cdot x_1 \cdot x_2 \cdots x_n)\\ - &=& \log(c^n) + \log(x_1) + \log(x_2) + \cdots + \log(x_n)\\\\ - &=& n \log(c) + \sum\limits_{i=1}^n \log (x_i)\\ -\end{eqnarray*} - -:::{#exm-log} - + \log \bigg(\prod\limits_{i=1}^n c x_i\bigg) &=& \log(cx_1 \cdot cx_2 \cdots cx_n)\\ + &=& \log(c^n \cdot x_1 \cdot x_2 \cdots x_n)\\ + &=& \log(c^n) + \log(x_1) + \log(x_2) + \cdots + \log(x_n)\\\\ + &=& n \log(c) + \sum\limits_{i=1}^n \log (x_i)\\ +\end{eqnarray*} +``` +::: {#exm-log} ## Logarithmic Functions Evaluate each of the following logarithms -1. $\log_4(16)$ +1. $\log_4(16)$ -2. $\log_2(16)$ +2. $\log_2(16)$ Simplify the following logarithm. By "simplify", we actually really mean - use as many of the logarithmic properties as you can. -3. $\log_4(x^3y^5)$ - +3. $\log_4(x^3y^5)$ ::: -:::{#exr-log1} - +::: {#exr-log1} ## Logarithmic Functions Evaluate each of the following logarithms -1. $\log_\frac{3}{2}(\frac{27}{8})$ +1. $\log_\frac{3}{2}(\frac{27}{8})$ Simplify each of the following logarithms. By "simplify", we actually really mean - use as many of the logarithmic properties as you can. -2. $\log(\frac{x^9y^5}{z^3})$ - -3. $\ln{\sqrt{xy}}$ +2. $\log(\frac{x^9y^5}{z^3})$ +3. $\ln{\sqrt{xy}}$ ::: ## Graphing Functions + What can a graph tell you about a function? -- Is the function increasing or decreasing? Over what part of the domain? -- How ``fast" does it increase or decrease? -- Are there global or local maxima and minima? Where? -- Are there inflection points? -- Is the function continuous? -- Is the function differentiable? -- Does the function tend to some limit? -- Other questions related to the substance of the problem at hand. +- Is the function increasing or decreasing? Over what part of the domain? +- How \`\`fast" does it increase or decrease? +- Are there global or local maxima and minima? Where? +- Are there inflection points? +- Is the function continuous? +- Is the function differentiable? +- Does the function tend to some limit? +- Other questions related to the substance of the problem at hand. ## Solving for Variables and Finding Roots Sometimes we're given a function $y=f(x)$ and we want to find how $x$ varies as a function of $y$. Use algebra to move $x$ to the left hand side (LHS) of the equation and so that the right hand side (RHS) is only a function of $y$. -:::{#exm-solvevar} - +::: {#exm-solvevar} ## Solving for Variables Solve for x: -1. $y=3x+2$ - -2. $y=e^x$ +1. $y=3x+2$ +2. $y=e^x$ ::: -Solving for variables is especially important when we want to find the __roots__ of an equation: those values of variables that cause an equation to equal zero. Especially important in finding equilibria and in doing maximum likelihood estimation. - -Procedure: Given $y=f(x)$, set $f(x)=0$. Solve for $x$. +Solving for variables is especially important when we want to find the **roots** of an equation: those values of variables that cause an equation to equal zero. Especially important in finding equilibria and in doing maximum likelihood estimation. -Multiple Roots: - $$f(x)=x^2 - 9 \quad\Longrightarrow\quad 0=x^2 - 9 \quad\Longrightarrow\quad 9=x^2 \quad\Longrightarrow\quad \pm \sqrt{9}=\sqrt{x^2} \quad\Longrightarrow\quad \pm 3=x$$ +Procedure: Given $y=f(x)$, set $f(x)=0$. Solve for $x$. -__Quadratic Formula:__ For quadratic equations $ax^2+bx+c=0$, use the quadratic formula: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ +Multiple Roots: $$f(x)=x^2 - 9 \quad\Longrightarrow\quad 0=x^2 - 9 \quad\Longrightarrow\quad 9=x^2 \quad\Longrightarrow\quad \pm \sqrt{9}=\sqrt{x^2} \quad\Longrightarrow\quad \pm 3=x$$ -:::{#exr-solvevar1} +**Quadratic Formula:** For quadratic equations $ax^2+bx+c=0$, use the quadratic formula: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ +::: {#exr-solvevar1} ## Finding Roots Solve for x: -1. $f(x)=3x+2 = 0$ +1. $f(x)=3x+2 = 0$ -2. $f(x)=x^2+3x-4=0$ - -3. $f(x)=e^{-x}-10 = 0$ +2. $f(x)=x^2+3x-4=0$ +3. $f(x)=e^{-x}-10 = 0$ ::: ## Sets -__Interior Point__: The point $\bf x$ is an interior point of the set $S$ if $\bf x$ is in $S$ and if there is some $\epsilon$-ball around $\bf x$ that contains only points in $S$. The __interior__ of $S$ is the collection of all interior points in $S$. The interior can also be defined as the union of all open sets in $S$. +**Interior Point**: The point $\bf x$ is an interior point of the set $S$ if $\bf x$ is in $S$ and if there is some $\epsilon$-ball around $\bf x$ that contains only points in $S$. The **interior** of $S$ is the collection of all interior points in $S$. The interior can also be defined as the union of all open sets in $S$. -- If the set $S$ is circular, the interior points are everything inside of the circle, but not on the circle's rim. -- Example: The interior of the set $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2< 4 \}$ . +- If the set $S$ is circular, the interior points are everything inside of the circle, but not on the circle's rim. +- Example: The interior of the set $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2< 4 \}$ . -__Boundary Point__: The point $\bf x$ is a boundary point of the set $S$ if every $\epsilon$-ball around $\bf x$ contains both points that are in $S$ and points that are outside $S$. The __boundary__ is the collection of all boundary points. +**Boundary Point**: The point $\bf x$ is a boundary point of the set $S$ if every $\epsilon$-ball around $\bf x$ contains both points that are in $S$ and points that are outside $S$. The **boundary** is the collection of all boundary points. -- If the set $S$ is circular, the boundary points are everything on the circle's rim. -- Example: The boundary of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 = 4 \}$. +- If the set $S$ is circular, the boundary points are everything on the circle's rim. +- Example: The boundary of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 = 4 \}$. -__Open__: A set $S$ is open if for each point $\bf x$ in $S$, there exists an open $\epsilon$-ball around $\bf x$ completely contained in $S$. +**Open**: A set $S$ is open if for each point $\bf x$ in $S$, there exists an open $\epsilon$-ball around $\bf x$ completely contained in $S$. -- If the set $S$ is circular and open, the points contained within the set get infinitely close to the circle's rim, but do not touch it. -- Example: $\{ (x,y) : x^2+y^2<4 \}$ +- If the set $S$ is circular and open, the points contained within the set get infinitely close to the circle's rim, but do not touch it. +- Example: $\{ (x,y) : x^2+y^2<4 \}$ -__Closed__: A set $S$ is closed if it contains all of its boundary points. +**Closed**: A set $S$ is closed if it contains all of its boundary points. -- Alternatively: A set is closed if its complement is open. -- If the set $S$ is circular and closed, the set contains all points within the rim as well as the rim itself. -- Example: $\{ (x,y) : x^2+y^2\le 4 \}$ -- Note: a set may be neither open nor closed. Example: $\{ (x,y) : 2 < x^2+y^2\le 4 \}$ +- Alternatively: A set is closed if its complement is open. +- If the set $S$ is circular and closed, the set contains all points within the rim as well as the rim itself. +- Example: $\{ (x,y) : x^2+y^2\le 4 \}$ +- Note: a set may be neither open nor closed. Example: $\{ (x,y) : 2 < x^2+y^2\le 4 \}$ -__Complement__: The complement of set $S$ is everything outside of $S$. +**Complement**: The complement of set $S$ is everything outside of $S$. -- If the set $S$ is circular, the complement of $S$ is everything outside of the circle. -- Example: The complement of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 > 4 \}$. +- If the set $S$ is circular, the complement of $S$ is everything outside of the circle. +- Example: The complement of $\{ (x,y) : x^2+y^2\le 4 \}$ is $\{ (x,y) : x^2+y^2 > 4 \}$. -__Empty__: The empty (or null) set is a unique set that has no elements, denoted by \{\} or $\emptyset$. +**Empty**: The empty (or null) set is a unique set that has no elements, denoted by {} or $\emptyset$. -- The empty set is an example of a set that is open and closed, or a "clopen" set. -- Examples: The set of squares with 5 sides; the set of countries south of the South Pole. +- The empty set is an example of a set that is open and closed, or a "clopen" set. +- Examples: The set of squares with 5 sides; the set of countries south of the South Pole. -## Answers to Examples and Exercises {.unnumbered} +## Answers to Examples and Exercises {.unnumbered} Answer to Example @exm-operators: - 1. 1 + 2 + 3 + 4 + 5 = 15 +1. 1 + 2 + 3 + 4 + 5 = 15 - 2. 1 * 2 * 3 * 4 * 5 = 120 +2. 1 \* 2 \* 3 \* 4 \* 5 = 120 - 3. 2 +3. 2 - 4. 4 * 3 * 2 * 1 = 24 +4. 4 \* 3 \* 2 \* 1 = 24 Answer to Exercise @exr-operators1: - 1. 7(4 + 3 + 7) = 98 +1. 7(4 + 3 + 7) = 98 - 2. 2 + 2 + 2 + 2 + 2 = 10 +2. 2 + 2 + 2 + 2 + 2 = 10 - 3. $2^3(7)(11)(2)$ = 1232 +3. $2^3(7)(11)(2)$ = 1232 Answer to Example @exm-functions: -1. one-to-one +1. one-to-one -2. many-to-one +2. many-to-one Answer to Exercise @exr-functions1: -1. many-to-one +1. many-to-one -2. one-to-one +2. one-to-one Answer to Example @exm-log: -1. 2 +1. 2 -2. 4 +2. 4 -3. $3\log_4(x) + 5\log_4(y)$ +3. $3\log_4(x) + 5\log_4(y)$ Answer to Exercise @exr-log1: -1. 3 +1. 3 -2. $9\log(x) + 5\log(y) - 3\log(z)$ +2. $9\log(x) + 5\log(y) - 3\log(z)$ -3. $\frac{1}{2}(\ln{x} + \ln{y})$ +3. $\frac{1}{2}(\ln{x} + \ln{y})$ Answer to Example @exm-solvevar: -1. $y=3x+2 \quad\Longrightarrow\quad -3x=2-y \quad\Longrightarrow\quad 3x=y-2 \quad\Longrightarrow\quad x=\frac{1}{3}(y-2)$ +1. $y=3x+2 \quad\Longrightarrow\quad -3x=2-y \quad\Longrightarrow\quad 3x=y-2 \quad\Longrightarrow\quad x=\frac{1}{3}(y-2)$ -2. $x = \ln{y}$ +2. $x = \ln{y}$ Answer to Exercise @exr-solvevar1: -1. $\frac{-2}{3}$ - -2. x = {1, -4} +1. $\frac{-2}{3}$ -3. x = - $\ln10$ +2. x = {1, -4} +3. x = - $\ln10$ diff --git a/03_limits.qmd b/03_limits.qmd index 74cdea6..4aba328 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -9,43 +9,41 @@ library(glue) library(patchwork) ``` -Solving limits, i.e. finding out the value of functions as its input moves closer to some value, is important for the social scientist's mathematical toolkit for two related tasks. The first is for the study of calculus, which will be in turn useful to show where certain functions are maximized or minimized. The second is for the study of statistical inference, which is the study of inferring things about things you cannot see by using things you can see. +Solving limits, i.e. finding out the value of functions as its input moves closer to some value, is important for the social scientist's mathematical toolkit for two related tasks. The first is for the study of calculus, which will be in turn useful to show where certain functions are maximized or minimized. The second is for the study of statistical inference, which is the study of inferring things about things you cannot see by using things you can see. ## Example: The Central Limit Theorem {.unnumbered} -Perhaps the most important theorem in statistics is the Central Limit Theorem, - -:::{#thm-clt-lim} +Perhaps the most important theorem in statistics is the Central Limit Theorem, +::: {#thm-clt-lim} ## Central Limit Theorem (i.i.d. case) -For any series of independent and identically distributed random variables $X_1, X_2, \cdots$, we know the distribution of its sum even if we do not know the distribution of $X$. The distribution of the sum is a Normal distribution. +For any series of independent and identically distributed random variables $X_1, X_2, \cdots$, we know the distribution of its sum even if we do not know the distribution of $X$. The distribution of the sum is a Normal distribution. $$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),$$ -where $\mu$ is the mean of $X$ and $\sigma$ is the standard deviation of $X$. The arrow is read as "converges in distribution to". $\text{Normal}(0, 1)$ indicates a Normal Distribution with mean 0 and variance 1. +where $\mu$ is the mean of $X$ and $\sigma$ is the standard deviation of $X$. The arrow is read as "converges in distribution to". $\text{Normal}(0, 1)$ indicates a Normal Distribution with mean 0 and variance 1. -That is, the limit of the distribution of the lefthand side is the distribution of the righthand side. +That is, the limit of the distribution of the lefthand side is the distribution of the righthand side. ::: -The sign of a limit is the arrow "$\rightarrow$". Although we have not yet covered probability (in Section @probability-theory) so we have not described what distributions and random variables are, it is worth foreshadowing the Central Limit Theorem. The Central Limit Theorem is powerful because it gives us a _guarantee_ of what would happen if $n \rightarrow \infty$, which in this case means we collected more data. +The sign of a limit is the arrow "$\rightarrow$". Although we have not yet covered probability (in Section @probability-theory) so we have not described what distributions and random variables are, it is worth foreshadowing the Central Limit Theorem. The Central Limit Theorem is powerful because it gives us a *guarantee* of what would happen if $n \rightarrow \infty$, which in this case means we collected more data. ## Example: The Law of Large Numbers {.unnumbered} A finding that perhaps rivals the Central Limit Theorem is the Law of Large Numbers: -:::{#thm-lln-lim} - +::: {#thm-lln-lim} ## (Weak) Law of Large Numbers For any draw of identically distributed independent variables with mean $\mu$, the sample average after $n$ draws, $\bar{X}_n$, converges in probability to the true mean as $n \rightarrow \infty$: $$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ -A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". +A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". ::: -Intuitively, the more data, the more accurate is your guess. For example, the Figure @fig-llnsim shows how the sample average from many coin tosses converges to the true value : 0.5. +Intuitively, the more data, the more accurate is your guess. For example, the Figure @fig-llnsim shows how the sample average from many coin tosses converges to the true value : 0.5. ```{r} #| label: fig-llnsim @@ -69,25 +67,23 @@ ggplot(tibble(n = 1:n, estimate = means), aes(x = n, y = estimate)) + ## Sequences -We need a couple of steps until we get to limit theorems in probability. First we will introduce a "sequence", then we will think about the limit of a sequence, then we will think about the limit of a _function_. +We need a couple of steps until we get to limit theorems in probability. First we will introduce a "sequence", then we will think about the limit of a sequence, then we will think about the limit of a *function*. A **sequence** $$\{x_n\}=\{x_1, x_2, x_3, \ldots, x_n\}$$ is an ordered set of real numbers, where $x_1$ is the first term in the sequence and $y_n$ is the $n$th term. Generally, a sequence is infinite, that is it extends to $n=\infty$. We can also write the sequence as $$\{x_n\}^\infty_{n=1}$$ where the subscript and superscript are read together as "from 1 to infinity." -:::{#exm-seqbehav} - +::: {#exm-seqbehav} ## Sequences How do these sequences behave? -1. $\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\}$ -2. $\{B_n\}=\left\{\frac{n^2+1}{n} \right\}$ -3. $\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\}$ - +1. $\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\}$ +2. $\{B_n\}=\left\{\frac{n^2+1}{n} \right\}$ +3. $\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\}$ ::: -We find the sequence by simply "plugging in" the integers into each $n$. The important thing is to get a sense of how these numbers are going to change. Example 1's numbers seem to come closer and closer to 2, but will it ever surpass 2? Example 2's numbers are also increasing each time, but will it hit a limit? What is the pattern in Example 3? Graphing helps you make this point more clearly. See the sequence of $n = 1, ...20$ for each of the three examples in Figure @fig-seqabc. +We find the sequence by simply "plugging in" the integers into each $n$. The important thing is to get a sense of how these numbers are going to change. Example 1's numbers seem to come closer and closer to 2, but will it ever surpass 2? Example 2's numbers are also increasing each time, but will it hit a limit? What is the pattern in Example 3? Graphing helps you make this point more clearly. See the sequence of $n = 1, ...20$ for each of the three examples in Figure @fig-seqabc. ```{r} #| label: fig-seqabc @@ -108,107 +104,98 @@ gA + gB + gC + plot_layout(nrow = 1) ## The Limit of a Sequence -The notion of "converging to a limit" is the behavior of the points in Example @exm-seqbehav. In some sense, that's the counterfactual we want to know. What happens as $n\rightarrow \infty$? +The notion of "converging to a limit" is the behavior of the points in Example @exm-seqbehav. In some sense, that's the counterfactual we want to know. What happens as $n\rightarrow \infty$? -1. Sequences like 1 above that converge to a limit. -2. Sequences like 2 above that increase without bound. -3. Sequences like 3 above that neither converge nor increase without bound --- alternating over the number line. +1. Sequences like 1 above that converge to a limit. +2. Sequences like 2 above that increase without bound. +3. Sequences like 3 above that neither converge nor increase without bound --- alternating over the number line. -:::{#def-limseq} -The sequence $\{y_n\}$ has the limit $L$, which we write as $$\lim\limits_{n \to \infty} y_n =L$$, if for any $\epsilon>0$ there is an integer $N$ (which depends on $\epsilon$) with the property that $|y_n -L|<\epsilon$ for each $n>N$. $\{y_n\}$ is said to converge to $L$. If the above does not hold, then $\{y_n\}$ diverges. +::: {#def-limseq} +The sequence $\{y_n\}$ has the limit $L$, which we write as $$\lim\limits_{n \to \infty} y_n =L$$, if for any $\epsilon>0$ there is an integer $N$ (which depends on $\epsilon$) with the property that $|y_n -L|<\epsilon$ for each $n>N$. $\{y_n\}$ is said to converge to $L$. If the above does not hold, then $\{y_n\}$ diverges. ::: We can also express the behavior of a sequence as bounded or not: -1. Bounded: if $|y_n|\le K$ for all $n$ -2. Monotonically Increasing: $y_{n+1}>y_n$ for all $n$ -3. Monotonically Decreasing: $y_{n+1}y_n$ for all $n$ +3. Monotonically Decreasing: $y_{n+1}0$ such that if $0<|x-c|<\delta$, then $|f(x)-L|<\epsilon$. +Let $f(x)$ be defined at each point in some open interval containing the point $c$. Then $L$ equals $\lim\limits_{x \to c} f(x)$ if for any (small positive) number $\epsilon$, there exists a corresponding number $\delta>0$ such that if $0<|x-c|<\delta$, then $|f(x)-L|<\epsilon$. ::: A neat, if subtle result is that $f(x)$ does not necessarily have to be defined at $c$ for $\lim\limits_{x \to c}$ to exist. Properties: Let $f$ and $g$ be functions with $\lim\limits_{x \to c} f(x)=k$ and $\lim\limits_{x \to c} g(x)=\ell$. -1. $\lim\limits_{x \to c}[f(x)+g(x)]=\lim\limits_{x \to c} f(x)+ \lim\limits_{x \to c} g(x)$ -2. $\lim\limits_{x \to c} kf(x) = k\lim\limits_{x \to c} f(x)$ -3. $\lim\limits_{x \to c} f(x) g(x) = \left[\lim\limits_{x \to c} f(x)\right]\cdot \left[\lim\limits_{x \to c} g(x)\right]$ -4. $\lim\limits_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)}$, provided $\lim\limits_{x \to c} g(x)\ne 0$. +1. $\lim\limits_{x \to c}[f(x)+g(x)]=\lim\limits_{x \to c} f(x)+ \lim\limits_{x \to c} g(x)$ +2. $\lim\limits_{x \to c} kf(x) = k\lim\limits_{x \to c} f(x)$ +3. $\lim\limits_{x \to c} f(x) g(x) = \left[\lim\limits_{x \to c} f(x)\right]\cdot \left[\lim\limits_{x \to c} g(x)\right]$ +4. $\lim\limits_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)}$, provided $\lim\limits_{x \to c} g(x)\ne 0$. Simple limits of functions can be solved as we did limits of sequences. Just be careful which part of the function is changing. -:::{#exm-limfun1} - +::: {#exm-limfun1} ## Limits of Functions -Find the limit of the following functions. - -1. $\lim_{x \to c} k$ -1. $\lim_{x \to c} x$ -1. $\lim_{x\to 2} (2x-3)$ -1. $\lim_{x \to c} x^n$ +Find the limit of the following functions. +1. $\lim_{x \to c} k$ +2. $\lim_{x \to c} x$ +3. $\lim_{x\to 2} (2x-3)$ +4. $\lim_{x \to c} x^n$ ::: -Limits can get more complex in roughly two ways. First, the functions may become large polynomials with many moving pieces. Second,the functions may become discontinuous. +Limits can get more complex in roughly two ways. First, the functions may become large polynomials with many moving pieces. Second,the functions may become discontinuous. -The function can be thought of as a more general or "smooth" version of sequences. For example, - -:::{#exr-limfunmax} +The function can be thought of as a more general or "smooth" version of sequences. For example, +::: {#exr-limfunmax} ## Limits of a Fraction of Functions Find the limit of @@ -218,21 +205,19 @@ $$\lim_{x\to\infty} \frac{(x^4 +3x−99)(2−x^5)}{(18x^7 +9x^6 −3x^2 −1)(x+ Now, the functions will become a bit more complex: -:::{#exr-discontlim} - +::: {#exr-discontlim} Solve the following limits of functions -1. $\lim\limits_{x\to 0} |x|$ -2. $\lim\limits_{x\to 0} \left(1+\frac{1}{x^2}\right)$ - +1. $\lim\limits_{x\to 0} |x|$ +2. $\lim\limits_{x\to 0} \left(1+\frac{1}{x^2}\right)$ ::: So there are a few more alternatives about what a limit of a function could be: -1. Right-hand limit: The value approached by $f(x)$ when you move from right to left. -2. Left-hand limit: The value approached by $f(x)$ when you move from left to right. -3. Infinity: The value approached by $f(x)$ as x grows infinitely large. Sometimes this may be a number; sometimes it might be $\infty$ or $-\infty$. -4. Negative infinity: The value approached by $f(x)$ as x grows infinitely negative. Sometimes this may be a number; sometimes it might be $\infty$ or $-\infty$. +1. Right-hand limit: The value approached by $f(x)$ when you move from right to left. +2. Left-hand limit: The value approached by $f(x)$ when you move from left to right. +3. Infinity: The value approached by $f(x)$ as x grows infinitely large. Sometimes this may be a number; sometimes it might be $\infty$ or $-\infty$. +4. Negative infinity: The value approached by $f(x)$ as x grows infinitely negative. Sometimes this may be a number; sometimes it might be $\infty$ or $-\infty$. The distinction between left and right becomes important when the function is not determined for some values of $x$. What are those cases in the examples below? @@ -258,31 +243,31 @@ fx1 + fx2 + plot_layout(nrow = 1) ## Continuity -To repeat a finding from the limits of functions: $f(x)$ does not necessarily have to be defined at $c$ for $\lim\limits_{x \to c}$ to exist. Functions that have breaks in their lines are called discontinuous. Functions that have no breaks are called continuous. Continuity is a concept that is more fundamental to, but related to that of "differentiability", which we will cover next in calculus. +To repeat a finding from the limits of functions: $f(x)$ does not necessarily have to be defined at $c$ for $\lim\limits_{x \to c}$ to exist. Functions that have breaks in their lines are called discontinuous. Functions that have no breaks are called continuous. Continuity is a concept that is more fundamental to, but related to that of "differentiability", which we will cover next in calculus. -:::{#def-continuity} +::: {#def-continuity} ## Continuity -Suppose that the domain of the function $f$ includes an open interval containing the point $c$. Then $f$ is continuous at $c$ if $\lim\limits_{x \to c} f(x)$ exists and if $\lim\limits_{x \to c} f(x)=f(c)$. Further, $f$ is continuous on an open interval $(a,b)$ if it is continuous at each point in the interval. + +Suppose that the domain of the function $f$ includes an open interval containing the point $c$. Then $f$ is continuous at $c$ if $\lim\limits_{x \to c} f(x)$ exists and if $\lim\limits_{x \to c} f(x)=f(c)$. Further, $f$ is continuous on an open interval $(a,b)$ if it is continuous at each point in the interval. ::: To prove that a function is continuous for all points is beyond this practical introduction to math, but the general intuition can be grasped by graphing. -:::{#exm-contdiscont} - +::: {#exm-contdiscont} ## Continuous and Discontinuous Functions -For each function, determine if it is continuous or discontinuous. +For each function, determine if it is continuous or discontinuous. - 1. $f(x) = \sqrt{x}$ - 2. $f(x) = e^x$ - 3. $f(x) = 1 + \frac{1}{x^2}$ - 4. $f(x) = \text{floor}(x)$. +1. $f(x) = \sqrt{x}$ +2. $f(x) = e^x$ +3. $f(x) = 1 + \frac{1}{x^2}$ +4. $f(x) = \text{floor}(x)$. -The floor is the smaller of the two integers bounding a number. So $\text{floor}(x = 2.999) = 2$, $\text{floor}(x = 2.0001) = 2$, and $\text{floor}(x = 2) = 2.$ +The floor is the smaller of the two integers bounding a number. So $\text{floor}(x = 2.999) = 2$, $\text{floor}(x = 2.0001) = 2$, and $\text{floor}(x = 2) = 2.$ ::: -:::{#sol-contdiscont} -In Figure @fig-contdiscont, we can see that the first two functions are continuous, and the next two are discontinuous. $f(x) = 1 + \frac{1}{x^2}$ is discontinuous at $x= 0$, and $f(x) = \text{floor}(x)$ is discontinuous at each whole number. +::: {#sol-contdiscont} +In Figure @fig-contdiscont, we can see that the first two functions are continuous, and the next two are discontinuous. $f(x) = 1 + \frac{1}{x^2}$ is discontinuous at $x= 0$, and $f(x) = \text{floor}(x)$ is discontinuous at each whole number. ::: ```{r} @@ -318,31 +303,27 @@ fx1 + fx2 + fx3 + fx4 + plot_layout(ncol = 2) Some properties of continuous functions: -1. If $f$ and $g$ are continuous at point $c$, then $f+g$, $f-g$, $f \cdot g$, $|f|$, and $\alpha f$ are continuous at point $c$ also. $f/g$ is continuous, provided $g(c)\ne 0$. -2. Boundedness: If $f$ is continuous on the closed bounded interval $[a,b]$, then there is a number $K$ such that $|f(x)|\le K$ for each $x$ in $[a,b]$. -3. Max/Min: If $f$ is continuous on the closed bounded interval $[a,b]$, then $f$ has a maximum and a minimum on $[a,b]$. They may be located at the end points. - -:::{#exr-discontdraw} +1. If $f$ and $g$ are continuous at point $c$, then $f+g$, $f-g$, $f \cdot g$, $|f|$, and $\alpha f$ are continuous at point $c$ also. $f/g$ is continuous, provided $g(c)\ne 0$. +2. Boundedness: If $f$ is continuous on the closed bounded interval $[a,b]$, then there is a number $K$ such that $|f(x)|\le K$ for each $x$ in $[a,b]$. +3. Max/Min: If $f$ is continuous on the closed bounded interval $[a,b]$, then $f$ has a maximum and a minimum on $[a,b]$. They may be located at the end points. +::: {#exr-discontdraw} ## Limit when Denominator converges to 0 Let $$f(x) = \frac{x^2 + 2x}{x}.$$ -1. Graph the function. Is it defined everywhere? -2. What is the functions limit at $x \rightarrow 0$? - +1. Graph the function. Is it defined everywhere? +2. What is the functions limit at $x \rightarrow 0$? ::: ## Answers to Examples {.unnumbered} Example @exm-seqbehav -:::{#sol-seqbehav} - - 1. $\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\} = \left\{1, \frac{7}{4}, \frac{17}{9}, \frac{31}{16}, \frac{49}{25}, \ldots\right\} = 2$ - 2. $\{B_n\}=\left\{\frac{n^2+1}{n} \right\} = \left\{2, \frac{5}{2}, \frac{10}{3}, \frac{17}{4}..., \right\}$ - 3. $\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\} = \left\{0, \frac{1}{2}, -\frac{2}{3}, \frac{3}{4}, -\frac{4}{5}\right\}$ - +::: {#sol-seqbehav} +1. $\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\} = \left\{1, \frac{7}{4}, \frac{17}{9}, \frac{31}{16}, \frac{49}{25}, \ldots\right\} = 2$ +2. $\{B_n\}=\left\{\frac{n^2+1}{n} \right\} = \left\{2, \frac{5}{2}, \frac{10}{3}, \frac{17}{4}..., \right\}$ +3. $\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\} = \left\{0, \frac{1}{2}, -\frac{2}{3}, \frac{3}{4}, -\frac{4}{5}\right\}$ ::: Exercise @exr-limseq2 @@ -353,36 +334,33 @@ Exercise @exr-limseq2 Example @exm-limfun1 -:::{#sol-limfun1} - - 1. $k$ - 2. $c$ - 3. $\lim_{x\to 2} (2x-3) = 2\lim\limits_{x\to 2} x - 3\lim\limits_{x\to 2} 1 = 1$ - 4. $\lim_{x \to c} x^n = \lim\limits_{x \to c} x \cdots[\lim\limits_{x \to c} x] = c\cdots c =c^n$ - +::: {#sol-limfun1} +1. $k$ +2. $c$ +3. $\lim_{x\to 2} (2x-3) = 2\lim\limits_{x\to 2} x - 3\lim\limits_{x\to 2} 1 = 1$ +4. $\lim_{x \to c} x^n = \lim\limits_{x \to c} x \cdots[\lim\limits_{x \to c} x] = c\cdots c =c^n$ ::: Exercise @exr-limfunmax -:::{#sol-} +::: {#sol-} Although this function seems large, the thing our eyes should focus on is where the highest order polynomial remains. That will grow the fastest, so if the highest order term is on the denominator, the fraction will converge to 0, if it is on the numerator it will converge to negative infinity. Previewing the multiplication by hand, we can see that the $-x^9$ on the numerator will be the largest power. So the answer will be $-\infty$. We can also confirm this by writing out fractions: +```{=tex} \begin{align*} & \lim_{x\to\infty}\frac{\left(1 + \frac{3}{x^3} - \frac{99}{4x^4}\right)\left(-\frac{2}{x^5} + 1\right)}{\left(1 + \frac{9}{18x} - \frac{3}{18x^5} - \frac{1}{18x^7} \right)\left(1 + \frac{1}{x}\right)} \\ &\times \frac{x^4}{1} \times -\frac{x^5}{1} \times \frac{1}{18x^7}\times \frac{1}{x}\\ =& 1 \times \lim_{-x\to\infty} \frac{x}{18} \end{align*} - +``` ::: Exercise @exr-discontdraw -:::{#sol-discontdraw} - +::: {#sol-discontdraw} See Figure @fig-hole-0. Divide each part by $x$, and we get $x + \frac{2}{x}$ on the numerator, $1$ on the denominator. So, without worrying about a function being not defined, we can say $\lim_{x\to 0}f(x) = 0$. - ::: ```{r} @@ -396,4 +374,3 @@ ggplot(range0, aes(x = x)) + labs(y = expression(f(x)), title = expression(f(x)==frac(x^2+2*x, x^2))) + geom_point(data = tibble(x = 0, y = 2), aes(y = y), pch = 21, fill = "white") ``` - diff --git a/04_calculus.qmd b/04_calculus.qmd index 793b892..5daed5b 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -12,19 +12,19 @@ Calculus is a fundamental part of any type of statistics exercise. Although you ## Example: The Mean is a Type of Integral {.unnumbered} -The average of a quantity is a type of weighted mean, where the potential values are weighted by their likelihood, loosely speaking. The integral is actually a general way to describe this weighted average when there are conceptually an infinite number of potential values. +The average of a quantity is a type of weighted mean, where the potential values are weighted by their likelihood, loosely speaking. The integral is actually a general way to describe this weighted average when there are conceptually an infinite number of potential values. -If $X$ is a continuous random variable, its expected value $E(X)$ -- the center of mass -- is given by +If $X$ is a continuous random variable, its expected value $E(X)$ -- the center of mass -- is given by $$E(X) = \int^{\infty}_{-\infty}x f(x) dx$$ -where $f(x)$ is the probability density function of $X$. +where $f(x)$ is the probability density function of $X$. -This is a continuous version of the case where $X$ is discrete, in which case +This is a continuous version of the case where $X$ is discrete, in which case $$E(X) = \sum^\infty_{j=1} x_j P(X = x_j)$$ -even more concretely, if the potential values of $X$ are finite, then we can write out the expected value as a weighted mean, where the weights is the probability that the value occurs. +even more concretely, if the potential values of $X$ are finite, then we can write out the expected value as a weighted mean, where the weights is the probability that the value occurs. $$E(X) = \large \sum_{x} \quad\left( \underbrace{x}_{\text{value}}\cdot \underbrace{P(X = x)}_{\text{weight, or PMF}}\right)$$ @@ -32,17 +32,17 @@ $$E(X) = \large \sum_{x} \quad\left( \underbrace{x}_{\text{value}}\cdot \underbr The derivative of $f$ at $x$ is its rate of change at $x$: how much $f(x)$ changes with a change in $x$. The rate of change is a fraction --- rise over run --- but because not all lines are straight and the rise over run formula will give us different values depending on the range we examine, we need to take a limit (Section @sec-limits-precalc). -:::{#def-derivative} +::: {#def-derivative} ## Derivative -Let $f$ be a function whose domain includes an open interval containing the point $x$. The derivative of $f$ at $x$ is given by + +Let $f$ be a function whose domain includes an open interval containing the point $x$. The derivative of $f$ at $x$ is given by $$\frac{d}{dx}f(x) =\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x} = \lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ There are a two main ways to denote a derivative: -* Leibniz Notation: $\frac{d}{dx}(f(x))$ -* Prime or Lagrange Notation: $f'(x)$ - +- Leibniz Notation: $\frac{d}{dx}(f(x))$ +- Prime or Lagrange Notation: $f'(x)$ ::: If $f(x)$ is a straight line, the derivative is the slope. For a curve, the slope changes by the values of $x$, so the derivative is the slope of the line tangent to the curve at $x$. See, For example, Figure @fig-derivsimple. @@ -75,46 +75,46 @@ gprimex <- fx0 + stat_function(fun = function(x) 3*(x^2), size = 0.5, linetype = fx + fprimex + gx + gprimex + plot_layout(ncol = 2) ``` -If $f'(x)$ exists at a point $x_0$, then $f$ is said to be __differentiable__ at $x_0$. That also implies that $f(x)$ is continuous at $x_0$. +If $f'(x)$ exists at a point $x_0$, then $f$ is said to be **differentiable** at $x_0$. That also implies that $f(x)$ is continuous at $x_0$. + +### Properties of derivatives {.unnumbered} -### Properties of derivatives {.unnumbered} -Suppose that $f$ and $g$ are differentiable at $x$ and that $\alpha$ is a constant. Then the functions $f\pm g$, $\alpha f$, $f g$, and $f/g$ (provided $g(x)\ne 0$) are also differentiable at $x$. Additionally, +Suppose that $f$ and $g$ are differentiable at $x$ and that $\alpha$ is a constant. Then the functions $f\pm g$, $\alpha f$, $f g$, and $f/g$ (provided $g(x)\ne 0$) are also differentiable at $x$. Additionally, -__Constant rule:__ $$\left[k f(x)\right]' = k f'(x)$$ +**Constant rule:** $$\left[k f(x)\right]' = k f'(x)$$ -__Sum rule:__ $$\left[f(x)\pm g(x)\right]' = f'(x)\pm g'(x)$$ +**Sum rule:** $$\left[f(x)\pm g(x)\right]' = f'(x)\pm g'(x)$$ With a bit more algebra, we can apply the definition of derivatives to get a formula for of the derivative of a product and a derivative of a quotient. -__Product rule:__ $$\left[f(x)g(x)\right]^\prime = f^\prime(x)g(x)+f(x)g^\prime(x)$$ +**Product rule:** $$\left[f(x)g(x)\right]^\prime = f^\prime(x)g(x)+f(x)g^\prime(x)$$ -__Quotient rule:__ $$\left[f(x)/g(x)\right]^\prime = \frac{f^\prime(x)g(x) - f(x)g^\prime(x)}{[g(x)]^2}, ~g(x)\neq 0$$ +**Quotient rule:** $$\left[f(x)/g(x)\right]^\prime = \frac{f^\prime(x)g(x) - f(x)g^\prime(x)}{[g(x)]^2}, ~g(x)\neq 0$$ -Finally, one way to think of the power of derivatives is that it takes a function a notch down in complexity. The power rule applies to any higher-order function: +Finally, one way to think of the power of derivatives is that it takes a function a notch down in complexity. The power rule applies to any higher-order function: -__Power rule:__ $$\left[x^k\right]^\prime = k x^{k-1}$$ +**Power rule:** $$\left[x^k\right]^\prime = k x^{k-1}$$ -For any real number $k$ (that is, both whole numbers and fractions). The power rule is proved __by induction__, a neat method of proof used in many fundamental applications to prove that a general statement holds for every possible case, even if there are countably infinite cases. We'll show a simple case where $k$ is an integer here. +For any real number $k$ (that is, both whole numbers and fractions). The power rule is proved **by induction**, a neat method of proof used in many fundamental applications to prove that a general statement holds for every possible case, even if there are countably infinite cases. We'll show a simple case where $k$ is an integer here. -:::{.proof} +::: proof ## Proof of Power Rule by Induction -We would like to prove that +We would like to prove that -$$\left[x^k\right]^\prime = k x^{k-1}$$ -for any integer $k$. +$$\left[x^k\right]^\prime = k x^{k-1}$$ for any integer $k$. -First, consider the first case (the base case) of $k = 1$. We can show by the definition of derivatives (setting $f(x) = x^1 = 1$) that +First, consider the first case (the base case) of $k = 1$. We can show by the definition of derivatives (setting $f(x) = x^1 = 1$) that $$[x^1]^\prime = \lim_{h \rightarrow 0}\frac{(x + h) - x}{(x + h) - x}= 1.$$ Because $1$ is also expressed as $1 x^{1- 1}$, the statement we want to prove holds for the case $k =1$. -Now, \emph{assume} that the statement holds for some integer $m$. That is, assume -$$\left[x^m\right]^\prime = m x^{m-1}$$ +Now, \emph{assume} that the statement holds for some integer $m$. That is, assume $$\left[x^m\right]^\prime = m x^{m-1}$$ Then, for the case $m + 1$, using the product rule above, we can simplify +```{=tex} \begin{align*} \left[x^{m + 1}\right]^\prime &= [x^{m}\cdot x]^\prime\\ &= (x^m)^\prime\cdot x + (x^m)\cdot (x)^\prime\\ @@ -123,32 +123,29 @@ Then, for the case $m + 1$, using the product rule above, we can simplify &= (m + 1)x^m\\ &= (m + 1)x^{(m + 1) - 1} \end{align*} - -Therefore, the rule holds for the case $k = m + 1$ once we have assumed it holds for $k = m$. Combined with the first case, this completes proof by induction -- we have now proved that the statement holds for all integers $k = 1, 2, 3, \cdots$. +``` +Therefore, the rule holds for the case $k = m + 1$ once we have assumed it holds for $k = m$. Combined with the first case, this completes proof by induction -- we have now proved that the statement holds for all integers $k = 1, 2, 3, \cdots$. To show that it holds for real fractions as well, we can prove expressing that exponent by a fraction of two integers. - ::: -These "rules" become apparent by applying the definition of the derivative above to each of the things to be "derived", but these come up so frequently that it is best to repeat until it is muscle memory. - -:::{#exr-introderivatives} +These "rules" become apparent by applying the definition of the derivative above to each of the things to be "derived", but these come up so frequently that it is best to repeat until it is muscle memory. +::: {#exr-introderivatives} ## Derivative of Polynomials -For each of the following functions, find the first-order derivative $f^\prime(x)$. - -1. $f(x)=c$ -1. $f(x)=x$ -1. $f(x)=x^2$ -1. $f(x)=x^3$ -1. $f(x)=\frac{1}{x^2}$ -1. $f(x)=(x^3)(2x^4)$ -1. $f(x) = x^4 - x^3 + x^2 - x + 1$ -1. $f(x) = (x^2 + 1)(x^3 - 1)$ -1. $f(x) = 3x^2 + 2x^{1/3}$ -1. $f(x)=\frac{x^2+1}{x^2-1}$ +For each of the following functions, find the first-order derivative $f^\prime(x)$. +1. $f(x)=c$ +2. $f(x)=x$ +3. $f(x)=x^2$ +4. $f(x)=x^3$ +5. $f(x)=\frac{1}{x^2}$ +6. $f(x)=(x^3)(2x^4)$ +7. $f(x) = x^4 - x^3 + x^2 - x + 1$ +8. $f(x) = (x^2 + 1)(x^3 - 1)$ +9. $f(x) = 3x^2 + 2x^{1/3}$ +10. $f(x)=\frac{x^2+1}{x^2-1}$ ::: ## Higher-Order Derivatives (Derivatives of Derivatives of Derivatives) {#derivpoly} @@ -157,14 +154,16 @@ The first derivative is applying the definition of derivatives on the function, $$f'(x), ~~ y', ~~ \frac{d}{dx}f(x), ~~ \frac{dy}{dx}$$ -We can keep applying the differentiation process to functions that are themselves derivatives. The derivative of $f'(x)$ with respect to $x$, would then be $$f''(x)=\lim\limits_{h\to 0}\frac{f'(x+h)-f'(x)}{h}$$ and we can therefore call it the __Second derivative:__ +We can keep applying the differentiation process to functions that are themselves derivatives. The derivative of $f'(x)$ with respect to $x$, would then be $$f''(x)=\lim\limits_{h\to 0}\frac{f'(x+h)-f'(x)}{h}$$ and we can therefore call it the **Second derivative:** $$f''(x), ~~ y'', ~~ \frac{d^2}{dx^2}f(x), ~~ \frac{d^2y}{dx^2}$$ -Similarly, the derivative of $f''(x)$ would be called the third derivative and is denoted $f'''(x)$. And by extension, the __nth derivative__ is expressed as $\frac{d^n}{dx^n}f(x)$, $\frac{d^ny}{dx^n}$. +Similarly, the derivative of $f''(x)$ would be called the third derivative and is denoted $f'''(x)$. And by extension, the **nth derivative** is expressed as $\frac{d^n}{dx^n}f(x)$, $\frac{d^ny}{dx^n}$. -:::{#exm-succession} +::: {#exm-succession} ## Succession of Derivatives + +```{=tex} \begin{align*} f(x) &=x^3\\ f^{\prime}(x) &=3x^2\\ @@ -172,42 +171,40 @@ f^{\prime\prime}(x) &=6x \\ f^{\prime\prime\prime}(x) &=6\\ f^{\prime\prime\prime\prime}(x) &=0\\ \end{align*} +``` ::: -Earlier, in Section @sec-derivintro, we said that if a function differentiable at a given point, then it must be continuous. Further, if $f'(x)$ is itself continuous, then $f(x)$ is called continuously differentiable. All of this matters because many of our findings about optimization (Section @sec-optim) rely on differentiation, and so we want our function to be differentiable in as many layers. A function that is continuously differentiable infinitly is called "smooth". Some examples: $f(x) = x^2$, $f(x) = e^x$. +Earlier, in Section @sec-derivintro, we said that if a function differentiable at a given point, then it must be continuous. Further, if $f'(x)$ is itself continuous, then $f(x)$ is called continuously differentiable. All of this matters because many of our findings about optimization (Section @sec-optim) rely on differentiation, and so we want our function to be differentiable in as many layers. A function that is continuously differentiable infinitly is called "smooth". Some examples: $f(x) = x^2$, $f(x) = e^x$. ## Composite Functions and the Chain Rule -As useful as the above rules are, many functions you'll see won't fit neatly in each case immediately. Instead, they will be functions of functions. For example, the difference between $x^2 + 1^2$ and $(x^2 + 1)^2$ may look trivial, but the sum rule can be easily applied to the former, while it's actually not obvious what do with the latter. +As useful as the above rules are, many functions you'll see won't fit neatly in each case immediately. Instead, they will be functions of functions. For example, the difference between $x^2 + 1^2$ and $(x^2 + 1)^2$ may look trivial, but the sum rule can be easily applied to the former, while it's actually not obvious what do with the latter. -__Composite functions__ are formed by substituting one function into another and are denoted by $$(f\circ g)(x)=f[g(x)].$$ To form $f[g(x)]$, the range of $g$ must be contained (at least in part) within the domain of $f$. The domain of $f\circ g$ consists of all the points in the domain of $g$ for which $g(x)$ is in the domain of $f$. +**Composite functions** are formed by substituting one function into another and are denoted by $$(f\circ g)(x)=f[g(x)].$$ To form $f[g(x)]$, the range of $g$ must be contained (at least in part) within the domain of $f$. The domain of $f\circ g$ consists of all the points in the domain of $g$ for which $g(x)$ is in the domain of $f$. -:::{#exm-composite} -Let $f(x)=\log x$ for $0 0$$ - +$$\left(e^{g(x)}\right)^\prime = e^{g(x)} \cdot g^\prime(x)$$ $$\left(\log g(x)\right)^\prime = \frac{g^\prime(x)}{g(x)}, ~~\text{if}~~ g(x) > 0$$ ::: -We will relegate the proofs to small excerpts. +We will relegate the proofs to small excerpts. ### Derivatives of natural exponential function ($e$) {.unnumbered} -To repeat the main rule in Theorem @thm-derivexplog, the intuition is that +To repeat the main rule in Theorem @thm-derivexplog, the intuition is that -1. Derivative of $e^x$ is itself: $\frac{d}{dx}e^x = e^x$ (See Figure @fig-derivexponent.) -2. Same thing if there were a constant in front: $\frac{d}{dx}\alpha e^x = \alpha e^x$ -3. Same thing no matter how many derivatives there are in front: $\frac{d^n}{dx^n} \alpha e^x = \alpha e^x$ -4. Chain Rule: When the exponent is a function of $x$, remember to take derivative of that function and add to product. $\frac{d}{dx}e^{g(x)}= e^{g(x)} g^\prime(x)$ +1. Derivative of $e^x$ is itself: $\frac{d}{dx}e^x = e^x$ (See Figure @fig-derivexponent.) +2. Same thing if there were a constant in front: $\frac{d}{dx}\alpha e^x = \alpha e^x$ +3. Same thing no matter how many derivatives there are in front: $\frac{d^n}{dx^n} \alpha e^x = \alpha e^x$ +4. Chain Rule: When the exponent is a function of $x$, remember to take derivative of that function and add to product. $\frac{d}{dx}e^{g(x)}= e^{g(x)} g^\prime(x)$ ```{r} #| label: fig-derivexponent @@ -267,25 +259,24 @@ fprimex <- fx0 + stat_function(fun = function(x) exp(x), size = 0.5) + fx + fprimex + plot_layout(nrow = 1) ``` -:::{#exm-exmderivexp} - +::: {#exm-exmderivexp} ## Derivative of exponents -Find the derivative for the following. -1. $f(x)=e^{-3x}$ -2. $f(x)=e^{x^2}$ -3. $f(x)=(x-1)e^x$ +Find the derivative for the following. +1. $f(x)=e^{-3x}$ +2. $f(x)=e^{x^2}$ +3. $f(x)=(x-1)e^x$ ::: ### Derivatives of $\log$ {.unnumbered} -The natural log is the mirror image of the natural exponent and has mirroring properties, again, to repeat the theorem, +The natural log is the mirror image of the natural exponent and has mirroring properties, again, to repeat the theorem, -1. log prime x is one over x: $\frac{d}{dx} \log x = \frac{1}{x}$ (Figure @fig-derivlog.) -2. Exponents become multiplicative constants: $\frac{d}{dx} \log x^k = \frac{d}{dx} k \log x = \frac{k}{x}$ -3. Chain rule again: $\frac{d}{dx} \log u(x) = \frac{u'(x)}{u(x)}\quad$ -4. For any positive base $b$, $\frac{d}{dx} b^x = (\log b)\left(b^x\right)$. +1. log prime x is one over x: $\frac{d}{dx} \log x = \frac{1}{x}$ (Figure @fig-derivlog.) +2. Exponents become multiplicative constants: $\frac{d}{dx} \log x^k = \frac{d}{dx} k \log x = \frac{k}{x}$ +3. Chain rule again: $\frac{d}{dx} \log u(x) = \frac{u'(x)}{u(x)}\quad$ +4. For any positive base $b$, $\frac{d}{dx} b^x = (\log b)\left(b^x\right)$. ```{r} #| label: fig-derivlog @@ -308,47 +299,46 @@ fprimex <- fx0 + stat_function(fun = function(x) ifelse(x <= 0, NA, 1/x), size = fx + fprimex + plot_layout(nrow = 1) ``` -:::{#exm-exmderivlog} - +::: {#exm-exmderivlog} ## Derivative of logs Find $dy/dx$ for the following. -1. $f(x)=\log(x^2+9)$ -2. $f(x)=\log(\log x)$ -3. $f(x)=(\log x)^2$ -4. $f(x)=\log e^x$ - +1. $f(x)=\log(x^2+9)$ +2. $f(x)=\log(\log x)$ +3. $f(x)=(\log x)^2$ +4. $f(x)=\log e^x$ ::: ### Outline of Proof {.unnumbered} -We actually show the derivative of the log first, and then the derivative of the exponential naturally follows. +We actually show the derivative of the log first, and then the derivative of the exponential naturally follows. The general derivative of the log at any base $a$ is solvable by the definition of derivatives. +```{=tex} \begin{align*} (\log_a x)^\prime = \lim\limits_{h\to 0} \frac{1}{h}\log_{a}\left(1 + \frac{h}{x}\right) \end{align*} - -Re-express $g = \frac{h}{x}$ and get -\begin{align*} +``` +Re-express $g = \frac{h}{x}$ and get \begin{align*} (\log_a x)^\prime &= \frac{1}{x}\lim_{g\to 0}\log_{a} (1 + g)^{\frac{1}{g}}\\ &= \frac{1}{x}\log_a e \end{align*} -By definition of $e$. As a special case, when $a = e$, then $(\log x)^\prime = \frac{1}{x}$. +By definition of $e$. As a special case, when $a = e$, then $(\log x)^\prime = \frac{1}{x}$. Now let's think about the inverse, taking the derivative of $y = a^x$. +```{=tex} \begin{align*} y &= a^x \\ \Rightarrow \log y &= x \log a\\ \Rightarrow \frac{y^\prime}{y} &= \log a\\ \Rightarrow y^\prime = y \log a\\ \end{align*} - -Then in the special case where $a = e$, +``` +Then in the special case where $a = e$, $$(e^x)^\prime = (e^x)$$ @@ -358,9 +348,9 @@ What happens when there's more than variable that is changing? > If you can do ordinary derivatives, you can do partial derivatives: just hold all the other input variables constant except for the one you're differentiating with respect to. (Joe Blitzstein's Math Notes) -Suppose we have a function $f$ now of two (or more) variables and we want to determine the rate of change relative to one of the variables. To do so, we would find its partial derivative, which is defined similar to the derivative of a function of one variable. +Suppose we have a function $f$ now of two (or more) variables and we want to determine the rate of change relative to one of the variables. To do so, we would find its partial derivative, which is defined similar to the derivative of a function of one variable. -__Partial Derivative__: Let $f$ be a function of the variables $(x_1,\ldots,x_n)$. The partial derivative of $f$ with respect to $x_i$ is +**Partial Derivative**: Let $f$ be a function of the variables $(x_1,\ldots,x_n)$. The partial derivative of $f$ with respect to $x_i$ is $$\frac{\partial f}{\partial x_i} (x_1,\ldots,x_n) = \lim\limits_{h\to 0} \frac{f(x_1,\ldots,x_i+h,\ldots,x_n)-f(x_1,\ldots,x_i,\ldots,x_n)}{h}$$ @@ -368,103 +358,106 @@ Only the $i$th variable changes --- the others are treated as constants. We can take higher-order partial derivatives, like we did with functions of a single variable, except now the higher-order partials can be with respect to multiple variables. -:::{#exm-partialtypes} +::: {#exm-partialtypes} ## More than one type of partial -Notice that you can take partials with regard to different variables. + +Notice that you can take partials with regard to different variables. Suppose $f(x,y)=x^2+y^2$. Then +```{=tex} \begin{align*} \frac{\partial f}{\partial x}(x,y) &=\\ \frac{\partial f}{\partial y}(x,y) &=\\ \frac{\partial^2 f}{\partial x^2}(x,y) &=\\ \frac{\partial^2 f}{\partial x \partial y}(x,y) &= \end{align*} +``` ::: -:::{#exr-partialderivs} +::: {#exr-partialderivs} Let $f(x,y)=x^3 y^4 +e^x -\log y$. What are the following partial derivaitves? +```{=tex} \begin{align*} \frac{\partial f}{\partial x}(x,y) &=\\ \frac{\partial f}{\partial y}(x,y) &=\\ \frac{\partial^2 f}{\partial x^2}(x,y) &=\\ \frac{\partial^2 f}{\partial x \partial y}(x,y) &= \end{align*} - +``` ::: ## Taylor Series Approximation {#taylorapprox} -A common form of approximation used in statistics involves derivatives. A Taylor series is a way to represent common functions as infinite series (a sum of infinite elements) of the function's derivatives at some point $a$. +A common form of approximation used in statistics involves derivatives. A Taylor series is a way to represent common functions as infinite series (a sum of infinite elements) of the function's derivatives at some point $a$. -For example, Taylor series are very helpful in representing nonlinear (read: difficult) functions as linear (read: manageable) functions. One can thus __approximate__ functions by using lower-order, finite series known as __Taylor polynomials__. If $a=0$, the series is called a Maclaurin series. +For example, Taylor series are very helpful in representing nonlinear (read: difficult) functions as linear (read: manageable) functions. One can thus **approximate** functions by using lower-order, finite series known as **Taylor polynomials**. If $a=0$, the series is called a Maclaurin series. -Specifically, a Taylor series of a real or complex function $f(x)$ that is infinitely differentiable in the neighborhood of point $a$ is: +Specifically, a Taylor series of a real or complex function $f(x)$ that is infinitely differentiable in the neighborhood of point $a$ is: +```{=tex} \begin{align*} - f(x) &= f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots\\ - &= \sum_{n=0}^\infty \frac{f^{(n)} (a)}{n!} (x-a)^n + f(x) &= f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots\\ + &= \sum_{n=0}^\infty \frac{f^{(n)} (a)}{n!} (x-a)^n \end{align*} - -__Taylor Approximation__: We can often approximate the curvature of a function $f(x)$ at point $a$ using a 2nd order Taylor polynomial around point $a$: +``` +**Taylor Approximation**: We can often approximate the curvature of a function $f(x)$ at point $a$ using a 2nd order Taylor polynomial around point $a$: $$f(x) = f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + R_2$$ -$R_2$ is the remainder (R for remainder, 2 for the fact that we took two derivatives) and often treated as negligible, -giving us: +$R_2$ is the remainder (R for remainder, 2 for the fact that we took two derivatives) and often treated as negligible, giving us: $$f(x) \approx f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2} (x-a)^2$$ -The more derivatives that are added, the smaller the remainder $R$ and the more accurate the approximation. Proofs involving limits guarantee that the remainder converges to 0 as the order of derivation increases. +The more derivatives that are added, the smaller the remainder $R$ and the more accurate the approximation. Proofs involving limits guarantee that the remainder converges to 0 as the order of derivation increases. ## The Indefinite Integration -So far, we've been interested in finding the derivative $f=F'$ of a function $F$. However, sometimes we're interested in exactly the reverse: finding the function $F$ for which $f$ is its derivative. We refer to $F$ as the antiderivative of $f$. +So far, we've been interested in finding the derivative $f=F'$ of a function $F$. However, sometimes we're interested in exactly the reverse: finding the function $F$ for which $f$ is its derivative. We refer to $F$ as the antiderivative of $f$. -:::{#def-antiderivative} +::: {#def-antiderivative} ## Antiderivative -The antiverivative of a function $f(x)$ is a differentiable function $F$ whose derivative is $f$. -$$F^\prime = f.$$ +The antiverivative of a function $f(x)$ is a differentiable function $F$ whose derivative is $f$. +$$F^\prime = f.$$ ::: -Another way to describe is through the inverse formula. Let $DF$ be the derivative of $F$. And let $DF(x)$ be the derivative of $F$ evaluated at $x$. Then the antiderivative is denoted by $D^{-1}$ (i.e., the inverse derivative). If $DF=f$, then $F=D^{-1}f$. +Another way to describe is through the inverse formula. Let $DF$ be the derivative of $F$. And let $DF(x)$ be the derivative of $F$ evaluated at $x$. Then the antiderivative is denoted by $D^{-1}$ (i.e., the inverse derivative). If $DF=f$, then $F=D^{-1}f$. -This definition bolsters the main takeaway about integrals and derivatives: They are inverses of each other. +This definition bolsters the main takeaway about integrals and derivatives: They are inverses of each other. -:::{#exr-antiderivatives} +::: {#exr-antiderivatives} ## Antiderivative -Find the antiderivative of the following: -1. $f(x) = \frac{1}{x^2}$ -2. $f(x) = 3e^{3x}$ +Find the antiderivative of the following: +1. $f(x) = \frac{1}{x^2}$ +2. $f(x) = 3e^{3x}$ ::: We know from derivatives how to manipulate $F$ to get $f$. But how do you express the procedure to manipulate $f$ to get $F$? For that, we need a new symbol, which we will call indefinite integration. -:::{#def-integral} +::: {#def-integral} ## Indefinite Integral -The indefinite integral of $f(x)$ is written -$$\int f(x) dx $$ +The indefinite integral of $f(x)$ is written -and is equal to the antiderivative of $f$. +$$\int f(x) dx $$ +and is equal to the antiderivative of $f$. ::: -:::{#exm-integralc} +::: {#exm-integralc} Draw the function $f(x)$ and its indefinite integral, $\int\limits f(x) dx$ $$f(x) = (x^2-4)$$ - ::: -:::{#sol-integralc} -The Indefinite Integral of the function $f(x) = (x^2-4)$ can, for example, be $F(x) = \frac{1}{3}x^3 - 4x.$ But it can also be $F(x) = \frac{1}{3}x^3 - 4x + 1$, because the constant 1 disappears when taking the derivative. +::: {#sol-integralc} +The Indefinite Integral of the function $f(x) = (x^2-4)$ can, for example, be $F(x) = \frac{1}{3}x^3 - 4x.$ But it can also be $F(x) = \frac{1}{3}x^3 - 4x + 1$, because the constant 1 disappears when taking the derivative. ::: Some of these functions are plotted in the bottom panel of Figure @fig-integralc as dotted lines. @@ -488,36 +481,36 @@ Fx <- ggplot(range1, aes(x = x)) + fx + Fx + plot_layout(ncol = 1) ``` -Notice from these examples that while there is only a single derivative for any function, there are multiple antiderivatives: one for any arbitrary constant $c$. $c$ just shifts the curve up or down on the $y$-axis. If more information is present about the antiderivative --- e.g., that it passes through a particular point --- then we can solve for a specific value of $c$. +Notice from these examples that while there is only a single derivative for any function, there are multiple antiderivatives: one for any arbitrary constant $c$. $c$ just shifts the curve up or down on the $y$-axis. If more information is present about the antiderivative --- e.g., that it passes through a particular point --- then we can solve for a specific value of $c$. ### Common Rules of Integration {.unnumbered} -Some common rules of integrals follow by virtue of being the inverse of a derivative. +Some common rules of integrals follow by virtue of being the inverse of a derivative. -1. Constants are allowed to slip out: $\int a f(x)dx = a\int f(x)dx$ -1. Integration of the sum is sum of integrations: $\int [f(x)+g(x)]dx=\int f(x)dx + \int g(x)dx$ -1. Reverse Power-rule: $\int x^n dx = \frac{1}{n+1} x^{n+1} + c$ -1. Exponents are still exponents: $\int e^x dx = e^x +c$ -1. Recall the derivative of $\log(x)$ is one over $x$, and so: $\int \frac{1}{x} dx = \log x + c$ -1. Reverse chain-rule: $\int e^{f(x)}f^\prime(x)dx = e^{f(x)}+c$ -1. More generally: $\int [f(x)]^n f'(x)dx = \frac{1}{n+1}[f(x)]^{n+1}+c$ -1. Remember the derivative of a log of a function: $\int \frac{f^\prime(x)}{f(x)}dx=\log f(x) + c$ +1. Constants are allowed to slip out: $\int a f(x)dx = a\int f(x)dx$ +2. Integration of the sum is sum of integrations: $\int [f(x)+g(x)]dx=\int f(x)dx + \int g(x)dx$ +3. Reverse Power-rule: $\int x^n dx = \frac{1}{n+1} x^{n+1} + c$ +4. Exponents are still exponents: $\int e^x dx = e^x +c$ +5. Recall the derivative of $\log(x)$ is one over $x$, and so: $\int \frac{1}{x} dx = \log x + c$ +6. Reverse chain-rule: $\int e^{f(x)}f^\prime(x)dx = e^{f(x)}+c$ +7. More generally: $\int [f(x)]^n f'(x)dx = \frac{1}{n+1}[f(x)]^{n+1}+c$ +8. Remember the derivative of a log of a function: $\int \frac{f^\prime(x)}{f(x)}dx=\log f(x) + c$ -:::{#exm-common-integration} +::: {#exm-common-integration} ## Common Integration -Simplify the following indefinite integrals: -* $\int 3x^2 dx$ -* $\int (2x+1)dx$ -* $\int e^x e^{e^x} dx$ +Simplify the following indefinite integrals: +- $\int 3x^2 dx$ +- $\int (2x+1)dx$ +- $\int e^x e^{e^x} dx$ ::: ## The Definite Integral: The Area under the Curve -If there is a indefinite integral, there _must_ be a definite integral. Indeed there is, but the notion of definite integrals comes from a different objective: finding the are a under a function. We will find, perhaps remarkably, that the formula we find to get the sum turns out to be expressible by the anti-derivative. +If there is a indefinite integral, there *must* be a definite integral. Indeed there is, but the notion of definite integrals comes from a different objective: finding the are a under a function. We will find, perhaps remarkably, that the formula we find to get the sum turns out to be expressible by the anti-derivative. -Suppose we want to determine the area $A(R)$ of a region $R$ defined by a curve $f(x)$ and some interval $a\le x \le b$. +Suppose we want to determine the area $A(R)$ of a region $R$ defined by a curve $f(x)$ and some interval $a\le x \le b$. ```{r} #| label: fig-defintfig @@ -544,59 +537,56 @@ g2 <- fx + geom_col(data = d2, aes(x, f), width = 0.1, fill = "gray", alpha = g1 + g2 + plot_layout(nrow = 1) ``` -One way to calculate the area would be to divide the interval $a\le x\le b$ into $n$ subintervals of length $\Delta x$ and then approximate the region with a series of rectangles, where the base of each rectangle is $\Delta x$ and the height is $f(x)$ at the midpoint of that interval. $A(R)$ would then be approximated by the area of the union of the rectangles, which is given by $$S(f,\Delta x)=\sum\limits_{i=1}^n f(x_i)\Delta x$$ and is called a __Riemann sum__. +One way to calculate the area would be to divide the interval $a\le x\le b$ into $n$ subintervals of length $\Delta x$ and then approximate the region with a series of rectangles, where the base of each rectangle is $\Delta x$ and the height is $f(x)$ at the midpoint of that interval. $A(R)$ would then be approximated by the area of the union of the rectangles, which is given by $$S(f,\Delta x)=\sum\limits_{i=1}^n f(x_i)\Delta x$$ and is called a **Riemann sum**. -As we decrease the size of the subintervals $\Delta x$, making the rectangles "thinner," we would expect our approximation of the area of the region to become closer to the true area. This allows us to express the area as a limit of a series: -$$A(R)=\lim\limits_{\Delta x\to 0}\sum\limits_{i=1}^n f(x_i)\Delta x$$ +As we decrease the size of the subintervals $\Delta x$, making the rectangles "thinner," we would expect our approximation of the area of the region to become closer to the true area. This allows us to express the area as a limit of a series: $$A(R)=\lim\limits_{\Delta x\to 0}\sum\limits_{i=1}^n f(x_i)\Delta x$$ -Figure @fig-defintfig shows that illustration. The curve depicted is $f(x) = -15(x - 5) + (x - 5)^3 + 50.$ We want approximate the area under the curve between the $x$ values of 0 and 10. We can do this in blocks of arbitrary width, where the sum of rectangles (the area of which is width times $f(x)$ evaluated at the midpoint of the bar) shows the Riemann Sum. As the width of the bars $\Delta x$ becomes smaller, the better the estimate of $A(R)$. +Figure @fig-defintfig shows that illustration. The curve depicted is $f(x) = -15(x - 5) + (x - 5)^3 + 50.$ We want approximate the area under the curve between the $x$ values of 0 and 10. We can do this in blocks of arbitrary width, where the sum of rectangles (the area of which is width times $f(x)$ evaluated at the midpoint of the bar) shows the Riemann Sum. As the width of the bars $\Delta x$ becomes smaller, the better the estimate of $A(R)$. This is how we define the "Definite" Integral: -:::{#def-defintegral} +::: {#def-defintegral} ## The Definite Integral (Riemann) -If for a given function $f$ the Riemann sum approaches a limit as $\Delta x \to 0$, then that limit is called the Riemann integral of $f$ from $a$ to $b$. We express this with the $\int$, symbol, and write $$\int\limits_a^b f(x) dx= \lim\limits_{\Delta x\to 0} \sum\limits_{i=1}^n f(x_i)\Delta x$$ -The most straightforward of a definite integral is the definite integral. That is, we read -$$\int\limits_a^b f(x) dx$$ as the definite integral of $f$ from $a$ to $b$ and we defined as the area under the "curve" $f(x)$ from point $x=a$ to $x=b$. +If for a given function $f$ the Riemann sum approaches a limit as $\Delta x \to 0$, then that limit is called the Riemann integral of $f$ from $a$ to $b$. We express this with the $\int$, symbol, and write $$\int\limits_a^b f(x) dx= \lim\limits_{\Delta x\to 0} \sum\limits_{i=1}^n f(x_i)\Delta x$$ + +The most straightforward of a definite integral is the definite integral. That is, we read $$\int\limits_a^b f(x) dx$$ as the definite integral of $f$ from $a$ to $b$ and we defined as the area under the "curve" $f(x)$ from point $x=a$ to $x=b$. ::: The fundamental theorem of calculus shows us that this sum is, in fact, the antiderivative. -:::{#thm-fftc} +::: {#thm-fftc} ## First Fundamental Theorem of Calculus -Let the function $f$ be bounded on $[a,b]$ and continuous on $(a,b)$. Then, suggestively, use the symbol $F(x)$ to denote the definite integral from $a$ to $x$: -$$F(x)=\int\limits_a^x f(t)dt, \quad a\le x\le b$$ -Then $F(x)$ has a derivative at each point in $(a,b)$ and $$F^\prime(x)=f(x), \quad a x) = 1 - P(X \le x)$. -:::{#exm-die-pmf} +::: {#exm-die-pmf} For a fair die with its value as $Y$, What are the following? -1. $P(Y\le 1)$ -2. $P(Y\le 3)$ -3. $P(Y\le 6)$ - +1. $P(Y\le 1)$ +2. $P(Y\le 3)$ +3. $P(Y\le 6)$ ::: ### Continuous Random Variables {.unnumbered} -We also have a similar definition for _continuous_ random variables. +We also have a similar definition for *continuous* random variables. -:::{#def-rv-continuous} +::: {#def-rv-continuous} ## Continuous Random Variable $X$ is a continuous random variable if there exists a nonnegative function $f(x)$ defined for all real $x\in (-\infty,\infty)$, such that for any interval $A$, $P(X\in A)=\int\limits_A f(x)dx$. Examples: age, income, GNP, temperature. - ::: -:::{#def-pdf} +::: {#def-pdf} ## Probability Density Function -The function $f$ above is called the probability density function (pdf) of $X$ and must satisfy -$$f(x)\ge 0$$ -$$\int\limits_{-\infty}^\infty f(x)dx=1$$ +The function $f$ above is called the probability density function (pdf) of $X$ and must satisfy $$f(x)\ge 0$$ $$\int\limits_{-\infty}^\infty f(x)dx=1$$ - Note also that $P(X = x)=0$ --- i.e., the probability of any point $y$ is zero. +Note also that $P(X = x)=0$ --- i.e., the probability of any point $y$ is zero. ::: +```{=tex} \begin{comment} \item[] \parbox[t]{4.5in}{Example: $f(y)=1, \quad 0\le y \le1$}\parbox{1.5in}{\hfill \epsffile{contpdf.eps}} \end{comment} +``` +For both discrete and continuous random variables, we have a unifying concept of another measure: the cumulative distribution: -For both discrete and continuous random variables, we have a unifying concept of another measure: the cumulative distribution: - -:::{#def-cdf} +::: {#def-cdf} ## Cumulative Density Function -Because the probability that a continuous random variable will assume any particular value is zero, we can only make statements about the probability of a continuous random variable being within an interval. The cumulative distribution gives the probability that $Y$ lies on the interval $(-\infty,y)$ and is defined as $$F(x)=P(X\le x)=\int\limits_{-\infty}^x f(s)ds$$ Note that $F(x)$ has similar properties with continuous distributions as it does with discrete - non-decreasing, continuous (not just right-continuous), and $\lim\limits_{x \to -\infty} F(x) = 0$ and $\lim\limits_{x \to \infty} F(x) = 1$. +Because the probability that a continuous random variable will assume any particular value is zero, we can only make statements about the probability of a continuous random variable being within an interval. The cumulative distribution gives the probability that $Y$ lies on the interval $(-\infty,y)$ and is defined as $$F(x)=P(X\le x)=\int\limits_{-\infty}^x f(s)ds$$ Note that $F(x)$ has similar properties with continuous distributions as it does with discrete - non-decreasing, continuous (not just right-continuous), and $\lim\limits_{x \to -\infty} F(x) = 0$ and $\lim\limits_{x \to \infty} F(x) = 1$. ::: -We can also make statements about the probability of $Y$ falling in an interval $a\le y\le b$. -$$P(a\le x\le b)=\int\limits_a^b f(x)dx$$ +We can also make statements about the probability of $Y$ falling in an interval $a\le y\le b$. $$P(a\le x\le b)=\int\limits_a^b f(x)dx$$ The PDF and CDF are linked by the integral: The CDF of the integral of the PDF: $$f(x) = F'(x)=\frac{dF(x)}{dx}$$ -:::{#exm-continuous-pdf} - -For $f(y)=1, \quad 0 0$ -* Continuous: $f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)},\quad f_X(x) > 0$ +- Discrete: $p_{Y|X}(y|x) = \frac{p(x,y)}{p_X(x)}, \quad p_X(x) > 0$ +- Continuous: $f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)},\quad f_X(x) > 0$ -:::{#exr-joint-distribution} +::: {#exr-joint-distribution} ## Discrete Outcomes -Suppose we are interested in the outcomes of flipping a coin and rolling a 6-sided die at the same time. The sample space for this process contains 12 elements: $$\{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}$$ We can define two random variables $X$ and $Y$ such that $X = 1$ if heads and $X = 0$ if tails, while $Y$ equals the number on the die. -We can then make statements about the joint distribution of $X$ and $Y$. What are the following? +Suppose we are interested in the outcomes of flipping a coin and rolling a 6-sided die at the same time. The sample space for this process contains 12 elements: $$\{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}$$ We can define two random variables $X$ and $Y$ such that $X = 1$ if heads and $X = 0$ if tails, while $Y$ equals the number on the die. -1. $P(X=x)$ -1. $P(Y=y)$ -1. $P(X=x, Y=y)$ -1. $P(X=x|Y=y)$ -1. Are X and Y independent? +We can then make statements about the joint distribution of $X$ and $Y$. What are the following? +1. $P(X=x)$ +2. $P(Y=y)$ +3. $P(X=x, Y=y)$ +4. $P(X=x|Y=y)$ +5. Are X and Y independent? ::: -## Expectation +## Expectation -We often want to summarize some characteristics of the distribution of a random variable. The most important summary is the expectation (or expected value, or mean), in which the possible values of a random variable are weighted by their probabilities. +We often want to summarize some characteristics of the distribution of a random variable. The most important summary is the expectation (or expected value, or mean), in which the possible values of a random variable are weighted by their probabilities. -:::{#def-expectation} +::: {#def-expectation} ## Expectation of a Discrete Random Variable -The expected value of a discrete random variable $Y$ is $$E(Y)=\sum\limits_{y} y P(Y=y)= \sum\limits_{y} y p(y)$$ -In words, it is the weighted average of all possible values of $Y$, weighted by the probability that $y$ occurs. It is not necessarily the number we would expect $Y$ to take on, but the average value of $Y$ after a large number of repetitions of an experiment. +The expected value of a discrete random variable $Y$ is $$E(Y)=\sum\limits_{y} y P(Y=y)= \sum\limits_{y} y p(y)$$\ +In words, it is the weighted average of all possible values of $Y$, weighted by the probability that $y$ occurs. It is not necessarily the number we would expect $Y$ to take on, but the average value of $Y$ after a large number of repetitions of an experiment. ::: -:::{#exm-expectdiscrete} +::: {#exm-expectdiscrete} ## Expectation of a Discrete Random Variable What is the expectation of a fair, six-sided die? - ::: -__Expectation of a Continuous Random Variable__: The expected -value of a continuous random variable is similar in concept to that of -the discrete random variable, except that instead of summing using -probabilities as weights, we integrate using the density to weight. -Hence, the expected value of the continuous variable $Y$ is defined by -$$E(Y)=\int\limits_{y} y f(y) dy$$ +**Expectation of a Continuous Random Variable**: The expected value of a continuous random variable is similar in concept to that of the discrete random variable, except that instead of summing using probabilities as weights, we integrate using the density to weight. Hence, the expected value of the continuous variable $Y$ is defined by $$E(Y)=\int\limits_{y} y f(y) dy$$ -:::{#exm-expectconti} +::: {#exm-expectconti} ## Expectation of a Continuous Random Variable Find $E(Y)$ for $f(y)=\frac{1}{1.5}, \quad 00$$ +A random variable $Y$ has a Poisson distribution if -The Poisson has the unusual feature that its expectation equals its variance: $E(Y)=\text{Var}(Y)=\lambda$. The Poisson distribution is often used to model rare event counts: counts of the number of events that occur during some unit of time. $\lambda$ is often called the "arrival rate." +$$P(Y = y)=\frac{\lambda^y}{y!}e^{-\lambda}, \quad y=0,1,2,\ldots, \quad \lambda>0$$ +The Poisson has the unusual feature that its expectation equals its variance: $E(Y)=\text{Var}(Y)=\lambda$. The Poisson distribution is often used to model rare event counts: counts of the number of events that occur during some unit of time. $\lambda$ is often called the "arrival rate." ::: -:::{#exm-pois} -Border disputes occur between two countries through a Poisson Distribution, at a rate of 2 per month. What is the probability of 0, 2, and less than 5 disputes occurring in a month? +::: {#exm-pois} +Border disputes occur between two countries through a Poisson Distribution, at a rate of 2 per month. What is the probability of 0, 2, and less than 5 disputes occurring in a month? ::: - \begin{comment} +```{=tex} +\begin{comment} \parbox{1.5in}{\hfill \epsffile{poispmf.eps}} \end{comment} +``` +Two *continuous* distributions used often are: -Two _continuous_ distributions used often are: - -:::{#def-unif} +::: {#def-unif} ## Uniform Distribution -A random variable $Y$ has a continuous uniform distribution on the interval $(\alpha,\beta)$ if its density is given by $$f(y)=\frac{1}{\beta-\alpha}, \quad \alpha\le y\le \beta$$ The mean and variance of $Y$ are $E(Y)=\frac{\alpha+\beta}{2}$ and $\text{Var}(Y)=\frac{(\beta-\alpha)^2}{12}$. + +A random variable $Y$ has a continuous uniform distribution on the interval $(\alpha,\beta)$ if its density is given by $$f(y)=\frac{1}{\beta-\alpha}, \quad \alpha\le y\le \beta$$ The mean and variance of $Y$ are $E(Y)=\frac{\alpha+\beta}{2}$ and $\text{Var}(Y)=\frac{(\beta-\alpha)^2}{12}$. ::: -:::{#exm-unif} +::: {#exm-unif} For $Y$ uniformly distributed over $(1,3)$, what are the following probabilities? -1. $P(Y=2)$ -1. Its density evaluated at 2, or $f(2)$ -1. $P(Y \le 2)$ -1. $P(Y > 2)$ - +1. $P(Y=2)$ +2. Its density evaluated at 2, or $f(2)$ +3. $P(Y \le 2)$ +4. $P(Y > 2)$ ::: +```{=tex} \begin{comment} \parbox{1.5in}{\hfill \epsffile{unifpdf.eps}} \end{comment} - -:::{#def-normal} +``` +::: {#def-normal} ## Normal Distribution -A random variable $Y$ is normally distributed with mean $E(Y)=\mu$ and variance $\text{Var}(Y)=\sigma^2$ if its density is +A random variable $Y$ is normally distributed with mean $E(Y)=\mu$ and variance $\text{Var}(Y)=\sigma^2$ if its density is $$f(y)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(y-\mu)^2}{2\sigma^2}}$$ - ::: -See Figure @fig-normaldens are various Normal Distributions with the same $\mu = 1$ and two versions of the variance. +See Figure @fig-normaldens are various Normal Distributions with the same $\mu = 1$ and two versions of the variance. ```{r} #| label: fig-normaldens @@ -800,12 +769,11 @@ fx ## Summarizing Observed Events (Data) -So far, we've talked about distributions in a theoretical sense, looking at different properties of random variables. We don't observe random variables; we observe realizations of the random variable. These realizations of events are roughly equivalent to what we mean by "data". +So far, we've talked about distributions in a theoretical sense, looking at different properties of random variables. We don't observe random variables; we observe realizations of the random variable. These realizations of events are roughly equivalent to what we mean by "data". -__Sample mean__: This is the most common measure of central tendency, calculated by summing across the observations and dividing by the number of observations. -$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i$$ -The sample mean is an _estimate_ of the expected value of a distribution. +**Sample mean**: This is the most common measure of central tendency, calculated by summing across the observations and dividing by the number of observations. $$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i$$ The sample mean is an *estimate* of the expected value of a distribution. +```{=tex} \begin{framed} Example: \begin{center} @@ -824,17 +792,16 @@ Y & 1 & 2 & 1 & 2 & 2 & 1 & 2 & 0 & 2 & 0\\ \item $m_x = $ \hspace{2.75cm} $m_y =$\\ \end{enumerate} \end{framed} +``` +**Dispersion**: We also typically want to know how spread out the data are relative to the center of the observed distribution. There are several ways to measure dispersion. -__Dispersion__: We also typically want to know how spread out the data are relative to the center of the observed distribution. There are several ways to measure dispersion. - -__Sample variance__: The sample variance is the sum of the squared deviations from the sample mean, divided by the number of observations minus 1. -$$ \hat{\text{Var}}(X) = \frac{1}{n-1}\sum_{i = 1}^n (x_i - \bar{x})^2$$ +**Sample variance**: The sample variance is the sum of the squared deviations from the sample mean, divided by the number of observations minus 1. $$ \hat{\text{Var}}(X) = \frac{1}{n-1}\sum_{i = 1}^n (x_i - \bar{x})^2$$ -Again, this is an _estimate_ of the variance of a random variable; we divide by $n - 1$ instead of $n$ in order to get an unbiased estimate. +Again, this is an *estimate* of the variance of a random variable; we divide by $n - 1$ instead of $n$ in order to get an unbiased estimate. -__Standard deviation__: The sample standard deviation is the square root of the sample variance. -$$ \hat{SD}(X) = \sqrt{\hat{\text{Var}}(X)} = \sqrt{\frac{1}{n-1}\sum_{i = 1}^n (x_i - \bar{x})^2}$$ +**Standard deviation**: The sample standard deviation is the square root of the sample variance. $$ \hat{SD}(X) = \sqrt{\hat{\text{Var}}(X)} = \sqrt{\frac{1}{n-1}\sum_{i = 1}^n (x_i - \bar{x})^2}$$ +```{=tex} \begin{framed} Example: Using table above, calculate: \begin{enumerate} @@ -842,144 +809,139 @@ Example: Using table above, calculate: \item $\SD(X) = $ \hspace{1.65cm} $\SD(Y) =$ \end{enumerate} \end{framed} +``` +**Covariance and Correlation**: Both of these quantities measure the degree to which two variables vary together, and are estimates of the covariance and correlation of two random variables as defined above. -__Covariance and Correlation__: Both of these quantities measure the degree to which two variables vary together, and are estimates of the covariance and correlation of two random variables as defined above. - -1. **Sample covariance**: $\hat{\text{Cov}}(X,Y) = \frac{1}{n-1}\sum_{i = 1}^n(x_i - \bar{x})(y_i - \bar{y})$ -2. **Sample correlation**: $\hat{\text{Corr}} = \frac{\hat{\text{Cov}}(X,Y)}{\sqrt{\hat{\text{Var}}(X)\hat{\text{Var}}(Y)}}$ - -:::{#exm-sample} -Example: Using the above table, calculate the sample versions of: +1. **Sample covariance**: $\hat{\text{Cov}}(X,Y) = \frac{1}{n-1}\sum_{i = 1}^n(x_i - \bar{x})(y_i - \bar{y})$ +2. **Sample correlation**: $\hat{\text{Corr}} = \frac{\hat{\text{Cov}}(X,Y)}{\sqrt{\hat{\text{Var}}(X)\hat{\text{Var}}(Y)}}$ -1. $\text{Cov}(X,Y)$ -2. $\text{Corr}(X, Y)$ +::: {#exm-sample} +Example: Using the above table, calculate the sample versions of: +1. $\text{Cov}(X,Y)$ +2. $\text{Corr}(X, Y)$ ::: ## Asymptotic Theory -In theoretical and applied research, asymptotic arguments are often made. In this section we briefly introduce some of this material. +In theoretical and applied research, asymptotic arguments are often made. In this section we briefly introduce some of this material. -What are asymptotics? In probability theory, asymptotic analysis is the study of limiting behavior. By limiting behavior, we mean the behavior of some random process as the number of observations gets larger and larger. +What are asymptotics? In probability theory, asymptotic analysis is the study of limiting behavior. By limiting behavior, we mean the behavior of some random process as the number of observations gets larger and larger. -Why is this important? We rarely know the true process governing the events we see in the social world. It is helpful to understand how such unknown processes theoretically must behave and asymptotic theory helps us do this. +Why is this important? We rarely know the true process governing the events we see in the social world. It is helpful to understand how such unknown processes theoretically must behave and asymptotic theory helps us do this. ### CLT and LLN We are now finally ready to revisit, with a bit more precise terms, the two pillars of statistical theory we motivated Section @sec-limitsfun with. -:::{#thm-clt} +::: {#thm-clt} ## Central Limit Theorem (i.i.d. case) -Let $\{X_n\} = \{X_1, X_2, \ldots\}$ be a sequence of i.i.d. random variables with finite mean ($\mu$) and variance ($\sigma^2$). Then, the sample mean $\bar{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n}$ increasingly converges into a Normal distribution. -$$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),$$ +Let $\{X_n\} = \{X_1, X_2, \ldots\}$ be a sequence of i.i.d. random variables with finite mean ($\mu$) and variance ($\sigma^2$). Then, the sample mean $\bar{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n}$ increasingly converges into a Normal distribution. +$$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),$$ ::: Another way to write this as a probability statement is that for all real numbers $a$, -$$P\left(\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \le a\right) \rightarrow \Phi(a)$$ -as $n\to \infty$, where $$\Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \, dx$$ is the CDF of a Normal distribution with mean 0 and variance 1. +$$P\left(\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \le a\right) \rightarrow \Phi(a)$$ as $n\to \infty$, where $$\Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \, dx$$ is the CDF of a Normal distribution with mean 0 and variance 1. -This result means that, as $n$ grows, the distribution of the sample mean $\bar X_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n)$ is approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt n}$, i.e., -$$\bar{X}_n \approx \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).$$ The standard deviation of $\bar X_n$ (which is roughly a measure of the precision of $\bar X_n$ as an estimator of $\mu$) decreases at the rate $1/\sqrt{n}$, so, for example, to increase its precision by $10$ (i.e., to get one more digit right), one needs to collect $10^2=100$ times more units of data. +This result means that, as $n$ grows, the distribution of the sample mean $\bar X_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n)$ is approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt n}$, i.e., $$\bar{X}_n \approx \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).$$ The standard deviation of $\bar X_n$ (which is roughly a measure of the precision of $\bar X_n$ as an estimator of $\mu$) decreases at the rate $1/\sqrt{n}$, so, for example, to increase its precision by $10$ (i.e., to get one more digit right), one needs to collect $10^2=100$ times more units of data. -Intuitively, this result also justifies that whenever a lot of small, independent processes somehow combine together to form the realized observations, practitioners often feel comfortable assuming Normality. - -:::{#thm-lln} +Intuitively, this result also justifies that whenever a lot of small, independent processes somehow combine together to form the realized observations, practitioners often feel comfortable assuming Normality. +::: {#thm-lln} ## Law of Large Numbers (LLN) -For any draw of independent random variables with the same mean $\mu$, the sample average after $n$ draws, $\bar{X}_n = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)$, converges in probability to the expected value of $X$, $\mu$ as $n \rightarrow \infty$: -$$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ +For any draw of independent random variables with the same mean $\mu$, the sample average after $n$ draws, $\bar{X}_n = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)$, converges in probability to the expected value of $X$, $\mu$ as $n \rightarrow \infty$: -A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". +$$\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0$$ +A shorthand of which is $\bar{X}_n \xrightarrow{p} \mu$, where the arrow is read as "converges in probability to". ::: -as $n\to \infty$. In other words, $P( \lim_{n\to\infty}\bar{X}_n = \mu) = 1$. This is an important motivation for the widespread use of the sample mean, as well as the intuition link between averages and expected values. +as $n\to \infty$. In other words, $P( \lim_{n\to\infty}\bar{X}_n = \mu) = 1$. This is an important motivation for the widespread use of the sample mean, as well as the intuition link between averages and expected values. -More precisely this version of the LLN is called the _weak_ law of large numbers because it leaves open the possibility that $|\bar{X}_n - \mu | > \varepsilon$ occurs many times. The _strong_ law of large numbers states that, under a few more conditions, the probability that the limit of the sample average is the true mean is 1 (and other possibilities occur with probability 0), but the difference is rarely consequential in practice. +More precisely this version of the LLN is called the *weak* law of large numbers because it leaves open the possibility that $|\bar{X}_n - \mu | > \varepsilon$ occurs many times. The *strong* law of large numbers states that, under a few more conditions, the probability that the limit of the sample average is the true mean is 1 (and other possibilities occur with probability 0), but the difference is rarely consequential in practice. -The Strong Law of Large Numbers holds so long as the expected value exists; no other assumptions are needed. However, the rate of convergence will differ greatly depending on the distribution underlying the observed data. When extreme observations occur often (i.e. kurtosis is large), the rate of convergence is much slower. Cf. The distribution of financial returns. +The Strong Law of Large Numbers holds so long as the expected value exists; no other assumptions are needed. However, the rate of convergence will differ greatly depending on the distribution underlying the observed data. When extreme observations occur often (i.e. kurtosis is large), the rate of convergence is much slower. Cf. The distribution of financial returns. ### Big $\mathcal{O}$ Notation -Some of you may encounter "big-OH''-notation. If $f, g$ are two functions, we say that $f = \mathcal{O}(g)$ if there exists some constant, $c$, such that $f(n) \leq c \times g(n)$ for large enough $n$. This notation is useful for simplifying complex problems in game theory, computer science, and statistics. +Some of you may encounter "big-OH''-notation. If $f, g$ are two functions, we say that $f = \mathcal{O}(g)$ if there exists some constant, $c$, such that $f(n) \leq c \times g(n)$ for large enough $n$. This notation is useful for simplifying complex problems in game theory, computer science, and statistics. -Example. +Example. -What is $\mathcal{O}( 5\exp(0.5 n) + n^2 + n / 2)$? Answer: $\exp(n)$. Why? Because, for large $n$, -$$\frac{ 5\exp(0.5 n) + n^2 + n / 2 }{ \exp(n)} \leq \frac{ c \exp(n) }{ \exp(n)} = c. $$ -whenever $n > 4$ and where $c = 1$. +What is $\mathcal{O}( 5\exp(0.5 n) + n^2 + n / 2)$? Answer: $\exp(n)$. Why? Because, for large $n$, $$\frac{ 5\exp(0.5 n) + n^2 + n / 2 }{ \exp(n)} \leq \frac{ c \exp(n) }{ \exp(n)} = c. $$ whenever $n > 4$ and where $c = 1$. +```{=tex} \begin{comment} n_seq <- 1:100; numerator <- (5 * exp(0.5 * n_seq) + n_seq^2 + n_seq / 2) denominator <- exp(n_seq) numerator / denominator <= (1 * denominator) / denominator \end{comment} - -## Answers to Examples and Exercises {.unnumbered} +``` +## Answers to Examples and Exercises {.unnumbered} Answer to Example @exm-counting: -1. $5 \times 5 \times 5 = 125$ +1. $5 \times 5 \times 5 = 125$ -2. $5 \times 4 \times 3 = 60$ +2. $5 \times 4 \times 3 = 60$ -3. $\binom{5}{3} = \frac{5!}{(5-3)!3!} = \frac{5 \times 4}{2 \times 1} = 10$ +3. $\binom{5}{3} = \frac{5!}{(5-3)!3!} = \frac{5 \times 4}{2 \times 1} = 10$ Answer to Exercise @exr-counting1: -1. $\binom{52}{4} = \frac{52!}{(52-4)!4!} = 270725$ +1. $\binom{52}{4} = \frac{52!}{(52-4)!4!} = 270725$ Answer to Example @exm-sets: -1. {1, 2, 3, 4, 5, 6} -2. {5, 6} -3. {1, 2, 7, 8, 9, 10} -4. {3, 4} +1. {1, 2, 3, 4, 5, 6} +2. {5, 6} +3. {1, 2, 7, 8, 9, 10} +4. {3, 4} Answer to Exercise @exr-sets1: Sample Space: {2, 3, 4, 5, 6, 7, 8} -1. {3, 4, 5, 6, 7} -2. {4, 5, 6} +1. {3, 4, 5, 6, 7} +2. {4, 5, 6} Answer to Example @exm-prob: -1. ${1, 2, 3, 4, 5, 6}$ +1. ${1, 2, 3, 4, 5, 6}$ -2. $\frac{1}{6}$ +2. $\frac{1}{6}$ -3. $0$ +3. $0$ -4. $\frac{1}{2}$ +4. $\frac{1}{2}$ -5. $\frac{4}{6} = \frac{2}{3}$ +5. $\frac{4}{6} = \frac{2}{3}$ -6. $1$ +6. $1$ -7. $A\cup B=\{1, 2, 3, 4, 6\}$, $A\cap B=\{2\}$, $\frac{5}{6}$ +7. $A\cup B=\{1, 2, 3, 4, 6\}$, $A\cap B=\{2\}$, $\frac{5}{6}$ Answer to Exercise @exr-prob1: -1. $P(X = 5) = \frac{4}{16}$, $P(X = 3) = \frac{2}{16}$, $P(X = 6) = \frac{3}{16}$ +1. $P(X = 5) = \frac{4}{16}$, $P(X = 3) = \frac{2}{16}$, $P(X = 6) = \frac{3}{16}$ -2. What is $P(X=5 \cup X = 3)^C = \frac{10}{16}$? +2. What is $P(X=5 \cup X = 3)^C = \frac{10}{16}$? Answer to Example @exm-condprobexm1: -1. $\frac{n_{ab} + n_{ab^c}}{N}$ +1. $\frac{n_{ab} + n_{ab^c}}{N}$ -2. $\frac{n_{ab} + n_{a^cb}}{N}$ +2. $\frac{n_{ab} + n_{a^cb}}{N}$ -3. $\frac{n_{ab}}{N}$ +3. $\frac{n_{ab}}{N}$ -4. $\frac{\frac{n_{ab}}{N}}{\frac{n_{ab} + n_{a^cb}}{N}} = \frac{n_{ab}}{n_{ab} + n_{a^cb}}$ +4. $\frac{\frac{n_{ab}}{N}}{\frac{n_{ab} + n_{a^cb}}{N}} = \frac{n_{ab}}{n_{ab} + n_{a^cb}}$ -5. $\frac{\frac{n_{ab}}{N}}{\frac{n_{ab} + n_{ab^c}}{N}} = \frac{n_{ab}}{n_{ab} + n_{ab^c}}$ +5. $\frac{\frac{n_{ab}}{N}}{\frac{n_{ab} + n_{ab^c}}{N}} = \frac{n_{ab}}{n_{ab} + n_{ab^c}}$ Answer to Example @exm-condprobexm2: @@ -987,12 +949,9 @@ $P(1|Odd) = \frac{P(1 \cap Odd)}{P(Odd)} = \frac{\frac{1}{6}}{\frac{1}{2}} = \fr Answer to Example @exm-bayesrule: -We are given that -$$P(D) = .4, P(D^c) = .6, P(S|D) = .5, P(S|D^c) = .9$$ -Using this, Bayes' Law and the Law of Total Probability, we know: +We are given that $$P(D) = .4, P(D^c) = .6, P(S|D) = .5, P(S|D^c) = .9$$ Using this, Bayes' Law and the Law of Total Probability, we know: -$$P(D|S) = \frac{P(D)P(S|D)}{P(D)P(S|D) + P(D^c)P(S|D^c)}$$ -$$P(D|S) = \frac{.4 \times .5}{.4 \times .5 + .6 \times .9 } = .27$$ +$$P(D|S) = \frac{P(D)P(S|D)}{P(D)P(S|D) + P(D^c)P(S|D^c)}$$ $$P(D|S) = \frac{.4 \times .5}{.4 \times .5 + .6 \times .9 } = .27$$ Answer to Exercise @exr-condprobexr: @@ -1004,12 +963,11 @@ $$P(M|C) = \frac{P(C|M)P(M)}{P(C)}$$ $$= \frac{P(C|M)P(M)}{P(C|M)P(M) + P(C|M^c)P(M^c)}$$ -$$= \frac{P(C|M)P(M)}{P(C|M)P(M) + [1-P(C^c|M^c)]P(M^c)}$$ -$$ = \frac{.95 \times .02}{.95 \times .02 + .03 \times .98} = .38$$ +$$= \frac{P(C|M)P(M)}{P(C|M)P(M) + [1-P(C^c|M^c)]P(M^c)}$$ $$ = \frac{.95 \times .02}{.95 \times .02 + .03 \times .98} = .38$$ Answer to Example @exm-expectdiscrete: -$E(Y)=7/2$ +$E(Y)=7/2$ We would never expect the result of a rolled die to be $7/2$, but that would be the average over a large number of rolls of the die. @@ -1023,32 +981,34 @@ $E(X) = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 2 \times \frac{3}{8} + 3 \ Since there is a 1 to 1 mapping from $X$ to $X^2:$ $E(X^2) = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 4 \times \frac{3}{8} + 9 \times \frac{1}{8} = \frac{24}{8} = 3$ +```{=tex} \begin{align*} \text{Var}(x) &= E(X^2) - E(x)^2\\ &= 3 - (\frac{3}{2})^2\\ &= \frac{3}{4} \end{align*} - +``` Answer to Exercise @exr-expvar: -1. $E(X) = -2(\frac{1}{5}) + -1(\frac{1}{6}) + 0(\frac{1}{5}) + 1(\frac{1}{15}) + 2(\frac{11}{30}) = \frac{7}{30}$ +1. $E(X) = -2(\frac{1}{5}) + -1(\frac{1}{6}) + 0(\frac{1}{5}) + 1(\frac{1}{15}) + 2(\frac{11}{30}) = \frac{7}{30}$ -2. $E(Y) = 0(\frac{1}{5}) + 1(\frac{7}{30}) + 4(\frac{17}{30}) = \frac{5}{2}$ +2. $E(Y) = 0(\frac{1}{5}) + 1(\frac{7}{30}) + 4(\frac{17}{30}) = \frac{5}{2}$ -3. +3. +```{=tex} \begin{align*} \text{Var}(X) &= E[X^2] - E[X]^2\\ &= E(Y) - E(X)^2\\ &= \frac{5}{2} - \frac{7}{30}^2 \approx 2.45 \end{align*} - +``` Answer to Exercise @exr-expvar2: -1. expectation = $\frac{6}{5}$, variance = $\frac{6}{25}$ +1. expectation = $\frac{6}{5}$, variance = $\frac{6}{25}$ Answer to Exercise @exr-expvar3: -1. mean = 2, standard deviation = $\sqrt(\frac{2}{3})$ +1. mean = 2, standard deviation = $\sqrt(\frac{2}{3})$ -2. $\frac{1}{8}(2 - \sqrt(\frac{2}{3}))^2$ +2. $\frac{1}{8}(2 - \sqrt(\frac{2}{3}))^2$ diff --git a/07_linear-algebra.qmd b/07_linear-algebra.qmd index bfcf6ab..0eb9559 100644 --- a/07_linear-algebra.qmd +++ b/07_linear-algebra.qmd @@ -2,17 +2,17 @@ Topics: -- Working with Vectors -- Linear Independence -- Basics of Matrix Algebra -- Square Matrices -- Linear Equations -- Systems of Linear Equations -- Systems of Equations as Matrices -- Solving Augmented Matrices and Systems of Equations -- Rank -- The Inverse of a Matrix -- Inverse of Larger Matrices +- Working with Vectors +- Linear Independence +- Basics of Matrix Algebra +- Square Matrices +- Linear Equations +- Systems of Linear Equations +- Systems of Equations as Matrices +- Solving Augmented Matrices and Systems of Equations +- Rank +- The Inverse of a Matrix +- Inverse of Larger Matrices ## Working with Vectors {#vector-def} @@ -29,30 +29,28 @@ $$ c{\bf v} = \begin{pmatrix} cv_1 & cv_2 & \dots & cv_n \end{pmatrix} $$ **Vector Norm**: The norm of a vector is a measure of its length. There are many different ways to calculate the norm, but the most common is the Euclidean norm (which corresponds to our usual conception of distance in three-dimensional space): $$ ||{\bf v}|| = \sqrt{{\bf v}\cdot{\bf v}} = \sqrt{ v_1v_1 + v_2v_2 + \cdots + v_nv_n}$$ -:::{#exm-vectors} +::: {#exm-vectors} ## Vector Algebra -Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{pmatrix}$. Calculate the following: +Let $a = \begin{pmatrix} 2&1&2\end{pmatrix}$, $b = \begin{pmatrix} 3&4&5 \end{pmatrix}$. Calculate the following: -1. $a - b$ - -2. $a \cdot b$ +1. $a - b$ +2. $a \cdot b$ ::: -:::{#exr-vectors1} +::: {#exr-vectors1} ## Vector Algebra -Let $u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}$, $v = \begin{pmatrix} 9&-3&2&8 \end{pmatrix}$, $w = \begin{pmatrix} 1&13& -7&2 &15 \end{pmatrix}$, and $c = 2$. Calculate the following: - - 1. $u-v$ +Let $u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}$, $v = \begin{pmatrix} 9&-3&2&8 \end{pmatrix}$, $w = \begin{pmatrix} 1&13& -7&2 &15 \end{pmatrix}$, and $c = 2$. Calculate the following: - 2. $cw$ +1. $u-v$ - 3. $u \cdot v$ +2. $cw$ - 4. $w \cdot v$ +3. $u \cdot v$ +4. $w \cdot v$ ::: ## Linear Independence {#linearindependence} @@ -69,25 +67,23 @@ Linear independence is only defined for sets of vectors with the same number of Since $\begin{pmatrix}9 & 13 & 17 \end{pmatrix}$ is a linear combination of $\begin{pmatrix}1 & 2 & 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 3& 4\end{pmatrix}$, and $\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}$, these 4 vectors constitute a linearly dependent set. -:::{#exm-linearindep} +::: {#exm-linearindep} ## Linear Independence Are the following sets of vectors linearly independent? - 1. $\begin{pmatrix}2 & 3 & 1 \end{pmatrix}$ and $\begin{pmatrix}4 & 6 & 1 \end{pmatrix}$ - 2. $\begin{pmatrix}1 & 0 & 0 \end{pmatrix}$, $\begin{pmatrix}0 & 5 & 0 \end{pmatrix}$, and $\begin{pmatrix}10 & 10 & 0 \end{pmatrix}$ - +1. $\begin{pmatrix}2 & 3 & 1 \end{pmatrix}$ and $\begin{pmatrix}4 & 6 & 1 \end{pmatrix}$ +2. $\begin{pmatrix}1 & 0 & 0 \end{pmatrix}$, $\begin{pmatrix}0 & 5 & 0 \end{pmatrix}$, and $\begin{pmatrix}10 & 10 & 0 \end{pmatrix}$ ::: -:::{#exr-linearindep1} +::: {#exr-linearindep1} ## Linear Independence Are the following sets of vectors linearly independent? - 1. $${\bf v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} $$ - - 2. $${\bf v}_1 = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} -4 \\ 6 \\ 5 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} -2 \\ 8 \\ 6 \end{pmatrix} $$ +1. $${\bf v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} $$ +2. $${\bf v}_1 = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} -4 \\ 6 \\ 5 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} -2 \\ 8 \\ 6 \end{pmatrix} $$ ::: ## Basics of Matrix Algebra {#matrixbasics} @@ -114,12 +110,9 @@ It's also useful to think of matrices as being made up of a collection of row or Note that matrices ${\bf A}$ and ${\bf B}$ must have the same dimensionality, in which case they are **conformable for addition**. -:::{#exm-matrixaddition} - +::: {#exm-matrixaddition} $${\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad - {\bf B}=\begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & 2 \end{pmatrix}$$ - ${\bf A+B}=$ - + {\bf B}=\begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & 2 \end{pmatrix}$$ ${\bf A+B}=$ ::: **Scalar Multiplication**: Given the scalar $s$, the scalar multiplication of $s {\bf A}$ is $$ s {\bf A}= s \begin{pmatrix} @@ -135,23 +128,19 @@ $${\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad s a_{m1} & s a_{m2} & \cdots & s a_{mn} \end{pmatrix}$$ -:::{#exm-scalarmulti} - +::: {#exm-scalarmulti} $s=2, \qquad {\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ $s {\bf A} =$ - ::: **Matrix Multiplication**: If ${\bf A}$ is an $m\times k$ matrix and $\bf B$ is a $k\times n$ matrix, then their product $\bf C = A B$ is the $m\times n$ matrix where $$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ik}b_{kj}$$ -:::{#exm-matrixmulti} - +::: {#exm-matrixmulti} 1. $\begin{pmatrix} a&b\\c&d\\e&f \end{pmatrix} \begin{pmatrix} A&B\\C&D \end{pmatrix} - =$ - -2. $\begin{pmatrix} 1&2&-1\\3&1&4 \end{pmatrix} \begin{pmatrix} -2&5\\4&-3\\2&1\end{pmatrix} =$ + =$ +2. $\begin{pmatrix} 1&2&-1\\3&1&4 \end{pmatrix} \begin{pmatrix} -2&5\\4&-3\\2&1\end{pmatrix} =$ ::: Note that the number of columns of the first matrix must equal the number of rows of the second matrix, in which case they are **conformable for multiplication**. The sizes of the matrices (including the resulting product) must be $$(m\times k)(k\times n)=(m\times n)$$ @@ -195,22 +184,21 @@ The following rules apply for transposed matrices: ``` Example of $({\bf AB})^T = {\bf B}^T{\bf A}^T$: $${\bf A}=\begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix}$$ $$ ({\bf AB})^T = \left[ \begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix} \begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix} \right]^T = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}$$ $$ {\bf B}^T{\bf A}^T= \begin{pmatrix} 0&2&3\\1&2&-1 \end{pmatrix} \begin{pmatrix} 1&2\\3&-1\\2&3 \end{pmatrix} = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}$$ -:::{#exr-matrixmulti1} +::: {#exr-matrixmulti1} ## Matrix Multiplication Let $$A = \begin{pmatrix} 2&0&-1&1\\1&2&0&1 \end{pmatrix}$$ $$B = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix} $$ - $$C = \begin{pmatrix} 3&2&-1\\0&4&6 \end{pmatrix}$$ +$$C = \begin{pmatrix} 3&2&-1\\0&4&6 \end{pmatrix}$$ Calculate the following: -1. $$AB$$ -2. $$BA$$ -3. $$(BC)^T$$ -4. $$BC^T$$ - +1. $$AB$$ +2. $$BA$$ +3. $$(BC)^T$$ +4. $$BC^T$$ ::: ## Systems of Linear Equations @@ -259,17 +247,16 @@ Methods to solve linear systems: 2. Elimination of variables 3. Matrix methods -:::{#exr-lineareq} +::: {#exr-lineareq} ## Linear Equations Provide a system of 2 equations with 2 unknowns that has -1. one solution - -2. no solution +1. one solution -3. infinite solutions +2. no solution +3. infinite solutions ::: ## Systems of Equations as Matrices @@ -301,15 +288,12 @@ The right hand side of the linear system is represented by the vector ${\bf b}=\ a_{m1} & a_{m2} & \cdots & a_{mn} & | & b_m \end{pmatrix}$$ -:::{#exr-augmatrix} +::: {#exr-augmatrix} ## Augmented Matrix Create an augmented matrix that represent the following system of equations: - $$2x_1 -7x_2 + 9x_3 -4x_4 = 8$$ - $$41x_2 + 9x_3 -5x_6 = 11$$ - $$x_1 -15x_2 -11x_5 = 9$$ - +$$2x_1 -7x_2 + 9x_3 -4x_4 = 8$$ $$41x_2 + 9x_3 -5x_6 = 11$$ $$x_1 -15x_2 -11x_5 = 9$$ ::: ## Finding Solutions to Augmented Matrices and Systems of Equations @@ -357,18 +341,16 @@ $$\begin{pmatrix} a_{11}+a_{21} & a_{12}+a_{22} & | & b_1+b_2 \end{pmatrix}$$ which represents a linear system equivalent to that represented by matrix $\widehat{\bf A}$. -:::{#exm-solvesys} - +::: {#exm-solvesys} Solve the following system of equations by using elementary row operations: $\begin{matrix} - x & - & 3y & = & -3\\ - 2x & + & y & = & 8 - \end{matrix}$ - + x & - & 3y & = & -3\\ + 2x & + & y & = & 8 + \end{matrix}$ ::: -:::{#exr-solvesys1} +::: {#exr-solvesys1} ## Solving Systems of Equations Put the following system of equations into augmented matrix form. Then, using Gaussian or Gauss-Jordan elimination, solve the system of equations by putting the matrix into row echelon or reduced row echelon form. @@ -388,7 +370,6 @@ $$ -x -4y + z = -6 \end{cases} $$ - ::: ## Rank --- and Whether a System Has One, Infinite, or No Solutions @@ -411,24 +392,23 @@ $\begin{pmatrix} 1 & 2 & 3 \\ Rank = 2 -:::{#exr-rank} +::: {#exr-rank} ## Rank of Matrices Find the rank of each matrix below: (Hint: transform the matrices into row echelon form. Remember that the number of nonzero rows of a matrix in row echelon form is the rank of that matrix) -1.$\begin{pmatrix} 1 & 1 & 2 \\ +1.$\begin{pmatrix} 1 & 1 & 2 \\ 2 & 1 & 3 \\ 1 & 2 & 3 \end{pmatrix}$ \bigskip -2.$\begin{pmatrix} 1 & 3 & 3 & -3 & 3\\ +2.$\begin{pmatrix} 1 & 3 & 3 & -3 & 3\\ 1 & 3 & 1 & 1 & 3 \\ 1 & 3 & 2 & -1 & -2 \\ 1 & 3 & 0 & 3 & -2 \end{pmatrix}$ - ::: Answer to Exercise @exr-rank: @@ -475,21 +455,18 @@ To summarize: To calculate the inverse of ${\bf A}$ b. If ${\bf C}\ne{\bf I}_n$, then $\bf C$ has a row of zeros. This means ${\bf A}$ is singular and ${\bf A}^{-1}$ does not exist. -:::{#exm-inverse} - +::: {#exm-inverse} Find the inverse of the following matricies: -1. ${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$ - +1. ${\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}$ ::: -:::{#exr-inverse1} +::: {#exr-inverse1} ## Finding the inverse of matrices Find the inverse of the following matrix: -1. ${\bf A}=\begin{pmatrix} 1&0&4\\0&2&0\\0&0&1 \end{pmatrix}$ - +1. ${\bf A}=\begin{pmatrix} 1&0&4\\0&2&0\\0&0&1 \end{pmatrix}$ ::: ## Linear Systems and Inverses @@ -510,27 +487,24 @@ If $\bf{A}$ is an $n\times n$ matrix,then $\bf{Ax}=\bf{b}$ is a system of $n$ eq ``` Hence, given $\bf{A}$ and $\bf{b}$ and given that $\bf{A}$ is nonsingular, then $\bf{x} = \bf{A}^{-1} \bf{b}$ is a unique solution to this system. -:::{#exr-invlinsys} - +::: {#exr-invlinsys} ## Solve linear system using inverses Use the inverse matrix to solve the following linear system: +```{=tex} \begin{align*} -3x + 4y &= 5 \\ 2x - y &= -10 \end{align*} +``` +***Hint: the linear system above can be written in the matrix form*** -___Hint: the linear system above can be written in the matrix form___ - -$\textbf{A}\textbf{z} = \textbf{b}$ - -given $$\textbf{A} = \begin{pmatrix} -3&4\\2&-1 \end{pmatrix},$$ +$\textbf{A}\textbf{z} = \textbf{b}$ -$$\textbf{z} = \begin{pmatrix} x\\y \end{pmatrix},$$ -and -$$\textbf{b} = \begin{pmatrix} 5\\-10 \end{pmatrix}$$ +given $$\textbf{A} = \begin{pmatrix} -3&4\\2&-1 \end{pmatrix},$$ +$$\textbf{z} = \begin{pmatrix} x\\y \end{pmatrix},$$ and $$\textbf{b} = \begin{pmatrix} 5\\-10 \end{pmatrix}$$ ::: ## Determinants @@ -572,23 +546,22 @@ For example, in figuring out whether the following matrix has an inverse? $${\bf &=& -8\nonumber \end{eqnarray} 2. Since $|{\bf A}|\ne 0$, we conclude that ${\bf A}$ has an inverse. -:::{#exr-determinants} +::: {#exr-determinants} ## Determinants and Inverses Determine whether the following matrices are nonsingular: - $$1. \begin{pmatrix} - 1 & 0 & 1\\ - 2 & 1 & 2\\ - 1 & 0 & -1 - \end{pmatrix}$$ +$$1. \begin{pmatrix} + 1 & 0 & 1\\ + 2 & 1 & 2\\ + 1 & 0 & -1 + \end{pmatrix}$$ $$2. \begin{pmatrix} - 2 & 1 & 2\\ - 1 & 0 & 1\\ - 4 & 1 & 4 - \end{pmatrix}$$ - + 2 & 1 & 2\\ + 1 & 0 & 1\\ + 4 & 1 & 4 + \end{pmatrix}$$ ::: ## Getting Inverse of a Matrix using its Determinant @@ -637,16 +610,15 @@ $$ \begin{pmatrix} \frac{2}{5} & \frac{3}{5}\\ \end{pmatrix}$$ -:::{#exr-calcinverse} +::: {#exr-calcinverse} ## Calculate Inverse using Determinant Formula Caculate the inverse of A $$A = \begin{pmatrix} - 3 & 5\\ - -7 & 2\\ - \end{pmatrix}$$ - + 3 & 5\\ + -7 & 2\\ + \end{pmatrix}$$ ::: ## Answers to Examples and Exercises {.unnumbered} From 756f29a76cf6a1b33609f2c2d11368ca4092e583 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 22:33:46 -0400 Subject: [PATCH 17/34] code files easier to fix 11-15 --- .gitignore | 1 + 11_data-handling_counting.qmd | 232 +++++++++++++---------- 12_matricies-manipulation.qmd | 105 +++++++---- 13_functions_obj_loops.qmd | 138 +++++++++----- 14_visualization.qmd | 161 +++++++++------- 15_project-dempeace.qmd | 68 ++++--- 17_non-wysiwyg.quarto_ipynb | 333 ---------------------------------- _quarto.yml | 7 + prefresher.Rproj | 2 + 9 files changed, 438 insertions(+), 609 deletions(-) delete mode 100644 17_non-wysiwyg.quarto_ipynb diff --git a/.gitignore b/.gitignore index 60561a8..76189d9 100644 --- a/.gitignore +++ b/.gitignore @@ -11,3 +11,4 @@ rsconnect prefresher\\.log /.quarto/ +*.quarto_ipynb diff --git a/11_data-handling_counting.qmd b/11_data-handling_counting.qmd index eb269a5..1480ec6 100644 --- a/11_data-handling_counting.qmd +++ b/11_data-handling_counting.qmd @@ -1,16 +1,18 @@ -# (PART) Programming {.unnumbered} +# Orientation and Reading in Data {#dataimport} -# Orientation and Reading in Data^[Module originally written by Shiro Kuriwaki] {#dataimport} +::: {.callout .callout-note} +Module originally written by Shiro Kuriwaki. +::: Welcome to the first in-class session for programming. Up till this point, you should have already: -* Completed the R Visualization and Programming primers (under "The Basics") on your own at , -* Made an account at RStudio Cloud and join the Math Prefresher 2019 Space, and -* Successfully signed up for the University wi-fi: (Access Harvard Secure with your HarvardKey. Try to get a HarvardKey as soon as possible.) +- Completed the R Visualization and Programming primers (under "The Basics") on your own at , +- Made an account at RStudio Cloud and join the Math Prefresher 2019 Space, and +- Successfully signed up for the University wi-fi: (Access Harvard Secure with your HarvardKey. Try to get a HarvardKey as soon as possible.) ## Motivation: Data and You {.unnumbered} -The modal social science project starts by importing existing datasets. Datasets come in all shapes and sizes. As you search for new data you may encounter dozens of file extensions -- csv, xlsx, dta, sav, por, Rdata, Rds, txt, xml, json, shp ... the list continues. Although these files can often be cumbersome, its a good to be able to find a way to encounter any file that your research may call for. +The modal social science project starts by importing existing datasets. Datasets come in all shapes and sizes. As you search for new data you may encounter dozens of file extensions -- csv, xlsx, dta, sav, por, Rdata, Rds, txt, xml, json, shp ... the list continues. Although these files can often be cumbersome, its a good to be able to find a way to encounter any file that your research may call for. Reviewing data import will allow us to get on the same page on how computer systems work. @@ -18,20 +20,20 @@ Reviewing data import will allow us to get on the same page on how computer syst Today we'll cover: -* What's what in RStudio -* What R is, at a high level -* How to read in data -* Comment on coding style on the way +- What's what in RStudio +- What R is, at a high level +- How to read in data +- Comment on coding style on the way -### Check your understanding {.unnumbered} +### Check your understanding {.unnumbered} -* What is the difference between a file and a folder? -* In the RStudio windows, what is the difference between the "Source" Pane and the "Console"? What is a "code chunk"? -* How do you read a R help page? What is the `Usage` section, the `Values` section, and the `Examples` section? -* What use is the "Environment" Pane? -* How would you read in a spreadsheet in R? -* How would you figure out what variables are in the data? size of the data? -* How would you read in a `csv` file, a `dta` file, a `sav` file? +- What is the difference between a file and a folder? +- In the RStudio windows, what is the difference between the "Source" Pane and the "Console"? What is a "code chunk"? +- How do you read a R help page? What is the `Usage` section, the `Values` section, and the `Examples` section? +- What use is the "Environment" Pane? +- How would you read in a spreadsheet in R? +- How would you figure out what variables are in the data? size of the data? +- How would you read in a `csv` file, a `dta` file, a `sav` file? ```{r} #| include: false @@ -43,53 +45,63 @@ library(fs) ## Orienting -1. We will be using a cloud version of RStudio at . You should join the Math Prefresher Space 2019 from the link that was emailed to you. Each day, click on the project with the day's date on it. +1. We will be using a cloud version of RStudio at . You should join the Math Prefresher Space 2019 from the link that was emailed to you. Each day, click on the project with the day's date on it. - Although most of you will probably doing your work on RStudio local rather than cloud, we are trying to use cloud because it makes it easier to standardize people's settings. + Although most of you will probably doing your work on RStudio local rather than cloud, we are trying to use cloud because it makes it easier to standardize people's settings. -2. RStudio (either cloud or desktop) is a __GUI__ and an IDE for the programming language R. A Graphical User Interface allows users to interface with the software (in this case R) using graphical aids like buttons and tabs. Often we don't think of GUIs because to most computer users, everything is a GUI (like Microsoft Word or your "Control Panel"), but it's always there! A Integrated Development Environment just says that the software to interface with R comes with useful useful bells and whistles to give you shortcuts. +2. RStudio (either cloud or desktop) is a **GUI** and an IDE for the programming language R. A Graphical User Interface allows users to interface with the software (in this case R) using graphical aids like buttons and tabs. Often we don't think of GUIs because to most computer users, everything is a GUI (like Microsoft Word or your "Control Panel"), but it's always there! A Integrated Development Environment just says that the software to interface with R comes with useful useful bells and whistles to give you shortcuts. - The __Console__ is kind of a the core window through which you see your GUI actually operating through R. It's not graphical so might not be as intuitive. But all your results, commands, errors, warnings.. you see them in here. A console tells you what's going on now. + The **Console** is kind of a the core window through which you see your GUI actually operating through R. It's not graphical so might not be as intuitive. But all your results, commands, errors, warnings.. you see them in here. A console tells you what's going on now. ![A Typical RStudio Window at Startup](images/11_1_rstudio-startup.png) -3. via the GUI, you the analyst needs to sends instructions, or __commands__, to the R application. The verb for this is "run" or "execute" the command. Computer programs ask users to provide instructions in very specific formats. While a English-speaking human can understand a sentence with a few typos in it by filling in the blanks, the same typo or misplaced character would halt a computer program. Each program has its own requirements for how commands should be typed; after all, each of these is its own language. We refer to the way a program needs its commands to be formatted as its __syntax__. +3. via the GUI, you the analyst needs to sends instructions, or **commands**, to the R application. The verb for this is "run" or "execute" the command. Computer programs ask users to provide instructions in very specific formats. While a English-speaking human can understand a sentence with a few typos in it by filling in the blanks, the same typo or misplaced character would halt a computer program. Each program has its own requirements for how commands should be typed; after all, each of these is its own language. We refer to the way a program needs its commands to be formatted as its **syntax**. -4. Theoretically, one could do all their work by typing in commands into the Console. But that would be a lot of work, because you'd have to give instructions each time you start your data analysis. Moreover, you'll have no record of what you did. That's why you need a __script__. This is a type of __code__. It can be referred to as a __source__ because that is the source of your commands. Source is also used as a verb; "source the script" just means execute it. RStudio doesn't start out with a script, so you can make one from "File > New" or the New file icon. +4. Theoretically, one could do all their work by typing in commands into the Console. But that would be a lot of work, because you'd have to give instructions each time you start your data analysis. Moreover, you'll have no record of what you did. That's why you need a **script**. This is a type of **code**. It can be referred to as a **source** because that is the source of your commands. Source is also used as a verb; "source the script" just means execute it. RStudio doesn't start out with a script, so you can make one from "File \> New" or the New file icon. ![Opening New Script (as opposed to the Console)](images/11_2_rstudio-script.png) -4. You can also open scripts that are in folders in your computer. A script is a type of File. Find your Files in the bottom-right "Files" pane. +4. You can also open scripts that are in folders in your computer. A script is a type of File. Find your Files in the bottom-right "Files" pane. - To load a dataset, you need to specify where that file is. Computer files (data, documents, programs) are organized hiearchically, like a branching tree. Folders can contain files, and also other folders. The GUI toolbar makes this lineaer and hiearchical relationship apparent. When we turn to locate the file in our commands, we need another set of syntax. Importantly, denote the hierarchy of a folder by the `/` (slash) symbol. `data/input/2018-08` indicates the `2018-08` folder, which is included in the `input` folder, which is in turn included in the `data` folder. + To load a dataset, you need to specify where that file is. Computer files (data, documents, programs) are organized hiearchically, like a branching tree. Folders can contain files, and also other folders. The GUI toolbar makes this lineaer and hiearchical relationship apparent. When we turn to locate the file in our commands, we need another set of syntax. Importantly, denote the hierarchy of a folder by the `/` (slash) symbol. `data/input/2018-08` indicates the `2018-08` folder, which is included in the `input` folder, which is in turn included in the `data` folder. Files (but not folders) have "file extensions" which you are probably familiar with already: `.docx`, `.pdf`, and `.pdf`. The file extensions you will see in a stats or quantitative social science class are: - * `.pdf`: PDF, a convenient format to view documents and slides in, regardless of Mac/Windows. - * `.csv`: A comma separated values file - * `.xlsx`: Microsoft Excel file - * `.dta`: Stata data - * `.sav`: SPSS data + - `.pdf`: PDF, a convenient format to view documents and slides in, regardless of Mac/Windows. - * `.R`: R code (script) - * `.Rmd`: Rmarkdown code (text + code) - * `.do`: Stata code (script) + - `.csv`: A comma separated values file + + - `.xlsx`: Microsoft Excel file + + - `.dta`: Stata data + + - `.sav`: SPSS data + + - `.R`: R code (script) + + - `.Rmd`: Rmarkdown code (text + code) + + - `.do`: Stata code (script) ![Opening an Existing Script from Files](images/11_3_rstudio-files.png) -5. In R, there are two main types of scripts. A classic `.R` file and a `.Rmd` file (for Rmarkdown). A .R file is just lines and lines of R code that is meant to be inserted right into the Console. A .Rmd tries to weave code and English together, to make it easier for users to create reports that interact with data and intersperse R code with explanation. For example, we built this book in Rmds. +5. In R, there are two main types of scripts. A classic `.R` file and a `.Rmd` file (for Rmarkdown). A .R file is just lines and lines of R code that is meant to be inserted right into the Console. A .Rmd tries to weave code and English together, to make it easier for users to create reports that interact with data and intersperse R code with explanation. For example, we built this book in Rmds. - The Rmarkdown facilitates is the use of __code chunks__, which are used here. These start and end with three back-ticks. In the beginning, we can add options in curly braces (`{}`). Specifying `r` in the beginning tells to render it as R code. Options like `echo = TRUE` switch between showing the code that was executed or not; `eval = TRUE` switch between evaluating the code. More about Rmarkdown in Section \@ref(nonwysiwyg). For example, this code chunk would evaluate `1 + 1` and show its output when compiled, but not display the code that was executed. +``` +The Rmarkdown facilitates is the use of __code chunks__, which are used here. These start and end with three back-ticks. In the beginning, we can add options in curly braces (`{}`). Specifying `r` in the beginning tells to render it as R code. Options like `echo = TRUE` switch between showing the code that was executed or not; `eval = TRUE` switch between evaluating the code. More about Rmarkdown in Section \@ref(nonwysiwyg). For example, this code chunk would evaluate `1 + 1` and show its output when compiled, but not display the code that was executed. +``` ![A code chunk in Rmarkdown (before rendering)](images/11_4_codechunk.png) \newpage + ## But what is R? -R is an object oriented programming language primarily used for statistical computing. An object oriented language is a programming language built around manipulating objects. -* In R, objects can be matrices, vectors, scalars, strings, and data frames, for example -* Objects contain different types of information -* Different objects have different allowable procedures: +R is an object oriented programming language primarily used for statistical computing. An object oriented language is a programming language built around manipulating objects. + +- In R, objects can be matrices, vectors, scalars, strings, and data frames, for example +- Objects contain different types of information +- Different objects have different allowable procedures: ```{r} # Adding a string and a string does not work because the '+' operator @@ -109,17 +121,19 @@ class('Harvard') Object oriented programming makes languages flexible and powerful: -* You can create custom functions and objects for your needs -* Other people can create great packages for everyone to use -* Many errors come from using the wrong data type and a lot of programming in R is getting data into the right format and type to work with. +- You can create custom functions and objects for your needs +- Other people can create great packages for everyone to use +- Many errors come from using the wrong data type and a lot of programming in R is getting data into the right format and type to work with. It is helpful to think in terms of object manipulation at a high level while programming in R, particularly at the beginning of tackling a new problem. Think about what objects you want to manipulate, what types they are, and how they fit together. Once you have the logic of your solution ready then you can write it in R. ## The Computer and You: Giving Instructions -We'll do the Peanut Butter and Jelly Exercise in class as an introduction to programming for those who are new.^[This Exercise is taken from Harvard's Introductory Undergraduate Class, CS50 (), and many other writeups.] +We'll do the Peanut Butter and Jelly Exercise in class as an introduction to programming for those who are new.[^11_data-handling_counting-1] + +[^11_data-handling_counting-1]: This Exercise is taken from Harvard's Introductory Undergraduate Class, CS50 (), and many other writeups. -Assignment: Take 5 minutes to write down on a piece of paper, how to make a peanut butter and jelly sandwich. Be as concise and unambiguous as possible so that a robot (who doesn't know what a PBJ is) would understand. You can assume that there will be loaf of sliced bread, a jar of jelly, a jar of peanut butter, and a knife. +Assignment: Take 5 minutes to write down on a piece of paper, how to make a peanut butter and jelly sandwich. Be as concise and unambiguous as possible so that a robot (who doesn't know what a PBJ is) would understand. You can assume that there will be loaf of sliced bread, a jar of jelly, a jar of peanut butter, and a knife. Simpler assignment: Say we just want a robot to be able to tell us if we have enough ingredients to make a peanut butter and jelly sandwich. Write down instructions so that if told how many slices of bread, servings of peanut butter, and servings of jelly you have, the robot can tell you if you can make a PBJ. @@ -134,13 +148,15 @@ n_jelly <- 9 ``` -## Base-R vs. tidyverse +## Base-R vs. tidyverse -One last thing before we jump into data. Many things in R and other open source packages have competing standards. A lecture on a technique inevitably biases one standard over another. Right now among R users in this area, there are two families of functions: base-R and tidyverse. R instructors thus face a dilemma about which to teach primarily.^[See for example this community discussion: https://community.rstudio.com/t/base-r-and-the-tidyverse/2965/17] +One last thing before we jump into data. Many things in R and other open source packages have competing standards. A lecture on a technique inevitably biases one standard over another. Right now among R users in this area, there are two families of functions: base-R and tidyverse. R instructors thus face a dilemma about which to teach primarily.[^11_data-handling_counting-2] -In this prefresher, we try our best to choose the one that is most useful to the modal task of social science researchers, and make use of the tidyverse functions in most applications. but feel free to suggest changes to us or to the booklet. +[^11_data-handling_counting-2]: See for example this community discussion: https://community.rstudio.com/t/base-r-and-the-tidyverse/2965/17 -Although you do not need to choose one over the other, for beginners it is confusing what is a tidyverse function and what is not. Many of the tidyverse _packages_ are covered in this 2017 graphic below, and the cheat-sheets that other programmers have written: https://www.rstudio.com/resources/cheatsheets/ +In this prefresher, we try our best to choose the one that is most useful to the modal task of social science researchers, and make use of the tidyverse functions in most applications. but feel free to suggest changes to us or to the booklet. + +Although you do not need to choose one over the other, for beginners it is confusing what is a tidyverse function and what is not. Many of the tidyverse *packages* are covered in this 2017 graphic below, and the cheat-sheets that other programmers have written: https://www.rstudio.com/resources/cheatsheets/ ![Names of Packages in the tidyverse Family](images/tidyverse-packages.png) @@ -148,50 +164,55 @@ The following side-by-side comparison of commands for a particular function comp ### Dataframe subsetting {.unnumbered} -| In order to ... | in tidyverse: | in base-R: | -|----------------------|:------------------------|:----------------------| -| Count each category | `count(df, var)` | `table(df$var)`| -| Filter rows by condition | `filter(df, var == "Female") ` | `df[df$var == "Female", ]` or `subset(df, var == "Female")` | -| Extract columns | `select(df, var1, var2)`| `df[, c("var1", "var2")]` | -| Extract a single column as a vector | `pull(df, var)` | `df[["var"]]` or `df[, "var"]` | -| Combine rows | `bind_rows()` | `rbind()` | -| Combine columns | `bind_cols()` | `cbind()` | -| Create a dataframe | `tibble(x = vec1, y = vec2)`| `data.frame(x = vec1, y = vec2)` | -| Turn a dataframe into a tidyverse dataframe | `tbl_df(df)` | | +| In order to ... | in tidyverse: | in base-R: | +|---------------------------------------------|:------------------------------|:------------------------------------------------------------| +| Count each category | `count(df, var)` | `table(df$var)` | +| Filter rows by condition | `filter(df, var == "Female")` | `df[df$var == "Female", ]` or `subset(df, var == "Female")` | +| Extract columns | `select(df, var1, var2)` | `df[, c("var1", "var2")]` | +| Extract a single column as a vector | `pull(df, var)` | `df[["var"]]` or `df[, "var"]` | +| Combine rows | `bind_rows()` | `rbind()` | +| Combine columns | `bind_cols()` | `cbind()` | +| Create a dataframe | `tibble(x = vec1, y = vec2)` | `data.frame(x = vec1, y = vec2)` | +| Turn a dataframe into a tidyverse dataframe | `tbl_df(df)` | | -Remember that tidyverse applies to _dataframes_ only, not vectors. For subsetting vectors, use the base-R functions with the square brackets. +Remember that tidyverse applies to *dataframes* only, not vectors. For subsetting vectors, use the base-R functions with the square brackets. ### Read data {.unnumbered} -Some non-tidyverse functions are not quite "base-R" but have similar relationships to tidyverse. For these, we recommend using the _tidyverse_ functions as a general rule due to their common format, simplicity, and scalability. +Some non-tidyverse functions are not quite "base-R" but have similar relationships to tidyverse. For these, we recommend using the *tidyverse* functions as a general rule due to their common format, simplicity, and scalability. -| In order to ... | in tidyverse: | in base-R: | -|----------------------|:------------------------|:----------------------| -| Read a Excel file | `read_excel()` | `read.xlsx()` | -| Read a csv | `read_csv()` | `read.csv()` | -| Read a Stata file | `read_dta()` | `read.dta()` | -| Substitute strings | `str_replace()` | `gsub()` | -| Return matching strings | `str_subset()` | `grep(., value = TRUE)` | -| Merge `data1` and `data2` on variables `x1` and `x2` | `left_join(data1, data2, by = c("x1", "x2"))` | `merge(data1, data2, by.x = "x1", by.y = "x2", all.x = TRUE)` | +| In order to ... | in tidyverse: | in base-R: | +|------------------------------------------------------|:----------------------------------------------|:--------------------------------------------------------------| +| Read a Excel file | `read_excel()` | `read.xlsx()` | +| Read a csv | `read_csv()` | `read.csv()` | +| Read a Stata file | `read_dta()` | `read.dta()` | +| Substitute strings | `str_replace()` | `gsub()` | +| Return matching strings | `str_subset()` | `grep(., value = TRUE)` | +| Merge `data1` and `data2` on variables `x1` and `x2` | `left_join(data1, data2, by = c("x1", "x2"))` | `merge(data1, data2, by.x = "x1", by.y = "x2", all.x = TRUE)` | ### Visualization {.unnumbered} Plotting by ggplot2 (from your tutorials) is also a tidyverse family. -| In order to ... | in tidyverse: | in base-R: | -|----------------------|:------------------------|:----------------------| -| Make a scatter plot | `ggplot(data, aes(x, y)) + geom_point()` | `plot(data$x, data$y)` | -| Make a line plot | `ggplot(data, aes(x, y)) + geom_line()` | `plot(data$x, data$y, type = "l")` | -| Make a histogram | `ggplot(data, aes(x, y)) + geom_histogram()` | `hist(data$x, data$y)` | -| Make a barplot | See Section \@ref(dataviz) | See Section \@ref(dataviz)| +| In order to ... | in tidyverse: | in base-R: | +|---------------------|:---------------------------------------------|:-----------------------------------| +| Make a scatter plot | `ggplot(data, aes(x, y)) + geom_point()` | `plot(data$x, data$y)` | +| Make a line plot | `ggplot(data, aes(x, y)) + geom_line()` | `plot(data$x, data$y, type = "l")` | +| Make a histogram | `ggplot(data, aes(x, y)) + geom_histogram()` | `hist(data$x, data$y)` | +| Make a barplot | See Section \@ref(dataviz) | See Section \@ref(dataviz) | ## A is for Athens -For our first dataset, let's try reading in a dataset on the Ancient Greek world. Political Theorists and Political Historians study the domestic systems, international wars, cultures and writing of this era to understand the first instance of democracy, the rise and overturning of tyranny, and the legacies of political institutions. +For our first dataset, let's try reading in a dataset on the Ancient Greek world. Political Theorists and Political Historians study the domestic systems, international wars, cultures and writing of this era to understand the first instance of democracy, the rise and overturning of tyranny, and the legacies of political institutions. + +This POLIS dataset was generously provided by Professor Josiah Ober of Stanford University. This dataset includes information on city states in the Ancient Greek world, parts of it collected by careful work by historians and archaeologists. It is part of his recent books on Greece (Ober 2015), "The Rise and Fall of Classical Greece"[^11_data-handling_counting-3] and Institutions in Ancient Athens (Ober 2010) , "Democracy and Knowledge: Innovation and Learning in Classical Athens."[^11_data-handling_counting-4] + +[^11_data-handling_counting-3]: [Ober, Josiah (2015). *The Rise and Fall of Classical Greece*. Princeton University Press.](https://press.princeton.edu/titles/10423.html) -This POLIS dataset was generously provided by Professor Josiah Ober of Stanford University. This dataset includes information on city states in the Ancient Greek world, parts of it collected by careful work by historians and archaeologists. It is part of his recent books on Greece (Ober 2015), "The Rise and Fall of Classical Greece"^[[Ober, Josiah (2015). _The Rise and Fall of Classical Greece_. Princeton University Press.](https://press.princeton.edu/titles/10423.html)] and Institutions in Ancient Athens (Ober 2010) , "Democracy and Knowledge: Innovation and Learning in Classical Athens."^[[Ober, Josiah (2010). _Democracy and Knowledge: Innovation and Learning in Classical Athens_. Princeton University Press.](https://press.princeton.edu/titles/8742.html)] +[^11_data-handling_counting-4]: [Ober, Josiah (2010). *Democracy and Knowledge: Innovation and Learning in Classical Athens*. Princeton University Press.](https://press.princeton.edu/titles/8742.html) + +### Locating the Data -### Locating the Data What files do we have in the `data/input` folder? ```{r} @@ -199,44 +220,45 @@ What files do we have in the `data/input` folder? dir_ls("data/input") ``` -A typical file format is Microsoft Excel. Although this is not usually the best format for R because of its highly formatted structure as opposed to plain text (more on this in Section \ref@(sec:wysiwyg)), recent packages have made this fairly easy. +A typical file format is Microsoft Excel. Although this is not usually the best format for R because of its highly formatted structure as opposed to plain text (more on this in Section \ref@(sec:wysiwyg)), recent packages have made this fairly easy. ### Reading in Data -In Rstudio, a good way to start is to use the GUI and the Import tool. Once you click a file, an option to "Import Dataset" comes up. RStudio picks the right function for you, and you can copy that code, but it's important to eventually be able to write that code yourself. +In Rstudio, a good way to start is to use the GUI and the Import tool. Once you click a file, an option to "Import Dataset" comes up. RStudio picks the right function for you, and you can copy that code, but it's important to eventually be able to write that code yourself. + +For the first time using an outside package, you first need to install it. -For the first time using an outside package, you first need to install it. ```{r} #| eval: false install.packages("readxl") ``` -After that, you don't need to install it again. But you __do__ need to load it each time. +After that, you don't need to install it again. But you **do** need to load it each time. ```{r} #| eval: false library(readxl) ``` -The package `readxl` has a website: https://readxl.tidyverse.org/. Other packages are not as user-friendly, but they have a help page with a table of contents of all their functions. +The package `readxl` has a website: https://readxl.tidyverse.org/. Other packages are not as user-friendly, but they have a help page with a table of contents of all their functions. ```{r} #| eval: false help(package = readxl) ``` -From the help page, we see that `read_excel()` is the function that we want to use. +From the help page, we see that `read_excel()` is the function that we want to use. -Let's try it. +Let's try it. ```{r} library(readxl) ober <- read_excel("data/input/ober_2018.xlsx") ``` -Review: what does the `/` mean? Why do we need the `data` term first? Does the argument need to be in quotes? +Review: what does the `/` mean? Why do we need the `data` term first? Does the argument need to be in quotes? -### Inspecting +### Inspecting For almost any dataset, you usually want to do a couple of standard checks first to understand what you loaded. @@ -255,26 +277,30 @@ ggplot(ober, aes(x = Fame)) + geom_histogram() ``` What about the distribution of fame by regime? + ```{r} ggplot(ober, aes(y = Fame, x = Regime, group = Regime)) + geom_boxplot() ``` -What do the 1's, 2's, and 3's stand for? +What do the 1's, 2's, and 3's stand for? ### Finding observations -These `tidyverse` commands from the `dplyr` package are newer and not built-in, but they are one of the increasingly more popular ways to wrangle data. +These `tidyverse` commands from the `dplyr` package are newer and not built-in, but they are one of the increasingly more popular ways to wrangle data. -* 80 percent of your data wrangling needs might be doable with these basic `dplyr` functions: `select`, `mutate`, `group_by`, `summarize`, and `arrange`. -* These verbs roughly correspond to the same commands in SQL, another important language in data science. -* The `%>%` symbol is a pipe. It takes the thing on the left side and pipes it down to the function on the right side. We could have done `count(cen10, race)` as `cen10 %>% count(race)`. That means take `cen10` and pass it on to the function `count`, which will count observations by race and return a collapsed dataset with the categories in its own variable and their respective counts in `n`. +- 80 percent of your data wrangling needs might be doable with these basic `dplyr` functions: `select`, `mutate`, `group_by`, `summarize`, and `arrange`. +- These verbs roughly correspond to the same commands in SQL, another important language in data science. +- The `%>%` symbol is a pipe. It takes the thing on the left side and pipes it down to the function on the right side. We could have done `count(cen10, race)` as `cen10 %>% count(race)`. That means take `cen10` and pass it on to the function `count`, which will count observations by race and return a collapsed dataset with the categories in its own variable and their respective counts in `n`. ### Extra: A sneak peak at Ober's data -Although this is a bit beyond our current stage, it's hard to resist the temptation to see what you can do with data like this. For example, you can map it.^[In mid-2018, changes in Google's services made it no longer possible to render maps on the fly. Therefore, the map is not currently rendered automatically (but can be rendered once the user registers their API). Instead, you now need to register with Google. See the [change](https://github.com/dkahle/ggmap/blob/e55c0b22b0d16a010b4b45dd2fce800ff0ef19b8/NEWS#L6-L12) to the pacakge ggmap.] +Although this is a bit beyond our current stage, it's hard to resist the temptation to see what you can do with data like this. For example, you can map it.[^11_data-handling_counting-5] + +[^11_data-handling_counting-5]: In mid-2018, changes in Google's services made it no longer possible to render maps on the fly. Therefore, the map is not currently rendered automatically (but can be rendered once the user registers their API). Instead, you now need to register with Google. See the [change](https://github.com/dkahle/ggmap/blob/e55c0b22b0d16a010b4b45dd2fce800ff0ef19b8/NEWS#L6-L12) to the pacakge ggmap. Using the `ggmap` package + ```{r} #| message: false #| warning: false @@ -283,6 +309,7 @@ library(ggmap) ``` First get a map of the Greek world. + ```{r} #| message: false #| warning: false @@ -293,11 +320,12 @@ greece <- get_map(location = c(lon = 22.6382849, lat = 39.543287), maptype = "toner") ggmap(greece) ``` + ![](images/ober_ggmap_default.png) I chose the specifications for arguments `zoom` and `maptype` by looking at the webpage and Googling some examples. -Ober's data has the latitude and longitude of each polis. Because the map of Greece has the same coordinates, we can add the polei on the same map. +Ober's data has the latitude and longitude of each polis. Because the map of Greece has the same coordinates, we can add the polei on the same map. ```{r} #| warning: false @@ -312,6 +340,7 @@ gg_ober + scale_y_continuous(limits = c(32, 44)) + theme_void() ``` + ![](images/ober_ggmap_polis.png) ## Exercises {.unnumbered} @@ -341,13 +370,16 @@ Make a scatterplot with the number of colonies on the x-axis and Fame on the y-a ``` ### 4 {.unnumbered} + Find the correct function to read the following datasets (available in your rstudio.cloud session) into your R window. -* `data/input/acs2015_1percent.csv`: A one percent sample of the American Community Survey -* `data/input/gapminder_wide.tab`: Country-level wealth and health from Gapminder^[Formatted and taken from ] -* `data/input/gapminder_wide.Rds`: A Rds version of the Gapminder (What is a Rds file? What's the difference?) -* `data/input/Nunn_Wantchekon_sample.dta`: A sample from the Afrobarometer survey (which we'll explore tomorrow). `.dta` is a Stata format. -* `data/input/german_credit.sav`: A hypothetical dataset on consumer credit. `.sav` is a SPSS format. +- `data/input/acs2015_1percent.csv`: A one percent sample of the American Community Survey +- `data/input/gapminder_wide.tab`: Country-level wealth and health from Gapminder[^11_data-handling_counting-6] +- `data/input/gapminder_wide.Rds`: A Rds version of the Gapminder (What is a Rds file? What's the difference?) +- `data/input/Nunn_Wantchekon_sample.dta`: A sample from the Afrobarometer survey (which we'll explore tomorrow). `.dta` is a Stata format. +- `data/input/german_credit.sav`: A hypothetical dataset on consumer credit. `.sav` is a SPSS format. + +[^11_data-handling_counting-6]: Formatted and taken from Our Recommendations: Look at the packages `haven` and `readr` @@ -356,6 +388,7 @@ Our Recommendations: Look at the packages `haven` and `readr` ``` ### 5 {.unnumbered} + Read Ober's codebook and find a variable that you think is interesting. Check the distribution of that variable in your data, get a couple of statistics, and summarize it in English. ```{r} @@ -363,6 +396,7 @@ Read Ober's codebook and find a variable that you think is interesting. Check th ``` ### 6 {.unnumbered} + This is day 1 and we covered a lot of material. Some of you might have found this completely new; others not so. Please click through this survey before you leave so we can adjust accordingly on the next few days. diff --git a/12_matricies-manipulation.qmd b/12_matricies-manipulation.qmd index 802fafa..f7b3973 100644 --- a/12_matricies-manipulation.qmd +++ b/12_matricies-manipulation.qmd @@ -1,4 +1,4 @@ -# Manipulating Vectors and Matrices^[Module originally written by Shiro Kuriwaki and Yon Soo Park] {#rmatrices} +# Manipulating Vectors and Matrices {#rmatrices} ```{r} #| include: false @@ -10,25 +10,31 @@ library(haven) library(ggplot2) ``` +::: {.callout .callout-note} +Module originally written by Shiro Kuriwaki and Yon Soo Park +::: + ### Motivation {.unnumbered} -[Nunn and Wantchekon (2011)](https://dash.harvard.edu/bitstream/handle/1/11986331/nunn-slave-trade.pdf) -- "The Slave Trade and the Origins of Mistrust in Africa"^[[Nunn, Nathan, and Leonard Wantchekon. 2011. “The Slave Trade and the Origins of Mistrust in Africa.” American Economic Review 101(7): 3221–52.](https://dash.harvard.edu/bitstream/handle/1/11986331/nunn-slave-trade.pdf)] -- argues that across African countries, the distrust of co-ethnics fueled by the slave trade has had long-lasting effects on modern day trust in these territories. They argued that the slave trade created distrust in these societies in part because as some African groups were employed by European traders to capture their neighbors and bring them to the slave ships. +[Nunn and Wantchekon (2011)](https://dash.harvard.edu/bitstream/handle/1/11986331/nunn-slave-trade.pdf) -- "The Slave Trade and the Origins of Mistrust in Africa"[^12_matricies-manipulation-1] -- argues that across African countries, the distrust of co-ethnics fueled by the slave trade has had long-lasting effects on modern day trust in these territories. They argued that the slave trade created distrust in these societies in part because as some African groups were employed by European traders to capture their neighbors and bring them to the slave ships. + +[^12_matricies-manipulation-1]: [Nunn, Nathan, and Leonard Wantchekon. 2011. “The Slave Trade and the Origins of Mistrust in Africa.” American Economic Review 101(7): 3221–52.](https://dash.harvard.edu/bitstream/handle/1/11986331/nunn-slave-trade.pdf) -Nunn and Wantchekon use a variety of statistical tools to make their case (adding controls, ordered logit, instrumental variables, falsification tests, causal mechanisms), many of which will be covered in future courses. In this module we will only touch on their first set of analysis that use Ordinary Least Squares (OLS). OLS is likely the most common application of linear algebra in the social sciences. We will cover some linear algebra, matrix manipulation, and vector manipulation from this data. +Nunn and Wantchekon use a variety of statistical tools to make their case (adding controls, ordered logit, instrumental variables, falsification tests, causal mechanisms), many of which will be covered in future courses. In this module we will only touch on their first set of analysis that use Ordinary Least Squares (OLS). OLS is likely the most common application of linear algebra in the social sciences. We will cover some linear algebra, matrix manipulation, and vector manipulation from this data. ### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: -* R basic programming -* Data Import -* Statistical Summaries. +- R basic programming +- Data Import +- Statistical Summaries. Today we'll cover -* Matrices & Dataframes in R -* Manipulating variables -* And other `R` tips +- Matrices & Dataframes in R +- Manipulating variables +- And other `R` tips ## Read Data @@ -38,12 +44,14 @@ nunn_full <- read_dta("data/input/Nunn_Wantchekon_AER_2011.dta") ``` Nunn and Wantchekon's main dataset has more than 20,000 observations. Each observation is a respondent from the Afrobarometer survey. + ```{r} head(nunn_full) colnames(nunn_full) ``` -First, let's consider a small subset of this dataset. +First, let's consider a small subset of this dataset. + ```{r} #| include: false #| eval: false @@ -62,12 +70,15 @@ nunn ``` ## data.frame vs. matricies -This is a `data.frame` object. + +This is a `data.frame` object. + ```{r} class(nunn) ``` But it can be also consider a matrix in the linear algebra sense. What are the dimensions of this matrix? + ```{r} nrow(nunn) ``` @@ -83,6 +94,7 @@ What is the difference between a `data.frame` and a matrix? A `data.frame` can h You can think of data frames maybe as matrices-plus, because a column can take on characters as well as numbers. As we just saw, this is often useful for real data analyses. Another way to think about data frames is that it is a type of list. Try the `str()` code below and notice how it is organized in slots. Each slot is a vector. They can be vectors of numbers or characters. + ```{r} #| eval: false # enter this on your console @@ -92,41 +104,50 @@ str(cen10) ## Handling matricies in `R` You can easily transpose a matrix + ```{r} X t(X) ``` -What are the values of all rows in the first column? +What are the values of all rows in the first column? + ```{r} X[, 1] ``` What are all the values of "exports"? (i.e. return the whole "exports" column) + ```{r} X[, "exports"] ``` What is the first observation (i.e. first row)? + ```{r} X[1, ] ``` What is the value of the first variable of the first observation? + ```{r} X[1, 1] ``` -Pause and consider the following problem on your own. What is the following code doing? +Pause and consider the following problem on your own. What is the following code doing? + ```{r} X[X[, "trust_neighbors"] == 0, "export_area"] ``` -Why does it give the same output as the following? + +Why does it give the same output as the following? + ```{r} X[which(X[, "trust_neighbors"] == 0), "export_area"] ``` -Some more manipulation +Some more manipulation + ```{r} X + X ``` @@ -153,20 +174,21 @@ colnames(X) ## Variable Transformations -`exports` is the total number of slaves that were taken from the individual's ethnic group between Africa's four slave trades between 1400-1900. +`exports` is the total number of slaves that were taken from the individual's ethnic group between Africa's four slave trades between 1400-1900. What is `ln_exports`? The article describes this as the natural log of one plus the `exports`. This is a transformation of one column by a particular function ```{r} log(1 + X[, "exports"]) ``` -Question for you: why add the 1? + +Question for you: why add the 1? Verify that this is the same as `X[, "ln_exports"]` ## Linear Combinations -In Table 1 we see "OLS Estimates". These are estimates of OLS coefficients and standard errors. You do not need to know what these are for now, but it doesn't hurt to getting used to seeing them. +In Table 1 we see "OLS Estimates". These are estimates of OLS coefficients and standard errors. You do not need to know what these are for now, but it doesn't hurt to getting used to seeing them. ![](images/nunn_wantchekon_table1.png) @@ -179,6 +201,7 @@ Take the first number in Table 1, which is -0.00068. Now, multiply this by `expo ``` Now, just one more step. Make a new matrix with just exports and the value 1 + ```{r} X2 <- cbind(1, X[, "exports"]) ``` @@ -194,6 +217,7 @@ colnames(X2) <- c("intercept", "exports") ``` What are the dimensions of the matrix `X2`? + ```{r} dim(X2) ``` @@ -205,11 +229,13 @@ B <- matrix(c(1.62, -0.00068)) ``` What are the dimensions of `B`? + ```{r} dim(B) ``` What is the product of `X2` and `B`? From the dimensions, can you tell if it will be conformable? + ```{r} X2 %*% B ``` @@ -237,6 +263,7 @@ head(cen10) ``` What is the dimension of this dataframe? What does the number of rows represent? What does the number of columns represent? + ```{r} #| message: false dim(cen10) @@ -251,7 +278,7 @@ What variables does this dataset hold? What kind of information does it have? colnames(cen10) ``` -We can access column vectors, or vectors that contain values of variables by using the $ sign +We can access column vectors, or vectors that contain values of variables by using the \$ sign ```{r} #| message: false @@ -267,7 +294,7 @@ We can look at a unique set of variable values by calling the unique function unique(cen10$state) ``` -How many different states are represented (this dataset includes DC as a state)? +How many different states are represented (this dataset includes DC as a state)? ```{r} #| message: false @@ -277,6 +304,7 @@ length(unique(cen10$state)) Matrices are rectangular structures of numbers (they have to be numbers, and they can't be characters). A cross-tab can be considered a matrix: + ```{r} table(cen10$race, cen10$sex) ``` @@ -287,7 +315,7 @@ dim(cross_tab) cross_tab[6, 2] ``` -But a subset of your data -- individual values-- can be considered a matrix too. +But a subset of your data -- individual values-- can be considered a matrix too. ```{r} #| warning: false @@ -304,7 +332,8 @@ cen10[1:20, c("race", "age")] cen10 %>% slice(1:20) %>% select(race, age) ``` -A vector is a special type of matrix with only one column or only one row +A vector is a special type of matrix with only one column or only one row + ```{r} # One column @@ -319,6 +348,7 @@ cen10 %>% slice(2) ``` What if we want a special subset of the data? For example, what if I only want the records of individuals in California? What if I just want the age and race of individuals in California? + ```{r} # subset for CA rows ca_subset <- cen10[cen10$state == "California", ] @@ -335,7 +365,8 @@ ca_subset_age_race_tidy <- cen10 %>% filter(state == "California") %>% select(ag all_equal(ca_subset_age_race, ca_subset_age_race_tidy) ``` -Some common operators that can be used to filter or to use as a condition. Remember, you can use the unique function to look at the set of all values a variable holds in the dataset. +Some common operators that can be used to filter or to use as a condition. Remember, you can use the unique function to look at the set of all values a variable holds in the dataset. + ```{r} # all individuals older than 30 and younger than 70 s1 <- cen10[cen10$age > 30 & cen10$age < 70, ] @@ -361,19 +392,25 @@ all_equal(s7, s8) ## Checkpoint {.unnumbered} ### 1 {.unnumbered} + Get the subset of cen10 for non-white individuals (Hint: look at the set of values for the race variable by using the unique function) + ```{r} # Enter here ``` ### 2 {.unnumbered} + Get the subset of cen10 for females over the age of 40 + ```{r} # Enter here ``` ### 3 {.unnumbered} + Get all the serial numbers for black, male individuals who don't live in Ohio or Nevada. + ```{r} # Enter here ``` @@ -382,15 +419,14 @@ Get all the serial numbers for black, male individuals who don't live in Ohio or ### 1 {.unnumbered} -Let -$$\mathbf{A} = \left[\begin{array} +Let $$\mathbf{A} = \left[\begin{array} {rrr} 0.6 & 0.2\\ 0.4 & 0.8\\ \end{array}\right] $$ -Use R to write code that will create the matrix $A$, and then consecutively multiply $A$ to itself 4 times. What is the value of $A^{4}$? +Use R to write code that will create the matrix $A$, and then consecutively multiply $A$ to itself 4 times. What is the value of $A^{4}$? ```{r} ## Enter yourself @@ -399,21 +435,22 @@ Use R to write code that will create the matrix $A$, and then consecutively mul Note that R notation of matrices is different from the math notation. Simply trying `X^n` where `X` is a matrix will only take the power of each element to `n`. Instead, this problem asks you to perform matrix multiplication. ### 2 {.unnumbered} + Let's apply what we learned about subsetting or filtering/selecting. Use the `nunn_full` dataset you have already loaded -a) First, show all observations (rows) that have a `"male"` variable higher than 0.5 +a) First, show all observations (rows) that have a `"male"` variable higher than 0.5 ```{r} ## Enter yourself ``` -b) Next, create a matrix / dataframe with only two columns: `"trust_neighbors"` and `"age"` +b) Next, create a matrix / dataframe with only two columns: `"trust_neighbors"` and `"age"` ```{r} ## Enter yourself ``` -c) Lastly, show all values of `"trust_neighbors"` and `"age"` for observations (rows) that have the "male" variable value that is higher than 0.5 +c) Lastly, show all values of `"trust_neighbors"` and `"age"` for observations (rows) that have the "male" variable value that is higher than 0.5 ```{r} ## Enter yourself @@ -425,11 +462,12 @@ Find a way to generate a vector of "column averages" of the matrix `X` from the ### 4 {.unnumbered} -Similarly, generate a vector of "column medians". +Similarly, generate a vector of "column medians". ### 5 {.unnumbered} Consider the regression that was run to generate Table 1: + ```{r} form <- "trust_neighbors ~ exports + age + age2 + male + urban_dum + factor(education) + factor(occupation) + factor(religion) + factor(living_conditions) + district_ethnic_frac + frac_ethnicity_in_district + isocode" lm_1_1 <- lm(as.formula(form), nunn_full) @@ -437,12 +475,15 @@ lm_1_1 <- lm(as.formula(form), nunn_full) # The below coef function returns a vector of OLS coefficiants coef(lm_1_1) ``` -First, get a small subset of the nunn_full dataset. This time, sample 20 rows and select for variables `exports`, `age`, `age2`, `male`, and `urban_dum`. To this small subset, add (`bind_cols()` in tidyverse or `cbind()` in base R) a column of 1's; this represents the intercept. If you need some guidance, look at how we sampled 10 rows selected for a different set of variables above in the lecture portion. + +First, get a small subset of the nunn_full dataset. This time, sample 20 rows and select for variables `exports`, `age`, `age2`, `male`, and `urban_dum`. To this small subset, add (`bind_cols()` in tidyverse or `cbind()` in base R) a column of 1's; this represents the intercept. If you need some guidance, look at how we sampled 10 rows selected for a different set of variables above in the lecture portion. + ```{r} # Enter here ``` -Next let's try calculating predicted values of levels of trust in neighbors by multiplying coefficients for the intercept, `exports`, `age`, `age2`, `male`, and `urban_dum` to the actual observed values for those variables in the small subset you've just created. +Next let's try calculating predicted values of levels of trust in neighbors by multiplying coefficients for the intercept, `exports`, `age`, `age2`, `male`, and `urban_dum` to the actual observed values for those variables in the small subset you've just created. + ```{r} # Hint: You can get just selected elements from the vector returned by coef(lm_1_1) diff --git a/13_functions_obj_loops.qmd b/13_functions_obj_loops.qmd index 0355db4..3537003 100644 --- a/13_functions_obj_loops.qmd +++ b/13_functions_obj_loops.qmd @@ -4,22 +4,22 @@ Up till now, you should have covered: -* R basic programming -* Data Import -* Statistical Summaries -* Visualization +- R basic programming +- Data Import +- Statistical Summaries +- Visualization Today we'll cover -* Objects -* Functions -* Loops +- Objects +- Functions +- Loops ## What is an object? -Now that we have covered some hands-on ways to use graphics, let's go into some fundamentals of the R language. +Now that we have covered some hands-on ways to use graphics, let's go into some fundamentals of the R language. -Let's first set up +Let's first set up ```{r} #| message: false @@ -34,23 +34,26 @@ library(ggplot2) cen10 <- read_csv("data/input/usc2010_001percent.csv", col_types = cols()) ``` -Objects are abstract symbols in which you store data. Here we will create an object from `copy`, and assign `cen10` to it. +Objects are abstract symbols in which you store data. Here we will create an object from `copy`, and assign `cen10` to it. ```{r} copy <- cen10 ``` This looks the same as the original dataset: + ```{r} copy ``` What happens if you do this next? + ```{r} copy <- "" ``` It got reassigned: + ```{r} copy ``` @@ -73,12 +76,13 @@ my_list ``` each slot can be anything. What are we doing here? We are defining the 1st slot of the list `my_list` to be a vector `c(1, 2, 3, 4, 5)` + ```{r} my_list[[1]] <- c(1, 2, 3, 4, 5) my_list ``` -You can even make nested lists. Let's say we want the 1st slot of the list to be another list of three elements. +You can even make nested lists. Let's say we want the 1st slot of the list to be another list of three elements. ```{r} my_list[[1]][[1]] <- "subitem 1 in slot 1 of my_list" @@ -89,35 +93,38 @@ my_list ``` ## Making your own objects -We've covered one type of object, which is a list. You saw it was quite flexible. How many types of objects are there? + +We've covered one type of object, which is a list. You saw it was quite flexible. How many types of objects are there? There are an infinite number of objects, because people make their own class of object. You can detect the type of the object (the class) by the function `class` Object can be said to be an instance of a class. -___Analogies___: +***Analogies***: -__class__ - Pokemon, __object__ - Pikachu +**class** - Pokemon, **object** - Pikachu -__class__ - Book, __object__ - To Kill a Mockingbird +**class** - Book, **object** - To Kill a Mockingbird -__class__ - DataFrame, __object__ - 2010 census data +**class** - DataFrame, **object** - 2010 census data -__class__ - Character, __object__ - "Programming is Fun" +**class** - Character, **object** - "Programming is Fun" What is type (class) of object is `cen10`? + ```{r} class(cen10) ``` -What about this text? +What about this text? + ```{r} class("some random text") ``` -To change or create the class of any object, you can _assign_ it. To do this, assign the name of your class to character to an object's `class()`. +To change or create the class of any object, you can *assign* it. To do this, assign the name of your class to character to an object's `class()`. -We can start from a simple list. For example, say we wanted to store data about pokemon. Because there is no pre-made package for this, we decide to make our own class. +We can start from a simple list. For example, say we wanted to store data about pokemon. Because there is no pre-made package for this, we decide to make our own class. ```{r} pikachu <- list(name = "Pikachu", @@ -126,7 +133,8 @@ pikachu <- list(name = "Pikachu", color = "Yellow") ``` -and we can give it any class name we want. +and we can give it any class name we want. + ```{r} class(pikachu) <- "Pokemon" str(pikachu) @@ -135,13 +143,15 @@ pikachu$type ``` ### Seeing R through objects + Most of the R objects that you will see as you advance are their own objects. For example, here's a linear regression object (which you will learn more about in Gov 2000): + ```{r} ols <- lm(mpg ~ wt + vs + gear + carb, mtcars) class(ols) ``` -Anything can be an object! Even graphs (in `ggplot`) can be assigned, re-assigned, and edited. +Anything can be an object! Even graphs (in `ggplot`) can be assigned, re-assigned, and edited. ```{r} #| warning: false @@ -161,11 +171,13 @@ gg_tab ``` You can change the orientation + ```{r} gg_tab<- gg_tab + coord_flip() ``` ### Parsing an object by `str()s` + It can be hard to understand an `R` object because it's contents are unknown. The function `str`, short for structure, is a quick way to look into the innards of an object ```{r} @@ -174,6 +186,7 @@ class(my_list) ``` Same for the object we just made + ```{r} str(pikachu) ``` @@ -187,17 +200,21 @@ str(gg_tab) ``` ## Types of variables -In the social science we often analyze variables. As you saw in the tutorial, different types of variables require different care. -A key link with what we just learned is that variables are also types of R objects. +In the social science we often analyze variables. As you saw in the tutorial, different types of variables require different care. + +A key link with what we just learned is that variables are also types of R objects. ### scalars + One number. How many people did we count in our Census sample? + ```{r} nrow(cen10) ``` Question: What proportion of our census sample is Native American? This number is also a scalar + ```{r} # Enter yourself unique(cen10$race) @@ -208,21 +225,23 @@ Hint: you can use the function `mean()` to calcualte the sample mean. The sample ### numeric vectors -A sequence of numbers. +A sequence of numbers. ```{r} grp_race_ordered$count class(grp_race_ordered$count) ``` -Or even, all the ages of the millions of people in our Census. Here are just the first few numbers of the list. +Or even, all the ages of the millions of people in our Census. Here are just the first few numbers of the list. + ```{r} head(cen10$age) ``` ### characters (aka strings) -This can be just one stretch of characters +This can be just one stretch of characters + ```{r} my_name <- "Meg" my_name @@ -230,6 +249,7 @@ class(my_name) ``` or more characters. Notice here that there's a difference between a vector of individual characters and a length-one object of characters. + ```{r} my_name_letters <- c("M","e","g") my_name_letters @@ -237,6 +257,7 @@ class(my_name_letters) ``` Finally, remember that lower vs. upper case matters in R! + ```{r} my_name2 <- "shiro" my_name == my_name2 @@ -246,20 +267,24 @@ my_name == my_name2 Most of what we do in R is executing a function. `read_csv()`, `nrow()`, `ggplot()` .. pretty much anything with a parentheses is a function. And even things like `<-` and `[` are functions as well. -A function is a set of instructions with specified ingredients. It takes an __input__, then __manipulates__ it -- changes it in some way -- and then returns the manipulated product. +A function is a set of instructions with specified ingredients. It takes an **input**, then **manipulates** it -- changes it in some way -- and then returns the manipulated product. + +One way to see what a function actually does is to enter it without parentheses. -One way to see what a function actually does is to enter it without parentheses. ```{r} #| eval: false # enter this on your console table ``` -You'll see below that the most basic functions are quite complicated internally. -You'll notice that functions contain other functions. _wrapper_ functions are functions that "wrap around" existing functions. This sounds redundant, but it's an important feature of programming. If you find yourself repeating a command more than two times, you should make your own function, rather than writing the same type of code. +You'll see below that the most basic functions are quite complicated internally. + +You'll notice that functions contain other functions. *wrapper* functions are functions that "wrap around" existing functions. This sounds redundant, but it's an important feature of programming. If you find yourself repeating a command more than two times, you should make your own function, rather than writing the same type of code. ### Write your own function + It's worth remembering the basic structure of a function. You create a new function, call it `my_fun` by this: + ```{r} #| eval: !expr F my_fun <- function() { @@ -279,6 +304,7 @@ count_men <- function(data) { ``` Then all we need to do is feed this function a dataset + ```{r} count_men(cen10) ``` @@ -305,6 +331,7 @@ nw_in_state <- function(data, state) { The last line is what gets generated from the function. To be more explicit you can wrap the last line around `return()`. (as in `return(nw.s/total.s`). `return()` is used when you want to break out of a function in the middle of it and not wait till the last line. Try it on your favorite state! + ```{r} nw_in_state(cen10, "Massachusetts") @@ -315,6 +342,7 @@ nw_in_state(cen10, "Massachusetts") ### 1 {.unnumbered} Try making your own function, `average_age_in_state`, that will give you the average age of people in a given state. + ```{r} # Enter on your own @@ -323,47 +351,51 @@ Try making your own function, `average_age_in_state`, that will give you the ave ### 2 {.unnumbered} Try making your own function, `asians_in_state`, that will give you the number of `Chinese`, `Japanese`, and `Other Asian or Pacific Islander` people in a given state. + ```{r} # Enter on your own ``` ### 3 {.unnumbered} -Try making your own function, 'top_10_oldest_cities', that will give you the names of cities whose population's average age is top 10 oldest. +Try making your own function, 'top_10_oldest_cities', that will give you the names of cities whose population's average age is top 10 oldest. ```{r} # Enter on your own ``` ## What is a package? -You can think of a package as a suite of functions that other people have already built for you to make your life easier. + +You can think of a package as a suite of functions that other people have already built for you to make your life easier. ```{r} help(package = "ggplot2") ``` -To use a package, you need to do two things: (1) install it, and then (2) load it. +To use a package, you need to do two things: (1) install it, and then (2) load it. Installing is a one-time thing + ```{r} #| eval: false install.packages("ggplot2") ``` -But you need to load each time you start a R instance. So always keep these commands on a script. +But you need to load each time you start a R instance. So always keep these commands on a script. + ```{r} library(ggplot2) ``` -In `rstudio.cloud`, we already installed a set of packages for you. But when you start your own R instance, you need to have installed the package at some point. +In `rstudio.cloud`, we already installed a set of packages for you. But when you start your own R instance, you need to have installed the package at some point. ## Conditionals -Sometimes, you want to execute a command only under certain conditions. This is done through the almost universal function, `if()`. Inside the `if` function we enter a logical statement. The line that is adjacent to, or follows, the `if()` statement only gets executed if the statement returns `TRUE`. +Sometimes, you want to execute a command only under certain conditions. This is done through the almost universal function, `if()`. Inside the `if` function we enter a logical statement. The line that is adjacent to, or follows, the `if()` statement only gets executed if the statement returns `TRUE`. -For example, +For example, -For example, +For example, ```{r} x <- 5 @@ -376,7 +408,7 @@ if (x >0) { } ``` -You can wrap that whole things in a function +You can wrap that whole things in a function ```{r} #| warning: false @@ -393,11 +425,13 @@ is_positive <- function(number) { is_positive(5) is_positive(-3) ``` + ## For-loops -Loops repeat the same statement, although the statement can be "the same" only in an abstract sense. Use the `for(x in X)` syntax to repeat the subsequent command as many times as there are elements in the right-hand object `X`. Each of these elements will be referred to the left-hand index `x` +Loops repeat the same statement, although the statement can be "the same" only in an abstract sense. Use the `for(x in X)` syntax to repeat the subsequent command as many times as there are elements in the right-hand object `X`. Each of these elements will be referred to the left-hand index `x` + +First, come up with a vector. -First, come up with a vector. ```{r} fruits <- c("apples", "oranges", "grapes") ``` @@ -408,7 +442,7 @@ Now we use the `fruits` vector in a `for` loop. for (fruit in fruits) { print(paste("I love", fruit)) } -``` +``` Here `for()` and `in` must be part of any for loop. The right hand side `fruits` must be a thing that exists. Finally the `left-hand` side object is "Pick your favor name." It is analogous to how we can index a sum with any letter. $\sum_{i=1}^{10}i$ and `sum_{j = 1}^{10}j` are in fact the same thing. @@ -416,7 +450,7 @@ Here `for()` and `in` must be part of any for loop. The right hand side `fruits` for (i in 1:length(fruits)) { print(paste("I love", fruits[i])) } -``` +``` ```{r} states_of_interest <- c("California", "Massachusetts", "New Hampshire", "Washington") @@ -431,9 +465,10 @@ for( state in states_of_interest){ } -``` +``` Instead of printing, you can store the information in a vector + ```{r} states_of_interest <- c("California", "Massachusetts", "New Hampshire", "Washington") male_percentages <- c() @@ -456,8 +491,7 @@ male_percentages ## Nested Loops -What if I want to calculate the population percentage of a race group for all race groups in states of interest? -You could probably use tidyverse functions to do this, but let's try using loops! +What if I want to calculate the population percentage of a race group for all race groups in states of interest? You could probably use tidyverse functions to do this, but let's try using loops! ```{r} @@ -471,27 +505,32 @@ for (state in states_of_interest) { } } ``` + ## Exercises {.unnumbered} ### Exercise 1: Write your own function {.unnumbered} + Write your own function that makes some task of data analysis simpler. Ideally, it would be a function that helps you do either of the previous tasks in fewer lines of code. You can use the three lines of code that was provided in exercise 1 to wrap that into another function too! + ```{r} # Enter yourself -``` +``` ### Exercise 2: Using Loops {.unnumbered} Using a loop, create a crosstab of sex and race for each state in the set "states_of_interest" + ```{r} states_of_interest <- c("California", "Massachusetts", "New Hampshire", "Washington") # Enter yourself -``` +``` ### Exercise 3: Storing information derived within loops in a global dataframe {.unnumbered} Recall the following nested loop + ```{r} states_of_interest <- c("California", "Massachusetts", "New Hampshire", "Washington") for (state in states_of_interest) { @@ -509,4 +548,3 @@ Instead of printing the percentage of each race in each state, create a datafram ```{r} ``` - diff --git a/14_visualization.qmd b/14_visualization.qmd index b62fcb1..67d371b 100644 --- a/14_visualization.qmd +++ b/14_visualization.qmd @@ -1,4 +1,8 @@ -# Visualization^[Module originally written by Shiro Kuriwaki] {#dataviz} +# Visualization {#dataviz} + +::: {.callout .callout-note} +Module originally written by Shiro Kuriwaki. +::: ```{r} #| include: false @@ -13,36 +17,42 @@ library(scales) ### Motivation: The Law of the Census {.unnumbered} -In this module, let's visualize some cross-sectional stats with an actual Census. Then, we'll do an example on time trends with Supreme Court ideal points. +In this module, let's visualize some cross-sectional stats with an actual Census. Then, we'll do an example on time trends with Supreme Court ideal points. + +Why care about the Census? The Census is one of the fundamental acts of a government. See the Law Review article by [Persily (2011)](http://cardozolawreview.com/Joomla1.5/content/32-3/Persily.32-3.pdf), "The Law of the Census."[^14_visualization-1] The Census is government's primary tool for apportionment (allocating seats to districts), appropriations (allocating federal funding), and tracking demographic change. See for example [Hochschild and Powell (2008)](https://dash.harvard.edu/bitstream/handle/1/3153295/hoschschild_racialreorganization.pdf?sequence=2) on how the categorizations of race in the Census during 1850-1930.[^14_visualization-2] Notice also that both of these pieces are not inherently "quantitative" --- the Persily article is a Law Review and the Hochschild and Powell article is on American Historical Development --- but data analysis would be certainly relevant. + +[^14_visualization-1]: [Persily, Nathaniel. 2011. "The Law of the Census: How to Count, What to Count, Whom to Count, and Where to Count Them.”](http://cardozolawreview.com/Joomla1.5/content/32-3/Persily.32-3.pdf). *Cardozo Law Review* 32(3): 755–91. -Why care about the Census? The Census is one of the fundamental acts of a government. See the Law Review article by [Persily (2011)](http://cardozolawreview.com/Joomla1.5/content/32-3/Persily.32-3.pdf), "The Law of the Census."^[[Persily, Nathaniel. 2011. "The Law of the Census: How to Count, What to Count, Whom to Count, and Where to Count Them.”](http://cardozolawreview.com/Joomla1.5/content/32-3/Persily.32-3.pdf). _Cardozo Law Review_ 32(3): 755–91.] The Census is government's primary tool for apportionment (allocating seats to districts), appropriations (allocating federal funding), and tracking demographic change. See for example [Hochschild and Powell (2008)](https://dash.harvard.edu/bitstream/handle/1/3153295/hoschschild_racialreorganization.pdf?sequence=2) on how the categorizations of race in the Census during 1850-1930.^[[Hochschild, Jennifer L., and Brenna Marea Powell. 2008. "Racial Reorganization and the United States Census 1850–1930: Mulattoes, Half-Breeds, Mixed Parentage, Hindoos, and the Mexican Race."](https://dash.harvard.edu/bitstream/handle/1/3153295/hoschschild_racialreorganization.pdf?sequence=2). _Studies in American Political Development_ 22(1): 59–96.] Notice also that both of these pieces are not inherently "quantitative" --- the Persily article is a Law Review and the Hochschild and Powell article is on American Historical Development --- but data analysis would be certainly relevant. +[^14_visualization-2]: [Hochschild, Jennifer L., and Brenna Marea Powell. 2008. "Racial Reorganization and the United States Census 1850–1930: Mulattoes, Half-Breeds, Mixed Parentage, Hindoos, and the Mexican Race."](https://dash.harvard.edu/bitstream/handle/1/3153295/hoschschild_racialreorganization.pdf?sequence=2). *Studies in American Political Development* 22(1): 59–96. -Time series data is a common form of data in social science data, and there is growing methodological work on making causal inferences with time series.^[[Blackwell, Matthew, and Adam Glynn. 2018. "How to Make Causal Inferences with Time-Series Cross-Sectional Data under Selection on Observables."](https://doi.org/10.1017/S0003055418000357) _American Political Science Review_] We will use the the ideological estimates of the Supreme court. +Time series data is a common form of data in social science data, and there is growing methodological work on making causal inferences with time series.[^14_visualization-3] We will use the the ideological estimates of the Supreme court. + +[^14_visualization-3]: [Blackwell, Matthew, and Adam Glynn. 2018. "How to Make Causal Inferences with Time-Series Cross-Sectional Data under Selection on Observables."](https://doi.org/10.1017/S0003055418000357) *American Political Science Review* ### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: -* The R Visualization and Programming primers at -* Reading and handling data -* Matrices and Vectors - * What does `:` mean in R? What about `==`? `,`?, `!=` , `&`, `|`, `%in% ` - * What does `%>%` do? +- The R Visualization and Programming primers at +- Reading and handling data +- Matrices and Vectors + - What does `:` mean in R? What about `==`? `,`?, `!=` , `&`, `|`, `%in%` + - What does `%>%` do? Today we'll cover: -* Visualization -* A bit of data wrangling +- Visualization +- A bit of data wrangling -### Check your understanding {.unnumbered} +### Check your understanding {.unnumbered} -* How do you make a barplot, in base-R and in ggplot? -* How do you add layers to a ggplot? -* How do you change the axes of a ggplot? -* How do you make a histogram? -* How do you make a graph that looks like this? +- How do you make a barplot, in base-R and in ggplot? +- How do you add layers to a ggplot? +- How do you change the axes of a ggplot? +- How do you make a histogram? +- How do you make a graph that looks like this? -![By Randy Schutt - Own work, CC BY-SA 3.0, [Wikimedia.]( https://commons.wikimedia.org/w/index.php?curid=29585342)](images/Martin-Quinn_Wikipedia.png) +![By Randy Schutt - Own work, CC BY-SA 3.0, [Wikimedia.](https://commons.wikimedia.org/w/index.php?curid=29585342)](images/Martin-Quinn_Wikipedia.png) ## Read data @@ -53,48 +63,56 @@ First, the census. Read in a subset of the 2010 Census that we looked at earlier cen10 <- readRDS("data/input/usc2010_001percent.Rds") ``` -The data comes from IPUMS^[[Ruggles, Steven, Katie Genadek, Ronald Goeken, Josiah Grover, and Matthew Sobek. 2015. Integrated Public Use Microdata Series: Version 6.0 dataset](http://doi.org/10.18128/D010.V6.0)], a great source to extract and analyze Census and Census-conducted survey (ACS, CPS) data. +The data comes from IPUMS[^14_visualization-4], a great source to extract and analyze Census and Census-conducted survey (ACS, CPS) data. + +[^14_visualization-4]: [Ruggles, Steven, Katie Genadek, Ronald Goeken, Josiah Grover, and Matthew Sobek. 2015. Integrated Public Use Microdata Series: Version 6.0 dataset](http://doi.org/10.18128/D010.V6.0) ## Counting + How many people are in your sample? ```{r} nrow(cen10) ``` -This and all subsequent tasks involve manipulating and summarizing data, sometimes called "wrangling". As per last time, there are both "base-R" and "tidyverse" approaches. +This and all subsequent tasks involve manipulating and summarizing data, sometimes called "wrangling". As per last time, there are both "base-R" and "tidyverse" approaches. We have already seen several functions from the tidyverse: -* `select` selects columns -* `filter` selects rows based on a logical (boolean) statement -* `slice` selects rows based on the row number -* `arrange` reordered the rows in descending order. +- `select` selects columns +- `filter` selects rows based on a logical (boolean) statement +- `slice` selects rows based on the row number +- `arrange` reordered the rows in descending order. -In this visualization section, we'll make use of the pair of functions `group_by()` and `summarize()`. +In this visualization section, we'll make use of the pair of functions `group_by()` and `summarize()`. ## Tabulating -Summarizing data is the key part of communication; good data viz gets the point across.^[[Kastellec, Jonathan P., and Eduardo L. Leoni. 2007. "Using Graphs Instead of Tables in Political Science."](http://www.princeton.edu/~jkastell/Tables2Graphs/graphs.pdf). _Perspectives on Politics_ 5 (4): 755–71.] Summaries of data come in two forms: tables and figures. +Summarizing data is the key part of communication; good data viz gets the point across.[^14_visualization-5] Summaries of data come in two forms: tables and figures. + +[^14_visualization-5]: [Kastellec, Jonathan P., and Eduardo L. Leoni. 2007. "Using Graphs Instead of Tables in Political Science."](http://www.princeton.edu/~jkastell/Tables2Graphs/graphs.pdf). *Perspectives on Politics* 5 (4): 755–71. Here are two ways to count by group, or to tabulate. In base-R Use the `table` function, that provides how many rows exist for an unique value of the vector (remember `unique` from yesterday?) + ```{r} table(cen10$race) ``` With tidyverse, a quick convenience function is `count`, with the variable to count on included. + ```{r} count(cen10, race) ``` We can check out the arguments of `count` and see that there is a `sort` option. What does this do? + ```{r} count(cen10, race, sort = TRUE) ``` -`count` is a kind of shorthand for `group_by()` and `summarize`. This code would have done the same. +`count` is a kind of shorthand for `group_by()` and `summarize`. This code would have done the same. ```{r} cen10 %>% @@ -102,7 +120,7 @@ cen10 %>% summarize(n = n()) ``` -If you are new to tidyverse, what would you _think_ each row did? Reading the function help page, verify if your intuition was correct. +If you are new to tidyverse, what would you *think* each row did? Reading the function help page, verify if your intuition was correct. where `n()` is a function that counts rows. @@ -112,15 +130,15 @@ Two prevalent ways of making graphing are referred to as "base-R" and "ggplot". ### base R -"Base-R" graphics are graphics that are made with R's default graphics commands. First, let's assign our tabulation to an object, -then put it in the `barplot()` function. +"Base-R" graphics are graphics that are made with R's default graphics commands. First, let's assign our tabulation to an object, then put it in the `barplot()` function. ```{r} barplot(table(cen10$race)) -``` +``` ### ggplot -A popular alternative a `ggplot` graphics, that you were introduced to in the tutorial. `gg` stands for grammar of graphics by Hadley Wickham, and it has a new semantics of explaining graphics in R. Again, first let's set up the data. + +A popular alternative a `ggplot` graphics, that you were introduced to in the tutorial. `gg` stands for grammar of graphics by Hadley Wickham, and it has a new semantics of explaining graphics in R. Again, first let's set up the data. Although the tutorial covered making scatter plots as the first cut, often data requires summaries before they made into graphs. @@ -131,16 +149,15 @@ For this example, let's group and count first like we just did. But assign it to grp_race <- count(cen10, race) ``` -We will now plot this grouped set of numbers. Recall that the `ggplot()` function takes two main arguments, `data` and `aes`. +We will now plot this grouped set of numbers. Recall that the `ggplot()` function takes two main arguments, `data` and `aes`. -1. First enter a single dataframe from which you will draw a plot. -2. Then enter the `aes`, or aesthetics. This defines which variable in the data the plotting functions should take for pre-set dimensions in graphics. The dimensions `x` and `y` are the most important. We will assign `race` and `count` to them, respectively, -3. After you close `ggplot()` .. add __layers__ by the plus sign. A `geom` is a layer of graphical representation, for example `geom_histogram` renders a histogram, `geom_point` renders a scatter plot. For a barplot, we can use `geom_col()` +1. First enter a single dataframe from which you will draw a plot. +2. Then enter the `aes`, or aesthetics. This defines which variable in the data the plotting functions should take for pre-set dimensions in graphics. The dimensions `x` and `y` are the most important. We will assign `race` and `count` to them, respectively, +3. After you close `ggplot()` .. add **layers** by the plus sign. A `geom` is a layer of graphical representation, for example `geom_histogram` renders a histogram, `geom_point` renders a scatter plot. For a barplot, we can use `geom_col()` -What is the right geometry layer to make a barplot? Turns out: +What is the right geometry layer to make a barplot? Turns out: ```{r} -#| fig.fullwidth: true ggplot(data = grp_race, aes(x = race, y = n)) + geom_col() ``` @@ -149,7 +166,6 @@ ggplot(data = grp_race, aes(x = race, y = n)) + geom_col() Adjusting your graphics to make the point clear is an important skill. Here is a base-R example of showing the same numbers but with a different design, in a way that aims to maximize the "data-to-ink ratio". ```{r} -#| fig.fullwidth: true par(oma = c(1, 11, 1, 1)) barplot(sort(table(cen10$race)), # sort numbers horiz = TRUE, # flip @@ -162,6 +178,7 @@ barplot(sort(table(cen10$race)), # sort numbers Notice that we applied the `sort()` function to order the bars in terms of their counts. The default ordering of a categorical variable / factor is alphabetical. Alphabetical ordering is uninformative and almost never the way you should order variables. In ggplot you might do this by: + ```{r} library(forcats) @@ -176,11 +193,13 @@ ggplot(data = grp_race_ordered, aes(x = race, y = n)) + caption = "Source: 2010 U.S. Census sample") ``` -The data ink ratio was popularized by Ed Tufte (originally a political economy scholar who has recently become well known for his data visualization work). See Tufte (2001), _The Visual Display of Quantitative Information_ and his website . For a R and ggplot focused example using social science examples, check out Healy (2018), _Data Visualization: A Practical Introduction_ with a draft at ^[Healy, Kieran. forthcoming. _Data Visualization: A Practical Introduction_. Princeton University Press]. There are a growing number of excellent books on data visualization. +The data ink ratio was popularized by Ed Tufte (originally a political economy scholar who has recently become well known for his data visualization work). See Tufte (2001), *The Visual Display of Quantitative Information* and his website . For a R and ggplot focused example using social science examples, check out Healy (2018), *Data Visualization: A Practical Introduction* with a draft at [^14_visualization-6]. There are a growing number of excellent books on data visualization. + +[^14_visualization-6]: Healy, Kieran. forthcoming. *Data Visualization: A Practical Introduction*. Princeton University Press ## Cross-tabs -Visualizations and Tables each have their strengths. A rule of thumb is that more than a dozen numbers on a table is too much to digest, but less than a dozen is too few for a figure to be worth it. Let's look at a table first. +Visualizations and Tables each have their strengths. A rule of thumb is that more than a dozen numbers on a table is too much to digest, but less than a dozen is too few for a figure to be worth it. Let's look at a table first. A cross-tab is counting with two types of variables, and is a simple and powerful tool to show the relationship between multiple variables. @@ -198,7 +217,7 @@ xtabs(~ state + race, cen10) What if we care about proportions within states, rather than counts? Say we'd like to compare the racial composition of a small state (like Delaware) and a large state (like California). In fact, most tasks of inference is about the unobserved population, not the observed data --- and proportions are estimates of a quantity in the population. -One way to transform a table of counts to a table of proportions is the function `prop.table`. Be careful what you want to take proportions of -- this is set by the `margin` argument. In R, the first margin (`margin = 1`) is _rows_ and the second (`margin = 2`) is _columns_. +One way to transform a table of counts to a table of proportions is the function `prop.table`. Be careful what you want to take proportions of -- this is set by the `margin` argument. In R, the first margin (`margin = 1`) is *rows* and the second (`margin = 2`) is *columns*. ```{r} ptab_race_state <- prop.table(xtab_race_state, margin = 2) @@ -208,7 +227,7 @@ Check out each of these table objects in your console and familiarize yourself w ## Composition Plots -How would you make the same figure with `ggplot()`? First, we want a count for each state $\times$ race combination. So group by those two factors and count how many observations are in each two-way categorization. `group_by()` can take any number of variables, separated by commas. +How would you make the same figure with `ggplot()`? First, we want a count for each state $\times$ race combination. So group by those two factors and count how many observations are in each two-way categorization. `group_by()` can take any number of variables, separated by commas. ```{r} grp_race_state <- cen10 %>% @@ -240,11 +259,13 @@ ggplot(data = grp_race_state, aes(x = state, y = n, fill = race)) + ## Line graphs -Line graphs are useful for plotting time trends. +Line graphs are useful for plotting time trends. The Census does not track individuals over time. So let's take up another example: The U.S. Supreme Court. Take the dataset `justices_court-median.csv`. -This data is adapted from the estimates of Martin and Quinn on their website .^[This exercise inspired from Princeton's R Camp Assignment.] +This data is adapted from the estimates of Martin and Quinn on their website .[^14_visualization-7] + +[^14_visualization-7]: This exercise inspired from Princeton's R Camp Assignment. ```{r} #| message: false @@ -257,30 +278,36 @@ What does the data look like? How do you think it is organized? What does each r justice ``` -As you might have guessed, these data can be shown in a time trend from the range of the `term` variable. As there are only nine justices at any given time and justices have life tenure, there times on the court are staggered. With a common measure of "preference", we can plot time trends of these justices ideal points on the same y-axis scale. +As you might have guessed, these data can be shown in a time trend from the range of the `term` variable. As there are only nine justices at any given time and justices have life tenure, there times on the court are staggered. With a common measure of "preference", we can plot time trends of these justices ideal points on the same y-axis scale. ```{r} ggplot(justice, aes(x = term, y = idealpt)) + geom_line() ``` + Why does the above graph not look like the the put in the beginning? Fix it by adding just one aesthetic to the graph. ```{r} # enter a correction that draws separate lines by group. ``` -If you got the right aesthetic, this seems to "work" off the shelf. But take a moment to see why the code was written as it is and how that maps on to the graphics. What is the `group` aesthetic doing for you? +If you got the right aesthetic, this seems to "work" off the shelf. But take a moment to see why the code was written as it is and how that maps on to the graphics. What is the `group` aesthetic doing for you? + +Now, this graphic already indicates a lot, but let's improve the graphics so people can actually read it. This is left for a Exercise. + +As social scientists, we should also not forget to ask ourselves whether these numerical measures are fit for what we care about, or actually succeeds in measuring what we'd like to measure. The estimation of these "ideal points" is a subfield of political methodology beyond this prefresher. For more reading, skim through the original paper by Martin and Quinn (2002).[^14_visualization-8] Also for a methodological discussion on the difficulty of measuring time series of preferences, check out Bailey (2013).[^14_visualization-9] -Now, this graphic already indicates a lot, but let's improve the graphics so people can actually read it. This is left for a Exercise. +[^14_visualization-8]: [Martin, Andrew D. and Kevin M. Quinn. 2002. "Dynamic Ideal Point Estimation via Markov Chain Monte Carlo for the U.S. Supreme Court, 1953-1999"](http://mqscores.lsa.umich.edu/media/pa02.pdf). *Political Analysis.* 10(2): 134-153. -As social scientists, we should also not forget to ask ourselves whether these numerical measures are fit for what we care about, or actually succeeds in measuring what we'd like to measure. The estimation of these "ideal points" is a subfield of political methodology beyond this prefresher. For more reading, skim through the original paper by Martin and Quinn (2002).^[[Martin, Andrew D. and Kevin M. Quinn. 2002. "Dynamic Ideal Point Estimation via Markov Chain Monte Carlo for the U.S. Supreme Court, 1953-1999"](http://mqscores.lsa.umich.edu/media/pa02.pdf). _Political Analysis._ 10(2): 134-153.] Also for a methodological discussion on the difficulty of measuring time series of preferences, check out Bailey (2013).^[[Bailey, Michael A. 2013. "Is Today’s Court the Most Conservative in Sixty Years? Challenges and Opportunities in Measuring Judicial Preferences." ](https://michaelbailey.georgetown.domains/wp-content/uploads/2018/05/JOP_proofs_June2013.pdf). _Journal of Politics_ 75(3): 821-834] +[^14_visualization-9]: [Bailey, Michael A. 2013. "Is Today’s Court the Most Conservative in Sixty Years? Challenges and Opportunities in Measuring Judicial Preferences."](https://michaelbailey.georgetown.domains/wp-content/uploads/2018/05/JOP_proofs_June2013.pdf). *Journal of Politics* 75(3): 821-834 ## Exercises {.unnumbered} -In the time remaining, try the following exercises. Order doesn't matter. + +In the time remaining, try the following exercises. Order doesn't matter. ### 1: Rural states {.unnumbered} -Make a well-labelled figure that plots the proportion of the state's population (as per the census) that is 65 years or older. Each state should be visualized as a point, rather than a bar, and there should be 51 points, ordered by their value. All labels should be readable. +Make a well-labelled figure that plots the proportion of the state's population (as per the census) that is 65 years or older. Each state should be visualized as a point, rather than a bar, and there should be 51 points, ordered by their value. All labels should be readable. ```{r} # Enter yourself @@ -289,30 +316,30 @@ Make a well-labelled figure that plots the proportion of the state's population ### 2: The swing justice {.unnumbered} -Using the `justices_court-median.csv` dataset and building off of the plot that was given, make an improved plot by implementing as many of the following changes (which hopefully improves the graph): +Using the `justices_court-median.csv` dataset and building off of the plot that was given, make an improved plot by implementing as many of the following changes (which hopefully improves the graph): -* Label axes -* Use a black-white background. -* Change the breaks of the x-axis to print numbers for every decade, not just every two decades. -* Plots each line in translucent gray, so the overlapping lines can be visualized clearly. (Hint: in ggplot the `alpha` argument controls the degree of transparency) -* Limit the scale of the y-axis to [-5, 5] so that the outlier justice in the 60s is trimmed and the rest of the data can be seen more easily (also, who is that justice?) -* Plot the ideal point of the justice who holds the "median" ideal point in a given term. To distinguish this with the others, plot this line separately in a very light red _below_ the individual justice's lines. -* Highlight the trend-line of only the nine justices who are _currently_ sitting on SCOTUS. Make sure this is clearer than the other past justices. -* Add the current nine justice's names to the right of the endpoint of the 2016 figure, alongside their ideal point. -* Make sure the text labels do not overlap with each other for readability using the `ggrepel` package. -* Extend the x-axis label to about 2020 so the text labels of justices are to the right of the trend-lines. -* Add a caption to your text describing the data briefly, as well as any features relevant for the reader (such as the median line and the trimming of the y-axis) +- Label axes +- Use a black-white background. +- Change the breaks of the x-axis to print numbers for every decade, not just every two decades. +- Plots each line in translucent gray, so the overlapping lines can be visualized clearly. (Hint: in ggplot the `alpha` argument controls the degree of transparency) +- Limit the scale of the y-axis to \[-5, 5\] so that the outlier justice in the 60s is trimmed and the rest of the data can be seen more easily (also, who is that justice?) +- Plot the ideal point of the justice who holds the "median" ideal point in a given term. To distinguish this with the others, plot this line separately in a very light red *below* the individual justice's lines. +- Highlight the trend-line of only the nine justices who are *currently* sitting on SCOTUS. Make sure this is clearer than the other past justices. +- Add the current nine justice's names to the right of the endpoint of the 2016 figure, alongside their ideal point. +- Make sure the text labels do not overlap with each other for readability using the `ggrepel` package. +- Extend the x-axis label to about 2020 so the text labels of justices are to the right of the trend-lines. +- Add a caption to your text describing the data briefly, as well as any features relevant for the reader (such as the median line and the trimming of the y-axis) ```{r} # Enter yourself -``` +``` ### 3: Don't sort by the alphabet {.unnumbered} -The Figure we made that shows racial composition by state has one notable shortcoming: it orders the states alphabetically, which is not particularly useful if you want see an overall pattern, without having particular states in mind. +The Figure we made that shows racial composition by state has one notable shortcoming: it orders the states alphabetically, which is not particularly useful if you want see an overall pattern, without having particular states in mind. -Find a way to modify the figures so that the states are ordered by the _proportion_ of White residents in the sample. +Find a way to modify the figures so that the states are ordered by the *proportion* of White residents in the sample. ```{r} # Enter yourself @@ -320,4 +347,4 @@ Find a way to modify the figures so that the states are ordered by the _proporti ### 4 What to show and how to show it {.unnumbered} -As a student of politics our goal is not necessarily to make pretty pictures, but rather make pictures that tell us something about politics, government, or society. If you could augment either the census dataset or the justices dataset in some way, what would be an substantively significant thing to show as a graphic? +As a student of politics our goal is not necessarily to make pretty pictures, but rather make pictures that tell us something about politics, government, or society. If you could augment either the census dataset or the justices dataset in some way, what would be an substantively significant thing to show as a graphic? diff --git a/15_project-dempeace.qmd b/15_project-dempeace.qmd index 29c53c5..c435c95 100644 --- a/15_project-dempeace.qmd +++ b/15_project-dempeace.qmd @@ -1,24 +1,29 @@ +# Joins and Merges, Wide and Long {#dempeace} -# Joins and Merges, Wide and Long^[Module originally written by Shiro Kuriwaki, Connor Jerzak, and Yon Soo Park] {#dempeace} +::: {.callout .callout-note} +Module originally written by Shiro Kuriwaki, Connor Jerzak, and Yon Soo Park. +::: ### Motivation {.unnumbered} -The "Democratic Peace" is one of the most widely discussed propositions in political science, covering the fields of International Relations and Comparative Politics, with insights to domestic politics of democracies (e.g. American Politics). The one-sentence idea is that democracies do not fight with each other. There have been much theoretical debate -- for example in earlier work, [Oneal and Russet (1999)](https://blackboard.angelo.edu/bbcswebdav/institution/LFA/CSS/Course%20Material/SEC6302/Readings/Lesson_3/Oneal-Russett.pdf) argue that the democratic peace is not due to the hegemony of strong democracies like the U.S. and attempt to distinguish between realist and what they call Kantian propositions (e.g. democratic governance, international organizations)^[[The Kantian Peace: The Pacific Benefits of Democracy, Interdependence, and International Organizations, 1885-1992. _World Politics_ 52(1):1-37](https://blackboard.angelo.edu/bbcswebdav/institution/LFA/CSS/Course%20Material/SEC6302/Readings/Lesson_3/Oneal-Russett.pdf)]. +The "Democratic Peace" is one of the most widely discussed propositions in political science, covering the fields of International Relations and Comparative Politics, with insights to domestic politics of democracies (e.g. American Politics). The one-sentence idea is that democracies do not fight with each other. There have been much theoretical debate -- for example in earlier work, [Oneal and Russet (1999)](https://blackboard.angelo.edu/bbcswebdav/institution/LFA/CSS/Course%20Material/SEC6302/Readings/Lesson_3/Oneal-Russett.pdf) argue that the democratic peace is not due to the hegemony of strong democracies like the U.S. and attempt to distinguish between realist and what they call Kantian propositions (e.g. democratic governance, international organizations)[^15_project-dempeace-1]. -An empirical demonstration of the democratic peace is also a good example of a __Time Series Cross Sectional__ (or panel) dataset, where the same units (in this case countries) are observed repeatedly for multiple time periods. Experience in assembling and analyzing a TSCS dataset will prepare you for any future research in this area. +[^15_project-dempeace-1]: [The Kantian Peace: The Pacific Benefits of Democracy, Interdependence, and International Organizations, 1885-1992. *World Politics* 52(1):1-37](https://blackboard.angelo.edu/bbcswebdav/institution/LFA/CSS/Course%20Material/SEC6302/Readings/Lesson_3/Oneal-Russett.pdf) + +An empirical demonstration of the democratic peace is also a good example of a **Time Series Cross Sectional** (or panel) dataset, where the same units (in this case countries) are observed repeatedly for multiple time periods. Experience in assembling and analyzing a TSCS dataset will prepare you for any future research in this area. ## Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: -* R basic programming -* Counting. -* Visualization. -* Objects and Classes. -* Matrix algebra in R -* Functions. +- R basic programming +- Counting. +- Visualization. +- Objects and Classes. +- Matrix algebra in R +- Functions. -Today you will work on your own, but feel free to ask a fellow classmate nearby or the instructor. The objective for this session is to get more experience using R, but in the process (a) test a prominent theory in the political science literature and (b) explore related ideas of interest to you. +Today you will work on your own, but feel free to ask a fellow classmate nearby or the instructor. The objective for this session is to get more experience using R, but in the process (a) test a prominent theory in the political science literature and (b) explore related ideas of interest to you. ## Setting up @@ -35,20 +40,22 @@ library(ggplot2) First start a directory for this project. This can be done manually or through RStudio's Project feature(`File > New Project...`) -Directories is the computer science / programming name for folders. While advice about how to structure your working directories might strike you as petty, we believe that starting from some well-tested guides will go a long way in improving the quality and efficiency of your work. +Directories is the computer science / programming name for folders. While advice about how to structure your working directories might strike you as petty, we believe that starting from some well-tested guides will go a long way in improving the quality and efficiency of your work. -Chapter 4 of Gentzkow and Shapiro's memo, [Code and Data for the Social Scientist](https://web.stanford.edu/~gentzkow/research/CodeAndData.pdf)] provides a good template. +Chapter 4 of Gentzkow and Shapiro's memo, [Code and Data for the Social Scientist](https://web.stanford.edu/~gentzkow/research/CodeAndData.pdf)\] provides a good template. ## Data Sources -Most projects you do will start with downloading data from elsewhere. For this task, you'll probably want to track down and download the following: -* __Correlates of war dataset (COW):__ Find and download the Militarized Interstate Disputes (MIDs) data from the Correlates of War website: . Or a dyad-version on dataverse: -* __PRIO Data on Armed Conflict:__ Find and download the Uppsala Conflict Data Program (UCDP) and PRIO dyad-year data on armed conflict() or this link to to the flat csv file (). -* __Polity:__ The Polity data can be downloaded from their website (). Look for the newest version of the time series that has the widest coverage. +Most projects you do will start with downloading data from elsewhere. For this task, you'll probably want to track down and download the following: + +- **Correlates of war dataset (COW):** Find and download the Militarized Interstate Disputes (MIDs) data from the Correlates of War website: . Or a dyad-version on dataverse: +- **PRIO Data on Armed Conflict:** Find and download the Uppsala Conflict Data Program (UCDP) and PRIO dyad-year data on armed conflict() or this link to to the flat csv file (). +- **Polity:** The Polity data can be downloaded from their website (). Look for the newest version of the time series that has the widest coverage. ## Example with 2 Datasets -Let's read in a sample dataset. +Let's read in a sample dataset. + ```{r} #| warning: false #| message: false @@ -57,6 +64,7 @@ mid <- read_csv("data/input/sample_mid.csv") ``` What does `polity` look like? + ```{r} unique(polity$country) ggplot(polity, aes(x = year, y = polity2)) + @@ -67,13 +75,14 @@ head(polity) ``` MID is a dataset that captures a `dispute` for a given country and year. + ```{r} mid ``` ## Loops -Notice that in the `mid` data, we have a start of a dispute vs. an end of a dispute.In order to combine this into the `polity` data, we want a way to give each of the interval years a row. +Notice that in the `mid` data, we have a start of a dispute vs. an end of a dispute.In order to combine this into the `polity` data, we want a way to give each of the interval years a row. There are many ways to do this, but one is a loop. We go through one row at a time, and then for each we make a new dataset. that has `year` as a sequence of each year. A lengthy loop like this is typically slow, and you'd want to recast the task so you can do things with functions. But, a loop is a good place to start. @@ -93,11 +102,13 @@ head(mid_year_by_year) ``` ## Merging + We want to combine these two datasets by merging. Base-R has a function called `merge`. `dplyr` has several types of `joins` (the same thing). Those names are based on SQL syntax. ![](images/dplyr-joins.png) Here we can do a `left_join` matching rows from `mid` to `polity`. We want to keep the rows in `polity` that do not match in `mid`, and label them as non-disputes. + ```{r} p_m <- left_join(polity, distinct(mid_year_by_year), @@ -113,6 +124,7 @@ p_m$dispute[is.na(p_m$dispute)] <- 0 ``` Reshape the dataset long to wide + ```{r} p_m_wide <- pivot_wider(p_m, id_cols = c(scode, ccode, country), @@ -128,7 +140,7 @@ Try building a panel that would be useful in answering the Democratic Peace Ques ### Task 1: Data Input and Standardization {.unnumbered} -Often, files we need are saved in the `.xls` or `xlsx` format. It is possible to read these files directly into `R`, but experience suggests that this process is slower than converting them first to `.csv` format and reading them in as `.csv` files. +Often, files we need are saved in the `.xls` or `xlsx` format. It is possible to read these files directly into `R`, but experience suggests that this process is slower than converting them first to `.csv` format and reading them in as `.csv` files. `readxl`/`readr`/`haven` packages() is constantly expanding to capture more file types. In day 1, we used the package `readxl`, using the `read_excel()` function. @@ -137,13 +149,14 @@ Often, files we need are saved in the `.xls` or `xlsx` format. It is possible to ``` ### Task 2: Data Merging {.unnumbered} -We will use data to test a version of the Democratic Peace Thesis (DPS). Democracies are said to go to war less because the leaders who wage wars are accountable to voters who have to bear the costs of war. Are democracies less likely to engage in militarized interstate disputes? -To start, let's download and merge some data. +We will use data to test a version of the Democratic Peace Thesis (DPS). Democracies are said to go to war less because the leaders who wage wars are accountable to voters who have to bear the costs of war. Are democracies less likely to engage in militarized interstate disputes? -* Load in the Militarized Interstate Dispute (MID) files. Militarized interstate disputes are hostile action between two formally recognized states. Examples of this would be threats to use force, threats to declare war, beginning war, fortifying a border with troops, and so on. -* Find a way to __merge__ the Polity IV dataset and the MID data. This process can be a bit tricky. -* An _advanced_ version of this task would be to download the dyadic form of the data and try merging that with polity. +To start, let's download and merge some data. + +- Load in the Militarized Interstate Dispute (MID) files. Militarized interstate disputes are hostile action between two formally recognized states. Examples of this would be threats to use force, threats to declare war, beginning war, fortifying a border with troops, and so on. +- Find a way to **merge** the Polity IV dataset and the MID data. This process can be a bit tricky. +- An *advanced* version of this task would be to download the dyadic form of the data and try merging that with polity. ```{r} @@ -152,11 +165,10 @@ To start, let's download and merge some data. ### Task 3: Tabulations and Visualization {.unnumbered} 1. Calculate the mean Polity2 score by year. Plot the result. Use graphical indicators of your choosing to show where key events fall in this timeline (such as 1914, 1929, 1939, 1989, 2008). Speculate on why the behavior from 1800 to 1920 seems to be qualitatively different than behavior afterwards. -2. Do the same but only among state-years that were invovled in a MID. Plot this line together with your results from 1. -3. Do the same but only among state years that were _not_ involved in a MID. -4. Arrive at a tentative conclusion for how well the Democratic Peace argument seems to hold up in this dataset. Visualize this conclusion. +2. Do the same but only among state-years that were invovled in a MID. Plot this line together with your results from 1. +3. Do the same but only among state years that were *not* involved in a MID. +4. Arrive at a tentative conclusion for how well the Democratic Peace argument seems to hold up in this dataset. Visualize this conclusion. ```{r} ``` - diff --git a/17_non-wysiwyg.quarto_ipynb b/17_non-wysiwyg.quarto_ipynb deleted file mode 100644 index ad04ee2..0000000 --- a/17_non-wysiwyg.quarto_ipynb +++ /dev/null @@ -1,333 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# LaTeX and markdown^[Module originally written by Shiro Kuriwaki] {#nonwysiwyg}\n", - "\n", - "\n", - "### Where are we? Where are we headed? {-}\n", - "\n", - "Up till now, you should have covered:\n", - "\n", - "* Statistical Programming in `R`\n", - "\n", - "This is only the beginning of `R` -- programming is like learning a language, so learn more as we use it. And yet `R` is of likely not the only programming language you will want to use. While we cannot introduce everything, we'll pick out a few that we think are particularly helpful.\n", - "\n", - "Here will cover\n", - "\n", - "* Markdown\n", - "* LaTeX (and BibTeX)\n", - "\n", - "as examples of a non-WYSIWYG editor\n", - "\n", - "and the next chapter (you can read it without reading this LaTeX chapter) covers\n", - "\n", - "* command-line\n", - "* git\n", - "\n", - "command-line are a basic set of tools that you may have to use from time to time. It also clarifies what more complicated programs are doing. Markdown is an example of compiling a plain text file. LaTeX is a typesetting program and git is a version control program -- both are useful for non-quantitative work as well.\n", - "\n", - "\n", - "\n", - "### Check your understanding {-}\n", - "\n", - "Check if you have an idea of how you might code the following tasks:\n", - "\n", - "* What does \"WYSIWYG\" stand for? How would a non-WYSIWYG format text?\n", - "* How do you start a header in markdown?\n", - "* What are some \"plain text\" editors?\n", - "* How do you start a document in `.tex`?\n", - "* How do you start a environment in `.tex`?\n", - "* How do you insert a figure in `.tex`?\n", - "* How do you reference a figure in `.tex`?\n", - "* What is a `.bib` file? \n", - "* Say you came across a interesting journal article. How would you want to maintain this reference so that you can refer to its citation in all your subsequent papers?\n", - "\n", - "\n", - "\n", - "\n", - "## Motivation\n", - "\n", - "Statistical programming is a fast-moving field. The beta version of `R` was released in 2000, `ggplot2` was released on 2005, and `RStudio` started around 2010. Of course, some programming technologies are quite \"old\": (`C` in 1969, `C++` around 1989, `TeX` in 1978, `Linux` in 1991, Mac OS in 1984). But it is easy to feel you are falling behind in the recent developments of programming. Today we will do a **brief** and rough overview of some fundamental and new tools other than `R`, with the general aim of having you break out of your comfort zone so you won't be shut out from learning these tools in the future.\n", - "\n", - "\n", - "## Markdown\n", - "\n", - "Markdown is the text we have been using throughout this course! At its core markdown is just plain text. Plain text does not have any formatting embedded in it. Instead, the formatting is coded up as text. Markdown is _not_ a WYSIWYG (What you see is what you get) text editor like Microsoft Word or Google Docs. This will mean that you need to explicitly code for `bold{text}` rather than hitting Command+B and making your text look __bold__ on your own computer. \n", - "\n", - "Markdown is known as a \"light-weight\" editor, which means that it is relatively easy to write code that will compile. It is quick and easy and satisfies most presentation purposes; you might want to try `LaTeX` for more involved papers.\n", - "\n", - "### markdown commands\n", - "For italic and bold, use either the asterisks or the underlines, \n", - "\n", - "```\n", - "*italic* **bold**\n", - "_italic_ __bold__\n", - "```\n", - "\n", - "And for headers use the hash symbols, \n", - "```\n", - "# Main Header\n", - "## Sub-headers\n", - "```\n", - "\n", - "### your own markdown \n", - "\n", - "RStudio makes it easy to compile your very first markdown file by giving you templates. Got to `New > R Markdown`, pick a document and click Ok. This will give you a skeleton of a document you can compile -- or \"knit\".\n", - "\n", - "Rmd is actually a slight modification of real markdown. It is a type of file that R reads and turns into a proper `md` file. Then, it uses a document-conversion called pandoc to compile your `md` into documents like PDF or HTML.\n", - "\n", - "![How Rmds become PDFs or HTMLs](images/RMarkdownFlow.png)\n", - "\n", - "### Quarto\n", - "\n", - "R Markdown (`.Rmd`) files have long been the go-to for reproducible writing workflows for R users.\n", - "In 2022, [Posit, PBC](https://posit.co/), who created R Markdown announced a new generation of markdown extensions, with Quarto.\n", - "Quarto (`.qmd`) files are a variation on R Markdown which allows for including R, python, Observable, Julia, and more within a document.\n", - "Quarto is largely compatible with older `.Rmd` files, just by changing the extension.\n", - "As such, you can integrate LaTeX and markdown seamlessly.\n", - "\n", - "Some benefits of using Quarto include:\n", - "\n", - "* [ease of customization with template partials](https://quarto.org/docs/journals/templates.html#template-partials)\n", - "* [journal submission templates for many journals](https://quarto.org/docs/extensions/listing-journals.html)\n", - "* [dozen of output types](https://quarto.org/docs/reference/)\n", - "* [the ability to make websites interacting only with Quarto](https://quarto.org/docs/websites/)\n", - "\n", - "\n", - "### A note on plain-text editors\n", - "\n", - "Multiple software exist where you can edit plain-text (roughly speaking, text that is not WYSIWYG). \n", - "\n", - "* [RStudio](https://posit.co/products/open-source/rstudio/) (especially for R-related links)\n", - "* TeXMaker, TeXShop (especially for TeX)\n", - "* [emacs](https://www.gnu.org/software/emacs/), aquamacs (general)\n", - "* [vim](http://www.vim.org/download.php) (general)\n", - "* [Sublime Text](https://www.sublimetext.com) (general)\n", - "\n", - "Each has their own keyboard shortcuts and special features. You can browse a couple and see which one(s) you like.\n", - "\n", - "Since June 2021, RStudio has offered a visual editor which tries to bridge the gap between plain-text and WYSIWYG.\n", - "While writing, it transforms plain markdown, RMarkdown, or Quarto documents into a \"WYSISWYM\" version, What You See Is What You Mean.\n", - "Formatting choices, like bold or italicized text are shown as **bold** or *italicized* text, rather than as intermediate markdown.\n", - "Lists, enumerations, and images are shown inline, rather than the code that includes them.\n", - "This is not a final form though, as styling still occurs when rendering the final document.\n", - "\n", - "## LaTeX\n", - "\n", - "LaTeX is a typesetting program. You'd engage with LaTeX much like you engage with your `R` code. You will interact with LaTeX in a text editor, and will writing code which will be interpreted by the LaTeX compiler and which will finally be parsed to form your final PDF.\n", - "\n", - "### compile online\n", - "\n", - "1. Go to \n", - "2. Scroll down and go to \"CREATE A NEW PAPER\" if you don't have an account.\n", - "3. Let's discuss the default template.\n", - "4. Make a new document, and set it as your main document. Then type in the Minimal Working Example (MWE):\n", - "\n", - "\n", - "\n", - "\n", - "```{bash, eval = FALSE}\n", - "\\documentclass{article}\n", - "\\begin{document}\n", - "Hello World\n", - "\\end{document}\n", - "```\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "### compile your first LaTeX document locally\n", - "LaTeX is a very stable system, and few changes to it have been made since the 1990s. The main benefit: better control over how your papers will look; better methods for writing equations or making tables; overall pleasing aesthetic. \n", - "\n", - "1. Open a plain text editor. Then type in the MWE\n", - "\n", - "\n", - "\n", - "\n", - "```{bash, eval = FALSE}\n", - "\\documentclass{article}\n", - "\\begin{document}\n", - "Hello World\n", - "\\end{document}\n", - "```\n", - "\n", - "\n", - "\n", - "\n", - "2. Save this as `hello_world.tex`. Make sure you get the file extension right. \n", - "3. Open this in your \"LaTeX\" editor. This can be `TeXMaker`, `Aqumacs`, etc..\n", - "4. Go through the click/dropdown interface and click compile.\n", - "\n", - "\n", - "### main LaTeX commands\n", - "LaTeX can cover most of your typesetting needs, to clean equations and intricate diagrams. \n", - "\n", - "Some main commands you'll be using are below, and a very concise cheat sheet here: \n", - "\n", - "Most involved features require that you begin a specific \"environment\" for that feature, clearly demarcating them by the notation `\\begin{figure}` and then `\\end{figure}`, e.g. in the case of figures.\n", - "\n", - "```\n", - "\\begin{figure}\n", - "\\includegraphics{histogram.pdf}\n", - "\\end{figure}\n", - "```\n", - "where `histogram.pdf` is a path to one of your files. \n", - "\n", - "Notice that each line starts with a backslash `\\` -- in LaTeX this is the symbol to run a command.\n", - "\n", - "The following syntax at the endpoints are shorthand for math equations.\n", - "```\n", - "\$$\\int x^2 dx\$$\n", - "```\n", - "these compile math symbols: $\\displaystyle \\int x^2 dx.$^[Enclosing with `$$` instead of `\$$` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.]\n", - "\n", - "\n", - "The `align` environment is useful to align your multi-line math, for example. \n", - "```\n", - "\\begin{align}\n", - "P(A \\mid B) &= \\frac{P(A \\cap B)}{P(B)}\\\\\n", - "&= \\frac{P(B \\mid A)P(A)}{P(B)}\n", - "\\end{align}\n", - "```\n", - "\\begin{align}\n", - "P(A \\mid B) &= \\frac{P(A \\cap B)}{P(B)}\\\\\n", - "&= \\frac{P(B \\mid A)P(A)}{P(B)}\n", - "\\end{align}\n", - "\n", - "Regression tables should be outputted as `.tex` files with packages like `xtable` and `stargazer`, and then called into LaTeX by `\\input{regression_table.tex}` where `regression_table.tex` is the path to your regression output.\n", - "\n", - "Figures and equations should be labelled with the tag (e.g. `label{tab:regression}` so that you can refer to them later with their tag `Table \\ref{tab:regression}`, instead of hard-coding `Table 2`).\n", - "\n", - "For some LaTeX commands you might need to load a separate package that someone else has written. Do this in your preamble (i.e. before `\\begin{document}`):\n", - "```\n", - "\\usepackage[options]{package}\n", - "```\n", - "where `package` is the name of the package and `options` are options specific to the package. \n", - "\n", - "\n", - "### Further Guides {-}\n", - "\n", - "For a more comprehensive listing of LaTeX commands, Mayya Komisarchik has a great tutorial set of folders: \n", - "\n", - "There is a version of LaTeX called Beamer, which is a popular way of making a slideshow. Slides in markdown is also a competitor. The language of Beamer is the same as LaTeX but has some special functions for slides.\n", - "\n", - "\n", - "## BibTeX\n", - "\n", - "BibTeX is a reference system for bibliographical tests. We have a `.bib` file separately on our computer. This is also a plain text file, but it encodes bibliographical resources with special syntax so that a program can rearrange parts accordingly for different citation systems. \n", - "\n", - "### what is a `.bib` file?\n", - "For example, here is the Nunn and Wantchekon article entry in `.bib` form.\n", - "\n", - "```{}\n", - "@article{nunn2011slave,\n", - " title={The Slave Trade and the Origins of Mistrust in Africa},\n", - " author={Nunn, Nathan and Wantchekon, Leonard},\n", - " journal={American Economic Review},\n", - " volume={101},\n", - " number={7},\n", - " pages={3221--3252},\n", - " year={2011}\n", - "}\n", - "```\n", - "\n", - "The first entry, `nunn2011slave`, is \"pick your favorite\" -- pick your own name for your reference system. The other slots in this `@article` entry are entries that refer to specific bibliographical text.\n", - "\n", - "### what does LaTeX do with .bib files?\n", - "Now, in LaTeX, if you type \n", - "\n", - " \\textcite{nunn2011slave} argue that current variation in the trust among citizens of African countries has historical roots in the European slave trade in the 1600s.\n", - " \n", - "as part of your text, then when the `.tex` file is compiled the PDF shows something like \n", - "\n", - "![](images/biblatex_inline.png)\n", - "\n", - "in whatever citation style (APSA, APA, Chicago) you pre-specified! \n", - "\n", - "\n", - "Also at the end of your paper you will have a bibliography with entries ordered and formatted in the appropriate citation.\n", - "\n", - "![](images/biblatex_bibliography.png)\n", - "\n", - "This is a much less frustrating way of keeping track of your references -- no need to hand-edit formatting the bibliography to conform to citation rules (which biblatex already knows) and no need to update your bibliography as you add and drop references (biblatex will only show entries that are used in the main text).\n", - "\n", - "\n", - "### stocking up on your .bib files\n", - "You should keep your own `.bib` file that has all your bibliographical resources. Storing entries is cheap (does not take much memory), so it is fine to keep all your references in one place (but you'll want to make a new one for collaborative projects where multiple people will compile a `.tex` file).\n", - "\n", - "For example, Gary's BibTeX file is here: \n", - "\n", - "Citation management software (Mendeley or Zotero) automatically generates .bib entries from your library of PDFs for you, provided you have the bibliography attributes right. \n", - "\n", - "## Exercise {-}\n", - "\n", - "Create a LaTeX document for a hypothetical research paper on your laptop and, once you've verified it compiles into a PDF, come show it to either one of the instructors. \n", - "\n", - "You can also use overleaf if you have preference for a cloud-based system. But don't swallow the built-in templates without understanding or testing them.\n", - "\n", - "Each student will have slightly different substantive interests, so we won't impose much of a standard. But at a minimum, the LaTeX document should have:\n", - "\n", - "* A title, author, date, and abstract\n", - "* Sections\n", - "* Italics and boldface\n", - "* A figure with a caption and in-text reference to it. \n", - "\n", - "\n", - "Depending on your subfield or interests, try to implement some of the following:\n", - "\n", - "* A bibliographical reference drawing from a separate `.bib` file\n", - "* A table\n", - "* A math expression\n", - "* A different font\n", - "* Different page margins\n", - "* Different line spacing\n", - "\n", - "\n", - "\n", - "\n", - "## Concluding the Prefresher {-}\n", - "\n", - "Math may not be the perfect tool for every aspiring political scientist, but hopefully it was useful background to have at the least:\n", - "\n", - "

Historians think this totally meaningless and nonsensical statistic is the product of an early-modern epistemological shift in which numbers and quantifiable data became revered above other kinds of knowledge as the most useful and credible form of truth https://t.co/wVFyAQGxEv

— Gina Anne Tam 譚吉娜 (@DGTam86) May 29, 2018
\n", - "\n", - "\n", - "\n", - "\n", - "But we should be aware that too much slant towards math and programming can miss the point:\n", - "\n", - "

To be clear, PhD training in Econ (first year) is often a disaster-- like how to prove the Central Limit Theorem (the LeBron James of Statistics) with polar-cooardinates. This is mostly a way to demoralize actual economists and select a bunch of unimaginative math jocks.

— Amitabh Chandra (@amitabhchandra2) August 14, 2018
\n", - "\n", - "\n", - "\n", - "\n", - "Keep on learning, trying new techniques to improve your work, and learn from others! \n", - "\n", - "

What #rstats tricks did it take you way too long to learn? One of mine is using readRDS and saveRDS instead of repeatedly loading from CSV

— Emily Riederer (@EmilyRiederer) August 19, 2017
\n", - "\n", - "\n", - "\n", - "### Your Feedback Matters {-}\n", - "\n", - "_Please tell us how we can improve the Prefresher_: The Prefresher is a work in progress, with material mainly driven by graduate students. Please tell us how we should change (or not change) each of its elements:\n", - "\n", - "https://harvard.az1.qualtrics.com/jfe/form/SV_esbzN8ZFAOPTqiV" - ], - "id": "86efe863" - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/_quarto.yml b/_quarto.yml index e7342c2..1a996be 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -18,6 +18,13 @@ book: - 05_optimization.qmd - 06_probability.qmd - 07_linear-algebra.qmd + - part: Programming + chapters: + - 11_data-handling_counting.qmd + - 12_matricies-manipulation.qmd + - 13_functions_obj_loops.qmd + - 14_visualization.qmd + - 15_project-dempeace.qmd delete_merged_file: true language: diff --git a/prefresher.Rproj b/prefresher.Rproj index 827cca1..588c680 100644 --- a/prefresher.Rproj +++ b/prefresher.Rproj @@ -13,3 +13,5 @@ RnwWeave: Sweave LaTeX: pdfLaTeX BuildType: Website + +MarkdownCanonical: Yes From 76d36a2f40bc544cce6e628e379b25b42233229d Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 22:38:15 -0400 Subject: [PATCH 18/34] rest of programming section --- 16_simulation.qmd | 149 +++++++++++++++++------------ 17_non-wysiwyg.qmd | 227 +++++++++++++++++++++++++++------------------ 18_text.qmd | 202 +++++++++++++++++++++------------------- _quarto.yml | 3 + 4 files changed, 337 insertions(+), 244 deletions(-) diff --git a/16_simulation.qmd b/16_simulation.qmd index 9b79edf..7791cfe 100644 --- a/16_simulation.qmd +++ b/16_simulation.qmd @@ -1,50 +1,61 @@ -# Simulation^[Module originally written by Connor Jerzak and Shiro Kuriwaki] {#simulation} +# Simulation {#simulation} ```{r} #| include: false library(dplyr) ``` +::: {.callout .callout-note} +Module originally written by Connor Jerzak and Shiro Kuriwaki. +::: + ### Motivation: Simulation as an Analytical Tool {.unnumbered} -An increasing amount of political science contributions now include a simulation. +An increasing amount of political science contributions now include a simulation. + +- [Axelrod (1977)](http://www-personal.umich.edu/~axe/research/Dissemination.pdf) demonstrated via simulation how atomized individuals evolve to be grouped in similar clusters or countries, a model of culture.[^16_simulation-1] +- [Chen and Rodden (2013)](http://www-personal.umich.edu/~jowei/florida.pdf) argued in a 2013 article that the vote-seat inequality in U.S. elections that is often attributed to intentional partisan gerrymandering can actually attributed to simply the reality of "human geography" -- Democratic voters tend to be concentrated in smaller area. Put another way, no feasible form of gerrymandering could spread out Democratic voters in such a way to equalize their vote-seat translation effectiveness. After demonstrating the empirical pattern of human geography, they advance their key claim by simulating thousands of redistricting plans and record the vote-seat ratio.[^16_simulation-2] +- [Gary King, James Honaker, and multiple other authors](https://gking.harvard.edu/files/abs/evil-abs.shtml) propose a way to analyze missing data with a method of multiple imputation, which uses a lot of simulation from a researcher's observed dataset.[^16_simulation-3] (Software: Amelia[^16_simulation-4]) + +[^16_simulation-1]: [Axelrod, Robert. 1997. "The Dissemination of Culture." *Journal of Conflict Resolution* 41(2): 203–26.](http://www-personal.umich.edu/~axe/research/Dissemination.pdf) + +[^16_simulation-2]: [Chen, Jowei, and Jonathan Rodden. "Unintentional Gerrymandering: Political Geography and Electoral Bias in Legislatures. *Quarterly Journal of Political Science*, 8:239-269"](http://www-personal.umich.edu/~jowei/florida.pdf) -* [Axelrod (1977)](http://www-personal.umich.edu/~axe/research/Dissemination.pdf) demonstrated via simulation how atomized individuals evolve to be grouped in similar clusters or countries, a model of culture.^[[Axelrod, Robert. 1997. "The Dissemination of Culture." _Journal of Conflict Resolution_ 41(2): 203–26.](http://www-personal.umich.edu/~axe/research/Dissemination.pdf)] -* [Chen and Rodden (2013)](http://www-personal.umich.edu/~jowei/florida.pdf) argued in a 2013 article that the vote-seat inequality in U.S. elections that is often attributed to intentional partisan gerrymandering can actually attributed to simply the reality of "human geography" -- Democratic voters tend to be concentrated in smaller area. Put another way, no feasible form of gerrymandering could spread out Democratic voters in such a way to equalize their vote-seat translation effectiveness. After demonstrating the empirical pattern of human geography, they advance their key claim by simulating thousands of redistricting plans and record the vote-seat ratio.^[[Chen, Jowei, and Jonathan Rodden. "Unintentional Gerrymandering: Political Geography and Electoral Bias in Legislatures. _Quarterly Journal of Political Science_, 8:239-269"](http://www-personal.umich.edu/~jowei/florida.pdf)] -* [Gary King, James Honaker, and multiple other authors](https://gking.harvard.edu/files/abs/evil-abs.shtml) propose a way to analyze missing data with a method of multiple imputation, which uses a lot of simulation from a researcher's observed dataset.^[[King, Gary, et al. "Analyzing Incomplete Political Science Data: An Alternative Algorithm for Multiple Imputation". _American Political Science Review_, 95: 49-69.](https://gking.harvard.edu/files/abs/evil-abs.shtml)] (Software: Amelia^[[James Honaker, Gary King, Matthew Blackwell (2011). Amelia II: A Program for Missing Data. Journal of - Statistical Software, 45(7), 1-47.](http://www.jstatsoft.org/v45/i07/)]) +[^16_simulation-3]: [King, Gary, et al. "Analyzing Incomplete Political Science Data: An Alternative Algorithm for Multiple Imputation". *American Political Science Review*, 95: 49-69.](https://gking.harvard.edu/files/abs/evil-abs.shtml) -Statistical methods also incorporate simulation: +[^16_simulation-4]: [James Honaker, Gary King, Matthew Blackwell (2011). Amelia II: A Program for Missing Data. Journal of Statistical Software, 45(7), 1-47.](http://www.jstatsoft.org/v45/i07/) -* The bootstrap: a statistical method for estimating uncertainty around some parameter by re-sampling observations. -* Bagging: a method for improving machine learning predictions by re-sampling observations, storing the estimate across many re-samples, and averaging these estimates to form the final estimate. A variance reduction technique. -* Statistical reasoning: if you are trying to understand a quantitative problem, a wonderful first-step to understand the problem better is to simulate it! The analytical solution is often very hard (or impossible), but the simulation is often much easier :-) +Statistical methods also incorporate simulation: + +- The bootstrap: a statistical method for estimating uncertainty around some parameter by re-sampling observations. +- Bagging: a method for improving machine learning predictions by re-sampling observations, storing the estimate across many re-samples, and averaging these estimates to form the final estimate. A variance reduction technique. +- Statistical reasoning: if you are trying to understand a quantitative problem, a wonderful first-step to understand the problem better is to simulate it! The analytical solution is often very hard (or impossible), but the simulation is often much easier :-) ### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: -* `R` basics -* Visualization -* Matrices and vectors -* Functions, objects, loops -* Joining real data +- `R` basics +- Visualization +- Matrices and vectors +- Functions, objects, loops +- Joining real data In this module, we will start to work with generating data within R, from thin air, as it were. Doing simulation also strengthens your understanding of Probability (Section \@ref{probability}). ### Check your Understanding {.unnumbered} -* What does the `sample()` function do? -* What does `runif()` stand for? -* What is a `seed`? -* What is a Monte Carlo? +- What does the `sample()` function do? +- What does `runif()` stand for? +- What is a `seed`? +- What is a Monte Carlo? Check if you have an idea of how you might code the following tasks: -* Simulate 100 rolls of a die -* Simulate one random ordering of 25 numbers -* Simulate 100 values of white noise (uniform random variables) -* Generate a "bootstrap" sample of an existing dataset +- Simulate 100 rolls of a die +- Simulate one random ordering of 25 numbers +- Simulate 100 values of white noise (uniform random variables) +- Generate a "bootstrap" sample of an existing dataset We're going to learn about this today! @@ -52,26 +63,30 @@ We're going to learn about this today! ## The `sample()` function -The core functions for coding up stochastic data revolves around several key functions, so we will simply review them here. +The core functions for coding up stochastic data revolves around several key functions, so we will simply review them here. Suppose you have a vector of values `x` and from it you want to randomly sample a sample of length `size`. For this, use the `sample` function + ```{r} sample(x = 1:10, size = 5) ``` There are two subtypes of sampling -- with and without replacement. -1. Sampling without replacement (`replace = FALSE`) means once an element of `x` is chosen, it will not be considered again: +1. Sampling without replacement (`replace = FALSE`) means once an element of `x` is chosen, it will not be considered again: + ```{r} sample(x = 1:10, size = 10, replace = FALSE) ## no number appears more than once ``` -2. Sampling with replacement (`replace = TRUE`) means that even if an element of `x` is chosen, it is put back in the pool and may be chosen again. +2. Sampling with replacement (`replace = TRUE`) means that even if an element of `x` is chosen, it is put back in the pool and may be chosen again. + ```{r} sample(x = 1:10, size = 10, replace = TRUE) ## any number can appear more than once ``` It follows then that you cannot sample without replacement a sample that is larger than the pool. + ```{r} #| error: true sample(x = 1:10, size = 100, replace = FALSE) @@ -80,24 +95,27 @@ sample(x = 1:10, size = 100, replace = FALSE) So far, every element in `x` has had an equal probability of being chosen. In some application, we want a sampling scheme where some elements are more likely to be chosen than others. The argument `prob` handles this. For example, this simulates 20 fair coin tosses (each outcome is equally likely to happen) + ```{r} sample(c("Head", "Tail"), size = 20, prob = c(0.5, 0.5), replace = TRUE) ``` But this simulates 20 biased coin tosses, where say the probability of Tails is 4 times more likely than the number of Heads + ```{r} sample(c("Head", "Tail"), size = 20, prob = c(0.2, 0.8), replace = TRUE) ``` ### Sampling rows from a dataframe -In tidyverse, there is a convenience function to sample rows randomly: `sample_n()` and `sample_frac()`. +In tidyverse, there is a convenience function to sample rows randomly: `sample_n()` and `sample_frac()`. For example, load the dataset on cars, `mtcars`, which has 32 observations. ```{r} mtcars ``` + sample_n picks a user-specified number of rows from the dataset: ```{r} @@ -112,26 +130,30 @@ sample_frac(mtcars, 0.10) As a side-note, these functions have very practical uses for any type of data analysis: -* Inspecting your dataset: using `head()` all the same time and looking over the first few rows might lead you to ignore any issues that end up in the bottom for whatever reason. -* Testing your analysis with a small sample: If running analyses on a dataset takes more than a handful of seconds, change your dataset upstream to a fraction of the size so the rest of the code runs in less than a second. Once verifying your analysis code runs, then re-do it with your full dataset (by simply removing the `sample_n` / `sample_frac` line of code in the beginning). While three seconds may not sound like much, they accumulate and eat up time. +- Inspecting your dataset: using `head()` all the same time and looking over the first few rows might lead you to ignore any issues that end up in the bottom for whatever reason. +- Testing your analysis with a small sample: If running analyses on a dataset takes more than a handful of seconds, change your dataset upstream to a fraction of the size so the rest of the code runs in less than a second. Once verifying your analysis code runs, then re-do it with your full dataset (by simply removing the `sample_n` / `sample_frac` line of code in the beginning). While three seconds may not sound like much, they accumulate and eat up time. ## Random numbers from specific distributions ### `rbinom()` {.unnumbered} -`rbinom` builds upon `sample` as a tool to help you answer the question -- what is the _total number of successes_ I would get if I sampled a binary (Bernoulli) result from a test with `size` number of trials each, with a event-wise probability of `prob`. The first argument `n` asks me how many such numbers I want. -For example, I want to know how many Heads I would get if I flipped a fair coin 100 times. +`rbinom` builds upon `sample` as a tool to help you answer the question -- what is the *total number of successes* I would get if I sampled a binary (Bernoulli) result from a test with `size` number of trials each, with a event-wise probability of `prob`. The first argument `n` asks me how many such numbers I want. + +For example, I want to know how many Heads I would get if I flipped a fair coin 100 times. + ```{r} rbinom(n = 1, size = 100, prob = 0.5) ``` Now imagine this I wanted to do this experiment 10 times, which would require I flip the coin 10 x 100 = 1000 times! Helpfully, we can do this in one line + ```{r} rbinom(n = 10, size = 100, prob = 0.5) ``` ### `runif()` {.unnumbered} -`runif` also simulates a stochastic scheme where each event has equal probability of getting chosen like `sample`, but is a continuous rather than discrete system. We will cover this more in the next math module. + +`runif` also simulates a stochastic scheme where each event has equal probability of getting chosen like `sample`, but is a continuous rather than discrete system. We will cover this more in the next math module. The intuition to emphasize here is that one can generate potentially infinite amounts (size `n`) of noise that is a essentially random @@ -160,11 +182,11 @@ hist(from_rnorm) ## r, p, and d -Each distribution can do more than generate random numbers (the prefix `r`). We can compute the cumulative probability by the function `pbinom()`, `punif()`, and `pnorm()`. Also the density -- the value of the PDF -- by `dbinom()`, `dunif()` and `dnorm()`. +Each distribution can do more than generate random numbers (the prefix `r`). We can compute the cumulative probability by the function `pbinom()`, `punif()`, and `pnorm()`. Also the density -- the value of the PDF -- by `dbinom()`, `dunif()` and `dnorm()`. ## `set.seed()` -`R` doesn't have the ability to generate truly random numbers! Random numbers are actually very hard to generate. (Think: flipping a coin --> can be perfectly predicted if I know wind speed, the angle the coin is flipped, etc.). Some people use random noise in the atmosphere or random behavior in quantum systems to generate "truly" (?) random numbers. Conversely, R uses deterministic algorithms which take as an input a "seed" and which then perform a series of operations to generate a sequence of random-seeming numbers (that is, numbers whose sequence is sufficiently hard to predict). +`R` doesn't have the ability to generate truly random numbers! Random numbers are actually very hard to generate. (Think: flipping a coin --\> can be perfectly predicted if I know wind speed, the angle the coin is flipped, etc.). Some people use random noise in the atmosphere or random behavior in quantum systems to generate "truly" (?) random numbers. Conversely, R uses deterministic algorithms which take as an input a "seed" and which then perform a series of operations to generate a sequence of random-seeming numbers (that is, numbers whose sequence is sufficiently hard to predict). Let's think about this another way. Sampling is a stochastic process, so every time you run `sample()` or `runif()` you are bound to get a different output (because different random seeds are used). This is intentional in some cases but you might want to avoid it in others. For example, you might want to diagnose a coding discrepancy by setting the random number generator to give the same number each time. To do this, use the function `set.seed()`. @@ -175,7 +197,7 @@ set.seed(02138) runif(n = 10) ``` -The random number generator should give you the exact same sequence of numbers if you precede the function by the same seed, +The random number generator should give you the exact same sequence of numbers if you precede the function by the same seed, ```{r} set.seed(02138) @@ -186,7 +208,9 @@ runif(n = 10) ### Census Sampling {.unnumbered} -What can we learn from surveys of populations, and how wrong do we get if our sampling is biased?^[This example is inspired from [Meng, Xiao-Li (2018). Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election. _Annals of Applied Statistics_ 12:2, 685–726. doi:10.1214/18-AOAS1161SF.](https://statistics.fas.harvard.edu/files/statistics-2/files/statistical_paradises_and_paradoxes.pdf)] Suppose we want to estimate the proportion of U.S. residents who are non-white (`race != "White"`). In reality, we do not have any population dataset to utilize and so we _only see the sample survey_. Here, however, to understand how sampling works, let's conveniently use the Census extract in some cases and pretend we didn't in others. +What can we learn from surveys of populations, and how wrong do we get if our sampling is biased?[^16_simulation-5] Suppose we want to estimate the proportion of U.S. residents who are non-white (`race != "White"`). In reality, we do not have any population dataset to utilize and so we *only see the sample survey*. Here, however, to understand how sampling works, let's conveniently use the Census extract in some cases and pretend we didn't in others. + +[^16_simulation-5]: This example is inspired from [Meng, Xiao-Li (2018). Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election. *Annals of Applied Statistics* 12:2, 685–726. doi:10.1214/18-AOAS1161SF.](https://statistics.fas.harvard.edu/files/statistics-2/files/statistical_paradises_and_paradoxes.pdf) (a) First, load `usc2010_001percent.csv` into your R session. After loading the `library(tidyverse)`, browse it. Although this is only a 0.01 percent extract, treat this as your population for pedagogical purposes. What is the population proportion of non-White residents? @@ -194,15 +218,15 @@ What can we learn from surveys of populations, and how wrong do we get if our sa ``` -(b) Setting a seed to `1669482`, sample 100 respondents from this sample. What is the proportion of non-White residents in this _particular_ sample? By how many percentage points are you off from (what we labelled as) the true proportion? +(b) Setting a seed to `1669482`, sample 100 respondents from this sample. What is the proportion of non-White residents in this *particular* sample? By how many percentage points are you off from (what we labelled as) the true proportion? ```{r} ``` -(c) Now imagine what you did above was one survey. What would we get if we did 20 surveys? +(c) Now imagine what you did above was one survey. What would we get if we did 20 surveys? -To simulate this, write a loop that does the same exercise 20 times, each time computing a sample proportion. Use the same seed at the top, but be careful to position the `set.seed` function such that it generates the same sequence of 20 samples, rather than 20 of the same sample. +To simulate this, write a loop that does the same exercise 20 times, each time computing a sample proportion. Use the same seed at the top, but be careful to position the `set.seed` function such that it generates the same sequence of 20 samples, rather than 20 of the same sample. Try doing this with a `for` loop and storing your sample proportions in a new length-20 vector. (Suggestion: make an empty vector first as a container). After running the loop, show a histogram of the 20 values. Also what is the average of the 20 sample estimates? @@ -210,7 +234,7 @@ Try doing this with a `for` loop and storing your sample proportions in a new le ``` -(d) Now, to make things more real, let's introduce some response bias. The goal here is not to correct response bias but to induce it and see how it affects our estimates. Suppose that non-White residents are 10 percent less likely to respond to enter your survey than White respondents. This is plausible if you think that the Census is from 2010 but you are polling in 2018, and racial minorities are more geographically mobile than Whites. Repeat the same exercise in (c) by modeling this behavior. +(d) Now, to make things more real, let's introduce some response bias. The goal here is not to correct response bias but to induce it and see how it affects our estimates. Suppose that non-White residents are 10 percent less likely to respond to enter your survey than White respondents. This is plausible if you think that the Census is from 2010 but you are polling in 2018, and racial minorities are more geographically mobile than Whites. Repeat the same exercise in (c) by modeling this behavior. You can do this by creating a variable, e.g. `propensity`, that is 0.9 for non-Whites and 1 otherwise. Then, you can refer to it in the propensity argument. @@ -218,13 +242,13 @@ You can do this by creating a variable, e.g. `propensity`, that is 0.9 for non-W ``` -(e) Finally, we want to see if more data ("Big Data") will improve our estimates. Using the same unequal response rates framework as (d), repeat the same exercise but instead of each poll collecting 100 responses, we collect 10,000. +(e) Finally, we want to see if more data ("Big Data") will improve our estimates. Using the same unequal response rates framework as (d), repeat the same exercise but instead of each poll collecting 100 responses, we collect 10,000. ```{r} ``` -(f) Optional - visualize your 2 pairs of 20 estimates, with a bar showing the "correct" population average. +(f) Optional - visualize your 2 pairs of 20 estimates, with a bar showing the "correct" population average. ```{r} @@ -232,67 +256,74 @@ You can do this by creating a variable, e.g. `propensity`, that is 0.9 for non-W ### Conditional Proportions {.unnumbered} -This example is not on simulation, but is meant to reinforce some of the probability discussion from math lecture. +This example is not on simulation, but is meant to reinforce some of the probability discussion from math lecture. Read in the Upshot Siena poll from Fall 2016, `data/input/upshot-siena-polls.csv`. -In addition to some standard demographic questions, we will focus on one called `vt_pres_2` in the csv. This is a two-way presidential vote question, asking respondents who they plan to vote for President if the election were held today -- Donald Trump, the Republican, or Hilary Clinton, the Democrat, with options for Other candidates as well. For this problem, use the two-way vote question rather than the 4-way vote question. +In addition to some standard demographic questions, we will focus on one called `vt_pres_2` in the csv. This is a two-way presidential vote question, asking respondents who they plan to vote for President if the election were held today -- Donald Trump, the Republican, or Hilary Clinton, the Democrat, with options for Other candidates as well. For this problem, use the two-way vote question rather than the 4-way vote question. -(a) Drop the the respondents who answered the November poll (i.e. those for which `poll == "November"`). We do this in order to ignore this November population in all subsequent parts of this question because they were not asked the Presidential vote question. +(a) Drop the the respondents who answered the November poll (i.e. those for which `poll == "November"`). We do this in order to ignore this November population in all subsequent parts of this question because they were not asked the Presidential vote question. ```{r} ``` -(b) Using the dataset after the procedure in (a), find the proportion of _poll respondents_ (those who are in the sample) who support Donald Trump. +(b) Using the dataset after the procedure in (a), find the proportion of *poll respondents* (those who are in the sample) who support Donald Trump. ```{r} ``` -(c) Among those who supported Donald Trump, what proportion of them has a Bachelor's degree or higher (i.e. have a Bachelor's, Graduate, or other Professional Degree)? +(c) Among those who supported Donald Trump, what proportion of them has a Bachelor's degree or higher (i.e. have a Bachelor's, Graduate, or other Professional Degree)? -(d) Among those who did not support Donald Trump (i.e. including supporters of Hilary Clinton, another candidate, or those who refused to answer the question), what proportion of them has a Bachelor's degree or higher? +(d) Among those who did not support Donald Trump (i.e. including supporters of Hilary Clinton, another candidate, or those who refused to answer the question), what proportion of them has a Bachelor's degree or higher? -(e) Express the numbers in the previous parts as probabilities of specified events. Define your own symbols: For example, we can let $T$ be the event that a randomly selected respondent in the poll supports Donald Trump, then the proportion in part (b) is the probability $P(T).$ +(e) Express the numbers in the previous parts as probabilities of specified events. Define your own symbols: For example, we can let $T$ be the event that a randomly selected respondent in the poll supports Donald Trump, then the proportion in part (b) is the probability $P(T).$ -(f) Suppose we randomly sampled a person who participated in the survey and found that he/she had a Bachelor's degree or higher. Given this evidence, what is the probability that the same person supports Donald Trump? Use Bayes Rule and show your work -- that is, do not use data or R to compute the quantity directly. Then, verify this is the case via R. +(f) Suppose we randomly sampled a person who participated in the survey and found that he/she had a Bachelor's degree or higher. Given this evidence, what is the probability that the same person supports Donald Trump? Use Bayes Rule and show your work -- that is, do not use data or R to compute the quantity directly. Then, verify this is the case via R. ### The Birthday problem {.unnumbered} -Write code that will answer the well-known birthday problem via simulation.^[This exercise draws from Imai (2017)] +Write code that will answer the well-known birthday problem via simulation.[^16_simulation-6] + +[^16_simulation-6]: This exercise draws from Imai (2017) The problem is fairly simple: Suppose $k$ people gather together in a room. What is the probability at least two people share the same birthday? -To simplify reality a bit, assume that (1) there are no leap years, and so there are always 365 days in a year, and (2) a given individual's birthday is randomly assigned and independent from each other. +To simplify reality a bit, assume that (1) there are no leap years, and so there are always 365 days in a year, and (2) a given individual's birthday is randomly assigned and independent from each other. + +*Step 1*: Set `k` to a concrete number. Pick a number from 1 to 365 randomly, `k` times to simulate birthdays (would this be with replacement or without?). -_Step 1_: Set `k` to a concrete number. Pick a number from 1 to 365 randomly, `k` times to simulate birthdays (would this be with replacement or without?). ```{r} # Your code ``` -_Step 2_: Write a line (or two) of code that gives a `TRUE` or `FALSE` statement of whether or not at least two people share the same birth date. +*Step 2*: Write a line (or two) of code that gives a `TRUE` or `FALSE` statement of whether or not at least two people share the same birth date. + ```{r} # Your code ``` -_Step 3_: The above steps will generate a `TRUE` or `FALSE` answer for your event of interest, but only for one realization of an event in the sample space. In order to estimate the _probability_ of your event happening, we need a "stochastic", as opposed to "deterministic", method. To do this, write a loop that does Steps 1 and 2 repeatedly for many times, call that number of times `sims`. For each of `sims` iteration, your code should give you a `TRUE` or `FALSE` answer. Code up a way to store these estimates. +*Step 3*: The above steps will generate a `TRUE` or `FALSE` answer for your event of interest, but only for one realization of an event in the sample space. In order to estimate the *probability* of your event happening, we need a "stochastic", as opposed to "deterministic", method. To do this, write a loop that does Steps 1 and 2 repeatedly for many times, call that number of times `sims`. For each of `sims` iteration, your code should give you a `TRUE` or `FALSE` answer. Code up a way to store these estimates. + ```{r} # Your code ``` -_Step 4_: Finally, generalize the function further by letting `k` be a user-defined number. You have now created a _Monte Carlo simulation_! +*Step 4*: Finally, generalize the function further by letting `k` be a user-defined number. You have now created a *Monte Carlo simulation*! + ```{r} # Your code ``` -_Step 5_: Generate a table or plot that shows how the probability of sharing a birthday changes by `k` (fixing `sims` at a large number like `1000`). Also generate a similar plot that shows how the probability of sharing a birthday changes by `sims` (fixing `k` at some arbitrary number like `10`). +*Step 5*: Generate a table or plot that shows how the probability of sharing a birthday changes by `k` (fixing `sims` at a large number like `1000`). Also generate a similar plot that shows how the probability of sharing a birthday changes by `sims` (fixing `k` at some arbitrary number like `10`). + ```{r} # Your code ``` -_Extra credit_: Give an "analytical" answer to this problem, that is an answer through deriving the mathematical expressions of the probability. +*Extra credit*: Give an "analytical" answer to this problem, that is an answer through deriving the mathematical expressions of the probability. + ```{r} # Your equations ``` - diff --git a/17_non-wysiwyg.qmd b/17_non-wysiwyg.qmd index 6d7c80d..4e1effd 100644 --- a/17_non-wysiwyg.qmd +++ b/17_non-wysiwyg.qmd @@ -1,24 +1,28 @@ -# LaTeX and markdown^[Module originally written by Shiro Kuriwaki] {#nonwysiwyg} +# LaTeX and markdown {#nonwysiwyg} + +::: {.callout .callout-note} +Module originally written by Shiro Kuriwaki. +::: ### Where are we? Where are we headed? {.unnumbered} Up till now, you should have covered: -* Statistical Programming in `R` +- Statistical Programming in `R` This is only the beginning of `R` -- programming is like learning a language, so learn more as we use it. And yet `R` is of likely not the only programming language you will want to use. While we cannot introduce everything, we'll pick out a few that we think are particularly helpful. Here will cover -* Markdown -* LaTeX (and BibTeX) +- Markdown +- LaTeX (and BibTeX) as examples of a non-WYSIWYG editor and the next chapter (you can read it without reading this LaTeX chapter) covers -* command-line -* git +- command-line +- git command-line are a basic set of tools that you may have to use from time to time. It also clarifies what more complicated programs are doing. Markdown is an example of compiling a plain text file. LaTeX is a typesetting program and git is a version control program -- both are useful for non-quantitative work as well. @@ -26,41 +30,43 @@ command-line are a basic set of tools that you may have to use from time to time Check if you have an idea of how you might code the following tasks: -* What does "WYSIWYG" stand for? How would a non-WYSIWYG format text? -* How do you start a header in markdown? -* What are some "plain text" editors? -* How do you start a document in `.tex`? -* How do you start a environment in `.tex`? -* How do you insert a figure in `.tex`? -* How do you reference a figure in `.tex`? -* What is a `.bib` file? -* Say you came across a interesting journal article. How would you want to maintain this reference so that you can refer to its citation in all your subsequent papers? +- What does "WYSIWYG" stand for? How would a non-WYSIWYG format text? +- How do you start a header in markdown? +- What are some "plain text" editors? +- How do you start a document in `.tex`? +- How do you start a environment in `.tex`? +- How do you insert a figure in `.tex`? +- How do you reference a figure in `.tex`? +- What is a `.bib` file? +- Say you came across a interesting journal article. How would you want to maintain this reference so that you can refer to its citation in all your subsequent papers? ## Motivation -Statistical programming is a fast-moving field. The beta version of `R` was released in 2000, `ggplot2` was released on 2005, and `RStudio` started around 2010. Of course, some programming technologies are quite "old": (`C` in 1969, `C++` around 1989, `TeX` in 1978, `Linux` in 1991, Mac OS in 1984). But it is easy to feel you are falling behind in the recent developments of programming. Today we will do a **brief** and rough overview of some fundamental and new tools other than `R`, with the general aim of having you break out of your comfort zone so you won't be shut out from learning these tools in the future. +Statistical programming is a fast-moving field. The beta version of `R` was released in 2000, `ggplot2` was released on 2005, and `RStudio` started around 2010. Of course, some programming technologies are quite "old": (`C` in 1969, `C++` around 1989, `TeX` in 1978, `Linux` in 1991, Mac OS in 1984). But it is easy to feel you are falling behind in the recent developments of programming. Today we will do a **brief** and rough overview of some fundamental and new tools other than `R`, with the general aim of having you break out of your comfort zone so you won't be shut out from learning these tools in the future. ## Markdown -Markdown is the text we have been using throughout this course! At its core markdown is just plain text. Plain text does not have any formatting embedded in it. Instead, the formatting is coded up as text. Markdown is _not_ a WYSIWYG (What you see is what you get) text editor like Microsoft Word or Google Docs. This will mean that you need to explicitly code for `bold{text}` rather than hitting Command+B and making your text look __bold__ on your own computer. +Markdown is the text we have been using throughout this course! At its core markdown is just plain text. Plain text does not have any formatting embedded in it. Instead, the formatting is coded up as text. Markdown is *not* a WYSIWYG (What you see is what you get) text editor like Microsoft Word or Google Docs. This will mean that you need to explicitly code for `bold{text}` rather than hitting Command+B and making your text look **bold** on your own computer. Markdown is known as a "light-weight" editor, which means that it is relatively easy to write code that will compile. It is quick and easy and satisfies most presentation purposes; you might want to try `LaTeX` for more involved papers. ### markdown commands -For italic and bold, use either the asterisks or the underlines, -``` +For italic and bold, use either the asterisks or the underlines, + +``` *italic* **bold** _italic_ __bold__ ``` -And for headers use the hash symbols, -``` +And for headers use the hash symbols, + +``` # Main Header ## Sub-headers ``` -### your own markdown +### your own markdown RStudio makes it easy to compile your very first markdown file by giving you templates. Got to `New > R Markdown`, pick a document and click Ok. This will give you a skeleton of a document you can compile -- or "knit". @@ -70,36 +76,28 @@ Rmd is actually a slight modification of real markdown. It is a type of file tha ### Quarto -R Markdown (`.Rmd`) files have long been the go-to for reproducible writing workflows for R users. -In 2022, [Posit, PBC](https://posit.co/), who created R Markdown announced a new generation of markdown extensions, with Quarto. -Quarto (`.qmd`) files are a variation on R Markdown which allows for including R, python, Observable, Julia, and more within a document. -Quarto is largely compatible with older `.Rmd` files, just by changing the extension. -As such, you can integrate LaTeX and markdown seamlessly. +R Markdown (`.Rmd`) files have long been the go-to for reproducible writing workflows for R users. In 2022, [Posit, PBC](https://posit.co/), who created R Markdown announced a new generation of markdown extensions, with Quarto. Quarto (`.qmd`) files are a variation on R Markdown which allows for including R, python, Observable, Julia, and more within a document. Quarto is largely compatible with older `.Rmd` files, just by changing the extension. As such, you can integrate LaTeX and markdown seamlessly. Some benefits of using Quarto include: -* [ease of customization with template partials](https://quarto.org/docs/journals/templates.html#template-partials) -* [journal submission templates for many journals](https://quarto.org/docs/extensions/listing-journals.html) -* [dozen of output types](https://quarto.org/docs/reference/) -* [the ability to make websites interacting only with Quarto](https://quarto.org/docs/websites/) +- [ease of customization with template partials](https://quarto.org/docs/journals/templates.html#template-partials) +- [journal submission templates for many journals](https://quarto.org/docs/extensions/listing-journals.html) +- [dozen of output types](https://quarto.org/docs/reference/) +- [the ability to make websites interacting only with Quarto](https://quarto.org/docs/websites/) ### A note on plain-text editors -Multiple software exist where you can edit plain-text (roughly speaking, text that is not WYSIWYG). +Multiple software exist where you can edit plain-text (roughly speaking, text that is not WYSIWYG). -* [RStudio](https://posit.co/products/open-source/rstudio/) (especially for R-related links) -* TeXMaker, TeXShop (especially for TeX) -* [emacs](https://www.gnu.org/software/emacs/), aquamacs (general) -* [vim](http://www.vim.org/download.php) (general) -* [Sublime Text](https://www.sublimetext.com) (general) +- [RStudio](https://posit.co/products/open-source/rstudio/) (especially for R-related links) +- TeXMaker, TeXShop (especially for TeX) +- [emacs](https://www.gnu.org/software/emacs/), aquamacs (general) +- [vim](http://www.vim.org/download.php) (general) +- [Sublime Text](https://www.sublimetext.com) (general) Each has their own keyboard shortcuts and special features. You can browse a couple and see which one(s) you like. -Since June 2021, RStudio has offered a visual editor which tries to bridge the gap between plain-text and WYSIWYG. -While writing, it transforms plain markdown, RMarkdown, or Quarto documents into a "WYSISWYM" version, What You See Is What You Mean. -Formatting choices, like bold or italicized text are shown as **bold** or *italicized* text, rather than as intermediate markdown. -Lists, enumerations, and images are shown inline, rather than the code that includes them. -This is not a final form though, as styling still occurs when rendering the final document. +Since June 2021, RStudio has offered a visual editor which tries to bridge the gap between plain-text and WYSIWYG. While writing, it transforms plain markdown, RMarkdown, or Quarto documents into a "WYSISWYM" version, What You See Is What You Mean. Formatting choices, like bold or italicized text are shown as **bold** or *italicized* text, rather than as intermediate markdown. Lists, enumerations, and images are shown inline, rather than the code that includes them. This is not a final form though, as styling still occurs when rendering the final document. ## LaTeX @@ -107,12 +105,12 @@ LaTeX is a typesetting program. You'd engage with LaTeX much like you engage wit ### compile online -1. Go to -2. Scroll down and go to "CREATE A NEW PAPER" if you don't have an account. -3. Let's discuss the default template. -4. Make a new document, and set it as your main document. Then type in the Minimal Working Example (MWE): +1. Go to +2. Scroll down and go to "CREATE A NEW PAPER" if you don't have an account. +3. Let's discuss the default template. +4. Make a new document, and set it as your main document. Then type in the Minimal Working Example (MWE): -```tex +``` tex \documentclass{article} \begin{document} Hello World @@ -120,64 +118,76 @@ Hello World ``` ### compile your first LaTeX document locally -LaTeX is a very stable system, and few changes to it have been made since the 1990s. The main benefit: better control over how your papers will look; better methods for writing equations or making tables; overall pleasing aesthetic. -1. Open a plain text editor. Then type in the MWE +LaTeX is a very stable system, and few changes to it have been made since the 1990s. The main benefit: better control over how your papers will look; better methods for writing equations or making tables; overall pleasing aesthetic. + +1. Open a plain text editor. Then type in the MWE -```tex +``` tex \documentclass{article} \begin{document} Hello World \end{document} ``` -2. Save this as `hello_world.tex`. Make sure you get the file extension right. -3. Open this in your "LaTeX" editor. This can be `TeXMaker`, `Aqumacs`, etc.. -4. Go through the click/dropdown interface and click compile. +2. Save this as `hello_world.tex`. Make sure you get the file extension right. +3. Open this in your "LaTeX" editor. This can be `TeXMaker`, `Aqumacs`, etc.. +4. Go through the click/dropdown interface and click compile. ### main LaTeX commands -LaTeX can cover most of your typesetting needs, to clean equations and intricate diagrams. + +LaTeX can cover most of your typesetting needs, to clean equations and intricate diagrams. Some main commands you'll be using are below, and a very concise cheat sheet here: Most involved features require that you begin a specific "environment" for that feature, clearly demarcating them by the notation `\begin{figure}` and then `\end{figure}`, e.g. in the case of figures. -``` +``` \begin{figure} \includegraphics{histogram.pdf} \end{figure} ``` -where `histogram.pdf` is a path to one of your files. + +where `histogram.pdf` is a path to one of your files. Notice that each line starts with a backslash `\` -- in LaTeX this is the symbol to run a command. The following syntax at the endpoints are shorthand for math equations. -``` + +``` $$\int x^2 dx$$ ``` -these compile math symbols: $\displaystyle \int x^2 dx.$^[Enclosing with `$$` instead of `$$` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.] -The `align` environment is useful to align your multi-line math, for example. -``` +these compile math symbols: $\displaystyle \int x^2 dx.$[^17_non-wysiwyg-1] + +[^17_non-wysiwyg-1]: Enclosing with `$$` instead of `$$` also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility. + +The `align` environment is useful to align your multi-line math, for example. + +``` \begin{align} P(A \mid B) &= \frac{P(A \cap B)}{P(B)}\\ &= \frac{P(B \mid A)P(A)}{P(B)} \end{align} ``` + +```{=tex} \begin{align} P(A \mid B) &= \frac{P(A \cap B)}{P(B)}\\ &= \frac{P(B \mid A)P(A)}{P(B)} \end{align} - +``` Regression tables should be outputted as `.tex` files with packages like `xtable` and `stargazer`, and then called into LaTeX by `\input{regression_table.tex}` where `regression_table.tex` is the path to your regression output. Figures and equations should be labelled with the tag (e.g. `label{tab:regression}` so that you can refer to them later with their tag `Table \ref{tab:regression}`, instead of hard-coding `Table 2`). For some LaTeX commands you might need to load a separate package that someone else has written. Do this in your preamble (i.e. before `\begin{document}`): -``` + +``` \usepackage[options]{package} ``` -where `package` is the name of the package and `options` are options specific to the package. + +where `package` is the name of the package and `options` are options specific to the package. ### Further Guides {.unnumbered} @@ -187,12 +197,13 @@ There is a version of LaTeX called Beamer, which is a popular way of making a sl ## BibTeX -BibTeX is a reference system for bibliographical tests. We have a `.bib` file separately on our computer. This is also a plain text file, but it encodes bibliographical resources with special syntax so that a program can rearrange parts accordingly for different citation systems. +BibTeX is a reference system for bibliographical tests. We have a `.bib` file separately on our computer. This is also a plain text file, but it encodes bibliographical resources with special syntax so that a program can rearrange parts accordingly for different citation systems. ### what is a `.bib` file? + For example, here is the Nunn and Wantchekon article entry in `.bib` form. -```tex +``` tex @article{nunn2011slave, title={The Slave Trade and the Origins of Mistrust in Africa}, author={Nunn, Nathan and Wantchekon, Leonard}, @@ -207,15 +218,18 @@ For example, here is the Nunn and Wantchekon article entry in `.bib` form. The first entry, `nunn2011slave`, is "pick your favorite" -- pick your own name for your reference system. The other slots in this `@article` entry are entries that refer to specific bibliographical text. ### what does LaTeX do with .bib files? -Now, in LaTeX, if you type - \textcite{nunn2011slave} argue that current variation in the trust among citizens of African countries has historical roots in the European slave trade in the 1600s. +Now, in LaTeX, if you type + +``` + \textcite{nunn2011slave} argue that current variation in the trust among citizens of African countries has historical roots in the European slave trade in the 1600s. +``` -as part of your text, then when the `.tex` file is compiled the PDF shows something like +as part of your text, then when the `.tex` file is compiled the PDF shows something like ![](images/biblatex_inline.png) -in whatever citation style (APSA, APA, Chicago) you pre-specified! +in whatever citation style (APSA, APA, Chicago) you pre-specified! Also at the end of your paper you will have a bibliography with entries ordered and formatted in the appropriate citation. @@ -224,53 +238,90 @@ Also at the end of your paper you will have a bibliography with entries ordered This is a much less frustrating way of keeping track of your references -- no need to hand-edit formatting the bibliography to conform to citation rules (which biblatex already knows) and no need to update your bibliography as you add and drop references (biblatex will only show entries that are used in the main text). ### stocking up on your .bib files + You should keep your own `.bib` file that has all your bibliographical resources. Storing entries is cheap (does not take much memory), so it is fine to keep all your references in one place (but you'll want to make a new one for collaborative projects where multiple people will compile a `.tex` file). For example, Gary's BibTeX file is here: -Citation management software (Mendeley or Zotero) automatically generates .bib entries from your library of PDFs for you, provided you have the bibliography attributes right. +Citation management software (Mendeley or Zotero) automatically generates .bib entries from your library of PDFs for you, provided you have the bibliography attributes right. ## Exercise {.unnumbered} -Create a LaTeX document for a hypothetical research paper on your laptop and, once you've verified it compiles into a PDF, come show it to either one of the instructors. +Create a LaTeX document for a hypothetical research paper on your laptop and, once you've verified it compiles into a PDF, come show it to either one of the instructors. You can also use overleaf if you have preference for a cloud-based system. But don't swallow the built-in templates without understanding or testing them. Each student will have slightly different substantive interests, so we won't impose much of a standard. But at a minimum, the LaTeX document should have: -* A title, author, date, and abstract -* Sections -* Italics and boldface -* A figure with a caption and in-text reference to it. +- A title, author, date, and abstract +- Sections +- Italics and boldface +- A figure with a caption and in-text reference to it. Depending on your subfield or interests, try to implement some of the following: -* A bibliographical reference drawing from a separate `.bib` file -* A table -* A math expression -* A different font -* Different page margins -* Different line spacing +- A bibliographical reference drawing from a separate `.bib` file +- A table +- A math expression +- A different font +- Different page margins +- Different line spacing ## Concluding the Prefresher {.unnumbered} Math may not be the perfect tool for every aspiring political scientist, but hopefully it was useful background to have at the least: - - + + +```{=html} +``` +But we should be aware that too much slant towards math and programming can miss the point: + + + +```{=html} +``` +Keep on learning, trying new techniques to improve your work, and learn from others! + + + +```{=html} + +``` ### Your Feedback Matters {.unnumbered} -_Please tell us how we can improve the Prefresher_: The Prefresher is a work in progress, with material mainly driven by graduate students. Please tell us how we should change (or not change) each of its elements: +*Please tell us how we can improve the Prefresher*: The Prefresher is a work in progress, with material mainly driven by graduate students. Please tell us how we should change (or not change) each of its elements: https://harvard.az1.qualtrics.com/jfe/form/SV_esbzN8ZFAOPTqiV diff --git a/18_text.qmd b/18_text.qmd index 21dda64..9e52bd5 100644 --- a/18_text.qmd +++ b/18_text.qmd @@ -1,4 +1,4 @@ -# Text^[Module originally written by Connor Jerzak] {#rtext} +# Text {#rtext} ```{r} #| include: false @@ -9,24 +9,36 @@ library(forcats) library(ggplot2) ``` +::: {.callout .callout-note} +Module originally written by Connor Jerzak. +::: + ## Where are we? Where are we headed? {.unnumbered} -Up till now, you should have covered: +Up till now, you should have covered: -* Loading in data; -* ```R``` notation; -* Matrix algebra. +- Loading in data; +- `R` notation; +- Matrix algebra. ## Review -* `"` and `'` are usually equivalent. -* `<-` and `=` are usually interchangeable^[Only equal signs are allowed to define the values of a functions' argument]. (`x <- 3` is equivalent to `x = 3`, although the former is more preferred because it explicitly states the assignment). -* Use `(` `)` when you are giving input to a function: +- `"` and `'` are usually equivalent. +- `<-` and `=` are usually interchangeable[^18_text-1]. (`x <- 3` is equivalent to `x = 3`, although the former is more preferred because it explicitly states the assignment). +- Use `(` `)` when you are giving input to a function: + +[^18_text-1]: Only equal signs are allowed to define the values of a functions' argument + ```{r} # my_results <- FunctionName(FunctionInputs) ``` - note `c(1,2,3)` is inputting three numbers in the function `c` -* Use `{` `}` when you are defining a function or writing a `for` loop: + +``` +note `c(1,2,3)` is inputting three numbers in the function `c` +``` + +- Use `{` `}` when you are defining a function or writing a `for` loop: + ```{r} #function MyFunction <- function(InputMatrix){ @@ -47,65 +59,66 @@ for(i in 1:20){ print(x) ``` -## Goals for today +## Goals for today -Today, we will learn more about using text data. Our objectives are: +Today, we will learn more about using text data. Our objectives are: -* Reading and writing in text in ```R```. -* To learn how to use paste and sprintf; -* To learn how to use regular expressions; -* To learn about other tools for representing + analyzing text in ```R```. +- Reading and writing in text in `R`. +- To learn how to use paste and sprintf; +- To learn how to use regular expressions; +- To learn about other tools for representing + analyzing text in `R`. -## Reading and writing text in R +## Reading and writing text in R -* To read in a text file, use readLines +- To read in a text file, use readLines -```` +``` readLines("~/Downloads/Carboxylic acid - Wikipedia.html") -```` +``` -* To write a text file, use: +- To write a text file, use: -```` +``` write.table(my_string_vector, "~/mydata.txt", sep="\t") -```` +``` ## `paste()` and `sprintf()` -paste and sprintf are useful commands in text processing, such as for automatically naming files or automatically performing a series of command over a subset of your data. Table making also will often need these commands. +paste and sprintf are useful commands in text processing, such as for automatically naming files or automatically performing a series of command over a subset of your data. Table making also will often need these commands. + +Paste concatenates vectors together. -Paste concatenates vectors together. -````{R} +```{R} #use collapse for inputs of length > 1 my_string <- c("Not", "one", "could", "equal") paste(my_string, collapse = " ") #use sep for inputs of length == 1 paste("Not", "one", "could", "equal", sep = " ") -```` +``` -For more sophisticated concatenation, use sprintf. This is very useful for automatically making tables. +For more sophisticated concatenation, use sprintf. This is very useful for automatically making tables. -````{R} +```{R} sprintf("Coefficient for %s: %.3f (%.2f)", "Gender", 1.52324, 0.03143) #%s is replaced by a character string #%.3f is replaced by a floating point digit with 3 decimal places #%.2f is replaced by a floating point digit with 2 decimal places -```` +``` -## Regular expressions +## Regular expressions -A regular expression is a special text string for describing a search pattern. -They are most often used in functions for detecting, locating, and replacing desired text in a corpus. +A regular expression is a special text string for describing a search pattern. They are most often used in functions for detecting, locating, and replacing desired text in a corpus. Use cases: -1. TEXT PARSING. E.g. I have 10000 congressional speaches. Find all those which mention Iran. -2. WEB SCRAPING. E.g. Parse html code in order to extract research information from an online table. -3. CLEANING DATA. E.g. After loading in a dataset, we might need to remove mistakes from the dataset, orsubset the data using regular expression tools. +1. TEXT PARSING. E.g. I have 10000 congressional speaches. Find all those which mention Iran. +2. WEB SCRAPING. E.g. Parse html code in order to extract research information from an online table. +3. CLEANING DATA. E.g. After loading in a dataset, we might need to remove mistakes from the dataset, orsubset the data using regular expression tools. + +Example in `R`. Extract the tweet mentioning Indonesia. -Example in ```R```. Extract the tweet mentioning Indonesia. ```{r} s1 <- "If only Bradley's arm was longer. RT" s2 <- "Share our love in Indonesia and in the World. RT if you agree." @@ -114,11 +127,11 @@ grepl(my_string, pattern = "Indonesia") my_string[ grepl(my_string, pattern = "Indonesia")] ``` -Key point: Many R commands use regular expressions. See ```?grepl```. Assume that ```x``` is a character vector and that ```pattern``` is the target pattern. In the earlier example, ```x``` could have been something like ```my_string``` and ```pattern``` would have been "```Indonesia```". Here are other key uses: +Key point: Many R commands use regular expressions. See `?grepl`. Assume that `x` is a character vector and that `pattern` is the target pattern. In the earlier example, `x` could have been something like `my_string` and `pattern` would have been "`Indonesia`". Here are other key uses: -1. DETECT PATTERNS. ```grepl(pattern, x)``` goes through all the entries of ```x``` and returns a string of TRUE and FALSE values of the same size as ```x```. It will return a ```TRUE``` whenever that string entry has the target pattern, and ```FALSE``` whenever it doesn't. +1. DETECT PATTERNS. `grepl(pattern, x)` goes through all the entries of `x` and returns a string of TRUE and FALSE values of the same size as `x`. It will return a `TRUE` whenever that string entry has the target pattern, and `FALSE` whenever it doesn't. -2. REPLACE PATTERNS. ```gsub(pattern, x, replacement)``` goes through all the entries of ```x``` replaces the ```pattern``` with ```replacement```. +2. REPLACE PATTERNS. `gsub(pattern, x, replacement)` goes through all the entries of `x` replaces the `pattern` with `replacement`. ```{r} gsub(x = my_string, @@ -126,7 +139,7 @@ gsub(x = my_string, replacement = "AAAA") ``` -3. LOCATE PATTERNS. ```regexpr(pattern, text)``` goes through each element of the character string. It returns a vector of the same length, with the entries of the vector corresponding to the location of the first pattern match, or a -1 if no match was obtained. +3. LOCATE PATTERNS. `regexpr(pattern, text)` goes through each element of the character string. It returns a vector of the same length, with the entries of the vector corresponding to the location of the first pattern match, or a -1 if no match was obtained. ```{r} regex_object <- regexpr(pattern = "was", text = my_string) @@ -138,21 +151,21 @@ regexpr(pattern = "was", text = my_string)[2] Seems simple? The problem: the patterns can get pretty complex! -### Character classes +### Character classes Some types of symbols are stand in for some more complex thing, rather than taken literally. -```[[:digit:]]``` Matches with all digits. +`[[:digit:]]` Matches with all digits. -```[[:lower:]]``` Matches with lower case letters. +`[[:lower:]]` Matches with lower case letters. -```[[:alpha:]]``` Matches with all alphabetic characters. +`[[:alpha:]]` Matches with all alphabetic characters. -```[[:punct:]]``` Matches with all punctuation characters. +`[[:punct:]]` Matches with all punctuation characters. -```[[:cntrl:]]``` Matches with "control" characters such as ```\n```, ```\r```, etc. +`[[:cntrl:]]` Matches with "control" characters such as `\n`, `\r`, etc. -Example in ```R```: +Example in `R`: ```{r} my_string <- "Do you think that 34% of apples are red?" @@ -160,11 +173,11 @@ gsub(my_string, pattern = "[[:digit:]]", replace ="DIGIT") gsub(my_string, pattern = "[[:alpha:]]", replace ="") ``` -### Special Characters. +### Special Characters. -Certain characters (such as ```., *, \```) have special meaning in the regular expressions framework (they are used to form conditional patterns as discussed below). Thus, when we want our pattern to explicitly include those characters as characters, we must "escape" them by using \\ or encoding them in \\Q...\\E. +Certain characters (such as `., *, \`) have special meaning in the regular expressions framework (they are used to form conditional patterns as discussed below). Thus, when we want our pattern to explicitly include those characters as characters, we must "escape" them by using \\ or encoding them in \\Q...\\E. -Example in ```R```: +Example in `R`: ```{r} my_string <- "Do *really* think he will win?" @@ -176,17 +189,17 @@ my_string <- "Now be brave! \n Dread what comrades say of you here in combat! " gsub(my_string, pattern = "\\\n", replace ="") ``` -### Conditional patterns +### Conditional patterns -```[]``` The target characters to match are located between the brackets. For example, ```[aAbB]``` will match with the characters ```a, A, b, B```. +`[]` The target characters to match are located between the brackets. For example, `[aAbB]` will match with the characters `a, A, b, B`. -```[^...]``` Matches with everything except the material between the brackets. For example, ```[^aAbB]``` will match with everything but the characters ```a, A, b, B```. +`[^...]` Matches with everything except the material between the brackets. For example, `[^aAbB]` will match with everything but the characters `a, A, b, B`. -```(?=)``` Lookahead -- match something that IS followed by the pattern. +`(?=)` Lookahead -- match something that IS followed by the pattern. -```(?!)``` Negative lookahead --- match something that is NOT followed by the pattern. +`(?!)` Negative lookahead --- match something that is NOT followed by the pattern. -```(?<=)``` Lookbehind -- match with something that follows the pattern. +`(?<=)` Lookbehind -- match with something that follows the pattern. ```{r} my_string <- "Do you think that 34%of the 23%of apples are red?" @@ -208,27 +221,24 @@ grepl(my_string, pattern = "^(?!presidential1).*\\.png", perl = TRUE) ``` -* Indicates which file names don't start with `presidential1` but do end in `.png` -* `^` indicates that the pattern should start at the beginning of the string. -* `?!` indicates negative lookahead -- we're looking for any pattern NOT following presidential1 which meets the subsequent conditions. (see below) -* The first `.` indicates that, following the negative lookahead, there can be any characters and the * says that it doesn't matter how many. Note that we have to escape the . in `.png`. (by writing `\\.` instead of just `.`) +- Indicates which file names don't start with `presidential1` but do end in `.png` +- `^` indicates that the pattern should start at the beginning of the string. +- `?!` indicates negative lookahead -- we're looking for any pattern NOT following presidential1 which meets the subsequent conditions. (see below) +- The first `.` indicates that, following the negative lookahead, there can be any characters and the \* says that it doesn't matter how many. Note that we have to escape the . in `.png`. (by writing `\\.` instead of just `.`) -You will have the chance to try out some regular expressions for yourself at the end! +You will have the chance to try out some regular expressions for yourself at the end! -## Representing Text +## Representing Text -In courses and research, we often want to analyze text, to extract meaning out of it. -One of the key decisions we need to make is how to represent the text as numbers. -Once the text is represented numerically, we can then apply a host of statistical -and machine learning methods to it. Those methods are discussed more in the Gov methods sequence (Gov 2000-2003). Here's a summary of the decisions you must make: +In courses and research, we often want to analyze text, to extract meaning out of it. One of the key decisions we need to make is how to represent the text as numbers. Once the text is represented numerically, we can then apply a host of statistical and machine learning methods to it. Those methods are discussed more in the Gov methods sequence (Gov 2000-2003). Here's a summary of the decisions you must make: -1. WHICH TEXT TO USE? Which text do I want to analyze? What is my universe of documents? -2. HOW TO REPRESENT THE TEXT NUMERICALLY? How do I use numbers to represent different things about the text? -3. HOW TO ANALYZE THE NUMERICAL REPRESENTATION? How do I extract meaning out of the numerical representation? +1. WHICH TEXT TO USE? Which text do I want to analyze? What is my universe of documents? +2. HOW TO REPRESENT THE TEXT NUMERICALLY? How do I use numbers to represent different things about the text? +3. HOW TO ANALYZE THE NUMERICAL REPRESENTATION? How do I extract meaning out of the numerical representation? -Representing text numerically. +Representing text numerically. -1. Document term matrix. The document term matrix (DTM) is a common method for representing text. The DTM is a matrix. Each row of this matrix corresponds to a document; each column corresponds to a word. It is often useful to look at summary statistics such as the percentage of speaches in which a Democratic lawmaker used the word "inequality" compared to a Republican; the DTM would be very helpful for this and other tasks. +1. Document term matrix. The document term matrix (DTM) is a common method for representing text. The DTM is a matrix. Each row of this matrix corresponds to a document; each column corresponds to a word. It is often useful to look at summary statistics such as the percentage of speaches in which a Democratic lawmaker used the word "inequality" compared to a Republican; the DTM would be very helpful for this and other tasks. ```{R} doc1 <- "Rage---Goddess, sing the rage of Peleus’ son Achilles, @@ -247,7 +257,7 @@ doc3 <- "Many cities of men he saw and learned their minds, DocVec <- c(doc1, doc2, doc3) ``` -Now we can use utility functions in the `tm` package: +Now we can use utility functions in the `tm` package: ```{R} #| eval: false @@ -257,6 +267,7 @@ DTM1 <- inspect( DocumentTermMatrix(DocCorpus) ) ``` Consider the effect of different "pre-processing" choices on the resulting DTM! + ```{r} #| eval: false DocVec <- tolower(DocVec) @@ -267,28 +278,25 @@ DTM2 <- inspect(DocumentTermMatrix(DocCorpus, control = list(stopwords = TRUE, stemming = TRUE))) ``` -Stemming is the process of reducing inflected/derived words to their word stem or base (e.g. stemming, stemmed, stemmer --> stem*) +Stemming is the process of reducing inflected/derived words to their word stem or base (e.g. stemming, stemmed, stemmer --\> stem\*) -## Important packages for parsing text +## Important packages for parsing text -1. rvest -- Useful for downloading and manipulating HTML and XM. -2. tm -- Useful for converting text into a numerical representation (forming DTMs). -3. stringr -- Useful for string parsing. +1. rvest -- Useful for downloading and manipulating HTML and XM. +2. tm -- Useful for converting text into a numerical representation (forming DTMs). +3. stringr -- Useful for string parsing. -## Exercises {.unnumbered} +## Exercises {.unnumbered} ### 1 {.unnumbered} -Figure out why this command does what it does: +Figure out why this command does what it does: -````{R} -sprintf("%s of spontaneous events are %s in the mind. - Really, %.2f?", - "15.03322123", "puzzles", 15.03322123) -``` +\``{R} sprintf("%s of spontaneous events are %s in the mind. Really, %.2f?", "15.03322123", "puzzles", 15.03322123)` ### 2 {.unnumbered} -Why does this command not work? + +Why does this command not work? ```{r} try(sprintf("%s of spontaneous events are %s in the mind. Really, %.2f?", @@ -297,7 +305,7 @@ try(sprintf("%s of spontaneous events are %s in the mind. Really, %.2f?", ### 3 {.unnumbered} -Using `grepl`, these materials, Google, and your friends, describe what the following command does. What changes when `value = FALSE`? +Using `grepl`, these materials, Google, and your friends, describe what the following command does. What changes when `value = FALSE`? ```{r} grep('\'', @@ -321,7 +329,8 @@ my_string <- c("legislative1_term1.png", ``` ### 5 {.unnumbered} -Using the same string as in the above, write code to automatically extract the file names that end in .pdf and that contain the text `term2`. + +Using the same string as in the above, write code to automatically extract the file names that end in .pdf and that contain the text `term2`. ```{r} # Your code here @@ -329,7 +338,7 @@ Using the same string as in the above, write code to automatically extract the ### 6 {.unnumbered} -Combine these two strings into a single string separated by a "-". Desired output: "The carbonyl group in aldehydes and ketones is an oxygen analog of the carbon–carbon double bond." +Combine these two strings into a single string separated by a "-". Desired output: "The carbonyl group in aldehydes and ketones is an oxygen analog of the carbon–carbon double bond." ```{r} string1 <- "The carbonyl group in aldehydes and ketones @@ -339,16 +348,15 @@ string2 <- "–carbon double bond." ### 7 {.unnumbered} -Challenge problem! Download this webpage +Challenge problem! Download this webpage -* Read the html file into your R workspace. -* Remove all of the htlm tags (you may need Google to help with this one). -* Remove all punctuation. -* Make all the characters lower case. -* Do this same process with this webpage (https://en.wikipedia.org/wiki/Iliad). -* Form a document term matrix from the two resulting text strings. +- Read the html file into your R workspace. +- Remove all of the htlm tags (you may need Google to help with this one). +- Remove all punctuation. +- Make all the characters lower case. +- Do this same process with this webpage (https://en.wikipedia.org/wiki/Iliad). +- Form a document term matrix from the two resulting text strings. ```{r} # Your code here ``` - diff --git a/_quarto.yml b/_quarto.yml index 1a996be..13cde02 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -25,6 +25,9 @@ book: - 13_functions_obj_loops.qmd - 14_visualization.qmd - 15_project-dempeace.qmd + - 16_simulation.qmd + - 17_non-wysiwyg.qmd + - 18_text.qmd delete_merged_file: true language: From ab41ea1fa533e627bc3c2605ec8a7accd9e7159b Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 22:39:38 -0400 Subject: [PATCH 19/34] git chapter --- 19_command-line_git.qmd | 98 +++++++++++++++++++++++------------------ _quarto.yml | 1 + 2 files changed, 55 insertions(+), 44 deletions(-) diff --git a/19_command-line_git.qmd b/19_command-line_git.qmd index 3f2d695..df449be 100644 --- a/19_command-line_git.qmd +++ b/19_command-line_git.qmd @@ -3,13 +3,17 @@ execute: eval: false --- -# Command-line, git^[Module originally written by Shiro Kuriwaki] {#commandline-git} +# Command-line, git {#commandline-git} + +::: {.callout .callout-note} +Module originally written by Shiro Kuriwaki +::: ## Where are we? Where are we headed? Up till now, you should have covered: -* Statistical Programming in `R` +- Statistical Programming in `R` In conjunction with the markdown/LaTeX chapter, which is mostly used for typesetting and presentation, here we'll introduce the command-line and git, more used for software extensions and version control @@ -17,41 +21,45 @@ In conjunction with the markdown/LaTeX chapter, which is mostly used for typeset Check if you have an idea of how you might code the following tasks: -* What is a GUI? -* What do the following commands stand for in shell: `ls` (or `dir` in Windows), `cd`, `rm`, `mv` (or `move` in windows), `cp` (or `copy` in Windows). -* What is the difference between a relative path and an absolute path? -* What paths do these refer to in shell/terminal: `~/`, `.`, `..` -* What is a _repository_ in github? -* What does it mean to "clone" a repository? +- What is a GUI? +- What do the following commands stand for in shell: `ls` (or `dir` in Windows), `cd`, `rm`, `mv` (or `move` in windows), `cp` (or `copy` in Windows). +- What is the difference between a relative path and an absolute path? +- What paths do these refer to in shell/terminal: `~/`, `.`, `..` +- What is a *repository* in github? +- What does it mean to "clone" a repository? ## command-line -Elementary programming operations are done on the command-line, or by entering commands into your computer. This is different from a UI or GUI -- graphical user-interface -- which are interfaces that allow you to click buttons and enter commands in more readable form. Although there are good enough GUIs for most of your needs, you still might need to go under the hood sometimes and run a command. +Elementary programming operations are done on the command-line, or by entering commands into your computer. This is different from a UI or GUI -- graphical user-interface -- which are interfaces that allow you to click buttons and enter commands in more readable form. Although there are good enough GUIs for most of your needs, you still might need to go under the hood sometimes and run a command. ### command-line commands + Open up `Terminal` in a Mac. (`Command Prompt` in Windows) Running this command in a Mac (`dir` in Windows) should show you a list of all files in the directory that you are currently in. -```bash + +``` bash ls ``` `pwd` stands for present working directory (`cd` in Windows) -```bash + +``` bash pwd ``` -`cd` means change directory. You need to give it what to change your current directory _to_. You can specify a name of another directory in your directory. +`cd` means change directory. You need to give it what to change your current directory *to*. You can specify a name of another directory in your directory. Or you can go up to your parent directory. The syntax for that are two periods, `..` . One period `.` refers to the current directory. -```bash +``` bash cd .. pwd ``` -`~/` stands for your home directory defined by your computer. -```bash +`~/` stands for your home directory defined by your computer. + +``` bash cd ~/ ls ``` @@ -61,23 +69,26 @@ Using `..` and `.` are "relative" to where you are currently at. So are things l Relative paths are nice if you have a shared Dropbox, for example, and I had `/Users/shirokuriwaki/mathcamp` but Connor's path to the same folder is `/Users/connorjerzak/mathcamp`. To run the same code in `mathcamp`, we should be using relative paths that start from "`mathcamp`". Relative paths are also shorter, and they are invariant to higher-level changes in your computer. ### running things via command-line + Suppose you have a simple Rscript, call it `hello_world.R`. This is simply a plain text file that contains -``` + +``` cat("Hello World") ``` Then in command-line, go to the directory that contains `hello_world.R` and enter -```bash +``` bash Rscript hello_world.R ``` -This should give you the output `Hello World`, which verifies that you "executed" the file with R via the command-line. +This should give you the output `Hello World`, which verifies that you "executed" the file with R via the command-line. ### why do command-line? + If you know exactly what you want to do your files and the changes are local, then command-line might be faster and be more sensible than navigating yourself through a GUI. For example, what if you wanted a single command that will run 10 R scripts successively at once (as Gentzkow and Shapiro suggest you should do in your research)? It is tedious to run each of your scripts on Rstudio, especially if running some take more than a few minutes. Instead you could write a "batch" script that you can run on the terminal, -```bash +``` bash Rscript 01_read_data.R Rscript 02_merge_data.R Rscript 03_run_regressions.R @@ -86,61 +97,60 @@ Rscript 05_maketable.R ``` Then run this single file, call it `run_all_Rscripts.sh`, on your terminal as -```bash + +``` bash sh run_all_Rscripts.sh ``` -On the other hand, command-line prompts may require more keystrokes, and is also less intuitive than a good GUI. It can also be dangerous for beginners, because it can allow you to make large irreversible changes inadvertently. For example, removing a file (`rm`) has no "Undo" feature. +On the other hand, command-line prompts may require more keystrokes, and is also less intuitive than a good GUI. It can also be dangerous for beginners, because it can allow you to make large irreversible changes inadvertently. For example, removing a file (`rm`) has no "Undo" feature. ## git -Git is a tool for version control. It comes pre-installed on Macs, you will probably need to install it yourself on Windows. +Git is a tool for version control. It comes pre-installed on Macs, you will probably need to install it yourself on Windows. ### why version control? -All version control software should be built to -* preserve all snapshots of your work -* and catalog them in such a way that you can refer back or even revert back your files to the past snapshot. -* makes it easy to see exactly which parts of your files you changed between directories. +All version control software should be built to + +- preserve all snapshots of your work +- and catalog them in such a way that you can refer back or even revert back your files to the past snapshot. +- makes it easy to see exactly which parts of your files you changed between directories. Further, git is most commonly used for collaborative work. -* maintains "branches", or parallel universes of your files that people can switch back and forth on, doing version control on each one -* makes it easy to "merge" a sub-branch to a master branch when it is ready. +- maintains "branches", or parallel universes of your files that people can switch back and forth on, doing version control on each one +- makes it easy to "merge" a sub-branch to a master branch when it is ready. -Note that Dropbox is useful for collaborative work too. But the added value of git's branches is that people can make different changes simultaneously on their computers and merge them to the master branch later. In Dropbox, there is only one copy of each thing so simultaneous editing is not possible. +Note that Dropbox is useful for collaborative work too. But the added value of git's branches is that people can make different changes simultaneously on their computers and merge them to the master branch later. In Dropbox, there is only one copy of each thing so simultaneous editing is not possible. ### open-source code at your fingertips + Some links to check out: -* -* -* +- +- +- GitHub is the GUI to git. Making an account there is free. Making an account will allow you to be a part of the collaborative programming community. It will also allow you to "fork" other people's "repositories". "Forking" is making your own copy of the project that forks off from the master project at a point in time. A "repository" is simply the name of your main project directory. -"cloning" someone else's repository is similar to forking -- it gives you your own copy. +"cloning" someone else's repository is similar to forking -- it gives you your own copy. ### commands in git + As you might have noticed from all the quoted terms, git uses a lot of its own terms that are not intuitive and hard to remember at first. The nuts and bolts of maintaining your version control further requires "adding", "committing", and "push"ing, sometimes "pull"ing. The tutorial is quite good. You'd want to have familiarity with command-line to fully understand this and use it in your work. -RStudio Projects has a great git GUI as well. +RStudio Projects has a great git GUI as well. ### GitHub Desktop -If you are using GitHub for managing git repositories, one option is to use a desktop version, [GitHub Desktop](https://desktop.github.com/). -It offers an in-between step to ease into the git lingo. -Repositories become a drop-down menu. -"push"ing, "pull"ing, and "fetch"ing all become a big button. -It also provides a visual difference interface, which shows the changes you are making to files before you "push" them. -It can't do everything, but it provides a way to become familiar with GitHub without the (potentially) intimidating aspects of diving full-on into the command line. +If you are using GitHub for managing git repositories, one option is to use a desktop version, [GitHub Desktop](https://desktop.github.com/). It offers an in-between step to ease into the git lingo. Repositories become a drop-down menu. "push"ing, "pull"ing, and "fetch"ing all become a big button. It also provides a visual difference interface, which shows the changes you are making to files before you "push" them. It can't do everything, but it provides a way to become familiar with GitHub without the (potentially) intimidating aspects of diving full-on into the command line. ### is git worth it? -While git is a powerful tool, you may choose to not use it for everything because +While git is a powerful tool, you may choose to not use it for everything because -* git is mainly for code, not data. It has a fairly strict limit on the size of your dataset that you cover. -* your collaborators might want to work with Dropbox -* unless you get a paid account, all your repositories will be public. +- git is mainly for code, not data. It has a fairly strict limit on the size of your dataset that you cover. +- your collaborators might want to work with Dropbox +- unless you get a paid account, all your repositories will be public. diff --git a/_quarto.yml b/_quarto.yml index 13cde02..7b36b09 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -28,6 +28,7 @@ book: - 16_simulation.qmd - 17_non-wysiwyg.qmd - 18_text.qmd + - 19_command-line_git.qmd delete_merged_file: true language: From 7842fddf00980d2650cdbfb1eef19bb06f65664b Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 22:57:51 -0400 Subject: [PATCH 20/34] solutions cleaning --- 21_solutions-warmup.qmd | 138 +++++++++++++++++------------------- 23_solution_programming.qmd | 12 ++-- _quarto.yml | 4 ++ 3 files changed, 78 insertions(+), 76 deletions(-) diff --git a/21_solutions-warmup.qmd b/21_solutions-warmup.qmd index 2f4c609..84c723d 100644 --- a/21_solutions-warmup.qmd +++ b/21_solutions-warmup.qmd @@ -1,93 +1,88 @@ -# (PART) Solutions {.unnumbered} - # Solutions to Warmup Questions {.unnumbered} ## Linear Algebra {.unnumbered} ### Vectors {.unnumbered} -Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pmatrix} 4\\5\\6 \end{pmatrix}$, and the scalar $c = 2$. +Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pmatrix} 4\\5\\6 \end{pmatrix}$, and the scalar $c = 2$. -1. $u + v = \begin{pmatrix}5\\7\\9\end{pmatrix}$ -2. $cv = \begin{pmatrix}8\\10\\12\end{pmatrix}$ -3. $u \cdot v = 1(4) + 2(5) + 3(6) = 32$ +1. $u + v = \begin{pmatrix}5\\7\\9\end{pmatrix}$ +2. $cv = \begin{pmatrix}8\\10\\12\end{pmatrix}$ +3. $u \cdot v = 1(4) + 2(5) + 3(6) = 32$ -If you are having trouble with these problems, please review Section \@ref(vector-def) "Working with Vectors" in Chapter \@ref(linearalgebra). +If you are having trouble with these problems, please review Section \@ref(vector-def) "Working with Vectors" in Chapter \@ref(linearalgebra). Are the following sets of vectors linearly independent? -1. $u = \begin{pmatrix} 1\\ 2\end{pmatrix}$, $v = \begin{pmatrix} 2\\4\end{pmatrix}$ +1. $u = \begin{pmatrix} 1\\ 2\end{pmatrix}$, $v = \begin{pmatrix} 2\\4\end{pmatrix}$ -$\leadsto$ No: $$2u = \begin{pmatrix} 2\\ 4\end{pmatrix}, v = \begin{pmatrix} 2\\ 4\end{pmatrix}$$ -so infinitely many linear combinations of $u$ and $v$ that amount to 0 exist. +$\leadsto$ No: $$2u = \begin{pmatrix} 2\\ 4\end{pmatrix}, v = \begin{pmatrix} 2\\ 4\end{pmatrix}$$ so infinitely many linear combinations of $u$ and $v$ that amount to 0 exist. -2. $u = \begin{pmatrix} 1\\ 2\\ 5 \end{pmatrix}$, $v = \begin{pmatrix} 3\\ 7\\ 9 \end{pmatrix}$ +2. $u = \begin{pmatrix} 1\\ 2\\ 5 \end{pmatrix}$, $v = \begin{pmatrix} 3\\ 7\\ 9 \end{pmatrix}$ -$\leadsto$ Yes: we cannot find linear combination of these two vectors that would amount to zero. +$\leadsto$ Yes: we cannot find linear combination of these two vectors that would amount to zero. -3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ +3. $a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$, $b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}$, $c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}$ -$\leadsto$ No: After playing around with some numbers, we can find that -$$-2a = \begin{pmatrix} -4\\ 2\\ -2 \end{pmatrix}, 3b = \begin{pmatrix} 9\\ -12\\ -6 \end{pmatrix}, -1c = \begin{pmatrix} -5\\ 10\\ 8 \end{pmatrix}$$ +$\leadsto$ No: After playing around with some numbers, we can find that $$-2a = \begin{pmatrix} -4\\ 2\\ -2 \end{pmatrix}, 3b = \begin{pmatrix} 9\\ -12\\ -6 \end{pmatrix}, -1c = \begin{pmatrix} -5\\ 10\\ 8 \end{pmatrix}$$ -So -$$-2a + 3b - c = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ +So $$-2a + 3b - c = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ i.e., a linear combination of these three vectors that would amount to zero exists. -If you are having trouble with these problems, please review Section \@ref(linearindependence). +If you are having trouble with these problems, please review Section \@ref(linearindependence). ### Matrices {.unnumbered} $${\bf A}=\begin{pmatrix} - 7 & 5 & 1 \\ - 11 & 9 & 3 \\ - 2 & 14 & 21 \\ - 4 & 1 & 5 - \end{pmatrix}$$ + 7 & 5 & 1 \\ + 11 & 9 & 3 \\ + 2 & 14 & 21 \\ + 4 & 1 & 5 + \end{pmatrix}$$ -What is the dimensionality of matrix ${\bf A}$? 4 $\times$ 3 +What is the dimensionality of matrix ${\bf A}$? 4 $\times$ 3 What is the element $a_{23}$ of ${\bf A}$? 3 -Given that +Given that $${\bf B}=\begin{pmatrix} - 1 & 2 & 8 \\ - 3 & 9 & 11 \\ - 4 & 7 & 5 \\ - 5 & 1 & 9 - \end{pmatrix}$$ + 1 & 2 & 8 \\ + 3 & 9 & 11 \\ + 4 & 7 & 5 \\ + 5 & 1 & 9 + \end{pmatrix}$$ $$\mathbf{A} + \mathbf{B} = \begin{pmatrix} - 8 & 7 & 9 \\ - 14 & 18 & 14 \\ - 6 & 21 & 26 \\ - 9 & 2 & 14 - \end{pmatrix}$$ + 8 & 7 & 9 \\ + 14 & 18 & 14 \\ + 6 & 21 & 26 \\ + 9 & 2 & 14 + \end{pmatrix}$$ -Given that +Given that $${\bf C}=\begin{pmatrix} - 1 & 2 & 8 \\ - 3 & 9 & 11 \\ - 4 & 7 & 5 \\ - \end{pmatrix}$$ + 1 & 2 & 8 \\ + 3 & 9 & 11 \\ + 4 & 7 & 5 \\ + \end{pmatrix}$$ $$\mathbf{A} + \mathbf{C} = \text{No solution, matrices non-conformable}$$ -Given that +Given that $$c = 2$$ $$c\textbf{A} = \begin{pmatrix} - 14 & 10 & 2 \\ - 22 & 18 & 6 \\ - 4 & 28 & 42 \\ - 8 & 2 & 10 - \end{pmatrix}$$ + 14 & 10 & 2 \\ + 22 & 18 & 6 \\ + 4 & 28 & 42 \\ + 8 & 2 & 10 + \end{pmatrix}$$ -If you are having trouble with these problems, please review Section \@ref(matrixbasics). +If you are having trouble with these problems, please review Section \@ref(matrixbasics). ## Operations {.unnumbered} @@ -95,17 +90,17 @@ If you are having trouble with these problems, please review Section \@ref(matri Simplify the following -1. $\sum\limits_{i = 1}^3 i = 1 + 2+ 3 = 6$ +1. $\sum\limits_{i = 1}^3 i = 1 + 2+ 3 = 6$ -2. $\sum\limits_{k = 1}^3(3k + 2) = 3\sum\limits_{k=1}^3k + \sum\limits_{k=1}^3 2= 3\times 6 + 3\times 2 = 24$ +2. $\sum\limits_{k = 1}^3(3k + 2) = 3\sum\limits_{k=1}^3k + \sum\limits_{k=1}^3 2= 3\times 6 + 3\times 2 = 24$ -3. $\sum\limits_{i= 1}^4 (3k + i + 2) = 3\sum\limits_{i= 1}^4k + \sum\limits_{i= 1}^4i + \sum\limits_{i= 1}^42 = 12k + 10 + 8 = 12k + 18$ +3. $\sum\limits_{i= 1}^4 (3k + i + 2) = 3\sum\limits_{i= 1}^4k + \sum\limits_{i= 1}^4i + \sum\limits_{i= 1}^42 = 12k + 10 + 8 = 12k + 18$ ### Products {.unnumbered} -1. $\prod\limits_{i= 1}^3 i = 1\cdot 2\cdot 3 = 6$ +1. $\prod\limits_{i= 1}^3 i = 1\cdot 2\cdot 3 = 6$ -2. $\prod\limits_{k=1}^3(3k + 2) = (3 + 2)\cdot (6 + 2)\cdot (9 + 2) = 440$ +2. $\prod\limits_{k=1}^3(3k + 2) = (3 + 2)\cdot (6 + 2)\cdot (9 + 2) = 440$ To review this material, please see Section \@ref(sum-notation). @@ -118,9 +113,9 @@ Simplify the following 3. $\log_{10}100 = \log_{10}10^2 = 2$ 4. $\log_{2}4 = \log_{2}2^2 = 2$ 5. when $\log$ is the natural log, $\log e = \log_{e} e^1 = 1$ -6. when $a, b, c$ are each constants, $e^a e^b e^c = e^{a + b + c}$, +6. when $a, b, c$ are each constants, $e^a e^b e^c = e^{a + b + c}$, 7. $\log 0 = \text{undefined}$ -- no exponentiation of anything will result in a 0. -8. $e^0 = 1$ -- any number raised to the 0 is always 1. +8. $e^0 = 1$ -- any number raised to the 0 is always 1. 9. $e^1 = e$ -- any number raised to the 1 is always itself 10. $\log e^2 = \log_e e^2 = 2$ @@ -130,9 +125,9 @@ To review this material, please see Section \@ref(logexponents) Find the limit of the following. -1. $\lim\limits_{x \to 2} (x - 1) = 1$ -2. $\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)} = 1$, though note that the original function $\frac{(x - 2) (x - 1)}{(x - 2)}$ would have been undefined at $x = 2$ because of a divide by zero problem; otherwise it would have been equal to $x - 1$. -3. $\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2} = 1$, same as above. +1. $\lim\limits_{x \to 2} (x - 1) = 1$ +2. $\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)} = 1$, though note that the original function $\frac{(x - 2) (x - 1)}{(x - 2)}$ would have been undefined at $x = 2$ because of a divide by zero problem; otherwise it would have been equal to $x - 1$. +3. $\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2} = 1$, same as above. To review this material please see Section \@ref(limitsfun) @@ -140,12 +135,12 @@ To review this material please see Section \@ref(limitsfun) For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\frac{d}{dx}f(x)$ -1. $f(x)=c$, $f'(x) = 0$ -2. $f(x)=x$, $f'(x) = 1$ -3. $f(x)=x^2$, $f'(x) = 2x$ -4. $f(x)=x^3$, $f'(x) = 3x^2$ -5. $f(x)=3x^2+2x^{1/3}$, $f'(x) = 6x + \frac{2}{3}x^{-2/3}$ -6. $f(x)=(x^3)(2x^4)$, $f'(x) = \frac{d}{dx}2x^7 = 14x^6$ +1. $f(x)=c$, $f'(x) = 0$ +2. $f(x)=x$, $f'(x) = 1$ +3. $f(x)=x^2$, $f'(x) = 2x$ +4. $f(x)=x^3$, $f'(x) = 3x^2$ +5. $f(x)=3x^2+2x^{1/3}$, $f'(x) = 6x + \frac{2}{3}x^{-2/3}$ +6. $f(x)=(x^3)(2x^4)$, $f'(x) = \frac{d}{dx}2x^7 = 14x^6$ For a review, please see Section \@ref(derivintro) - \@ref(derivpoly) @@ -153,19 +148,20 @@ For a review, please see Section \@ref(derivintro) - \@ref(derivpoly) For each of the followng functions $f(x)$, does a maximum and minimum exist in the domain $x \in \mathbf{R}$? If so, for what are those values and for which values of $x$? -1. $f(x) = x$ $\leadsto$ neither exists. -2. $f(x) = x^2$ $\leadsto$ a minimum $f(x) = 0$ exists at $x = 0$, but not a maximum. -3. $f(x) = -(x - 2)^2$ $\leadsto$ a maximum $f(x) = 0$ exists at $x = 2$, but not a minimum. +1. $f(x) = x$ $\leadsto$ neither exists. +2. $f(x) = x^2$ $\leadsto$ a minimum $f(x) = 0$ exists at $x = 0$, but not a maximum. +3. $f(x) = -(x - 2)^2$ $\leadsto$ a maximum $f(x) = 0$ exists at $x = 2$, but not a minimum. If you are stuck, please try sketching out a picture of each of the functions. ## Probability {.unnumbered} -1. If there are 12 cards, numbered 1 to 12, and 4 cards are chosen, $\binom{12}{4} = \frac{12\cdot 11\cdot 10\cdot 9}{4!} = 495$ possible hands exist (unordered, without replacement) . -2. Let $A = \{1,3,5,7,8\}$ and $B = \{2,4,7,8,12,13\}$. Then $A \cup B = \{1, 2, 3, 4, 5, 7, 8, 12, 13\}$, $A \cap B = \{7, 8\}$? If $A$ is a subset of the Sample Space $S = \{1,2,3,4,5,6,7,8,9,10\}$, then the complement $A^C = \{2, 4, 6, 9, 10\}$ +1. If there are 12 cards, numbered 1 to 12, and 4 cards are chosen, $\binom{12}{4} = \frac{12\cdot 11\cdot 10\cdot 9}{4!} = 495$ possible hands exist (unordered, without replacement) . -3. If we roll two fair dice, what is the probability that their sum would be 11? $\leadsto \frac{1}{18}$ -4. If we roll two fair dice, what is the probability that their sum would be 12? $\leadsto \frac{1}{36}$. There are two independent dice, so $6^2 = 36$ options in total. While the previous question had two possibilities for a sum of 11 (5,6 and 6,5), there is only one possibility out of 36 for a sum of 12 (6,6). +2. Let $A = \{1,3,5,7,8\}$ and $B = \{2,4,7,8,12,13\}$. Then $A \cup B = \{1, 2, 3, 4, 5, 7, 8, 12, 13\}$, $A \cap B = \{7, 8\}$? If $A$ is a subset of the Sample Space $S = \{1,2,3,4,5,6,7,8,9,10\}$, then the complement $A^C = \{2, 4, 6, 9, 10\}$ -For a review, please see Sections \@ref(setoper) - \@ref(probdef) +3. If we roll two fair dice, what is the probability that their sum would be 11? $\leadsto \frac{1}{18}$ +4. If we roll two fair dice, what is the probability that their sum would be 12? $\leadsto \frac{1}{36}$. There are two independent dice, so $6^2 = 36$ options in total. While the previous question had two possibilities for a sum of 11 (5,6 and 6,5), there is only one possibility out of 36 for a sum of 12 (6,6). + +For a review, please see Sections \@ref(setoper) - \@ref(probdef) diff --git a/23_solution_programming.qmd b/23_solution_programming.qmd index c29d9a5..508bd9f 100644 --- a/23_solution_programming.qmd +++ b/23_solution_programming.qmd @@ -10,7 +10,7 @@ library(forcats) library(scales) ``` -## Chapter \@ref(dataviz): Visualization +## Chapter @sec-dataviz: Visualization ### 1 State Proportions {.unnumbered} @@ -29,6 +29,7 @@ grp_st <- cen10 %>% ``` Plot points + ```{r} ggplot(grp_st, aes(x = state, y = prop)) + geom_point() + @@ -48,7 +49,8 @@ ggplot(grp_st, aes(x = state, y = prop)) + justices <- read_csv("data/input/justices_court-median.csv") ``` -Keep justices who are in the dataset in 2016, +Keep justices who are in the dataset in 2016, + ```{r} in_2017 <- justices %>% filter(term >= 2016) %>% @@ -59,7 +61,8 @@ df_indicator <- justices %>% left_join(in_2017) ``` -All together +All together + ```{r} ggplot(df_indicator, aes(x = term, y = idealpt, group = justice_id)) + geom_line(aes(y = median_idealpt), color = "red", size = 2, alpha = 0.1) + @@ -122,7 +125,7 @@ proportion_d <- c() answer <- data.frame(race_d, state_d, proportion_d) ``` -Then +Then ```{r} for (state in states_of_interest) { @@ -254,4 +257,3 @@ for (i in 1:20) { mean(ests) ``` - diff --git a/_quarto.yml b/_quarto.yml index 7b36b09..a3838df 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -29,6 +29,10 @@ book: - 17_non-wysiwyg.qmd - 18_text.qmd - 19_command-line_git.qmd + - part: Solutions + chapters: + - 21_solutions-warmup.qmd + - 23_solution_programming.qmd delete_merged_file: true language: From 34ec28acffc62d66bd067ba4fc7f12df7609a66b Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 22:59:17 -0400 Subject: [PATCH 21/34] Delete prefresher.log --- prefresher.log | 3043 ------------------------------------------------ 1 file 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Font Info: Font shape `TU/latinmodern-math.otf(3)/m/n' will be -(Font) scaled to size 4.99947pt on input line 5. -[3] [4] [5] [6]) -\tf@toc=\write4 -\openout4 = `prefresher.toc'. - -[7] [8 - -] [9] [10 - -] [11] [12] [13] [14] [15 - -] [16] -Chapter 1. -LaTeX Font Info: Font shape `TU/latinmodern-math.otf(1)/m/n' will be -(Font) scaled to size 14.4pt on input line 481. -LaTeX Font Info: Font shape `TU/latinmodern-math.otf(2)/m/n' will be -(Font) scaled to size 14.40152pt on input line 481. -LaTeX Font Info: Font shape `TU/latinmodern-math.otf(3)/m/n' will be -(Font) scaled to size 14.39845pt on input line 481. -[17 - -] [18] [19] [20] [21] [22] [23] [24] [25] [26 - -] -Chapter 2. -[27] -File: prefresher_files/figure-latex/llnsim-1.pdf Graphic file (type pdf) - -[28] -File: prefresher_files/figure-latex/seqabc-1.pdf Graphic file (type pdf) - -[29] [30] [31] -File: prefresher_files/figure-latex/unnamed-chunk-6-1.pdf Graphic file (type pd -f) - -[32] [33] -File: 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[lmroman10-bold]:mapping=tex-text;! -[81] [82 - -] -Chapter 5. -[83] [84] -File: images/probability.pdf Graphic file (type pdf) - -[85] [86] [87] [88] [89] -File: images/rv.pdf Graphic file (type pdf) - -[90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] -File: prefresher_files/figure-latex/normaldens-1.pdf Graphic file (type pdf) - -[101] -Underfull \hbox (badness 10000) in paragraph at lines 4283--4284 - - [] - -[102] [103] [104] [105] [106] [107] [108 - -] -Chapter 6. - -Underfull \hbox (badness 10000) in paragraph at lines 4603--4604 - - [] - -[109] -Underfull \hbox (badness 1817) in paragraph at lines 4628--4629 -[][]\TU/lmr/bx/n/10 Exercise 6.1 \TU/lmr/m/n/10 (Vector Algebra)\TU/lmr/bx/n/10 - . 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Size Colonies Regime[] - [] - - -Overfull \hbox (33.015pt too wide) in paragraph at lines 6046--6046 -[]\TU/lmtt/m/n/10 ## - [] - [] - - -Overfull \hbox (27.765pt too wide) in paragraph at lines 6047--6047 -[]\TU/lmtt/m/n/10 ## 1 1 Alal~ 42.1 9.51 most Greek 1.12 - 100-~ 0 [] - [] - - -Overfull \hbox (27.765pt too wide) in paragraph at lines 6048--6048 -[]\TU/lmtt/m/n/10 ## 2 2 Empo~ 42.1 3.11 most barba~ 2.12 - 25-1~ 0 [] - [] - - -Overfull \hbox (38.265pt too wide) in paragraph at lines 6049--6049 -[]\TU/lmtt/m/n/10 ## 3 3 Mass~ 43.3 5.38 most Greek 4 - 25-1~ 2 no ev~[] - [] - - -Overfull \hbox (27.765pt too wide) in paragraph at lines 6050--6050 -[]\TU/lmtt/m/n/10 ## 4 4 Rhode 42.3 3.17 most Greek 0.87 - 0 [] - [] - - -Overfull \hbox (27.765pt too wide) in paragraph at lines 6051--6051 -[]\TU/lmtt/m/n/10 ## 5 5 Abak~ 38.1 15.1 most barba~ 1 - 0 [] - [] - - -Overfull \hbox (27.765pt too wide) in paragraph at lines 6052--6052 -[]\TU/lmtt/m/n/10 ## 6 6 Adra~ 37.7 14.8 most Greek 1 - 0 [] - 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RT if yAAAAu agree."[] - [] - -[228] [229] [230] -Overfull \hbox (85.515pt too wide) in paragraph at lines 10958--10958 -[]\TU/lmtt/m/n/10 ## [1] "15.03322123 of spontaneous events are puzzles in the -mind. \n Really, 15.03?" - [] - -[231] [232] -Chapter 15. -[233 - -] -Underfull \vbox (badness 3250) has occurred while \output is active [] - -[234] -Overfull \hbox (4.227pt too wide) in paragraph at lines 11217--11218 -\TU/lmr/m/n/10 called relative paths. 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From c60474e30a2728bcaa9ab4b690786dfbfcbd8b77 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 22:59:33 -0400 Subject: [PATCH 22/34] Update .gitignore --- .gitignore | 1 + 1 file changed, 1 insertion(+) diff --git a/.gitignore b/.gitignore index 76189d9..4b2902b 100644 --- a/.gitignore +++ b/.gitignore @@ -12,3 +12,4 @@ rsconnect prefresher\\.log /.quarto/ *.quarto_ipynb +*.log From 2c0495a91d61af7ddd94e971f4cf91ed89393da0 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 23:00:08 -0400 Subject: [PATCH 23/34] set visual --- README.md | 34 +++++++++++++++------------------- 1 file changed, 15 insertions(+), 19 deletions(-) diff --git a/README.md b/README.md index b78850a..acf4eb7 100644 --- a/README.md +++ b/README.md @@ -1,38 +1,34 @@ -# Math Prefresher Text -![](https://travis-ci.org/IQSS/prefresher.svg?branch=master) [![Github All Releases](https://img.shields.io/github/downloads/IQSS/prefresher/total.svg)]() +# Math Prefresher Text -_(view this book on a [browser](https://iqss.github.io/prefresher/) or in a [PDF](https://github.com/IQSS/prefresher/releases))_ +![](https://travis-ci.org/IQSS/prefresher.svg?branch=master) [![Github All Releases](https://img.shields.io/github/downloads/IQSS/prefresher/total.svg)]() +*(view this book on a [browser](https://iqss.github.io/prefresher/) or in a [PDF](https://github.com/IQSS/prefresher/releases))* -The [__Harvard Gov Prefresher__](https://projects.iq.harvard.edu/prefresher) is held each year in August. All relevant information is on our website, including the day-to-day schedule and the current instructors. +The [**Harvard Gov Prefresher**](https://projects.iq.harvard.edu/prefresher) is held each year in August. All relevant information is on our website, including the day-to-day schedule and the current instructors. The booklet maintained in this repository is the text for the Prefresher, and is the accumulation of continuous improvements by previous instructors for the past 25 years (See the [title page](https://iqss.github.io/prefresher/) for the full list). Christopher T. Kenny is the repository maintainer. [Shiro Kuriwaki](https://github.com/kuriwaki) was the repository maintainer from 2018 to 2023. - ## Technical: How the book is compiled -The text for the book is originally written in RMarkdown following the `bookdown` template, which allows interweaving prose and code. RMarkdown is converted into markdown (with generated code output) and again converted into a book by pandoc. Two formats are provided: a HTML version and a PDF (generated by TeX). The HTML version is hosted on a website listed above. +The text for the book is originally written in RMarkdown following the `bookdown` template, which allows interweaving prose and code. RMarkdown is converted into markdown (with generated code output) and again converted into a book by pandoc. Two formats are provided: a HTML version and a PDF (generated by TeX). The HTML version is hosted on a website listed above. -The repository is also associated with a set of scripts that automatically compiles and deploys the new book to the URL (https://iqss.github.io/prefresher/). Here is basically how it works: - -1. A [Travis-CI app](https://travis-ci.org/IQSS/prefresher) is linked to this repository. The `.travis.yml` file in the repository is what Travis runs. -2. First, Travis creates a virtual machine, downloads all the materials on the `master` branch of the repository, starts a R session, and installs the R packages listed in `DESCRIPTION`. -3. Then, according to the `.travis.yml` file, Travis runs the shell scripts `_build.sh`. This compiles the Rmd files and images into a html book format (in the virtual machine). -4. Then, Travis runs the second script, `_deploy.sh`. This shell script clones the `gh-pages` branch of the repo into a _sub-directory_ of the repository (called `book-deploy`). Then, it removes the content of that cloned copy and then copies the compiled html (in `_book`) into `book-deploy`. It tracks that contents of `book-deploy` and pushes it back up to the `gh-pages branch`. -5. The repository is separately setup with Github Pages, which is designed to notice any `index.html` and associated html files in the `gh-pages` branch and upload the html into a book format at https://iqss.github.io/prefresher/. -6. Every time a commit is pushed to a branch, it will trigger Travis to run procedures 1 through 6. This is the same for pull requests to merge those branches into master. Travis will check beforehand if the merged branch will compile. This is useful to test out the compilation in the cloud before you alter master. - -_This system follows the guidelines outline in the [bookdown manual](https://bookdown.org/yihui/bookdown/github.html)._ +The repository is also associated with a set of scripts that automatically compiles and deploys the new book to the URL (https://iqss.github.io/prefresher/). Here is basically how it works: +1. A [Travis-CI app](https://travis-ci.org/IQSS/prefresher) is linked to this repository. The `.travis.yml` file in the repository is what Travis runs. +2. First, Travis creates a virtual machine, downloads all the materials on the `master` branch of the repository, starts a R session, and installs the R packages listed in `DESCRIPTION`. +3. Then, according to the `.travis.yml` file, Travis runs the shell scripts `_build.sh`. This compiles the Rmd files and images into a html book format (in the virtual machine). +4. Then, Travis runs the second script, `_deploy.sh`. This shell script clones the `gh-pages` branch of the repo into a *sub-directory* of the repository (called `book-deploy`). Then, it removes the content of that cloned copy and then copies the compiled html (in `_book`) into `book-deploy`. It tracks that contents of `book-deploy` and pushes it back up to the `gh-pages branch`. +5. The repository is separately setup with Github Pages, which is designed to notice any `index.html` and associated html files in the `gh-pages` branch and upload the html into a book format at https://iqss.github.io/prefresher/. +6. Every time a commit is pushed to a branch, it will trigger Travis to run procedures 1 through 6. This is the same for pull requests to merge those branches into master. Travis will check beforehand if the merged branch will compile. This is useful to test out the compilation in the cloud before you alter master. +*This system follows the guidelines outline in the [bookdown manual](https://bookdown.org/yihui/bookdown/github.html).* ## Contributing -This material is maintained under a GPL License, and other insturctors are welcome to fork, clone, or make copies of the material. Comments and suggestions are also always welcome. +This material is maintained under a GPL License, and other insturctors are welcome to fork, clone, or make copies of the material. Comments and suggestions are also always welcome. ![](images/readme-license.png) - -You may also be interested in a paper about the prefresher: +You may also be interested in a paper about the prefresher: > [The "Math Prefresher" and The Collective Future of Political Science Graduate Training](https://gking.harvard.edu/prefresher), by Gary King, Shiro Kuriwaki, and Yon Soo Park. From 2391ce880f6ea39d299bdd63e49d78f19cf5d657 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 23:16:49 -0400 Subject: [PATCH 24/34] fix more refs --- 11_data-handling_counting.qmd | 10 ++++------ 16_simulation.qmd | 2 +- 21_solutions-warmup.qmd | 16 ++++++++-------- 23_solution_programming.qmd | 6 +++--- R_exercises/01_data-handling_counting.R | 2 +- _quarto.yml | 5 +++++ 6 files changed, 22 insertions(+), 19 deletions(-) diff --git a/11_data-handling_counting.qmd b/11_data-handling_counting.qmd index 1480ec6..460a3fe 100644 --- a/11_data-handling_counting.qmd +++ b/11_data-handling_counting.qmd @@ -51,7 +51,7 @@ library(fs) 2. RStudio (either cloud or desktop) is a **GUI** and an IDE for the programming language R. A Graphical User Interface allows users to interface with the software (in this case R) using graphical aids like buttons and tabs. Often we don't think of GUIs because to most computer users, everything is a GUI (like Microsoft Word or your "Control Panel"), but it's always there! A Integrated Development Environment just says that the software to interface with R comes with useful useful bells and whistles to give you shortcuts. - The **Console** is kind of a the core window through which you see your GUI actually operating through R. It's not graphical so might not be as intuitive. But all your results, commands, errors, warnings.. you see them in here. A console tells you what's going on now. +The **Console** is kind of a the core window through which you see your GUI actually operating through R. It's not graphical so might not be as intuitive. But all your results, commands, errors, warnings.. you see them in here. A console tells you what's going on now. ![A Typical RStudio Window at Startup](images/11_1_rstudio-startup.png) @@ -87,9 +87,7 @@ library(fs) 5. In R, there are two main types of scripts. A classic `.R` file and a `.Rmd` file (for Rmarkdown). A .R file is just lines and lines of R code that is meant to be inserted right into the Console. A .Rmd tries to weave code and English together, to make it easier for users to create reports that interact with data and intersperse R code with explanation. For example, we built this book in Rmds. -``` -The Rmarkdown facilitates is the use of __code chunks__, which are used here. These start and end with three back-ticks. In the beginning, we can add options in curly braces (`{}`). Specifying `r` in the beginning tells to render it as R code. Options like `echo = TRUE` switch between showing the code that was executed or not; `eval = TRUE` switch between evaluating the code. More about Rmarkdown in Section \@ref(nonwysiwyg). For example, this code chunk would evaluate `1 + 1` and show its output when compiled, but not display the code that was executed. -``` +The Rmarkdown facilitates is the use of **code chunks**, which are used here. These start and end with three back-ticks. In the beginning, we can add options in curly braces (`{}`). Specifying `r` in the beginning tells to render it as R code. Options like `echo = TRUE` switch between showing the code that was executed or not; `eval = TRUE` switch between evaluating the code. More about Rmarkdown in Section @sec-nonwysiwyg. For example, this code chunk would evaluate `1 + 1` and show its output when compiled, but not display the code that was executed. ![A code chunk in Rmarkdown (before rendering)](images/11_4_codechunk.png) @@ -199,7 +197,7 @@ Plotting by ggplot2 (from your tutorials) is also a tidyverse family. | Make a scatter plot | `ggplot(data, aes(x, y)) + geom_point()` | `plot(data$x, data$y)` | | Make a line plot | `ggplot(data, aes(x, y)) + geom_line()` | `plot(data$x, data$y, type = "l")` | | Make a histogram | `ggplot(data, aes(x, y)) + geom_histogram()` | `hist(data$x, data$y)` | -| Make a barplot | See Section \@ref(dataviz) | See Section \@ref(dataviz) | +| Make a barplot | See Section @sec-dataviz | See Section @sec-dataviz | ## A is for Athens @@ -270,7 +268,7 @@ ober dim(ober) ``` -From your tutorials, you also know how to do graphics! Graphics are useful for grasping your data, but we will cover them more deeply in Chapter \@ref(dataviz). +From your tutorials, you also know how to do graphics! Graphics are useful for grasping your data, but we will cover them more deeply in Chapter @sec-dataviz. ```{r} ggplot(ober, aes(x = Fame)) + geom_histogram() diff --git a/16_simulation.qmd b/16_simulation.qmd index 7791cfe..2aac1bf 100644 --- a/16_simulation.qmd +++ b/16_simulation.qmd @@ -41,7 +41,7 @@ Up till now, you should have covered: - Functions, objects, loops - Joining real data -In this module, we will start to work with generating data within R, from thin air, as it were. Doing simulation also strengthens your understanding of Probability (Section \@ref{probability}). +In this module, we will start to work with generating data within R, from thin air, as it were. Doing simulation also strengthens your understanding of Probability (Section @sec-probability). ### Check your Understanding {.unnumbered} diff --git a/21_solutions-warmup.qmd b/21_solutions-warmup.qmd index 84c723d..a09788c 100644 --- a/21_solutions-warmup.qmd +++ b/21_solutions-warmup.qmd @@ -10,7 +10,7 @@ Define the vectors $u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}$, $v = \begin{pm 2. $cv = \begin{pmatrix}8\\10\\12\end{pmatrix}$ 3. $u \cdot v = 1(4) + 2(5) + 3(6) = 32$ -If you are having trouble with these problems, please review Section \@ref(vector-def) "Working with Vectors" in Chapter \@ref(linearalgebra). +If you are having trouble with these problems, please review Section @sec-vector-def "Working with Vectors" in Chapter @sec-linearalgebra. Are the following sets of vectors linearly independent? @@ -30,7 +30,7 @@ So $$-2a + 3b - c = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ i.e., a linear combination of these three vectors that would amount to zero exists. -If you are having trouble with these problems, please review Section \@ref(linearindependence). +If you are having trouble with these problems, please review Section @sec-linearindependence. ### Matrices {.unnumbered} @@ -82,7 +82,7 @@ $$c\textbf{A} = \begin{pmatrix} 8 & 2 & 10 \end{pmatrix}$$ -If you are having trouble with these problems, please review Section \@ref(matrixbasics). +If you are having trouble with these problems, please review Section @sec-matrixbasics. ## Operations {.unnumbered} @@ -102,7 +102,7 @@ Simplify the following 2. $\prod\limits_{k=1}^3(3k + 2) = (3 + 2)\cdot (6 + 2)\cdot (9 + 2) = 440$ -To review this material, please see Section \@ref(sum-notation). +To review this material, please see Section @ref-sum-notation. ### Logs and exponents {.unnumbered} @@ -119,7 +119,7 @@ Simplify the following 9. $e^1 = e$ -- any number raised to the 1 is always itself 10. $\log e^2 = \log_e e^2 = 2$ -To review this material, please see Section \@ref(logexponents) +To review this material, please see Section @sec-logexponents ## Limits {.unnumbered} @@ -129,7 +129,7 @@ Find the limit of the following. 2. $\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)} = 1$, though note that the original function $\frac{(x - 2) (x - 1)}{(x - 2)}$ would have been undefined at $x = 2$ because of a divide by zero problem; otherwise it would have been equal to $x - 1$. 3. $\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2} = 1$, same as above. -To review this material please see Section \@ref(limitsfun) +To review this material please see Section @sec-limitsfun ## Calculus {.unnumbered} @@ -142,7 +142,7 @@ For each of the following functions $f(x)$, find the derivative $f'(x)$ or $\fra 5. $f(x)=3x^2+2x^{1/3}$, $f'(x) = 6x + \frac{2}{3}x^{-2/3}$ 6. $f(x)=(x^3)(2x^4)$, $f'(x) = \frac{d}{dx}2x^7 = 14x^6$ -For a review, please see Section \@ref(derivintro) - \@ref(derivpoly) +For a review, please see Section @sec-derivintro - @sec-derivpoly ## Optimization {.unnumbered} @@ -164,4 +164,4 @@ If you are stuck, please try sketching out a picture of each of the functions. 4. If we roll two fair dice, what is the probability that their sum would be 12? $\leadsto \frac{1}{36}$. There are two independent dice, so $6^2 = 36$ options in total. While the previous question had two possibilities for a sum of 11 (5,6 and 6,5), there is only one possibility out of 36 for a sum of 12 (6,6). -For a review, please see Sections \@ref(setoper) - \@ref(probdef) +For a review, please see Sections @sec-setoper - @sec-probdef diff --git a/23_solution_programming.qmd b/23_solution_programming.qmd index 508bd9f..7390fe2 100644 --- a/23_solution_programming.qmd +++ b/23_solution_programming.qmd @@ -84,7 +84,7 @@ ggplot(df_indicator, aes(x = term, y = idealpt, group = justice_id)) + theme_bw() ``` -## Chapter \@ref(robjloops): Objects and Loops +## Chapter @sec-robjloops: Objects and Loops ```{r} #| message: false @@ -139,7 +139,7 @@ for (state in states_of_interest) { } ``` -## Chapter \@ref(dempeace): Demoratic Peace Project +## Chapter @sec-dempeace: Demoratic Peace Project ### Task 1: Data Input and Standardization {.unnumbered} @@ -197,7 +197,7 @@ answer + geom_line(data =mean_polity_no_mid, aes(x = year, y = mean_polity_mid), ``` -## Chapter \@ref(simulation): Simulation +## Chapter @sec-simulation: Simulation ### Census Sampling diff --git a/R_exercises/01_data-handling_counting.R b/R_exercises/01_data-handling_counting.R index e38f528..71bd055 100644 --- a/R_exercises/01_data-handling_counting.R +++ b/R_exercises/01_data-handling_counting.R @@ -83,7 +83,7 @@ ober dim(ober) # From your tutorials, you also know how to do graphics! -# Graphics are useful for grasping your data, but we will cover them more deeply in Chapter \@ref(dataviz). +# Graphics are useful for grasping your data, but we will cover them more deeply in Chapter @sec-dataviz. ggplot(ober, aes(x = Fame)) + geom_histogram() diff --git a/_quarto.yml b/_quarto.yml index a3838df..9875d46 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -34,6 +34,11 @@ book: - 21_solutions-warmup.qmd - 23_solution_programming.qmd +format: + html: + theme: + - cosmo + delete_merged_file: true language: ui: From 56fac3f5023e8375c9a75b384d1b7ce40d8da70a Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 23:25:24 -0400 Subject: [PATCH 25/34] fix sec, full re-render --- 02_functions.qmd | 2 +- 03_limits.qmd | 2 +- 04_calculus.qmd | 2 +- 05_optimization.qmd | 6 +- 06_probability.qmd | 2 +- 07_linear-algebra.qmd | 8 +- 11_data-handling_counting.qmd | 2 +- 12_matricies-manipulation.qmd | 2 +- 13_functions_obj_loops.qmd | 2 +- 14_visualization.qmd | 2 +- 15_project-dempeace.qmd | 2 +- 16_simulation.qmd | 2 +- 17_non-wysiwyg.qmd | 2 +- 18_text.qmd | 2 +- 19_command-line_git.qmd | 2 +- _book/images/1_rstudio_startup.png | Bin 948645 -> 0 bytes _book/images/2_rstudio_script.png | Bin 989498 -> 0 bytes _book/images/3_rstudio_files.png | Bin 1161872 -> 0 bytes _book/prefresher.tex | 12033 --------------------------- _book/style.css | 14 - 20 files changed, 18 insertions(+), 12069 deletions(-) delete mode 100644 _book/images/1_rstudio_startup.png delete mode 100644 _book/images/2_rstudio_script.png delete mode 100644 _book/images/3_rstudio_files.png delete mode 100644 _book/prefresher.tex delete mode 100644 _book/style.css diff --git a/02_functions.qmd b/02_functions.qmd index 10f5503..ac98124 100644 --- a/02_functions.qmd +++ b/02_functions.qmd @@ -4,7 +4,7 @@ title: Functions and Operations **Topics** Dimensionality; Interval Notation for ${\bf R}^1$; Neighborhoods: Intervals, Disks, and Balls; Introduction to Functions; Domain and Range; Some General Types of Functions; $\log$, $\ln$, and $\exp$; Other Useful Functions; Graphing Functions; Solving for Variables; Finding Roots; Limit of a Function; Continuity; Sets, Sets, and More Sets. -## Summation Operators $\sum$ and $\prod$ {#sum-notation} +## Summation Operators $\sum$ and $\prod$ {#sec-sum-notation} Addition (+), Subtraction (-), multiplication and division are basic operations of arithmetic -- combining numbers. In statistics and calculus, we want to add a *sequence* of numbers that can be expressed as a pattern without needing to write down all its components. For example, how would we express the sum of all numbers from 1 to 100 without writing a hundred numbers? diff --git a/03_limits.qmd b/03_limits.qmd index 4aba328..f0c0d75 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -1,4 +1,4 @@ -# Limits {#limits-precalc} +# Limits {#sec-limits-precalc} ```{r} #| include: false diff --git a/04_calculus.qmd b/04_calculus.qmd index 5daed5b..eb9b28a 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -1,4 +1,4 @@ -# Calculus {#derivatives} +# Calculus {#sec-derivatives} ```{r} #| include: false diff --git a/05_optimization.qmd b/05_optimization.qmd index 959e3d4..1144bcd 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -1,4 +1,4 @@ -# Optimization {#optim} +# Optimization {#sec-optim} ```{r} #| include: false @@ -638,7 +638,6 @@ $s_1 \neq 0$ $\lambda_1 \neq 0$ & \multicolumn{4}{l|}{No solution} &\\ \end{tabular} \end{center} ``` - This shows that there are two critical points: $(0,0)$ and $(\frac{8}{3},\frac{4}{3})$. 5. Find maximum: Looking at the values of $f(x_1,x_2)$ at the critical points, we see that $f(x_1,x_2)$ is maximized at $x_1^* = 0$ and $x_2^*=0$. @@ -688,7 +687,6 @@ $s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{No solution}& \\ \end{tabular} \end{center} ``` - This shows that there are four critical points: $(0,0)$, $(4,0)$, $(0,4)$, and $(\frac{8}{3},\frac{4}{3})$ 4. Find maximum: Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem. @@ -803,7 +801,6 @@ $x_1 \neq 0, x_2 \neq 0$ & $-\frac{16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\ \end{tabular} \end{center} ``` - 4. Find Maximum: Three of the critical points violate the requirement that $\lambda \geq 0$, so the point $(0,0,0)$ is the maximum. ::: {#exr-ktlogs} @@ -851,7 +848,6 @@ $x_1 \neq 0, x_2 \neq 0$ & & $\frac{4}{3}$ & $\frac{4}{3}$ & $\log\frac{7}{3}$\ \end{tabular} \end{center} ``` - 4. Find Maximum: Three of the critical points violate the constraints, so the point $(\frac{4}{3},\frac{4}{3})$ is the maximum.\\ diff --git a/06_probability.qmd b/06_probability.qmd index 4f355e0..89104eb 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -1,4 +1,4 @@ -# Probability Theory {#probability-theory} +# Probability Theory {#sec-probability-theory} Probability and Inferences are mirror images of each other, and both are integral to social science. Probability quantifies uncertainty, which is important because many things in the social world are at first uncertain. Inference is then the study of how to learn about facts you don't observe from facts you do observe. diff --git a/07_linear-algebra.qmd b/07_linear-algebra.qmd index 0eb9559..759e648 100644 --- a/07_linear-algebra.qmd +++ b/07_linear-algebra.qmd @@ -1,4 +1,4 @@ -# Linear Algebra {#linearalgebra} +# Linear Algebra {#sec-linearalgebra} Topics: @@ -14,7 +14,7 @@ Topics: - The Inverse of a Matrix - Inverse of Larger Matrices -## Working with Vectors {#vector-def} +## Working with Vectors {#sec-vector-def} **Vector**: A vector in $n$-space is an ordered list of $n$ numbers. These numbers can be represented as either a row vector or a column vector: $$ {\bf v} \begin{pmatrix} v_1 & v_2 & \dots & v_n\end{pmatrix} , {\bf v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$$ @@ -53,7 +53,7 @@ Let $u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}$, $v = \begin{pmatrix} 9&-3&2&8 \ 4. $w \cdot v$ ::: -## Linear Independence {#linearindependence} +## Linear Independence {#sec-linearindependence} **Linear combinations**: The vector ${\bf u}$ is a linear combination of the vectors ${\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k$ if $${\bf u} = c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k$$ @@ -86,7 +86,7 @@ Are the following sets of vectors linearly independent? 2. $${\bf v}_1 = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} -4 \\ 6 \\ 5 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} -2 \\ 8 \\ 6 \end{pmatrix} $$ ::: -## Basics of Matrix Algebra {#matrixbasics} +## Basics of Matrix Algebra {#sec-matrixbasics} **Matrix**: A matrix is an array of real numbers arranged in $m$ rows by $n$ columns. The dimensionality of the matrix is defined as the number of rows by the number of columns, $m \times n$. diff --git a/11_data-handling_counting.qmd b/11_data-handling_counting.qmd index 460a3fe..762cf5e 100644 --- a/11_data-handling_counting.qmd +++ b/11_data-handling_counting.qmd @@ -1,4 +1,4 @@ -# Orientation and Reading in Data {#dataimport} +# Orientation and Reading in Data {#sec-dataimport} ::: {.callout .callout-note} Module originally written by Shiro Kuriwaki. diff --git a/12_matricies-manipulation.qmd b/12_matricies-manipulation.qmd index f7b3973..687cb7a 100644 --- a/12_matricies-manipulation.qmd +++ b/12_matricies-manipulation.qmd @@ -1,4 +1,4 @@ -# Manipulating Vectors and Matrices {#rmatrices} +# Manipulating Vectors and Matrices {#sec-rmatrices} ```{r} #| include: false diff --git a/13_functions_obj_loops.qmd b/13_functions_obj_loops.qmd index 3537003..0637113 100644 --- a/13_functions_obj_loops.qmd +++ b/13_functions_obj_loops.qmd @@ -1,4 +1,4 @@ -# Objects, Functions, Loops {#robjloops} +# Objects, Functions, Loops {#sec-robjloops} ### Where are we? Where are we headed? {.unnumbered} diff --git a/14_visualization.qmd b/14_visualization.qmd index 67d371b..45a79fe 100644 --- a/14_visualization.qmd +++ b/14_visualization.qmd @@ -1,4 +1,4 @@ -# Visualization {#dataviz} +# Visualization {#sec-dataviz} ::: {.callout .callout-note} Module originally written by Shiro Kuriwaki. diff --git a/15_project-dempeace.qmd b/15_project-dempeace.qmd index c435c95..0a607fd 100644 --- a/15_project-dempeace.qmd +++ b/15_project-dempeace.qmd @@ -1,4 +1,4 @@ -# Joins and Merges, Wide and Long {#dempeace} +# Joins and Merges, Wide and Long {#sec-dempeace} ::: {.callout .callout-note} Module originally written by Shiro Kuriwaki, Connor Jerzak, and Yon Soo Park. diff --git a/16_simulation.qmd b/16_simulation.qmd index 2aac1bf..283fd1f 100644 --- a/16_simulation.qmd +++ b/16_simulation.qmd @@ -1,4 +1,4 @@ -# Simulation {#simulation} +# Simulation {#sec-simulation} ```{r} #| include: false diff --git a/17_non-wysiwyg.qmd b/17_non-wysiwyg.qmd index 4e1effd..9d654cc 100644 --- a/17_non-wysiwyg.qmd +++ b/17_non-wysiwyg.qmd @@ -1,4 +1,4 @@ -# LaTeX and markdown {#nonwysiwyg} +# LaTeX and markdown {#sec-nonwysiwyg} ::: {.callout .callout-note} Module originally written by Shiro Kuriwaki. diff --git a/18_text.qmd b/18_text.qmd index 9e52bd5..e4e572c 100644 --- a/18_text.qmd +++ b/18_text.qmd @@ -1,4 +1,4 @@ -# Text {#rtext} +# Text {#sec-rtext} ```{r} #| include: false diff --git a/19_command-line_git.qmd b/19_command-line_git.qmd index df449be..3536e17 100644 --- a/19_command-line_git.qmd +++ b/19_command-line_git.qmd @@ -3,7 +3,7 @@ execute: eval: false --- -# Command-line, git {#commandline-git} +# Command-line, git {#sec-commandline-git} ::: {.callout .callout-note} Module originally written by Shiro Kuriwaki diff --git a/_book/images/1_rstudio_startup.png b/_book/images/1_rstudio_startup.png deleted file mode 100644 index 496ed4ccaeb3b0cb260e46492ba6eebe35a64f7b..0000000000000000000000000000000000000000 GIT binary patch literal 0 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zZWP3*`$c(__vxqYgNV#EBiMv^HbE61_*Cc;8=2Ey^|$%%wOIe5y0LTqPofWMUz6v% zC%p5i5?-8k2U}b8vlGrgSEm7oQm?yj#h)|Fa==p&av0@osG{rJ(1-WeG`x%dI?W&r ztBsaGDHl5x9%kqWGJJ+r6N2}#%HnW;@$-AOTY%)IFR`|iOZq^LQ{F}Q);3TXA$r=4 zcoRpHQF@rzhAEh}art}{*=Rv4 zr*6xfAZr(|A*kH`QDgB%aDXmjs`oLsCvPX=`EJ3dgU(+c)>M&=3qACFX<$D(6_*#X zura{@LVR=c=^z}k#HUvo<@Vg1t_sj86qX@qdVbnXxCbFGE@dJjNPi>sjV>NSP4DqLq}d0*P%3h@-XMAlD;T06Gi*TpC*b2FH`#N%)3=W zAwGfe)V0KBLRO3?^~5HN4<~9Wj-(v~Tca}uy*kBkb&4@Oj?wBlkt_Jaf>Lw01{(Ga u9T)JQOuy Date: Fri, 5 Jul 2024 01:21:35 -0400 Subject: [PATCH 29/34] remove files that aren't used for quarto or GHA --- .travis.yml | 10 ---------- _book/prefresher.pdf | Bin 4398620 -> 4398492 bytes _build.sh | 9 --------- _deploy.sh | 22 ---------------------- _quarto.yml | 1 + old-output.yml | 16 ---------------- style.css | 14 -------------- 7 files changed, 1 insertion(+), 71 deletions(-) delete mode 100644 .travis.yml delete mode 100644 _build.sh delete mode 100644 _deploy.sh delete mode 100644 old-output.yml delete 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z%^CHJ4#aqBL%9xS8Q>RPcY<0}{Ad2}uc5%)o_G=Lm+YP%)ini3i*ta*$)Cm86mnt~ zq{TX5TE{&g|FvT0qG8@Netkvg6Iofq)!|hS<>7JhDWGB?8vcK!kJ`FfJ6`khw(|7$ Z_q4OW#waBzCnql>e@#F@RY&dG{|D*~>MH;M diff --git a/_build.sh b/_build.sh deleted file mode 100644 index d1c0d9a..0000000 --- a/_build.sh +++ /dev/null @@ -1,9 +0,0 @@ -#!/bin/sh - -# Called by .travis.yml. Compiles bookdown in HTML form. All R packages needed to -# be listed in the separate DESCRIPTION file to be installed. - -set -ev - -Rscript -e "bookdown::render_book('index.Rmd', 'bookdown::gitbook')" -# Rscript -e "bookdown::render_book('index.Rmd', 'bookdown::pdf_book')" \ No newline at end of file diff --git a/_deploy.sh b/_deploy.sh deleted file mode 100644 index 112d80d..0000000 --- a/_deploy.sh +++ /dev/null @@ -1,22 +0,0 @@ -#!/bin/sh - -# Called by .travis.yml. Copies the compiled html into a separate directory, -# book-deploy, and pushes that up to the gh-pages branch of the main repository. -# This then gets automatically recognized by Github Pages and is deployed. - -set -e - -[ -z "${GITHUB_PAT}" ] && exit 0 -[ "${TRAVIS_BRANCH}" != "master" ] && exit 0 - -git config --global user.email "shirokuriwaki@gmail.com" -git config --global user.name "Shiro Kuriwaki" - -# clone the repository to the book-deploy directory -git clone -b gh-pages https://${GITHUB_PAT}@github.com/${TRAVIS_REPO_SLUG}.git book-deploy -cd book-deploy -git rm -rf * -cp -r ../_book/* ./ -git add --all * -git commit -m "copy to deployment via travis" || true -git push -q origin gh-pages diff --git a/_quarto.yml b/_quarto.yml index 535f1a2..daa788d 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -11,6 +11,7 @@ book: site-url: https://iqss.github.io/prefresher/ cover-image: "images/logo.png" repo-url: "https://github.com/IQSS/prefresher" + downloads: [pdf, epub] chapters: - index.qmd - 01_warmup.qmd diff --git a/old-output.yml b/old-output.yml deleted file mode 100644 index c725da0..0000000 --- a/old-output.yml +++ /dev/null @@ -1,16 +0,0 @@ -bookdown::gitbook: - css: style.css - config: - toc: - collapse: section - scroll_highlight: yes - before: | - - download: ["pdf", "epub"] - edit : https://github.com/IQSS/prefresher/edit/master/%s -bookdown::pdf_book: - includes: - in_header: preamble.tex - latex_engine: xelatex - citation_package: natbib - keep_tex: yes diff --git a/style.css b/style.css deleted file mode 100644 index f317b43..0000000 --- a/style.css +++ /dev/null @@ -1,14 +0,0 @@ -p.caption { - color: #777; - margin-top: 10px; -} -p code { - white-space: inherit; -} -pre { - word-break: normal; - word-wrap: normal; -} -pre code { - white-space: inherit; -} From 715ccf6ca91c342a59a216e8a4cab1c65abcc637 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Fri, 5 Jul 2024 01:38:04 -0400 Subject: [PATCH 30/34] fix colors --- _quarto.yml | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/_quarto.yml b/_quarto.yml index daa788d..b1ac962 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -50,4 +50,7 @@ format: geometry: "margin=1.5in" link-citations: true output-file: "prefresher.pdf" - linkcolor: "Crimson" + linkcolor: "crimson" + toccolor: "crimson" + urlcolor: "crimson" + citecolor: "crimson" From 1608764013be75bf05157bc0a1b08c6b8d034322 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Fri, 5 Jul 2024 13:29:07 -0400 Subject: [PATCH 31/34] fix downloading --- _quarto.yml | 3 +-- rename.R | 4 ---- 2 files changed, 1 insertion(+), 6 deletions(-) delete mode 100644 rename.R diff --git a/_quarto.yml b/_quarto.yml index b1ac962..440fbb8 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -1,7 +1,6 @@ project: type: book output-dir: _book - post-render: rename.R book: title: "Math Prefresher for Political Scientists" @@ -11,7 +10,7 @@ book: site-url: https://iqss.github.io/prefresher/ cover-image: "images/logo.png" repo-url: "https://github.com/IQSS/prefresher" - downloads: [pdf, epub] + downloads: [pdf] chapters: - index.qmd - 01_warmup.qmd diff --git a/rename.R b/rename.R deleted file mode 100644 index 58fbc98..0000000 --- a/rename.R +++ /dev/null @@ -1,4 +0,0 @@ -# output-name is ignored -cat('Renaming PDF output\n') -file.rename('_book/Math-Prefresher-for-Political-Scientists.pdf', - '_book/prefresher.pdf') \ No newline at end of file From 61362a7a81296762ea7d7ec4a90f124cf5c9875d Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Fri, 5 Jul 2024 13:29:33 -0400 Subject: [PATCH 32/34] move book name --- .gitignore | 2 +- ...h-Prefresher-for-Political-Scientists.pdf} | Bin 4398492 -> 4398195 bytes 2 files changed, 1 insertion(+), 1 deletion(-) rename _book/{prefresher.pdf => Math-Prefresher-for-Political-Scientists.pdf} (95%) diff --git a/.gitignore b/.gitignore index 4b2902b..b77101a 100644 --- a/.gitignore +++ b/.gitignore @@ -2,7 +2,7 @@ .Rhistory .RData .Ruserdata -_book + 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zVzY*?V^M^jJ4`r6gzXlXwGFV&23;V!Pwf6=;*GpvCOyFvSI+c269SUwR+jS07QLMO z_qZ3nE(nW6B2f*^e-hr`$=`|GJn9W<5Wk+eMGhCOgloi97-X9ZS(U`Nn+-hkH{$Xc zr}0U^WJ)YqS~F@hgWR<|je@gWxwB5-d@+ii$b)&`qo@=<>Q0M!` z!mlApt^Q}$O!m)zXJ*c>$2sJHyh1%1yH6+1&*-)UH%t&FzahFeeG0l?A%yU_$#1Ca zA2$DVze=iOD3u!=GcR+{@87HPJj2YV>WQx-Tw#p Cdp+U+ From c32f0d07926d881a718a572ad7e1a1e4cde8826b Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Fri, 5 Jul 2024 13:29:39 -0400 Subject: [PATCH 33/34] Update .gitignore --- .gitignore | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.gitignore b/.gitignore index b77101a..4b2902b 100644 --- a/.gitignore +++ b/.gitignore @@ -2,7 +2,7 @@ .Rhistory .RData .Ruserdata - +_book _bookdown_files rsconnect From 8d5628b934e4c986c3934e8b057354c4c02c2d68 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Fri, 5 Jul 2024 13:52:09 -0400 Subject: [PATCH 34/34] clean up TOCs --- 11_data-handling_counting.qmd | 12 ++++++------ 12_matricies-manipulation.qmd | 16 ++++++++-------- 13_functions_obj_loops.qmd | 6 +++--- 14_visualization.qmd | 8 ++++---- 15_project-dempeace.qmd | 6 +++--- 16_simulation.qmd | 6 +++--- 17_non-wysiwyg.qmd | 16 ++++++++-------- 18_text.qmd | 16 ++++++++-------- 19_command-line_git.qmd | 10 +++++----- ...th-Prefresher-for-Political-Scientists.pdf | Bin 4398195 -> 4394375 bytes 10 files changed, 48 insertions(+), 48 deletions(-) diff --git a/11_data-handling_counting.qmd b/11_data-handling_counting.qmd index 762cf5e..651e96e 100644 --- a/11_data-handling_counting.qmd +++ b/11_data-handling_counting.qmd @@ -343,7 +343,7 @@ gg_ober + ## Exercises {.unnumbered} -### 1 {.unnumbered} +#### 1 {.unnumbered} What is the Fame value of Delphoi? @@ -351,7 +351,7 @@ What is the Fame value of Delphoi? # Enter here ``` -### 2 {.unnumbered} +#### 2 {.unnumbered} Find the polis with the top 10 Fame values. @@ -359,7 +359,7 @@ Find the polis with the top 10 Fame values. # Enter here ``` -### 3 {.unnumbered} +#### 3 {.unnumbered} Make a scatterplot with the number of colonies on the x-axis and Fame on the y-axis. @@ -367,7 +367,7 @@ Make a scatterplot with the number of colonies on the x-axis and Fame on the y-a # Enter here ``` -### 4 {.unnumbered} +#### 4 {.unnumbered} Find the correct function to read the following datasets (available in your rstudio.cloud session) into your R window. @@ -385,7 +385,7 @@ Our Recommendations: Look at the packages `haven` and `readr` # Enter here, perhaps making a chunk for each file. ``` -### 5 {.unnumbered} +#### 5 {.unnumbered} Read Ober's codebook and find a variable that you think is interesting. Check the distribution of that variable in your data, get a couple of statistics, and summarize it in English. @@ -393,7 +393,7 @@ Read Ober's codebook and find a variable that you think is interesting. Check th # Enter here ``` -### 6 {.unnumbered} +#### 6 {.unnumbered} This is day 1 and we covered a lot of material. Some of you might have found this completely new; others not so. Please click through this survey before you leave so we can adjust accordingly on the next few days. diff --git a/12_matricies-manipulation.qmd b/12_matricies-manipulation.qmd index 687cb7a..f73eb39 100644 --- a/12_matricies-manipulation.qmd +++ b/12_matricies-manipulation.qmd @@ -391,7 +391,7 @@ all_equal(s7, s8) ## Checkpoint {.unnumbered} -### 1 {.unnumbered} +#### 1 {.unnumbered} Get the subset of cen10 for non-white individuals (Hint: look at the set of values for the race variable by using the unique function) @@ -399,7 +399,7 @@ Get the subset of cen10 for non-white individuals (Hint: look at the set of valu # Enter here ``` -### 2 {.unnumbered} +#### 2 {.unnumbered} Get the subset of cen10 for females over the age of 40 @@ -407,7 +407,7 @@ Get the subset of cen10 for females over the age of 40 # Enter here ``` -### 3 {.unnumbered} +#### 3 {.unnumbered} Get all the serial numbers for black, male individuals who don't live in Ohio or Nevada. @@ -417,7 +417,7 @@ Get all the serial numbers for black, male individuals who don't live in Ohio or ## Exercises {.unnumbered} -### 1 {.unnumbered} +#### 1 {.unnumbered} Let $$\mathbf{A} = \left[\begin{array} {rrr} @@ -434,7 +434,7 @@ Use R to write code that will create the matrix $A$, and then consecutively mult Note that R notation of matrices is different from the math notation. Simply trying `X^n` where `X` is a matrix will only take the power of each element to `n`. Instead, this problem asks you to perform matrix multiplication. -### 2 {.unnumbered} +#### 2 {.unnumbered} Let's apply what we learned about subsetting or filtering/selecting. Use the `nunn_full` dataset you have already loaded @@ -456,15 +456,15 @@ c) Lastly, show all values of `"trust_neighbors"` and `"age"` for observations ## Enter yourself ``` -### 3 {.unnumbered} +#### 3 {.unnumbered} Find a way to generate a vector of "column averages" of the matrix `X` from the Nunn and Wantchekon data in one line of code. Each entry in the vector should contain the sample average of the values in the column. So a 100 by 4 matrix should generate a length-4 matrix. -### 4 {.unnumbered} +#### 4 {.unnumbered} Similarly, generate a vector of "column medians". -### 5 {.unnumbered} +#### 5 {.unnumbered} Consider the regression that was run to generate Table 1: diff --git a/13_functions_obj_loops.qmd b/13_functions_obj_loops.qmd index be15aee..df0caa3 100644 --- a/13_functions_obj_loops.qmd +++ b/13_functions_obj_loops.qmd @@ -337,7 +337,7 @@ nw_in_state(cen10, "Massachusetts") ## Checkpoint {.unnumbered} -### 1 {.unnumbered} +#### 1 {.unnumbered} Try making your own function, `average_age_in_state`, that will give you the average age of people in a given state. @@ -346,7 +346,7 @@ Try making your own function, `average_age_in_state`, that will give you the ave ``` -### 2 {.unnumbered} +#### 2 {.unnumbered} Try making your own function, `asians_in_state`, that will give you the number of `Chinese`, `Japanese`, and `Other Asian or Pacific Islander` people in a given state. @@ -354,7 +354,7 @@ Try making your own function, `asians_in_state`, that will give you the number o # Enter on your own ``` -### 3 {.unnumbered} +#### 3 {.unnumbered} Try making your own function, 'top_10_oldest_cities', that will give you the names of cities whose population's average age is top 10 oldest. diff --git a/14_visualization.qmd b/14_visualization.qmd index 45a79fe..0f246f1 100644 --- a/14_visualization.qmd +++ b/14_visualization.qmd @@ -305,7 +305,7 @@ As social scientists, we should also not forget to ask ourselves whether these n In the time remaining, try the following exercises. Order doesn't matter. -### 1: Rural states {.unnumbered} +#### 1: Rural states {.unnumbered} Make a well-labelled figure that plots the proportion of the state's population (as per the census) that is 65 years or older. Each state should be visualized as a point, rather than a bar, and there should be 51 points, ordered by their value. All labels should be readable. @@ -314,7 +314,7 @@ Make a well-labelled figure that plots the proportion of the state's population ``` -### 2: The swing justice {.unnumbered} +#### 2: The swing justice {.unnumbered} Using the `justices_court-median.csv` dataset and building off of the plot that was given, make an improved plot by implementing as many of the following changes (which hopefully improves the graph): @@ -335,7 +335,7 @@ Using the `justices_court-median.csv` dataset and building off of the plot that ``` -### 3: Don't sort by the alphabet {.unnumbered} +#### 3: Don't sort by the alphabet {.unnumbered} The Figure we made that shows racial composition by state has one notable shortcoming: it orders the states alphabetically, which is not particularly useful if you want see an overall pattern, without having particular states in mind. @@ -345,6 +345,6 @@ Find a way to modify the figures so that the states are ordered by the *proporti # Enter yourself ``` -### 4 What to show and how to show it {.unnumbered} +#### 4 What to show and how to show it {.unnumbered} As a student of politics our goal is not necessarily to make pretty pictures, but rather make pictures that tell us something about politics, government, or society. If you could augment either the census dataset or the justices dataset in some way, what would be an substantively significant thing to show as a graphic? diff --git a/15_project-dempeace.qmd b/15_project-dempeace.qmd index 0a607fd..352fd8c 100644 --- a/15_project-dempeace.qmd +++ b/15_project-dempeace.qmd @@ -138,7 +138,7 @@ select(p_m_wide, 1:10) Try building a panel that would be useful in answering the Democratic Peace Question, perhaps in these steps. -### Task 1: Data Input and Standardization {.unnumbered} +#### Task 1: Data Input and Standardization {.unnumbered} Often, files we need are saved in the `.xls` or `xlsx` format. It is possible to read these files directly into `R`, but experience suggests that this process is slower than converting them first to `.csv` format and reading them in as `.csv` files. @@ -148,7 +148,7 @@ Often, files we need are saved in the `.xls` or `xlsx` format. It is possible to ``` -### Task 2: Data Merging {.unnumbered} +#### Task 2: Data Merging {.unnumbered} We will use data to test a version of the Democratic Peace Thesis (DPS). Democracies are said to go to war less because the leaders who wage wars are accountable to voters who have to bear the costs of war. Are democracies less likely to engage in militarized interstate disputes? @@ -162,7 +162,7 @@ To start, let's download and merge some data. ``` -### Task 3: Tabulations and Visualization {.unnumbered} +#### Task 3: Tabulations and Visualization {.unnumbered} 1. Calculate the mean Polity2 score by year. Plot the result. Use graphical indicators of your choosing to show where key events fall in this timeline (such as 1914, 1929, 1939, 1989, 2008). Speculate on why the behavior from 1800 to 1920 seems to be qualitatively different than behavior afterwards. 2. Do the same but only among state-years that were invovled in a MID. Plot this line together with your results from 1. diff --git a/16_simulation.qmd b/16_simulation.qmd index 984602b..5711512 100644 --- a/16_simulation.qmd +++ b/16_simulation.qmd @@ -206,7 +206,7 @@ runif(n = 10) ## Exercises {.unnumbered} -### Census Sampling {.unnumbered} +#### Census Sampling {.unnumbered} What can we learn from surveys of populations, and how wrong do we get if our sampling is biased?[^16_simulation-5] Suppose we want to estimate the proportion of U.S. residents who are non-white (`race != "White"`). In reality, we do not have any population dataset to utilize and so we *only see the sample survey*. Here, however, to understand how sampling works, let's conveniently use the Census extract in some cases and pretend we didn't in others. @@ -254,7 +254,7 @@ You can do this by creating a variable, e.g. `propensity`, that is 0.9 for non-W ``` -### Conditional Proportions {.unnumbered} +#### Conditional Proportions {.unnumbered} This example is not on simulation, but is meant to reinforce some of the probability discussion from math lecture. @@ -282,7 +282,7 @@ In addition to some standard demographic questions, we will focus on one called (f) Suppose we randomly sampled a person who participated in the survey and found that he/she had a Bachelor's degree or higher. Given this evidence, what is the probability that the same person supports Donald Trump? Use Bayes Rule and show your work -- that is, do not use data or R to compute the quantity directly. Then, verify this is the case via R. -### The Birthday problem {.unnumbered} +#### The Birthday problem {.unnumbered} Write code that will answer the well-known birthday problem via simulation.[^16_simulation-6] diff --git a/17_non-wysiwyg.qmd b/17_non-wysiwyg.qmd index 9d654cc..5a4723a 100644 --- a/17_non-wysiwyg.qmd +++ b/17_non-wysiwyg.qmd @@ -50,7 +50,7 @@ Markdown is the text we have been using throughout this course! At its core mark Markdown is known as a "light-weight" editor, which means that it is relatively easy to write code that will compile. It is quick and easy and satisfies most presentation purposes; you might want to try `LaTeX` for more involved papers. -### markdown commands +### Markdown commands For italic and bold, use either the asterisks or the underlines, @@ -66,7 +66,7 @@ And for headers use the hash symbols, ## Sub-headers ``` -### your own markdown +### Your own markdown RStudio makes it easy to compile your very first markdown file by giving you templates. Got to `New > R Markdown`, pick a document and click Ok. This will give you a skeleton of a document you can compile -- or "knit". @@ -103,7 +103,7 @@ Since June 2021, RStudio has offered a visual editor which tries to bridge the g LaTeX is a typesetting program. You'd engage with LaTeX much like you engage with your `R` code. You will interact with LaTeX in a text editor, and will writing code which will be interpreted by the LaTeX compiler and which will finally be parsed to form your final PDF. -### compile online +### Compile online 1. Go to 2. Scroll down and go to "CREATE A NEW PAPER" if you don't have an account. @@ -117,7 +117,7 @@ Hello World \end{document} ``` -### compile your first LaTeX document locally +### Compile your first LaTeX document locally LaTeX is a very stable system, and few changes to it have been made since the 1990s. The main benefit: better control over how your papers will look; better methods for writing equations or making tables; overall pleasing aesthetic. @@ -134,7 +134,7 @@ Hello World 3. Open this in your "LaTeX" editor. This can be `TeXMaker`, `Aqumacs`, etc.. 4. Go through the click/dropdown interface and click compile. -### main LaTeX commands +### Main LaTeX commands LaTeX can cover most of your typesetting needs, to clean equations and intricate diagrams. @@ -199,7 +199,7 @@ There is a version of LaTeX called Beamer, which is a popular way of making a sl BibTeX is a reference system for bibliographical tests. We have a `.bib` file separately on our computer. This is also a plain text file, but it encodes bibliographical resources with special syntax so that a program can rearrange parts accordingly for different citation systems. -### what is a `.bib` file? +### What is a `.bib` file? For example, here is the Nunn and Wantchekon article entry in `.bib` form. @@ -217,7 +217,7 @@ For example, here is the Nunn and Wantchekon article entry in `.bib` form. The first entry, `nunn2011slave`, is "pick your favorite" -- pick your own name for your reference system. The other slots in this `@article` entry are entries that refer to specific bibliographical text. -### what does LaTeX do with .bib files? +### What does LaTeX do with .bib files? Now, in LaTeX, if you type @@ -237,7 +237,7 @@ Also at the end of your paper you will have a bibliography with entries ordered This is a much less frustrating way of keeping track of your references -- no need to hand-edit formatting the bibliography to conform to citation rules (which biblatex already knows) and no need to update your bibliography as you add and drop references (biblatex will only show entries that are used in the main text). -### stocking up on your .bib files +### Stocking up on your .bib files You should keep your own `.bib` file that has all your bibliographical resources. Storing entries is cheap (does not take much memory), so it is fine to keep all your references in one place (but you'll want to make a new one for collaborative projects where multiple people will compile a `.tex` file). diff --git a/18_text.qmd b/18_text.qmd index e4e572c..6e956be 100644 --- a/18_text.qmd +++ b/18_text.qmd @@ -288,13 +288,13 @@ Stemming is the process of reducing inflected/derived words to their word stem o ## Exercises {.unnumbered} -### 1 {.unnumbered} +#### 1 {.unnumbered} Figure out why this command does what it does: -\``{R} sprintf("%s of spontaneous events are %s in the mind. Really, %.2f?", "15.03322123", "puzzles", 15.03322123)` +`r sprintf("%s of spontaneous events are %s in the mind. Really, %.2f?", "15.03322123", "puzzles", 15.03322123)` -### 2 {.unnumbered} +#### 2 {.unnumbered} Why does this command not work? @@ -303,7 +303,7 @@ try(sprintf("%s of spontaneous events are %s in the mind. Really, %.2f?", "15.03322123", "puzzles", "15.03322123" ), TRUE) ``` -### 3 {.unnumbered} +#### 3 {.unnumbered} Using `grepl`, these materials, Google, and your friends, describe what the following command does. What changes when `value = FALSE`? @@ -312,7 +312,7 @@ grep('\'', c("To dare is to lose one's footing momentarily.", "To not dare is to lose oneself."), value = TRUE) ``` -### 4 {.unnumbered} +#### 4 {.unnumbered} Write code to automatically extract the file names that DO end start with presidential and DO end in .pdf @@ -328,7 +328,7 @@ my_string <- c("legislative1_term1.png", "presidential1_term2.pdf") ``` -### 5 {.unnumbered} +#### 5 {.unnumbered} Using the same string as in the above, write code to automatically extract the file names that end in .pdf and that contain the text `term2`. @@ -336,7 +336,7 @@ Using the same string as in the above, write code to automatically extract the f # Your code here ``` -### 6 {.unnumbered} +#### 6 {.unnumbered} Combine these two strings into a single string separated by a "-". Desired output: "The carbonyl group in aldehydes and ketones is an oxygen analog of the carbon–carbon double bond." @@ -346,7 +346,7 @@ string1 <- "The carbonyl group in aldehydes and ketones string2 <- "–carbon double bond." ``` -### 7 {.unnumbered} +#### 7 {.unnumbered} Challenge problem! Download this webpage diff --git a/19_command-line_git.qmd b/19_command-line_git.qmd index 3536e17..f33e07a 100644 --- a/19_command-line_git.qmd +++ b/19_command-line_git.qmd @@ -68,7 +68,7 @@ Using `..` and `.` are "relative" to where you are currently at. So are things l Relative paths are nice if you have a shared Dropbox, for example, and I had `/Users/shirokuriwaki/mathcamp` but Connor's path to the same folder is `/Users/connorjerzak/mathcamp`. To run the same code in `mathcamp`, we should be using relative paths that start from "`mathcamp`". Relative paths are also shorter, and they are invariant to higher-level changes in your computer. -### running things via command-line +### Running things via command-line Suppose you have a simple Rscript, call it `hello_world.R`. This is simply a plain text file that contains @@ -108,7 +108,7 @@ On the other hand, command-line prompts may require more keystrokes, and is also Git is a tool for version control. It comes pre-installed on Macs, you will probably need to install it yourself on Windows. -### why version control? +### Why version control? All version control software should be built to @@ -123,7 +123,7 @@ Further, git is most commonly used for collaborative work. Note that Dropbox is useful for collaborative work too. But the added value of git's branches is that people can make different changes simultaneously on their computers and merge them to the master branch later. In Dropbox, there is only one copy of each thing so simultaneous editing is not possible. -### open-source code at your fingertips +### Open-source code at your fingertips Some links to check out: @@ -135,7 +135,7 @@ GitHub is the GUI to git. Making an account there is free. "cloning" someone else's repository is similar to forking -- it gives you your own copy. -### commands in git +### Commands in git As you might have noticed from all the quoted terms, git uses a lot of its own terms that are not intuitive and hard to remember at first. The nuts and bolts of maintaining your version control further requires "adding", "committing", and "push"ing, sometimes "pull"ing. @@ -147,7 +147,7 @@ RStudio Projects has a great git GUI as well. If you are using GitHub for managing git repositories, one option is to use a desktop version, [GitHub Desktop](https://desktop.github.com/). It offers an in-between step to ease into the git lingo. Repositories become a drop-down menu. "push"ing, "pull"ing, and "fetch"ing all become a big button. It also provides a visual difference interface, which shows the changes you are making to files before you "push" them. It can't do everything, but it provides a way to become familiar with GitHub without the (potentially) intimidating aspects of diving full-on into the command line. -### is git worth it? +### Is git worth it? 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    1. Gov Prefresher

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-\newcommand{\Var}{\mathrm{Var}} -\newcommand{\SD}{\mathrm{SD}} -\newcommand{\Cov}{\mathrm{Cov}} -\newcommand{\fx}{f({\bf x})} -\newcommand\R{{\textsf R~}} -\newcommand\Rst{\textsf{RStudio}} - -% spacing between environments -\usepackage{amsthm} -\makeatletter -\def\thm@space@setup{% - \thm@preskip=15pt plus 2pt minus 4pt - \thm@postskip=\thm@preskip -} -\makeatother - -% link colors in pdf? -\usepackage{xcolor} -\definecolor{crimson}{RGB}{204,0,0} -\hypersetup{ - colorlinks = true, - urlcolor = crimson, - linkbordercolor = {white}, - linkcolor = crimson -} - - -% Title format -\usepackage{titling} -\pretitle{\Huge\sffamily} -\posttitle{\par\vskip 1em} -\predate{\LARGE\sffamily} -\postdate{\par} - -\urlstyle{tt} -\usepackage[]{natbib} -\bibliographystyle{apalike} - -\title{Math Prefresher for Political Scientists} -\author{} -\date{\vspace{-2.5em}June 2020} - -\usepackage{amsthm} -\newtheorem{theorem}{Theorem}[chapter] -\newtheorem{lemma}{Lemma}[chapter] -\newtheorem{corollary}{Corollary}[chapter] -\newtheorem{proposition}{Proposition}[chapter] -\newtheorem{conjecture}{Conjecture}[chapter] -\theoremstyle{definition} -\newtheorem{definition}{Definition}[chapter] -\theoremstyle{definition} -\newtheorem{example}{Example}[chapter] -\theoremstyle{definition} -\newtheorem{exercise}{Exercise}[chapter] -\theoremstyle{remark} -\newtheorem*{remark}{Remark} -\newtheorem*{solution}{Solution} -\begin{document} -\maketitle - -{ -\setcounter{tocdepth}{1} -\tableofcontents -} -\hypertarget{about-this-booklet}{% -\chapter*{About this Booklet}\label{about-this-booklet}} -\addcontentsline{toc}{chapter}{About this Booklet} - -The \href{https://projects.iq.harvard.edu/prefresher}{\textbf{Harvard Gov Prefresher}} is held each year in August. All relevant information is on our website, including the day-to-day schedule. The 2019 Prefresher instructors are \href{https://wcfia.harvard.edu/shannon-lynn-parker}{Shannon Parker} and \href{http://schwenzfeier.github.io/}{Meg Schwenzfeier}, and the faculty sponsor is \href{https://gking.harvard.edu}{Gary King}. - -This booklet serves as the text for the Prefresher, available as a \href{https://iqss.github.io/prefresher/}{webpage} and as a \href{https://github.com/IQSS/prefresher/releases}{printable PDF}. It is the product of generations of Prefresher instructors. See below for a full list of instructors and contributors. - -For information about the role of the prefresher (or ``math camp'') as a introduction to graduate school, you may also be interested in \href{https://gking.harvard.edu/prefresher}{``The Math Prefresher and The Collective Future of Political Science Graduate Training''}, in \emph{PS: Political Science \& Politics}, by Gary King, Shiro Kuriwaki, and Yon Soo Park. - -\hypertarget{authors-and-contributors}{% -\subsection*{Authors and Contributors}\label{authors-and-contributors}} -\addcontentsline{toc}{subsection}{Authors and Contributors} - -\begin{itemize} -\tightlist -\item - Authors and Instructors: Curt Signorino 1996-1997; Ken Scheve 1997-1998; Eric Dickson 1998-2000; Orit Kedar 1999; James Fowler 2000-2001; Kosuke Imai 2001-2002; Jacob Kline 2002; Dan Epstein 2003; Ben Ansell 2003-2004; Ryan Moore 2004-2005; Mike Kellermann 2005-2006; Ellie Powell 2006-2007; Jen Katkin 2007-2008; Patrick Lam 2008-2009; Viridiana Rios 2009-2010; Jennifer Pan 2010-2011; Konstantin Kashin 2011-2012; Soledad Prillaman 2013; Stephen Pettigrew 2013-2014; Anton Strezhnev 2014-2015; Mayya Komisarchik 2015-2016; Connor Jerzak 2016-2017; Shiro Kuriwaki 2017-2018; Yon Soo Park 2018 -\item - Repository Maintainer: Shiro Kuriwaki (\href{https://github.com/kuriwaki}{kuriwaki}) -\item - Contributors: Thanks to Juan Dodyk (\href{https://github.com/juandodyk}{juandodyk}), Hunter Rendleman (\href{https://github.com/hrendleman}{hrendleman}), and Tyler Simko (\href{https://github.com/tylersimko}{tylersimko}) for contributing to the booklet for corrections and improvements as students. -\end{itemize} - -\hypertarget{contributing}{% -\subsection*{Contributing}\label{contributing}} -\addcontentsline{toc}{subsection}{Contributing} - -We transitioned the booklet into a bookdown \href{https://github.com/IQSS/prefresher}{github repository} in 2018. As we update this version, we appreciate any bug reports or fixes appreciated. - -All changes should be made in the \texttt{.Rmd} files in the project root. Changes pushed to the repository will be checked for compilation by Travis-CI. To contribute a change, please make a pull request and set the repository maintainer as the reviewer. - -\hypertarget{pre-prefresher-exercises}{% -\chapter*{Pre-Prefresher Exercises}\label{pre-prefresher-exercises}} -\addcontentsline{toc}{chapter}{Pre-Prefresher Exercises} - -Before our first meeting, please try solving these questions. They are a sample of the very beginning of each math section. We have provided links to the parts of the book you can read if the concepts are new to you. - -The goal of this ``pre''-prefresher assignment is not to intimidate you but to set common expectations so you can make the most out of the actual Prefresher. Even if you do not understand some or all of these questions after skimming through the linked sections, your effort will pay off and you will be better prepared for the math prefresher. We are also open to adjusting these expectations based on feedback (this class is for \emph{you}), so please do not hesitate to write to the instructors for feedback. - -\hypertarget{linear-algebra}{% -\section*{Linear Algebra}\label{linear-algebra}} -\addcontentsline{toc}{section}{Linear Algebra} - -\hypertarget{vectors}{% -\subsection*{Vectors}\label{vectors}} -\addcontentsline{toc}{subsection}{Vectors} - -Define the vectors \(u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}\), \(v = \begin{pmatrix} 4\\5\\6 \end{pmatrix}\), and the scalar \(c = 2\). Calculate the following: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(u + v\) -\item - \(cv\) -\item - \(u \cdot v\) -\end{enumerate} - -If you are having trouble with these problems, please review Section \ref{vector-def} ``Working with Vectors'' in Chapter \ref{linearalgebra}. - -Are the following sets of vectors linearly independent? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(u = \begin{pmatrix} 1\\ 2\end{pmatrix}\), \(v = \begin{pmatrix} 2\\4\end{pmatrix}\) -\item - \(u = \begin{pmatrix} 1\\ 2\\ 5 \end{pmatrix}\), \(v = \begin{pmatrix} 3\\ 7\\ 9 \end{pmatrix}\) -\item - \(a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}\), \(b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}\), \(c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}\) (this requires some guesswork) -\end{enumerate} - -If you are having trouble with these problems, please review Section \ref{linearindependence}. - -\hypertarget{matrices}{% -\subsection*{Matrices}\label{matrices}} -\addcontentsline{toc}{subsection}{Matrices} - -\[{\bf A}=\begin{pmatrix} - 7 & 5 & 1 \\ - 11 & 9 & 3 \\ - 2 & 14 & 21 \\ - 4 & 1 & 5 - \end{pmatrix}\] - -What is the dimensionality of matrix \({\bf A}\)? - -What is the element \(a_{23}\) of \({\bf A}\)? - -Given that - -\[{\bf B}=\begin{pmatrix} - 1 & 2 & 8 \\ - 3 & 9 & 11 \\ - 4 & 7 & 5 \\ - 5 & 1 & 9 - \end{pmatrix}\] - -What is \({\bf A}\) + \({\bf B}\)? - -Given that - -\[{\bf C}=\begin{pmatrix} - 1 & 2 & 8 \\ - 3 & 9 & 11 \\ - 4 & 7 & 5 \\ - \end{pmatrix}\] - -What is \({\bf A}\) + \({\bf C}\)? - -Given that - -\[c = 2\] - -What is \(c\)\({\bf A}\)? - -If you are having trouble with these problems, please review Section \ref{matrixbasics}. - -\hypertarget{operations}{% -\section*{Operations}\label{operations}} -\addcontentsline{toc}{section}{Operations} - -\hypertarget{summation}{% -\subsection*{Summation}\label{summation}} -\addcontentsline{toc}{subsection}{Summation} - -Simplify the following - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\sum\limits_{i = 1}^3 i\) -\item - \(\sum\limits_{k = 1}^3(3k + 2)\) -\item - \(\sum\limits_{i= 1}^4 (3k + i + 2)\) -\end{enumerate} - -\hypertarget{products}{% -\subsection*{Products}\label{products}} -\addcontentsline{toc}{subsection}{Products} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\prod\limits_{i= 1}^3 i\) -\item - \(\prod\limits_{k=1}^3(3k + 2)\) -\end{enumerate} - -To review this material, please see Section \ref{sum-notation}. - -\hypertarget{logs-and-exponents}{% -\subsection*{Logs and exponents}\label{logs-and-exponents}} -\addcontentsline{toc}{subsection}{Logs and exponents} - -Simplify the following - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(4^2\) -\item - \(4^2 2^3\) -\item - \(\log_{10}100\) -\item - \(\log_{2}4\) -\item - \(\log e\), where \(\log\) is the natural log (also written as \(\ln\)) -- a log with base \(e\), and \(e\) is Euler's constant -\item - \(e^a e^b e^c\), where \(a, b, c\) are each constants -\item - \(\log 0\) -\item - \(e^0\) -\item - \(e^1\) -\item - \(\log e^2\) -\end{enumerate} - -To review this material, please see Section \ref{logexponents} - -\hypertarget{limits}{% -\section*{Limits}\label{limits}} -\addcontentsline{toc}{section}{Limits} - -Find the limit of the following. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\lim\limits_{x \to 2} (x - 1)\) -\item - \(\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)}\) -\item - \(\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2}\) -\end{enumerate} - -To review this material please see Section \ref{limitsfun} - -\hypertarget{calculus}{% -\section*{Calculus}\label{calculus}} -\addcontentsline{toc}{section}{Calculus} - -For each of the following functions \(f(x)\), find the derivative \(f'(x)\) or \(\frac{d}{dx}f(x)\) - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x)=c\) -\item - \(f(x)=x\) -\item - \(f(x)=x^2\) -\item - \(f(x)=x^3\) -\item - \(f(x)=3x^2+2x^{1/3}\) -\item - \(f(x)=(x^3)(2x^4)\) -\end{enumerate} - -For a review, please see Section \ref{derivintro} - \ref{derivpoly} - -\hypertarget{optimization}{% -\section*{Optimization}\label{optimization}} -\addcontentsline{toc}{section}{Optimization} - -For each of the followng functions \(f(x)\), does a maximum and minimum exist in the domain \(x \in \mathbf{R}\)? If so, for what are those values and for which values of \(x\)? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x) = x\) -\item - \(f(x) = x^2\) -\item - \(f(x) = -(x - 2)^2\) -\end{enumerate} - -If you are stuck, please try sketching out a picture of each of the functions. - -\hypertarget{probability}{% -\section*{Probability}\label{probability}} -\addcontentsline{toc}{section}{Probability} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - If there are 12 cards, numbered 1 to 12, and 4 cards are chosen, how many distinct possible choices are there? (unordered, without replacement) -\item - Let \(A = \{1,3,5,7,8\}\) and \(B = \{2,4,7,8,12,13\}\). What is \(A \cup B\)? What is \(A \cap B\)? If \(A\) is a subset of the Sample Space \(S = \{1,2,3,4,5,6,7,8,9,10\}\), what is the complement \(A^C\)? -\item - If we roll two fair dice, what is the probability that their sum would be 11? -\item - If we roll two fair dice, what is the probability that their sum would be 12? -\end{enumerate} - -For a review, please see Sections \ref{setoper} - \ref{probdef}. - -\hypertarget{part-math}{% -\part{Math}\label{part-math}} - -\hypertarget{functions-and-operations}{% -\chapter{Functions and Operations}\label{functions-and-operations}} - -\textbf{Topics} -Dimensionality; -Interval Notation for \({\bf R}^1\); -Neighborhoods: Intervals, Disks, and Balls; Introduction to Functions; -Domain and Range; -Some General Types of Functions; -\(\log\), \(\ln\), and \(\exp\); -Other Useful Functions; -Graphing Functions; -Solving for Variables; -Finding Roots; -Limit of a Function; -Continuity; Sets, Sets, and More Sets. - -\hypertarget{sum-notation}{% -\section{\texorpdfstring{Summation Operators \(\sum\) and \(\prod\)}{Summation Operators \textbackslash sum and \textbackslash prod}}\label{sum-notation}} - -Addition (+), Subtraction (-), multiplication and division are basic operations of arithmetic -- combining numbers. In statistics and calculus, we want to add a \emph{sequence} of numbers that can be expressed as a pattern without needing to write down all its components. For example, how would we express the sum of all numbers from 1 to 100 without writing a hundred numbers? - -For this we use the summation operator \(\sum\) and the product operator \(\prod\). - -\textbf{Summation:} - -\[\sum\limits_{i=1}^{100} x_i = x_1+x_2+x_3+\cdots+x_{100}\] - -The bottom of the \(\sum\) symbol indicates an index (here, \(i\)), and its start value \(1\). At the top is where the index ends. The notion of ``addition'' is part of the \(\sum\) symbol. The content to the right of the summation is the meat of what we add. While you can pick your favorite index, start, and end values, the content must also have the index. - -\begin{itemize} -\tightlist -\item - \(\sum\limits_{i=1}^n c x_i = c \sum\limits_{i=1}^n x_i\) -\item - \(\sum\limits_{i=1}^n (x_i + y_i) = \sum\limits_{i=1}^n x_i + \sum\limits_{i=1}^n y_i\) -\item - \(\sum\limits_{i=1}^n c = n c\) -\end{itemize} - -\textbf{Product:} - -\[\prod\limits_{i=1}^n x_i = x_1 x_2 x_3 \cdots x_n\] - -Properties: - -\begin{itemize} -\tightlist -\item - \(\prod\limits_{i=1}^n c x_i = c^n \prod\limits_{i=1}^n x_i\) -\item - \(\prod\limits_{i=k}^n c x_k = c^{n-k} \prod\limits_{i=k}^n x_i\) -\item - \(\prod\limits_{i=1}^n (x_i + y_i) =\) a total mess -\item - \(\prod\limits_{i=1}^n c = c^n\) -\end{itemize} - -Other Useful Functions - -\textbf{Factorials!:} - -\[x! = x\cdot (x-1) \cdot (x-2) \cdots (1)\] - -\textbf{Modulo:} Tells you the remainder when you divide the first number by the second. - -\begin{itemize} -\tightlist -\item - \(17 \mod 3 = 2\) -\item - \(100 \ \% \ 30 = 10\) -\end{itemize} - -\begin{example}[Operators] -\protect\hypertarget{exm:operators}{}{\label{exm:operators} \iffalse (Operators) \fi{} } - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\sum\limits_{i=1}^{5} i =\) -\item - \(\prod\limits_{i=1}^{5} i =\) -\item - \(14 \mod 4 =\) -\item - \(4! =\) -\end{enumerate} -\end{example} - -\begin{exercise}[Operators] -\protect\hypertarget{exr:operators1}{}{\label{exr:operators1} \iffalse (Operators) \fi{} }Let \(x_1 = 4, x_2 = 3, x_3 = 7, x_4 = 11, x_5 = 2\) - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\sum\limits_{i=1}^{3} (7)x_i\) -\item - \(\sum\limits_{i=1}^{5} 2\) -\item - \(\prod\limits_{i=3}^{5} (2)x_i\) -\end{enumerate} -\end{exercise} - -\hypertarget{introduction-to-functions}{% -\section{Introduction to Functions}\label{introduction-to-functions}} - -A \textbf{function} (in \({\bf R}^1\)) is a mapping, or transformation, that relates members of one set to members of another set. For instance, if you have two sets: set \(A\) and set \(B\), a function from \(A\) to \(B\) maps every value \(a\) in set \(A\) such that \(f(a) \in B\). Functions can be ``many-to-one'', where many values or combinations of values from set \(A\) produce a single output in set \(B\), or they can be ``one-to-one'', where each value in set \(A\) corresponds to a single value in set \(B\). A function by definition has a single function value for each element of its domain. This means, there cannot be ``one-to-many'' mapping. - -\textbf{Dimensionality}: \({\bf R}^1\) is the set of all real numbers extending from \(-\infty\) to \(+\infty\) --- i.e., the real number line. \({\bf R}^n\) is an \(n\)-dimensional space, where each of the \(n\) axes extends from \(-\infty\) to \(+\infty\). - -\begin{itemize} -\tightlist -\item - \({\bf R}^1\) is a one dimensional line. -\item - \({\bf R}^2\) is a two dimensional plane. -\item - \({\bf R}^3\) is a three dimensional space. -\end{itemize} - -Points in \({\bf R}^n\) are ordered \(n\)-tuples (just means an combination of \(n\) elements where order matters), where each element of the \(n\)-tuple represents the coordinate along that dimension. - -For example: - -\begin{itemize} -\tightlist -\item - \({\bf R}^1\): (3) -\item - \({\bf R}^2\): (-15, 5) -\item - \({\bf R}^3\): (86, 4, 0) -\end{itemize} - -Examples of mapping notation: - -Function of one variable: \(f:{\bf R}^1\to{\bf R}^1\) - -\begin{itemize} -\tightlist -\item - \(f(x)=x+1\). For each \(x\) in \({\bf R}^1\), \(f(x)\) assigns the number \(x+1\). -\end{itemize} - -Function of two variables: \(f: {\bf R}^2\to{\bf R}^1\). - -\begin{itemize} -\tightlist -\item - \(f(x,y)=x^2+y^2\). For each ordered pair \((x,y)\) in \({\bf R}^2\), \(f(x,y)\) assigns the number \(x^2+y^2\). -\end{itemize} - -We often use variable \(x\) as input and another \(y\) as output, e.g.~\(y=x+1\) - -\begin{example}[Functions] -\protect\hypertarget{exm:functions}{}{\label{exm:functions} \iffalse (Functions) \fi{} } -For each of the following, state whether they are one-to-one or many-to-one functions. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - For \(x \in [0,\infty]\), \(f : x \rightarrow x^2\) (this could also be written as \(f(x) = x^2\)). -\item - For \(x \in [-\infty, \infty]\), \(f: x \rightarrow x^2\). -\end{enumerate} -\end{example} - -\begin{exercise}[Functions] -\protect\hypertarget{exr:functions1}{}{\label{exr:functions1} \iffalse (Functions) \fi{} } -For each of the following, state whether they are one-to-one or many-to-one functions. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - For \(x \in [-3, \infty]\), \(f: x \rightarrow x^2\). -\item - For \(x \in [0, \infty]\), \(f: x \rightarrow \sqrt{x}\) -\end{enumerate} -\end{exercise} - -Some functions are defined only on proper subsets of \({\bf R}^n\). - -\begin{itemize} -\tightlist -\item - \textbf{Domain}: the set of numbers in \(X\) at which \(f(x)\) is defined. -\item - \textbf{Range}: elements of \(Y\) assigned by \(f(x)\) to elements of \(X\), or \[f(X)=\{ y : y=f(x), x\in X\}\] - Most often used when talking about a function \(f:{\bf R}^1\to{\bf R}^1\). -\item - \textbf{Image}: same as range, but more often used when talking about a function \(f:{\bf R}^n\to{\bf R}^1\). -\end{itemize} - -Some General Types of Functions - -\textbf{Monomials}: \(f(x)=a x^k\) - -\(a\) is the coefficient. \(k\) is the degree. - -Examples: \(y=x^2\), \(y=-\frac{1}{2}x^3\) - -\textbf{Polynomials}: sum of monomials. - -Examples: \(y=-\frac{1}{2}x^3+x^2\), \(y=3x+5\) - -The degree of a polynomial is the highest degree of its monomial terms. Also, it's often a good idea to write polynomials with terms in decreasing degree. - -\textbf{Exponential Functions}: Example: \(y=2^x\) - -\hypertarget{logexponents}{% -\section{\texorpdfstring{\(\log\) and \(\exp\)}{\textbackslash log and \textbackslash exp}}\label{logexponents}} - -\textbf{Relationship of logarithmic and exponential functions}: -\[y=\log_a(x) \iff a^y=x\] - -The log function can be thought of as an inverse for exponential functions. \(a\) is referred to as the ``base'' of the logarithm. - -\textbf{Common Bases}: The two most common logarithms are base 10 and base \(e\). - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Base 10: \(\quad y=\log_{10}(x) \iff 10^y=x\). The base 10 logarithm is often simply written as ``\(\log(x)\)'' with no base denoted. -\item - Base \(e\): \(\quad y=\log_e(x) \iff e^y=x\). The base \(e\) logarithm is referred to as the ``natural'' logarithm and is written as ``\(\ln(x)\)". -\end{enumerate} - -\begin{comment} - {\includegraphics[width=1in, angle = 270]{ln.eps}} \, {\includegraphics[width=1in, angle = 270]{exp.eps}} - \end{comment} - -\textbf{Properties of exponential functions:} - -\begin{itemize} -\tightlist -\item - \(a^x a^y = a^{x+y}\) -\item - \(a^{-x} = 1/a^x\) -\item - \(a^x/a^y = a^{x-y}\) -\item - \((a^x)^y = a^{x y}\) -\item - \(a^0 = 1\) -\end{itemize} - -\textbf{Properties of logarithmic functions} (any base): - -Generally, when statisticians or social scientists write \(\log(x)\) they mean \(\log_e(x)\). In other words: \(\log_e(x) \equiv \ln(x) \equiv \log(x)\) - -\[\log_a(a^x)=x\] and -\[a^{\log_a(x)}=x\] - -\begin{itemize} -\tightlist -\item - \(\log(x y)=\log(x)+\log(y)\) -\item - \(\log(x^y)=y\log(x)\) -\item - \(\log(1/x)=\log(x^{-1})=-\log(x)\) -\item - \(\log(x/y)=\log(x\cdot y^{-1})=\log(x)+\log(y^{-1})=\log(x)-\log(y)\) -\item - \(\log(1)=\log(e^0)=0\) -\end{itemize} - -\textbf{Change of Base Formula}: Use the change of base formula to switch bases as necessary: -\[\log_b(x) = \frac{\log_a(x)}{\log_a(b)}\] - -Example: \[\log_{10}(x) = \frac{\ln(x)}{\ln(10)}\] - -You can use logs to go between sum and product notation. This will be particularly important when you're learning maximum likelihood estimation. - -\begin{eqnarray*} - \log \bigg(\prod\limits_{i=1}^n x_i \bigg) &=& \log(x_1 \cdot x_2 \cdot x_3 \cdots \cdot x_n)\\ - &=& \log(x_1) + \log(x_2) + \log(x_3) + \cdots + \log(x_n)\\ - &=& \sum\limits_{i=1}^n \log (x_i) -\end{eqnarray*} - -Therefore, you can see that the log of a product is equal to the sum of the logs. We can write this more generally by adding in a constant, \(c\): - -\begin{eqnarray*} - \log \bigg(\prod\limits_{i=1}^n c x_i\bigg) &=& \log(cx_1 \cdot cx_2 \cdots cx_n)\\ - &=& \log(c^n \cdot x_1 \cdot x_2 \cdots x_n)\\ - &=& \log(c^n) + \log(x_1) + \log(x_2) + \cdots + \log(x_n)\\\\ - &=& n \log(c) + \sum\limits_{i=1}^n \log (x_i)\\ -\end{eqnarray*} - -\begin{example}[Logarithmic Functions] -\protect\hypertarget{exm:log}{}{\label{exm:log} \iffalse (Logarithmic Functions) \fi{} } -Evaluate each of the following logarithms - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\log_4(16)\) -\item - \(\log_2(16)\) -\end{enumerate} - -Simplify the following logarithm. By ``simplify'', we actually really mean - use as many of the logarithmic properties as you can. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{2} -\tightlist -\item - \(\log_4(x^3y^5)\) -\end{enumerate} -\end{example} - -\begin{exercise}[Logarithmic Functions] -\protect\hypertarget{exr:log1}{}{\label{exr:log1} \iffalse (Logarithmic Functions) \fi{} } - -Evaluate each of the following logarithms - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\log_\frac{3}{2}(\frac{27}{8})\) -\end{enumerate} - -Simplify each of the following logarithms. By ``simplify'', we actually really mean - use as many of the logarithmic properties as you can. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{1} -\item - \(\log(\frac{x^9y^5}{z^3})\) -\item - \(\ln{\sqrt{xy}}\) -\end{enumerate} -\end{exercise} - -\hypertarget{graphing-functions}{% -\section{Graphing Functions}\label{graphing-functions}} - -What can a graph tell you about a function? - -\begin{itemize} -\tightlist -\item - Is the function increasing or decreasing? Over what part of the domain? -\item - How ``fast" does it increase or decrease? -\item - Are there global or local maxima and minima? Where? -\item - Are there inflection points? -\item - Is the function continuous? -\item - Is the function differentiable? -\item - Does the function tend to some limit? -\item - Other questions related to the substance of the problem at hand. -\end{itemize} - -\hypertarget{solving-for-variables-and-finding-roots}{% -\section{Solving for Variables and Finding Roots}\label{solving-for-variables-and-finding-roots}} - -Sometimes we're given a function \(y=f(x)\) and we want to find how \(x\) varies as a function of \(y\). Use algebra to move \(x\) to the left hand side (LHS) of the equation and so that the right hand side (RHS) is only a function of \(y\). - -\begin{example}[Solving for Variables] -\protect\hypertarget{exm:solvevar}{}{\label{exm:solvevar} \iffalse (Solving for Variables) \fi{} } - -Solve for x: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(y=3x+2\) -\item - \(y=e^x\) -\end{enumerate} -\end{example} - -Solving for variables is especially important when we want to find the \textbf{roots} of an equation: those values of variables that cause an equation to equal zero. Especially important in finding equilibria and in doing maximum likelihood estimation. - -Procedure: Given \(y=f(x)\), set \(f(x)=0\). Solve for \(x\). - -Multiple Roots: -\[f(x)=x^2 - 9 \quad\Longrightarrow\quad 0=x^2 - 9 \quad\Longrightarrow\quad 9=x^2 \quad\Longrightarrow\quad \pm \sqrt{9}=\sqrt{x^2} \quad\Longrightarrow\quad \pm 3=x\] - -\textbf{Quadratic Formula:} For quadratic equations \(ax^2+bx+c=0\), use the quadratic formula: \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\] - -\begin{exercise}[Finding Roots] -\protect\hypertarget{exr:solvevar1}{}{\label{exr:solvevar1} \iffalse (Finding Roots) \fi{} } -Solve for x: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(f(x)=3x+2 = 0\) -\item - \(f(x)=x^2+3x-4=0\) -\item - \(f(x)=e^{-x}-10 = 0\) -\end{enumerate} -\end{exercise} - -\hypertarget{sets}{% -\section{Sets}\label{sets}} - -\textbf{Interior Point}: The point \(\bf x\) is an interior point of the set \(S\) if \(\bf x\) is in \(S\) and if there is some \(\epsilon\)-ball around \(\bf x\) that contains only points in \(S\). The \textbf{interior} of \(S\) is the collection of all interior points in \(S\). The interior can also be defined as the union of all open sets in \(S\). - -\begin{itemize} -\tightlist -\item - If the set \(S\) is circular, the interior points are everything inside of the circle, but not on the circle's rim. -\item - Example: The interior of the set \(\{ (x,y) : x^2+y^2\le 4 \}\) is \(\{ (x,y) : x^2+y^2< 4 \}\) . -\end{itemize} - -\textbf{Boundary Point}: The point \(\bf x\) is a boundary point of the set \(S\) if every \(\epsilon\)-ball around \(\bf x\) contains both points that are in \(S\) and points that are outside \(S\). The \textbf{boundary} is the collection of all boundary points. - -\begin{itemize} -\tightlist -\item - If the set \(S\) is circular, the boundary points are everything on the circle's rim. -\item - Example: The boundary of \(\{ (x,y) : x^2+y^2\le 4 \}\) is \(\{ (x,y) : x^2+y^2 = 4 \}\). -\end{itemize} - -\textbf{Open}: A set \(S\) is open if for each point \(\bf x\) in \(S\), there exists an open \(\epsilon\)-ball around \(\bf x\) completely contained in \(S\). - -\begin{itemize} -\tightlist -\item - If the set \(S\) is circular and open, the points contained within the set get infinitely close to the circle's rim, but do not touch it. -\item - Example: \(\{ (x,y) : x^2+y^2<4 \}\) -\end{itemize} - -\textbf{Closed}: A set \(S\) is closed if it contains all of its boundary points. - -\begin{itemize} -\tightlist -\item - Alternatively: A set is closed if its complement is open. -\item - If the set \(S\) is circular and closed, the set contains all points within the rim as well as the rim itself. -\item - Example: \(\{ (x,y) : x^2+y^2\le 4 \}\) -\item - Note: a set may be neither open nor closed. Example: \(\{ (x,y) : 2 < x^2+y^2\le 4 \}\) -\end{itemize} - -\textbf{Complement}: The complement of set \(S\) is everything outside of \(S\). - -\begin{itemize} -\tightlist -\item - If the set \(S\) is circular, the complement of \(S\) is everything outside of the circle. -\item - Example: The complement of \(\{ (x,y) : x^2+y^2\le 4 \}\) is \(\{ (x,y) : x^2+y^2 > 4 \}\). -\end{itemize} - -\textbf{Empty}: The empty (or null) set is a unique set that has no elements, denoted by \{\} or \(\emptyset\). - -\begin{itemize} -\tightlist -\item - The empty set is an example of a set that is open and closed, or a ``clopen'' set. -\item - Examples: The set of squares with 5 sides; the set of countries south of the South Pole. -\end{itemize} - -\hypertarget{answers-to-examples-and-exercises}{% -\section*{Answers to Examples and Exercises}\label{answers-to-examples-and-exercises}} -\addcontentsline{toc}{section}{Answers to Examples and Exercises} - -Answer to Example \ref{exm:operators}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - 1 + 2 + 3 + 4 + 5 = 15 -\item - 1 * 2 * 3 * 4 * 5 = 120 -\item - 2 -\item - 4 * 3 * 2 * 1 = 24 -\end{enumerate} - -Answer to Exercise \ref{exr:operators1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - 7(4 + 3 + 7) = 98 -\item - 2 + 2 + 2 + 2 + 2 = 10 -\item - \(2^3(7)(11)(2)\) = 1232 -\end{enumerate} - -Answer to Example \ref{exm:functions}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - one-to-one -\item - many-to-one -\end{enumerate} - -Answer to Exercise \ref{exr:functions1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - many-to-one -\item - one-to-one -\end{enumerate} - -Answer to Example \ref{exm:log}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - 2 -\item - 4 -\item - \(3\log_4(x) + 5\log_4(y)\) -\end{enumerate} - -Answer to Exercise \ref{exr:log1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - 3 -\item - \(9\log(x) + 5\log(y) - 3\log(z)\) -\item - \(\frac{1}{2}(\ln{x} + \ln{y})\) -\end{enumerate} - -Answer to Example \ref{exm:solvevar}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(y=3x+2 \quad\Longrightarrow\quad -3x=2-y \quad\Longrightarrow\quad 3x=y-2 \quad\Longrightarrow\quad x=\frac{1}{3}(y-2)\) -\item - \(x = \ln{y}\) -\end{enumerate} - -Answer to Exercise \ref{exr:solvevar1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\frac{-2}{3}\) -\item - x = \{1, -4\} -\item - x = - \(\ln10\) -\end{enumerate} - -\hypertarget{limits-precalc}{% -\chapter{Limits}\label{limits-precalc}} - -Solving limits, i.e.~finding out the value of functions as its input moves closer to some value, is important for the social scientist's mathematical toolkit for two related tasks. The first is for the study of calculus, which will be in turn useful to show where certain functions are maximized or minimized. The second is for the study of statistical inference, which is the study of inferring things about things you cannot see by using things you can see. - -\hypertarget{example-the-central-limit-theorem}{% -\section*{Example: The Central Limit Theorem}\label{example-the-central-limit-theorem}} -\addcontentsline{toc}{section}{Example: The Central Limit Theorem} - -Perhaps the most important theorem in statistics is the Central Limit Theorem, - -\begin{theorem}[Central Limit Theorem (i.i.d. case)] -\protect\hypertarget{thm:clt-lim}{}{\label{thm:clt-lim} \iffalse (Central Limit Theorem (i.i.d. case)) \fi{} }For any series of independent and identically distributed random variables \(X_1, X_2, \cdots\), we know the distribution of its sum even if we do not know the distribution of \(X\). The distribution of the sum is a Normal distribution. - -\[\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),\] - -where \(\mu\) is the mean of \(X\) and \(\sigma\) is the standard deviation of \(X\). The arrow is read as ``converges in distribution to''. \(\text{Normal}(0, 1)\) indicates a Normal Distribution with mean 0 and variance 1. - -That is, the limit of the distribution of the lefthand side is the distribution of the righthand side. -\end{theorem} - -The sign of a limit is the arrow ``\(\rightarrow\)''. Although we have not yet covered probability (in Section \ref{probability-theory}) so we have not described what distributions and random variables are, it is worth foreshadowing the Central Limit Theorem. The Central Limit Theorem is powerful because it gives us a \emph{guarantee} of what would happen if \(n \rightarrow \infty\), which in this case means we collected more data. - -\hypertarget{example-the-law-of-large-numbers}{% -\section*{Example: The Law of Large Numbers}\label{example-the-law-of-large-numbers}} -\addcontentsline{toc}{section}{Example: The Law of Large Numbers} - -A finding that perhaps rivals the Central Limit Theorem is the Law of Large Numbers: - -\begin{theorem}[(Weak) Law of Large Numbers] -\protect\hypertarget{thm:lln-lim}{}{\label{thm:lln-lim} \iffalse ((Weak) Law of Large Numbers) \fi{} }For any draw of identically distributed independent variables with mean \(\mu\), the sample average after \(n\) draws, \(\bar{X}_n\), converges in probability to the true mean as \(n \rightarrow \infty\): - -\[\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0\] - -A shorthand of which is \(\bar{X}_n \xrightarrow{p} \mu\), where the arrow is read as ``converges in probability to''. -\end{theorem} - -Intuitively, the more data, the more accurate is your guess. For example, the Figure \ref{fig:llnsim} shows how the sample average from many coin tosses converges to the true value : 0.5. - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/llnsim-1.pdf} -\caption{\label{fig:llnsim}As the number of coin tosses goes to infinity, the average probabiity of heads converges to 0.5} -\end{figure} - -\hypertarget{sequences}{% -\section{Sequences}\label{sequences}} - -We need a couple of steps until we get to limit theorems in probability. First we will introduce a ``sequence'', then we will think about the limit of a sequence, then we will think about the limit of a \emph{function}. - -A \textbf{sequence} \[\{x_n\}=\{x_1, x_2, x_3, \ldots, x_n\}\] is an ordered set of real numbers, where \(x_1\) is the first term in the sequence and \(y_n\) is the \(n\)th term. Generally, a sequence is infinite, that is it extends to \(n=\infty\). We can also write the sequence as \[\{x_n\}^\infty_{n=1}\] - -where the subscript and superscript are read together as ``from 1 to infinity.'' - -\begin{example}[Sequences] -\protect\hypertarget{exm:seqbehav}{}{\label{exm:seqbehav} \iffalse (Sequences) \fi{} } -How does these sequence behave? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\}\) -\item - \(\{B_n\}=\left\{\frac{n^2+1}{n} \right\}\) -\item - \(\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\}\) -\end{enumerate} -\end{example} - -We find the sequence by simply ``plugging in'' the integers into each \(n\). The important thing is to get a sense of how these numbers are going to change. Example 1's numbers seem to come closer and closer to 2, but will it ever surpass 2? Example 2's numbers are also increasing each time, but will it hit a limit? What is the pattern in Example 3? Graphing helps you make this point more clearly. See the sequence of \(n = 1, ...20\) for each of the three examples in Figure \ref{fig:seqabc}. - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/seqabc-1.pdf} -\caption{\label{fig:seqabc}Behavior of Some Sequences} -\end{figure} - -\hypertarget{the-limit-of-a-sequence}{% -\section{The Limit of a Sequence}\label{the-limit-of-a-sequence}} - -The notion of ``converging to a limit'' is the behavior of the points in Example \ref{exm:seqbehav}. In some sense, that's the counterfactual we want to know. What happens as \(n\rightarrow \infty\)? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Sequences like 1 above that converge to a limit. -\item - Sequences like 2 above that increase without bound. -\item - Sequences like 3 above that neither converge nor increase without bound --- alternating over the number line. -\end{enumerate} - -\begin{definition} -\protect\hypertarget{def:unnamed-chunk-2}{}{\label{def:unnamed-chunk-2} }The sequence \(\{y_n\}\) has the limit \(L\), which we write as \[\lim\limits_{n \to \infty} y_n =L\], if for any \(\epsilon>0\) there is an integer \(N\) (which depends on \(\epsilon\)) with the property that \(|y_n -L|<\epsilon\) for each \(n>N\). \(\{y_n\}\) is said to converge to \(L\). If the above does not hold, then \(\{y_n\}\) diverges. -\end{definition} - -We can also express the behavior of a sequence as bounded or not: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Bounded: if \(|y_n|\le K\) for all \(n\) -\item - Monotonically Increasing: \(y_{n+1}>y_n\) for all \(n\) -\item - Monotonically Decreasing: \(y_{n+1}0\) such that if \(0<|x-c|<\delta\), then \(|f(x)-L|<\epsilon\). -\end{definition} - -A neat, if subtle result is that \(f(x)\) does not necessarily have to be defined at \(c\) for \(\lim\limits_{x \to c}\) to exist. - -Properties: Let \(f\) and \(g\) be functions with \(\lim\limits_{x \to c} f(x)=k\) and \(\lim\limits_{x \to c} g(x)=\ell\). - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\lim\limits_{x \to c}[f(x)+g(x)]=\lim\limits_{x \to c} f(x)+ \lim\limits_{x \to c} g(x)\) -\item - \(\lim\limits_{x \to c} kf(x) = k\lim\limits_{x \to c} f(x)\) -\item - \(\lim\limits_{x \to c} f(x) g(x) = \left[\lim\limits_{x \to c} f(x)\right]\cdot \left[\lim\limits_{x \to c} g(x)\right]\) -\item - \(\lim\limits_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)}\), provided \(\lim\limits_{x \to c} g(x)\ne 0\). -\end{enumerate} - -Simple limits of functions can be solved as we did limits of sequences. Just be careful which part of the function is changing. - -\begin{example}[Limits of Functions] -\protect\hypertarget{exm:limfun1}{}{\label{exm:limfun1} \iffalse (Limits of Functions) \fi{} }Find the limit of the following functions. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\lim_{x \to c} k\) -\item - \(\lim_{x \to c} x\) -\item - \(\lim_{x\to 2} (2x-3)\) -\item - \(\lim_{x \to c} x^n\) -\end{enumerate} -\end{example} - -Limits can get more complex in roughly two ways. First, the functions may become large polynomials with many moving pieces. Second,the functions may become discontinuous. - -The function can be thought of as a more general or ``smooth'' version of sequences. For example, - -\begin{exercise}[Limits of a Fraction of Functions] -\protect\hypertarget{exr:limfunmax}{}{\label{exr:limfunmax} \iffalse (Limits of a Fraction of Functions) \fi{} } -Find the limit of - -\[\lim_{x\to\infty} \frac{(x^4 +3x−99)(2−x^5)}{(18x^7 +9x^6 −3x^2 −1)(x+1)}\] -\end{exercise} - -Now, the functions will become a bit more complex: - -\begin{exercise} -\protect\hypertarget{exr:discontlim}{}{\label{exr:discontlim} }Solve the following limits of functions - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\lim\limits_{x\to 0} |x|\) -\item - \(\lim\limits_{x\to 0} \left(1+\frac{1}{x^2}\right)\) -\end{enumerate} -\end{exercise} - -So there are a few more alternatives about what a limit of a function could be: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Right-hand limit: The value approached by \(f(x)\) when you move from right to left. -\item - Left-hand limit: The value approached by \(f(x)\) when you move from left to right. -\item - Infinity: The value approached by \(f(x)\) as x grows infinitely large. Sometimes this may be a number; sometimes it might be \(\infty\) or \(-\infty\). -\item - Negative infinity: The value approached by \(f(x)\) as x grows infinitely negative. Sometimes this may be a number; sometimes it might be \(\infty\) or \(-\infty\). -\end{enumerate} - -The distinction between left and right becomes important when the function is not determined for some values of \(x\). What are those cases in the examples below? - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-6-1.pdf} -\caption{\label{fig:unnamed-chunk-6}Functions which are not defined in some areas} -\end{figure} - -\hypertarget{continuity}{% -\section{Continuity}\label{continuity}} - -To repeat a finding from the limits of functions: \(f(x)\) does not necessarily have to be defined at \(c\) for \(\lim\limits_{x \to c}\) to exist. Functions that have breaks in their lines are called discontinuous. Functions that have no breaks are called continuous. Continuity is a concept that is more fundamental to, but related to that of ``differentiability'', which we will cover next in calculus. - -\begin{definition}[Continuity] -\protect\hypertarget{def:unnamed-chunk-7}{}{\label{def:unnamed-chunk-7} \iffalse (Continuity) \fi{} }Suppose that the domain of the function \(f\) includes an open interval containing the point \(c\). Then \(f\) is continuous at \(c\) if \(\lim\limits_{x \to c} f(x)\) exists and if \(\lim\limits_{x \to c} f(x)=f(c)\). Further, \(f\) is continuous on an open interval \((a,b)\) if it is continuous at each point in the interval. -\end{definition} - -To prove that a function is continuous for all points is beyond this practical introduction to math, but the general intuition can be grasped by graphing. - -\begin{example}[Continuous and Discontinuous Functions] -\protect\hypertarget{exm:contdiscont}{}{\label{exm:contdiscont} \iffalse (Continuous and Discontinuous Functions) \fi{} } -For each function, determine if it is continuous or discontinuous. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x) = \sqrt{x}\) -\item - \(f(x) = e^x\) -\item - \(f(x) = 1 + \frac{1}{x^2}\) -\item - \(f(x) = \text{floor}(x)\). -\end{enumerate} - -The floor is the smaller of the two integers bounding a number. So \(\text{floor}(x = 2.999) = 2\), \(\text{floor}(x = 2.0001) = 2\), and \(\text{floor}(x = 2) = 2.\) -\end{example} - -\begin{solution} -\iffalse{} {Solution. } \fi{}In Figure \ref{fig:fig-contdiscont}, we can see that the first two functions are continuous, and the next two are discontinuous. \(f(x) = 1 + \frac{1}{x^2}\) is discontinuous at \(x= 0\), and \(f(x) = \text{floor}(x)\) is discontinuous at each whole number. -\end{solution} - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/fig-contdiscont-1.pdf} -\caption{\label{fig:fig-contdiscont}Continuous and Discontinuous Functions} -\end{figure} - -Some properties of continuous functions: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - If \(f\) and \(g\) are continuous at point \(c\), then \(f+g\), \(f-g\), \(f \cdot g\), \(|f|\), and \(\alpha f\) are continuous at point \(c\) also. \(f/g\) is continuous, provided \(g(c)\ne 0\). -\item - Boundedness: If \(f\) is continuous on the closed bounded interval \([a,b]\), then there is a number \(K\) such that \(|f(x)|\le K\) for each \(x\) in \([a,b]\). -\item - Max/Min: If \(f\) is continuous on the closed bounded interval \([a,b]\), then \(f\) has a maximum and a minimum on \([a,b]\). They may be located at the end points. -\end{enumerate} - -\begin{exercise}[Limit when Denominator converges to 0] -\protect\hypertarget{exr:discontdraw}{}{\label{exr:discontdraw} \iffalse (Limit when Denominator converges to 0) \fi{} } -Let \[f(x) = \frac{x^2 + 2x}{x}.\] - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Graph the function. Is it defined everywhere? -\item - What is the functions limit at \(x \rightarrow 0\)? -\end{enumerate} -\end{exercise} - -\hypertarget{answers-to-examples}{% -\section*{Answers to Examples}\label{answers-to-examples}} -\addcontentsline{toc}{section}{Answers to Examples} - -Example \ref{exm:seqbehav} - -\begin{solution} -\iffalse{} {Solution. } \fi{} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\{A_n\}=\left\{ 2-\frac{1}{n^2} \right\} = \left\{1, \frac{7}{4}, \frac{17}{9}, \frac{31}{16}, \frac{49}{25}, \ldots\right\} = 2\) -\item - \(\{B_n\}=\left\{\frac{n^2+1}{n} \right\} = \left\{2, \frac{5}{2}, \frac{10}{3}, \frac{17}{4}..., \right\}\) -\item - \(\{C_n\}=\left\{(-1)^n \left(1-\frac{1}{n}\right) \right\} = \left\{0, \frac{1}{2}, -\frac{2}{3}, \frac{3}{4}, -\frac{4}{5}\right\}\) -\end{enumerate} -\end{solution} - -Exercise \ref{exr:limseq2} - -Example \ref{exm:limfun1} - -\begin{solution} -\iffalse{} {Solution. } \fi{} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(k\) -\item - \(c\) -\item - \(\lim_{x\to 2} (2x-3) = 2\lim\limits_{x\to 2} x - 3\lim\limits_{x\to 2} 1 = 1\) -\item - \(\lim_{x \to c} x^n = \lim\limits_{x \to c} x \cdots[\lim\limits_{x \to c} x] = c\cdots c =c^n\) -\end{enumerate} -\end{solution} - -Exercise \ref{exr:limfunmax} - -\begin{solution} -\iffalse{} {Solution. } \fi{}Although this function seems large, the thing our eyes should focus on is where the highest order polynomial remains. That will grow the fastest, so if the highest order term is on the denominator, the fraction will converge to 0, if it is on the numerator it will converge to negative infinity. Previewing the multiplication by hand, we can see that the \(-x^9\) on the numerator will be the largest power. So the answer will be \(-\infty\). We can also confirm this by writing out fractions: - -\begin{align*} -& \lim_{x\to\infty}\frac{\left(1 + \frac{3}{x^3} - \frac{99}{4x^4}\right)\left(-\frac{2}{x^5} + 1\right)}{\left(1 + \frac{9}{18x} - \frac{3}{18x^5} - \frac{1}{18x^7} \right)\left(1 + \frac{1}{x}\right)} \\ -&\times \frac{x^4}{1} \times -\frac{x^5}{1} \times \frac{1}{18x^7}\times \frac{1}{x}\\ -=& 1 \times \lim_{-x\to\infty} \frac{x}{18} -\end{align*} -\end{solution} - -Exercise \ref{exr:discontdraw} - -\begin{solution} -\iffalse{} {Solution. } \fi{} -See Figure \ref{fig:fig-hole-0}. - -Divide each part by \(x\), and we get \(x + \frac{2}{x}\) on the numerator, \(1\) on the denominator. So, without worrying about a function being not defined, we can say \(\lim_{x\to 0}f(x) = 0\). -\end{solution} - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/fig-hole-0-1.pdf} -\caption{\label{fig:fig-hole-0}A function undedefined at x = 0} -\end{figure} - -\hypertarget{derivatives}{% -\chapter{Calculus}\label{derivatives}} - -Calculus is a fundamental part of any type of statistics exercise. Although you may not be taking derivatives and integral in your daily work as an analyst, calculus undergirds many concepts we use: maximization, expectation, and cumulative probability. - -\hypertarget{example-the-mean-is-a-type-of-integral}{% -\section*{Example: The Mean is a Type of Integral}\label{example-the-mean-is-a-type-of-integral}} -\addcontentsline{toc}{section}{Example: The Mean is a Type of Integral} - -The average of a quantity is a type of weighted mean, where the potential values are weighted by their likelihood, loosely speaking. The integral is actually a general way to describe this weighted average when there are conceptually an infinite number of potential values. - -If \(X\) is a continuous random variable, its expected value \(E(X)\) -- the center of mass -- is given by - -\[E(X) = \int^{\infty}_{-\infty}x f(x) dx\] - -where \(f(x)\) is the probability density function of \(X\). - -This is a continuous version of the case where \(X\) is discrete, in which case - -\[E(X) = \sum^\infty_{j=1} x_j P(X = x_j)\] - -even more concretely, if the potential values of \(X\) are finite, then we can write out the expected value as a weighted mean, where the weights is the probability that the value occurs. - -\[E(X) = \large \sum_{x} \quad\left( \underbrace{x}_{\text{value}}\cdot \underbrace{P(X = x)}_{\text{weight, or PMF}}\right)\] - -\hypertarget{derivintro}{% -\section{Derivatives}\label{derivintro}} - -The derivative of \(f\) at \(x\) is its rate of change at \(x\): how much \(f(x)\) changes with a change in \(x\). The rate of change is a fraction --- rise over run --- but because not all lines are straight and the rise over run formula will give us different values depending on the range we examine, we need to take a limit (Section \ref{limits-precalc}). - -\begin{definition}[Derivative] -\protect\hypertarget{def:unnamed-chunk-15}{}{\label{def:unnamed-chunk-15} \iffalse (Derivative) \fi{} }Let \(f\) be a function whose domain includes an open interval containing the point \(x\). The derivative of \(f\) at \(x\) is given by - -\[\frac{d}{dx}f(x) =\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x} = \lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h} -\] - -There are a two main ways to denote a derivate: - -\begin{itemize} -\tightlist -\item - Leibniz Notation: \(\frac{d}{dx}(f(x))\) -\item - Prime or Lagrange Notation: \(f'(x)\) -\end{itemize} -\end{definition} - -If \(f(x)\) is a straight line, the derivative is the slope. For a curve, the slope changes by the values of \(x\), so the derivative is the slope of the line tangent to the curve at \(x\). See, For example, Figure \ref{fig:derivsimple}. - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/derivsimple-1.pdf} -\caption{\label{fig:derivsimple}The Derivative as a Slope} -\end{figure} - -If \(f'(x)\) exists at a point \(x_0\), then \(f\) is said to be \textbf{differentiable} at \(x_0\). That also implies that \(f(x)\) is continuous at \(x_0\). - -\hypertarget{properties-of-derivatives}{% -\subsection*{Properties of derivatives}\label{properties-of-derivatives}} -\addcontentsline{toc}{subsection}{Properties of derivatives} - -Suppose that \(f\) and \(g\) are differentiable at \(x\) and that \(\alpha\) is a constant. Then the functions \(f\pm g\), \(\alpha f\), \(f g\), and \(f/g\) (provided \(g(x)\ne 0\)) are also differentiable at \(x\). Additionally, - -\textbf{Constant rule:} \[\left[k f(x)\right]' = k f'(x)\] - -\textbf{Sum rule:} \[\left[f(x)\pm g(x)\right]' = f'(x)\pm g'(x)\] - -With a bit more algebra, we can apply the definition of derivatives to get a formula for of the derivative of a product and a derivative of a quotient. - -\textbf{Product rule:} \[\left[f(x)g(x)\right]^\prime = f^\prime(x)g(x)+f(x)g^\prime(x)\] - -\textbf{Quotient rule:} \[\left[f(x)/g(x)\right]^\prime = \frac{f^\prime(x)g(x) - f(x)g^\prime(x)}{[g(x)]^2}, ~g(x)\neq 0\] - -Finally, one way to think of the power of derivatives is that it takes a function a notch down in complexity. The power rule applies to any higher-order function: - -\textbf{Power rule:} \[\left[x^k\right]^\prime = k x^{k-1}\] - -For any real number \(k\) (that is, both whole numbers and fractions). The power rule is proved \textbf{by induction}, a neat method of proof used in many fundamental applications to prove that a general statement holds for every possible case, even if there are countably infinite cases. We'll show a simple case where \(k\) is an integer here. - -\begin{proof}[Proof of Power Rule by Induction] -\iffalse{} {Proof (Proof of Power Rule by Induction). } \fi{} -We would like to prove that - -\[\left[x^k\right]^\prime = k x^{k-1}\] -for any integer \(k\). - -First, consider the first case (the base case) of \(k = 1\). We can show by the definition of derivatives (setting \(f(x) = x^1 = 1\)) that - -\[[x^1]^\prime = \lim_{h \rightarrow 0}\frac{(x + h) - x}{(x + h) - x}= 1.\] - -Because \(1\) is also expressed as \(1 x^{1- 1}\), the statement we want to prove holds for the case \(k =1\). - -Now, \emph{assume} that the statement holds for some integer \(m\). That is, assume -\[\left[x^m\right]^\prime = m x^{m-1}\] - -Then, for the case \(m + 1\), using the product rule above, we can simplify - -\begin{align*} -\left[x^{m + 1}\right]^\prime &= [x^{m}\cdot x]^\prime\\ -&= (x^m)^\prime\cdot x + (x^m)\cdot (x)^\prime\\ -&= m x^{m - 1}\cdot x + x^m ~~\because \text{by previous assumption}\\ -&= mx^m + x^m\\ -&= (m + 1)x^m\\ -&= (m + 1)x^{(m + 1) - 1} -\end{align*} - -Therefore, the rule holds for the case \(k = m + 1\) once we have assumed it holds for \(k = m\). Combined with the first case, this completes proof by induction -- we have now proved that the statement holds for all integers \(k = 1, 2, 3, \cdots\). - -To show that it holds for real fractions as well, we can prove expressing that exponent by a fraction of two integers. -\end{proof} - -These ``rules'' become apparent by applying the definition of the derivative above to each of the things to be ``derived'', but these come up so frequently that it is best to repeat until it is muscle memory. - -\begin{exercise}[Derivative of Polynomials] -\protect\hypertarget{exr:introderivatives}{}{\label{exr:introderivatives} \iffalse (Derivative of Polynomials) \fi{} } -For each of the following functions, find the first-order derivative \(f^\prime(x)\). - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x)=c\) -\item - \(f(x)=x\) -\item - \(f(x)=x^2\) -\item - \(f(x)=x^3\) -\item - \(f(x)=\frac{1}{x^2}\) -\item - \(f(x)=(x^3)(2x^4)\) -\item - \(f(x) = x^4 - x^3 + x^2 - x + 1\) -\item - \(f(x) = (x^2 + 1)(x^3 - 1)\) -\item - \(f(x) = 3x^2 + 2x^{1/3}\) -\item - \(f(x)=\frac{x^2+1}{x^2-1}\) -\end{enumerate} -\end{exercise} - -\hypertarget{derivpoly}{% -\section{Higher-Order Derivatives (Derivatives of Derivatives of Derivatives)}\label{derivpoly}} - -The first derivative is applying the definition of derivatives on the function, and it can be expressed as - -\[f'(x), ~~ y', ~~ \frac{d}{dx}f(x), ~~ \frac{dy}{dx}\] - -We can keep applying the differentiation process to functions that are themselves derivatives. The derivative of \(f'(x)\) with respect to \(x\), would then be \[f''(x)=\lim\limits_{h\to 0}\frac{f'(x+h)-f'(x)}{h}\] and we can therefore call it the \textbf{Second derivative:} - -\[f''(x), ~~ y'', ~~ \frac{d^2}{dx^2}f(x), ~~ \frac{d^2y}{dx^2}\] - -Similarly, the derivative of \(f''(x)\) would be called the third derivative and is denoted \(f'''(x)\). And by extension, the \textbf{nth derivative} is expressed as \(\frac{d^n}{dx^n}f(x)\), \(\frac{d^ny}{dx^n}\). - -\begin{example}[Succession of Derivatives] -\protect\hypertarget{exm:unnamed-chunk-16}{}{\label{exm:unnamed-chunk-16} \iffalse (Succession of Derivatives) \fi{} }\begin{align*} -f(x) &=x^3\\ -f^{\prime}(x) &=3x^2\\ -f^{\prime\prime}(x) &=6x \\ -f^{\prime\prime\prime}(x) &=6\\ -f^{\prime\prime\prime\prime}(x) &=0\\ -\end{align*} -\end{example} - -Earlier, in Section \ref{derivintro}, we said that if a function differentiable at a given point, then it must be continuous. Further, if \(f'(x)\) is itself continuous, then \(f(x)\) is called continuously differentiable. All of this matters because many of our findings about optimization (Section \ref{optim}) rely on differentiation, and so we want our function to be differentiable in as many layers. A function that is continuously differentiable infinitly is called ``smooth''. Some examples: \(f(x) = x^2\), \(f(x) = e^x\). - -\hypertarget{composite-functions-and-the-chain-rule}{% -\section{Composite Functions and the Chain Rule}\label{composite-functions-and-the-chain-rule}} - -As useful as the above rules are, many functions you'll see won't fit neatly in each case immediately. Instead, they will be functions of functions. For example, the difference between \(x^2 + 1^2\) and \((x^2 + 1)^2\) may look trivial, but the sum rule can be easily applied to the former, while it's actually not obvious what do with the latter. - -\textbf{Composite functions} are formed by substituting one function into another and are denoted by \[(f\circ g)(x)=f[g(x)].\] To form \(f[g(x)]\), the range of \(g\) must be contained (at least in part) within the domain of \(f\). The domain of \(f\circ g\) consists of all the points in the domain of \(g\) for which \(g(x)\) is in the domain of \(f\). - -\begin{example} -\protect\hypertarget{exm:unnamed-chunk-17}{}{\label{exm:unnamed-chunk-17} }Let \(f(x)=\log x\) for \(0 0\] -\end{theorem} - -We will relegate the proofs to small excerpts. - -\hypertarget{derivatives-of-natural-exponential-function-e}{% -\subsection*{\texorpdfstring{Derivatives of natural exponential function (\(e\))}{Derivatives of natural exponential function (e)}}\label{derivatives-of-natural-exponential-function-e}} -\addcontentsline{toc}{subsection}{Derivatives of natural exponential function (\(e\))} - -To repeat the main rule in Theorem \ref{thm:derivexplog}, the intuition is that - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Derivative of \(e^x\) is itself: \(\frac{d}{dx}e^x = e^x\) (See Figure \ref{fig:fig-derivexponent}) -\item - Same thing if there were a constant in front: \(\frac{d}{dx}\alpha e^x = \alpha e^x\) -\item - Same thing no matter how many derivatives there are in front: \(\frac{d^n}{dx^n} \alpha e^x = \alpha e^x\) -\item - Chain Rule: When the exponent is a function of \(x\), remember to take derivative of that function and add to product. \(\frac{d}{dx}e^{g(x)}= e^{g(x)} g^\prime(x)\) -\end{enumerate} - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/fig-derivexponent-1.pdf} -\caption{\label{fig:fig-derivexponent}Derivative of the Exponential Function} -\end{figure} - -\begin{example}[Derivative of exponents] -\protect\hypertarget{exm:exmderivexp}{}{\label{exm:exmderivexp} \iffalse (Derivative of exponents) \fi{} }Find the derivative for the following. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x)=e^{-3x}\) -\item - \(f(x)=e^{x^2}\) -\item - \(f(x)=(x-1)e^x\) -\end{enumerate} -\end{example} - -\hypertarget{derivatives-of-log}{% -\subsection*{\texorpdfstring{Derivatives of \(\log\)}{Derivatives of \textbackslash log}}\label{derivatives-of-log}} -\addcontentsline{toc}{subsection}{Derivatives of \(\log\)} - -The natural log is the mirror image of the natural exponent and has mirroring properties, again, to repeat the theorem, - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - log prime x is one over x: \(\frac{d}{dx} \log x = \frac{1}{x}\) (Figure \ref{fig:fig-derivlog}) -\item - Exponents become multiplicative constants: \(\frac{d}{dx} \log x^k = \frac{d}{dx} k \log x = \frac{k}{x}\) -\item - Chain rule again: \(\frac{d}{dx} \log u(x) = \frac{u'(x)}{u(x)}\quad\) -\item - For any positive base \(b\), \(\frac{d}{dx} b^x = (\log b)\left(b^x\right)\). -\end{enumerate} - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/fig-derivlog-1.pdf} -\caption{\label{fig:fig-derivlog}Derivative of the Natural Log} -\end{figure} - -\begin{example}[Derivative of logs] -\protect\hypertarget{exm:exmderivlog}{}{\label{exm:exmderivlog} \iffalse (Derivative of logs) \fi{} }Find \(dy/dx\) for the following. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x)=\log(x^2+9)\) -\item - \(f(x)=\log(\log x)\) -\item - \(f(x)=(\log x)^2\) -\item - \(f(x)=\log e^x\) -\end{enumerate} -\end{example} - -\hypertarget{outline-of-proof}{% -\subsection*{Outline of Proof}\label{outline-of-proof}} -\addcontentsline{toc}{subsection}{Outline of Proof} - -We actually show the derivative of the log first, and then the derivative of the exponential naturally follows. - -The general derivative of the log at any base \(a\) is solvable by the definition of derivatives. - -\begin{align*} -(\log_a x)^\prime = \lim\limits_{h\to 0} \frac{1}{h}\log_{a}\left(1 + \frac{h}{x}\right) -\end{align*} - -Re-express \(g = \frac{h}{x}\) and get -\begin{align*} -(\log_a x)^\prime &= \frac{1}{x}\lim_{g\to 0}\log_{a} (1 + g)^{\frac{1}{g}}\\ -&= \frac{1}{x}\log_a e -\end{align*} - -By definition of \(e\). As a special case, when \(a = e\), then \((\log x)^\prime = \frac{1}{x}\). - -Now let's think about the inverse, taking the derivative of \(y = a^x\). - -\begin{align*} -y &= a^x \\ -\Rightarrow \log y &= x \log a\\ -\Rightarrow \frac{y^\prime}{y} &= \log a\\ -\Rightarrow y^\prime = y \log a\\ -\end{align*} - -Then in the special case where \(a = e\), - -\[(e^x)^\prime = (e^x)\] - -\hypertarget{partial-derivatives}{% -\section{Partial Derivatives}\label{partial-derivatives}} - -What happens when there's more than variable that is changing? - -\begin{quote} -If you can do ordinary derivatives, you can do partial derivatives: just hold all the other input variables constant except for the one you're differentiating with respect to. (Joe Blitzstein's Math Notes) -\end{quote} - -Suppose we have a function \(f\) now of two (or more) variables and we want to determine the rate of change relative to one of the variables. To do so, we would find its partial derivative, which is defined similar to the derivative of a function of one variable. - -\textbf{Partial Derivative}: Let \(f\) be a function of the variables \((x_1,\ldots,x_n)\). The partial derivative of \(f\) with respect to \(x_i\) is - -\[\frac{\partial f}{\partial x_i} (x_1,\ldots,x_n) = \lim\limits_{h\to 0} \frac{f(x_1,\ldots,x_i+h,\ldots,x_n)-f(x_1,\ldots,x_i,\ldots,x_n)}{h}\] - -Only the \(i\)th variable changes --- the others are treated as constants. - -We can take higher-order partial derivatives, like we did with functions of a single variable, except now the higher-order partials can be with respect to multiple variables. - -\begin{example}[More than one type of partial] -\protect\hypertarget{exm:unnamed-chunk-18}{}{\label{exm:unnamed-chunk-18} \iffalse (More than one type of partial) \fi{} }Notice that you can take partials with regard to different variables. - -Suppose \(f(x,y)=x^2+y^2\). Then - -\begin{align*} -\frac{\partial f}{\partial x}(x,y) &=\\ -\frac{\partial f}{\partial y}(x,y) &=\\ -\frac{\partial^2 f}{\partial x^2}(x,y) &=\\ -\frac{\partial^2 f}{\partial x \partial y}(x,y) &= -\end{align*} -\end{example} - -\begin{exercise} -\protect\hypertarget{exr:unnamed-chunk-19}{}{\label{exr:unnamed-chunk-19} }Let \(f(x,y)=x^3 y^4 +e^x -\log y\). What are the following partial derivaitves? - -\begin{align*} -\frac{\partial f}{\partial x}(x,y) &=\\ -\frac{\partial f}{\partial y}(x,y) &=\\ -\frac{\partial^2 f}{\partial x^2}(x,y) &=\\ -\frac{\partial^2 f}{\partial x \partial y}(x,y) &= -\end{align*} -\end{exercise} - -\hypertarget{taylorapprox}{% -\section{Taylor Series Approximation}\label{taylorapprox}} - -A common form of approximation used in statistics involves derivatives. A Taylor series is a way to represent common functions as infinite series (a sum of infinite elements) of the function's derivatives at some point \(a\). - -For example, Taylor series are very helpful in representing nonlinear (read: difficult) functions as linear (read: manageable) functions. One can thus \textbf{approximate} functions by using lower-order, finite series known as \textbf{Taylor polynomials}. If \(a=0\), the series is called a Maclaurin series. - -Specifically, a Taylor series of a real or complex function \(f(x)\) that is infinitely differentiable in the neighborhood of point \(a\) is: - -\begin{align*} - f(x) &= f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots\\ - &= \sum_{n=0}^\infty \frac{f^{(n)} (a)}{n!} (x-a)^n -\end{align*} - -\textbf{Taylor Approximation}: We can often approximate the curvature of a function \(f(x)\) at point \(a\) using a 2nd order Taylor polynomial around point \(a\): - -\[f(x) = f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 -+ R_2\] - -\(R_2\) is the remainder (R for remainder, 2 for the fact that we took two derivatives) and often treated as negligible, -giving us: - -\[f(x) \approx f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2} (x-a)^2\] - -The more derivatives that are added, the smaller the remainder \(R\) and the more accurate the approximation. Proofs involving limits guarantee that the remainder converges to 0 as the order of derivation increases. - -\hypertarget{the-indefinite-integration}{% -\section{The Indefinite Integration}\label{the-indefinite-integration}} - -So far, we've been interested in finding the derivative \(f=F'\) of a function \(F\). However, sometimes we're interested in exactly the reverse: finding the function \(F\) for which \(f\) is its derivative. We refer to \(F\) as the antiderivative of \(f\). - -\begin{definition}[Antiderivative] -\protect\hypertarget{def:unnamed-chunk-20}{}{\label{def:unnamed-chunk-20} \iffalse (Antiderivative) \fi{} }The antiverivative of a function \(f(x)\) is a differentiable function \(F\) whose derivative is \(f\). - -\[F^\prime = f.\] -\end{definition} - -Another way to describe is through the inverse formula. Let \(DF\) be the derivative of \(F\). And let \(DF(x)\) be the derivative of \(F\) evaluated at \(x\). Then the antiderivative is denoted by \(D^{-1}\) (i.e., the inverse derivative). If \(DF=f\), then \(F=D^{-1}f\). - -This definition bolsters the main takeaway about integrals and derivatives: They are inverses of each other. - -\begin{exercise}[Antiderivative] -\protect\hypertarget{exr:unnamed-chunk-21}{}{\label{exr:unnamed-chunk-21} \iffalse (Antiderivative) \fi{} }Find the antiderivative of the following: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x) = \frac{1}{x^2}\) -\item - \(f(x) = 3e^{3x}\) -\end{enumerate} -\end{exercise} - -We know from derivatives how to manipulate \(F\) to get \(f\). But how do you express the procedure to manipulate \(f\) to get \(F\)? For that, we need a new symbol, which we will call indefinite integration. - -\begin{definition}[Indefinite Integral] -\protect\hypertarget{def:unnamed-chunk-22}{}{\label{def:unnamed-chunk-22} \iffalse (Indefinite Integral) \fi{} }The indefinite integral of \(f(x)\) is written - -\[\int f(x) dx \] - -and is equal to the antiderivative of \(f\). -\end{definition} - -\begin{example} -\protect\hypertarget{exm:unnamed-chunk-23}{}{\label{exm:unnamed-chunk-23} }Draw the function \(f(x)\) and its indefinite integral, \(\int\limits f(x) dx\) - -\[f(x) = (x^2-4)\] -\end{example} - -\begin{solution} -\iffalse{} {Solution. } \fi{}The Indefinite Integral of the function \(f(x) = (x^2-4)\) can, for example, be \(F(x) = \frac{1}{3}x^3 - 4x.\) But it can also be \(F(x) = \frac{1}{3}x^3 - 4x + 1\), because the constant 1 disappears when taking the derivative. -\end{solution} - -Some of these functions are plotted in the bottom panel of Figure \ref{fig:integralc} as dotted lines. - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/integralc-1.pdf} -\caption{\label{fig:integralc}The Many Indefinite Integrals of a Function} -\end{figure} - -Notice from these examples that while there is only a single derivative for any function, there are multiple antiderivatives: one for any arbitrary constant \(c\). \(c\) just shifts the curve up or down on the \(y\)-axis. If more information is present about the antiderivative --- e.g., that it passes through a particular point --- then we can solve for a specific value of \(c\). - -\hypertarget{common-rules-of-integration}{% -\subsection*{Common Rules of Integration}\label{common-rules-of-integration}} -\addcontentsline{toc}{subsection}{Common Rules of Integration} - -Some common rules of integrals follow by virtue of being the inverse of a derivative. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Constants are allowed to slip out: \(\int a f(x)dx = a\int f(x)dx\) -\item - Integration of the sum is sum of integrations: \(\int [f(x)+g(x)]dx=\int f(x)dx + \int g(x)dx\) -\item - Reverse Power-rule: \(\int x^n dx = \frac{1}{n+1} x^{n+1} + c\) -\item - Exponents are still exponents: \(\int e^x dx = e^x +c\) -\item - Recall the derivative of \(\log(x)\) is one over \(x\), and so: \(\int \frac{1}{x} dx = \log x + c\) -\item - Reverse chain-rule: \(\int e^{f(x)}f^\prime(x)dx = e^{f(x)}+c\) -\item - More generally: \(\int [f(x)]^n f'(x)dx = \frac{1}{n+1}[f(x)]^{n+1}+c\) -\item - Remember the derivative of a log of a function: \(\int \frac{f^\prime(x)}{f(x)}dx=\log f(x) + c\) -\end{enumerate} - -\begin{example}[Common Integration] -\protect\hypertarget{exm:unnamed-chunk-25}{}{\label{exm:unnamed-chunk-25} \iffalse (Common Integration) \fi{} }Simplify the following indefinite integrals: - -\begin{itemize} -\tightlist -\item - \(\int 3x^2 dx\) -\item - \(\int (2x+1)dx\) -\item - \(\int e^x e^{e^x} dx\) -\end{itemize} -\end{example} - -\hypertarget{the-definite-integral-the-area-under-the-curve}{% -\section{The Definite Integral: The Area under the Curve}\label{the-definite-integral-the-area-under-the-curve}} - -If there is a indefinite integral, there \emph{must} be a definite integral. Indeed there is, but the notion of definite integrals comes from a different objective: finding the are a under a function. We will find, perhaps remarkably, that the formula we find to get the sum turns out to be expressible by the anti-derivative. - -Suppose we want to determine the area \(A(R)\) of a region \(R\) defined by a curve \(f(x)\) and some interval \(a\le x \le b\). - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/defintfig-1.pdf} -\caption{\label{fig:defintfig}The Riemann Integral as a Sum of Evaluations} -\end{figure} - -One way to calculate the area would be to divide the interval \(a\le x\le b\) into \(n\) subintervals of length \(\Delta x\) and then approximate the region with a series of rectangles, where the base of each rectangle is \(\Delta x\) and the height is \(f(x)\) at the midpoint of that interval. \(A(R)\) would then be approximated by the area of the union of the rectangles, which is given by \[S(f,\Delta x)=\sum\limits_{i=1}^n f(x_i)\Delta x\] and is called a \textbf{Riemann sum}. - -As we decrease the size of the subintervals \(\Delta x\), making the rectangles ``thinner,'' we would expect our approximation of the area of the region to become closer to the true area. This allows us to express the area as a limit of a series: -\[A(R)=\lim\limits_{\Delta x\to 0}\sum\limits_{i=1}^n f(x_i)\Delta x\] - -Figure \ref{fig:defintfig} shows that illustration. The curve depicted is \(f(x) = -15(x - 5) + (x - 5)^3 + 50.\) We want approximate the area under the curve between the \(x\) values of 0 and 10. We can do this in blocks of arbitrary width, where the sum of rectangles (the area of which is width times \(f(x)\) evaluated at the midpoint of the bar) shows the Riemann Sum. As the width of the bars \(\Delta x\) becomes smaller, the better the estimate of \(A(R)\). - -This is how we define the ``Definite'' Integral: - -\begin{definition}[The Definite Integral (Riemann)] -\protect\hypertarget{def:unnamed-chunk-26}{}{\label{def:unnamed-chunk-26} \iffalse (The Definite Integral (Riemann)) \fi{} }If for a given function \(f\) the Riemann sum approaches a limit as \(\Delta x \to 0\), then that limit is called the Riemann integral of \(f\) from \(a\) to \(b\). We express this with the \(\int\), symbol, and write \[\int\limits_a^b f(x) dx= \lim\limits_{\Delta x\to 0} \sum\limits_{i=1}^n f(x_i)\Delta x\] - -The most straightforward of a definite integral is the definite integral. That is, we read -\[\int\limits_a^b f(x) dx\] as the definite integral of \(f\) from \(a\) to \(b\) and we defined as the area under the ``curve'' \(f(x)\) from point \(x=a\) to \(x=b\). -\end{definition} - -The fundamental theorem of calculus shows us that this sum is, in fact, the antiderivative. - -\begin{theorem}[First Fundamental Theorem of Calculus] -\protect\hypertarget{thm:unnamed-chunk-27}{}{\label{thm:unnamed-chunk-27} \iffalse (First Fundamental Theorem of Calculus) \fi{} }Let the function \(f\) be bounded on \([a,b]\) and continuous on \((a,b)\). Then, suggestively, use the symbol \(F(x)\) to denote the definite integral from \(a\) to \(x\): -\[F(x)=\int\limits_a^x f(t)dt, \quad a\le x\le b\] - -Then \(F(x)\) has a derivative at each point in \((a,b)\) and \[F^\prime(x)=f(x), \quad a0\) -\item - \textbf{Decreasing:} \(f'(x)<0\) -\item - \textbf{Neither increasing nor decreasing}: \(f'(x)=0\) - i.e.~a maximum, minimum, or saddle point -\end{enumerate} - -So for example, \(f(x) = x^2 + 2\) and \(f^\prime(x) = 2x\) - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-43-1.pdf} -\caption{\label{fig:unnamed-chunk-43}Maxima and Minima} -\end{figure} - -\begin{exercise}[Plotting a mazimum and minimum] -\protect\hypertarget{exr:unnamed-chunk-44}{}{\label{exr:unnamed-chunk-44} \iffalse (Plotting a mazimum and minimum) \fi{} }Plot \(f(x)=x^3+ x^2 + 2\), plot its derivative, and identifiy where the derivative is zero. Is there a maximum or minimum? -\end{exercise} - -The second derivative \(f''(x)\) identifies whether the function \(f(x)\) at the point \(x\) is - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Concave down: \(f''(x)<0\) -\item - Concave up: \(f''(x)>0\) -\end{enumerate} - -\textbf{Maximum (Minimum)}: \(x_0\) is a \textbf{local maximum (minimum)} if \(f(x_0)>f(x)\) (\(f(x_0)f(x)\) (\(f(x_0)0\) -\item - Need more info: \(f'(x)=0\) and \(f''(x)=0\) -\end{enumerate} - -\textbf{Global Maxima and Minima} Sometimes no global max or min exists --- e.g., \(f(x)\) not bounded above or below. However, there are three situations where we can fairly easily identify global max or min. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \textbf{Functions with only one critical point.} If \(x_0\) is a local max or min of \(f\) and it is the only critical point, then it is the global max or min. -\item - \textbf{Globally concave up or concave down functions.} If \(f''(x)\) is never zero, then there is at most one critical point. That critical point is a global maximum if \(f''<0\) and a global minimum if \(f''>0\). -\item - \textbf{Functions over closed and bounded intervals} must have both a global maximum and a global minimum. -\end{enumerate} - -\begin{example}[Maxima and Minima by drawing] -\protect\hypertarget{exm:unnamed-chunk-46}{}{\label{exm:unnamed-chunk-46} \iffalse (Maxima and Minima by drawing) \fi{} } -Find any critical points and identify whether they are a max, min, or saddle point: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x)=x^2+2\) -\item - \(f(x)=x^3+2\) -\item - \(f(x)=|x^2-1|\), \(x\in [-2,2]\) -\end{enumerate} -\end{example} - -\hypertarget{concavity-of-a-function}{% -\section{Concavity of a Function}\label{concavity-of-a-function}} - -Concavity helps identify the curvature of a function, \(f(x)\), in 2 dimensional space. - -\begin{definition}[Concave Function] -\protect\hypertarget{def:unnamed-chunk-47}{}{\label{def:unnamed-chunk-47} \iffalse (Concave Function) \fi{} }A function \(f\) is strictly concave over the set S \underline{if} \(\forall x_1,x_2 \in S\) and \(\forall a \in (0,1)\), \[f(ax_1 + (1-a)x_2) > af(x_1) + (1-a)f(x_2)\] -\textit{Any} line connecting two points on a concave function will lie \textit{below} the function. -\end{definition} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-48-1.pdf} - -\begin{definition}[Convex Function] -\protect\hypertarget{def:unnamed-chunk-49}{}{\label{def:unnamed-chunk-49} \iffalse (Convex Function) \fi{} }Convex: A function f is strictly convex over the set S \underline{if} \(\forall x_1,x_2 \in S\) and \(\forall a \in (0,1)\), \[f(ax_1 + (1-a)x_2) < af(x_1) + (1-a)f(x_2)\] - -Any line connecting two points on a convex function will lie above the function. -\end{definition} - -\begin{comment} - \parbox{2in}{\includegraphics[scale=.4]{Convex.pdf}} -\end{comment} - -Sometimes, concavity and convexity are strict of a requirement. For most purposes of getting solutions, what we call quasi-concavity is enough. - -\begin{definition}[Quasiconcave Function] -\protect\hypertarget{def:unnamed-chunk-50}{}{\label{def:unnamed-chunk-50} \iffalse (Quasiconcave Function) \fi{} }A function f is quasiconcave over the set S if \(\forall x_1,x_2 \in S\) and \(\forall a \in (0,1)\), \[f(ax_1 + (1-a)x_2) \ge \min(f(x_1),f(x_2))\] - -No matter what two points you select, the \textit{lowest} valued point will always be an end point. -\end{definition} - -\begin{comment} - \parbox{2in}{\includegraphics[scale=.4]{Quasiconcave.pdf}} -\end{comment} - -\begin{definition}[Quasiconvex] -\protect\hypertarget{def:unnamed-chunk-51}{}{\label{def:unnamed-chunk-51} \iffalse (Quasiconvex) \fi{} }A function f is quasiconvex over the set \(S\) if \(\forall x_1,x_2 \in S\) and \(\forall a \in (0,1)\), \[f(ax_1 + (1-a)x_2) \le \max(f(x_1),f(x_2))\] -No matter what two points you select, the \textit{highest} valued point will always be an end point. -\end{definition} - -\begin{comment} - \parbox{1.8in}{\includegraphics[scale=.4]{Quasiconvex.pdf}} -\end{comment} - -\textbf{Second Derivative Test of Concavity}: The second derivative can be used to understand concavity. - -If -\[\begin{array}{lll} -f''(x) < 0 & \Rightarrow & \text{Concave}\\ -f''(x) > 0 & \Rightarrow & \text{Convex} -\end{array}\] - -\hypertarget{quadratic-forms}{% -\subsection*{Quadratic Forms}\label{quadratic-forms}} -\addcontentsline{toc}{subsection}{Quadratic Forms} - -Quadratic forms is shorthand for a way to summarize a function. This is important for finding concavity because - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Approximates local curvature around a point --- e.g., used to - identify max vs min vs saddle point. -\item - They are simple to express even in \(n\) dimensions: -\item - Have a matrix representation. -\end{enumerate} - -\textbf{Quadratic Form}: A polynomial where each term is a monomial -of degree 2 in any number of variables: - -\begin{align*} -\text{One variable: }& Q(x_1) = a_{11}x_1^2\\ -\text{Two variables: }& Q(x_1,x_2) = a_{11}x_1^2 + a_{12}x_1x_2 + a_{22}x_2^2\\ -\text{N variables: }& Q(x_1,\cdots,x_n)=\sum\limits_{i\le j} a_{ij}x_i x_j -\end{align*} - -which can be written in matrix terms: - -One variable - -\[Q(\mathbf{x}) = x_1^\top a_{11} x_1\] - -N variables: -\begin{align*} -Q(\mathbf{x}) &=\begin{pmatrix} x_1 & x_2 & \cdots & x_n \end{pmatrix}\begin{pmatrix} -a_{11}&\frac{1}{2}a_{12}&\cdots&\frac{1}{2}a_{1n}\\ -\frac{1}{2}a_{12}&a_{22}&\cdots&\frac{1}{2}a_{2n}\\ -\vdots&\vdots&\ddots&\vdots\\ -\frac{1}{2}a_{1n}&\frac{1}{2}a_{2n}&\cdots&a_{nn} -\end{pmatrix} -\begin{pmatrix} x_1\\x_2\\\vdots\\x_n\end{pmatrix}\\ -&= \mathbf{x}^\top\mathbf{Ax} -\end{align*} - -For example, the Quadratic on \(\mathbf{R}^2\): -\begin{align*} - Q(x_1,x_2)&=\begin{pmatrix} x_1& x_2 \end{pmatrix} \begin{pmatrix} a_{11}&\frac{1}{2} a_{12}\\ - \frac{1}{2}a_{12}&a_{22}\end{pmatrix} \begin{pmatrix} x_1\\x_2 \end{pmatrix} \\ - &= a_{11}x_1^2 + a_{12}x_1x_2 + a_{22}x_2^2 -\end{align*} - -\hypertarget{definiteness-of-quadratic-forms}{% -\subsection*{Definiteness of Quadratic Forms}\label{definiteness-of-quadratic-forms}} -\addcontentsline{toc}{subsection}{Definiteness of Quadratic Forms} - -When the function \(f(\mathbf{x})\) has more than two inputs, determining whether it has a maxima and minima (remember, functions may have many inputs but they have only one output) is a bit more tedious. Definiteness helps identify the curvature of a function, \(Q(\textbf{x})\), in n dimensional space. - -\textbf{Definiteness}: By definition, a quadratic form always takes on the value of zero when \(x = 0\), \(Q(\textbf{x})=0\) at \(\textbf{x}=0\). The definiteness of the matrix \(\textbf{A}\) is determined by whether the -quadratic form \(Q(\textbf{x})=\textbf{x}^\top\textbf{A}\textbf{x}\) is greater than zero, less than -zero, or sometimes both over all \(\mathbf{x}\ne 0\). - -\hypertarget{foc-and-soc}{% -\section{FOC and SOC}\label{foc-and-soc}} - -We can see from a graphical representation that if a point is a local maxima or minima, it must meet certain conditions regarding its derivative. These are so commonly used that we refer these to ``First Order Conditions'' (FOCs) and ``Second Order Conditions'' (SOCs) in the economic tradition. - -\hypertarget{first-order-conditions}{% -\subsection*{First Order Conditions}\label{first-order-conditions}} -\addcontentsline{toc}{subsection}{First Order Conditions} - -When we examined functions of one variable \(x\), we found critical points by taking the first derivative, setting it to zero, and solving for \(x\). For functions of \(n\) variables, the critical points are found in much the same way, except now we set the partial derivatives equal to zero. Note: We will only consider critical points on the interior of a function's domain. - -In a derivative, we only took the derivative with respect to one variable at a time. When we take the derivative separately with respect to all variables in the elements of \(\mathbf{x}\) and then express the result as a vector, we use the term Gradient and Hessian. - -\begin{definition}[Gradient] -\protect\hypertarget{def:unnamed-chunk-52}{}{\label{def:unnamed-chunk-52} \iffalse (Gradient) \fi{} } -Given a function \(f(\textbf{x})\) in \(n\) variables, the gradient \(\nabla f(\mathbf{x})\) (the greek letter nabla ) is a column vector, where the \(i\)th element is the partial derivative of \(f(\textbf{x})\) with respect to \(x_i\): - -\[\nabla f(\mathbf{x}) = \begin{pmatrix} -\frac{\partial f(\mathbf{x})}{\partial x_1}\\ \frac{\partial f(\mathbf{x})}{\partial x_2}\\ - \vdots \\ \frac{\partial f(\mathbf{x})}{\partial x_n} \end{pmatrix}\] -\end{definition} - -Before we know whether a point is a maxima or minima, if it meets the FOC it is a ``Critical Point''. - -\begin{definition}[Critical Point] -\protect\hypertarget{def:unnamed-chunk-53}{}{\label{def:unnamed-chunk-53} \iffalse (Critical Point) \fi{} } -\(\mathbf{x}^*\) is a critical point if and only if \(\nabla f(\mathbf{x}^*)=0\). If the partial derivative of f(x) with respect to \(x^*\) is 0, then \(\mathbf{x}^*\) is a critical point. To solve for \(\mathbf{x}^*\), find the gradient, set each element equal to 0, and solve the system of equations. \[\mathbf{x}^* = \begin{pmatrix} x_1^*\\x_2^*\\ \vdots \\ x_n^*\end{pmatrix}\] -\end{definition} - -\begin{example} -\protect\hypertarget{exm:unnamed-chunk-54}{}{\label{exm:unnamed-chunk-54} } -Example: Given a function \(f(\mathbf{x})=(x_1-1)^2+x_2^2+1\), find the (1) Gradient and (2) Critical point of \(f(\mathbf{x})\). -\end{example} - -\begin{solution} -\iffalse{} {Solution. } \fi{}Gradient - -\begin{align*} -\nabla f(\mathbf{x}) &= \begin{pmatrix}\frac{\partial f(\mathbf{x})}{\partial x_1}\\ \frac{\partial f(\mathbf{x})}{\partial x_2} \end{pmatrix}\\ -&= \begin{pmatrix} 2(x_1-1)\\ 2x_2 \end{pmatrix} -\end{align*} - -Critical Point \(\mathbf{x}^* =\) - -\begin{align*} -&\frac{\partial f(\mathbf{x})}{\partial x_1} = 2(x_1-1) = 0\\ -&\Rightarrow x_1^* = 1\\ -&\frac{\partial f(\mathbf{x})}{\partial x_2} = 2x_2 = 0\\ -&\Rightarrow x_2^* = 0\\ -\end{align*} - -So \[\mathbf{x}^* = (1,0)\] -\end{solution} - -\hypertarget{second-order-conditions}{% -\subsection*{Second Order Conditions}\label{second-order-conditions}} -\addcontentsline{toc}{subsection}{Second Order Conditions} - -When we found a critical point for a function of one variable, we used the second derivative as a indicator of the curvature at the point in order to determine whether the point was a min, max, or saddle (second derivative test of concavity). For functions of \(n\) variables, we use \emph{second order partial derivatives} as an indicator of curvature. - -\begin{definition}[Hessian] -\protect\hypertarget{def:unnamed-chunk-56}{}{\label{def:unnamed-chunk-56} \iffalse (Hessian) \fi{} } -Given a function \(f(\mathbf{x})\) in \(n\) variables, the hessian \(\mathbf{H(x)}\) is -an \(n\times n\) matrix, where the \((i,j)\)th element is the second order -partial derivative of \(f(\mathbf{x})\) with respect to \(x_i\) and \(x_j\): - -\[\mathbf{H(x)}=\begin{pmatrix} -\frac{\partial^2 f(\mathbf{x})}{\partial x_1^2}&\frac{\partial^2f(\mathbf{x})}{\partial x_1 \partial x_2}& -\cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_1 \partial x_n}\\[9pt] -\frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_2^2}& -\cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_2 \partial x_n}\\ -\vdots & \vdots & \ddots & \vdots \\[3pt] -\frac{\partial^2 f(\mathbf{x})}{\partial x_n \partial x_1}&\frac{\partial^2f(\mathbf{x})}{\partial x_n \partial x_2}& -\cdots & \frac{\partial^2 f(\mathbf{x})}{\partial x_n^2}\end{pmatrix}\] -\end{definition} - -Note that the hessian will be a symmetric matrix because \(\frac{\partial f(\mathbf{x})}{\partial x_1\partial x_2} = \frac{\partial f(\mathbf{x})}{\partial x_2\partial x_1}\). - -Also note that given that \(f(\mathbf{x})\) is of quadratic form, each element of the hessian will be a constant. - -These definitions will be employed when we determine the \textbf{Second Order Conditions} of a function: - -Given a function \(f(\mathbf{x})\) and a point \(\mathbf{x}^*\) such that \(\nabla f(\mathbf{x}^*)=0\), - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Hessian is Positive Definite \(\quad \Longrightarrow \quad\) Strict Local Min -\item - Hessian is Positive Semidefinite \(\forall \mathbf{x}\in B(\mathbf{x}^*,\epsilon)\)\} \(\quad \Longrightarrow \quad\) Local Min -\item - Hessian is Negative Definite \(\quad \Longrightarrow \quad\) Strict Local Max -\item - Hessian is Negative Semidefinite \(\forall \mathbf{x}\in B(\mathbf{x}^*,\epsilon)\)\} \(\quad \Longrightarrow \quad\) Local Max -\item - Hessian is Indefinite \(\quad \Longrightarrow \quad\) Saddle Point -\end{enumerate} - -\begin{example}[Max and min with two dimensions] -\protect\hypertarget{exm:unnamed-chunk-57}{}{\label{exm:unnamed-chunk-57} \iffalse (Max and min with two dimensions) \fi{} } -We found that the only critical point of -\(f(\mathbf{x})=(x_1-1)^2+x_2^2+1\) is at \(\mathbf{x}^*=(1,0)\). Is it a min, max, or -saddle point? -\end{example} - -\begin{solution} -\iffalse{} {Solution. } \fi{} -The Hessian is -\begin{align*} -\mathbf{H(x)} &= \begin{pmatrix} 2&0\\0&2 \end{pmatrix} -\end{align*} -The Leading principal minors of the Hessian are \(M_1=2; M_2=4\). Now we consider Definiteness. Since both leading principal minors are positive, the Hessian is positive definite. - -Maxima, Minima, or Saddle Point? Since the Hessian is positive definite and the gradient equals 0, \(x^\star = (1,0)\) is a strict local minimum. - -Note: Alternate check of definiteness. Is \(\mathbf{H(x^*)} \geq \leq 0 \quad \forall \quad \mathbf{x}\ne 0\) - -\begin{align*} -\mathbf{x}^\top H(\mathbf{x}^*) \mathbf{x} &= \begin{pmatrix} x_1 & x_2 \end{pmatrix}\\ -&= \begin{pmatrix} 2&0\\0&2 \end{pmatrix}\\ -\begin{pmatrix} x_1\\x_2\end{pmatrix} &= 2x_1^2+2x_2^2 -\end{align*} - -For any \(\mathbf{x}\ne 0\), \(2(x_1^2+x_2^2)>0\), so the Hessian is positive definite and \(\mathbf{x}^*\) is a strict local minimum. -\end{solution} - -\hypertarget{definiteness-and-concavity}{% -\subsection*{Definiteness and Concavity}\label{definiteness-and-concavity}} -\addcontentsline{toc}{subsection}{Definiteness and Concavity} - -Although definiteness helps us to understand the curvature of an n-dimensional function, it does not necessarily tell us whether the function is globally concave or convex. - -We need to know whether a function is globally concave or convex to determine whether a critical point is a global min or max. We can use the definiteness of the Hessian to determine whether a function is globally concave or convex: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Hessian is Positive Semidefinite \(\forall \mathbf{x}\)\} \(\quad \Longrightarrow \quad\) Globally Convex -\item - Hessian is Negative Semidefinite \(\forall \mathbf{x}\)\} \(\quad \Longrightarrow \quad\) Globally Concave -\end{enumerate} - -Notice that the definiteness conditions must be satisfied over the entire domain. - -\hypertarget{global-maxima-and-minima}{% -\section{Global Maxima and Minima}\label{global-maxima-and-minima}} - -\textbf{Global Max/Min Conditions}: Given a function \(f(\mathbf{x})\) and a point \(\mathbf{x}^*\) such that \(\nabla f(\mathbf{x}^*)=0\), - -\begin{enumerate} - \item \parbox[t]{2in}{$f(\mathbf{x})$ Globally Convex} $\quad -\Longrightarrow \quad$ Global Min - \item \parbox[t]{2in}{$f(\mathbf{x})$ Globally Concave} $\quad -\Longrightarrow \quad$ Global Max -\end{enumerate} - -Note that showing that \(\mathbf{H(x^*)}\) is negative semidefinite is -not enough to guarantee \(\mathbf{x}^*\) is a local max. However, showing that -\(\mathbf{H(x)}\) is negative semidefinite for all \(\mathbf{x}\) guarantees that \(x^*\) -is a global max. (The same goes for positive semidefinite and minima.)\textbackslash{} - -Example: Take \(f_1(x)=x^4\) and \(f_2(x)=-x^4\). Both have \(x=0\) as -a critical point. Unfortunately, \(f''_1(0)=0\) and \(f''_2(0)=0\), so we -can't tell whether \(x=0\) is a min or max for either. However, -\(f''_1(x)=12x^2\) and \(f''_2(x)=-12x^2\). For all \(x\), \(f''_1(x)\ge 0\) -and \(f''_2(x)\le 0\) --- i.e., \(f_1(x)\) is globally convex and \(f_2(x)\) -is globally concave. So \(x=0\) is a global min of \(f_1(x)\) and a -global max of \(f_2(x)\). - -\begin{exercise} -\protect\hypertarget{exr:unnamed-chunk-59}{}{\label{exr:unnamed-chunk-59} }Given \(f(\mathbf{x})=x_1^3-x_2^3+9x_1x_2\), find any maxima or minima. -\end{exercise} - -\begin{enumerate} - \item First order conditions. - \begin{enumerate} - \item Gradient $\nabla f(\mathbf{x}) = $ - $$\phantom{\begin{pmatrix} \frac{\partial f}{\partial x_1} \\ - \frac{\partial f}{\partial x_2}\end{pmatrix} = - \begin{pmatrix} 3x_1^2+9x_2 \\ -3x_2^2+9x_1 \end{pmatrix}}$$ - \item Critical Points $\mathbf{x^*} =$\\ - \phantom{Set the gradient equal to zero and solve for $x_1$ and - $x_2$.We have two equations and two unknowns. Solving for $x_1$ - and $x_2$, we get two critical points: $\mathbf{x}_1^*=(0,0)$ and - $\mathbf{x}_2^*=(3,-3)$.} - $$\phantom{3x_1^2 + 9x_2 = 0 \quad \Rightarrow \quad 9x_2 = - -3x_1^2 \quad \Rightarrow \quad x_2 = -\frac{1}{3}x_1^2}$$ - $$\phantom{-3x_2^2 + 9x_1 = 0 \quad \Rightarrow \quad - -3(-\frac{1} {3}x_1^2)^2 + 9x_1 = 0}$$ - $$\phantom{\Rightarrow \quad -\frac{1}{3}x_1^4 - + 9x_1 = 0 \quad \Rightarrow \quad x_1^3 = 27x_1 \quad - \Rightarrow \quad x_1 = 3}$$ - $$\phantom{3(3)^2 + 9x_2 = 0 \quad \Rightarrow \quad x_2 = -3}$$ - \end{enumerate} - - \item Second order conditions. - \begin{enumerate} - \item Hessian $\mathbf{H(x)} = $ - $$\phantom{\begin{pmatrix} 6x_1&9\\9&-6x_2 \end{pmatrix}}$$ - \item Hessian $\mathbf{H(x_1^*)} = $ - $$\phantom{\begin{pmatrix} 0&9\\9&0\end{pmatrix}}$$ - \item Leading principal minors of $\mathbf{H(x_1^*)} = $ - $$\phantom{M_1=0; M_2=-81}$$\\ - \item Definiteness of $\mathbf{H(x_1^*)}$?\\ - \phantom{$\mathbf{H(x_1^*)}$ is indefinite}\\ - \item Maxima, Minima, or Saddle Point for $\mathbf{x_1^*}$?\\ - \phantom{Since $\mathbf{H(x_1^*)}$ is indefinite, $\mathbf{x}_1^*=(0,0)$ - is a saddle point.}\\ - \item Hessian $\mathbf{H(x_2^*)} = $ - $$\phantom{\begin{pmatrix} 18&9\\9&18\end{pmatrix}}$$ - \item Leading principal minors of $\mathbf{H(x_2^*)} = $ - $$\phantom{M_1=18; M_2=243}$$\\ - \item Definiteness of $\mathbf{H(x_2^*)}$?\\ - \phantom{$\mathbf{H(x_2^*)}$ is positive definite}\\ - \item Maxima, Minima, or Saddle Point for $\mathbf{x}_2^*$?\\ - \phantom{Since $\mathbf{H(x_2^*)}$ is positive definite, $\mathbf{x}_1^*=(3,-3)$ is a strict local minimum}\\ - \end{enumerate} - - \item Global concavity/convexity. - \begin{enumerate} - \item Is f(x) globally concave/convex?\\ - \phantom{No. In evaluating the Hessians for $\mathbf{x}_1^*$ - and $\mathbf{x}_2^*$ we saw that the Hessian is not positive - semidefinite at x $=$ (0,0).}\\ - \item Are any $\mathbf{x^*}$ global minima or maxima?\\ - \phantom{No. Since the function is not globally concave/convex, - we can't infer that $\mathbf{x}_2^*=(3,-3)$ is a global minimum. - In fact, if we set $x_1=0$, the $f(\mathbf{x})=-x_2^3$, which will - go to $-\infty$ as $x_2\to \infty$.}\\ - \end{enumerate} -\end{enumerate} - -\hypertarget{constrained-optimization}{% -\section{Constrained Optimization}\label{constrained-optimization}} - -We have already looked at optimizing a function in one or more dimensions over the whole domain of the function. Often, however, we want to find the maximum or minimum of a function over some restricted part of its domain. - -ex: Maximizing utility subject to a budget constraint - -\begin{figure} -\centering -\includegraphics{images/constraint.png} -\caption{A typical Utility Function with a Budget Constraint} -\end{figure} - -\textbf{Types of Constraints}: For a function \(f(x_1, \dots, x_n)\), there are two types of constraints that can be imposed: - -\begin{enumerate} -\item \textbf{Equality constraints:} constraints of the form $c(x_1,\dots, x_n) = r$. -Budget constraints are the classic example of equality constraints in social science. -\item \textbf{Inequality constraints:} constraints of the form $c(x_1, \dots, x_n) \leq r$. These might arise from non-negativity constraints or other threshold effects. -\end{enumerate} - -In any constrained optimization problem, the constrained maximum will always be less than or equal to the unconstrained maximum. If the constrained maximum is less than the unconstrained maximum, then the constraint is binding. Essentially, this means that you can treat your constraint as an equality constraint rather than an inequality constraint. - -For example, the budget constraint binds when you spend your entire budget. This generally happens because we believe that utility is strictly increasing in consumption, i.e.~you always want more so you spend everything you have. - -Any number of constraints can be placed on an optimization problem. When working with multiple constraints, always make sure that the set of constraints are not pathological; it must be possible for all of the constraints to be satisfied simultaneously. - -\textbf{Set-up for Constrained Optimization:} -\[\max_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)\] -\[\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)\] -This tells us to maximize/minimize our function, \(f(x_1,x_2)\), with respect to the choice variables, \(x_1,x_2\), subject to the constraint. - -Example: -\[\max_{x_1,x_2} f(x_1, x_2) = -(x_1^2 + 2x_2^2) \text{ s.t. }x_1 + x_2 = 4\] -It is easy to see that the \textit{unconstrained} maximum occurs at \((x_1, x_2) = (0,0)\), but that does not satisfy the constraint. How should we proceed? - -\hypertarget{equality-constraints}{% -\subsection*{Equality Constraints}\label{equality-constraints}} -\addcontentsline{toc}{subsection}{Equality Constraints} - -Equality constraints are the easiest to deal with because we know that the maximum or minimum has to lie on the (intersection of the) constraint(s). - -The trick is to change the problem from a constrained optimization problem in \(n\) variables to an unconstrained optimization problem in \(n + k\) variables, adding \emph{one} variable for \emph{each} equality constraint. We do this using a lagrangian multiplier. - -\textbf{Lagrangian function}: The Lagrangian function allows us to combine the function we want to optimize and the constraint function into a single function. Once we have this single function, we can proceed as if this were an \emph{unconstrained} optimization problem. - -For each constraint, we must include a \textbf{Lagrange multiplier} (\(\lambda_i\)) as an additional variable in the analysis. These terms are the link between the constraint and the Lagrangian function. - -Given a \emph{two dimensional} set-up: -\[\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) = a\] - -We define the Lagrangian function \(L(x_1,x_2,\lambda_1)\) as follows: -\[L(x_1,x_2,\lambda_1) = f(x_1,x_2) - \lambda_1 (c(x_1,x_2) - a)\] - -More generally, in \emph{n dimensions}: -\[ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) - \sum_{i=1}^k\lambda_i(c_i(x_1,\dots, x_n) - r_i)\] - -\textbf{Getting the sign right:} Note that above we subtract the lagrangian term \emph{and} we subtract the constraint constant from the constraint function. Occasionally, you may see the following alternative form of the Lagrangian, which is \emph{equivalent}: -\[ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) + \sum_{i=1}^k\lambda_i(r_i - c_i(x_1,\dots, x_n))\] -Here we add the lagrangian term \emph{and} we subtract the constraining function from the constraint constant. - -\textbf{Using the Lagrangian to Find the Critical Points}: To find the critical points, we take the partial derivatives of lagrangian function, \(L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k)\), with respect to each of its variables (all choice variables \(\mathbf{x}\) \emph{and} all lagrangian multipliers \(\mathbf{\lambda}\)). At a critical point, each of these partial derivatives must be equal to zero, so we obtain a system of \(n + k\) equations in \(n + k\) unknowns: - -\begin{align*} -\frac{\partial L}{\partial x_1} &= \frac{\partial f}{\partial x_1} - \sum_{i = 1}^k\lambda_i\frac{\partial c_i}{\partial x_1} = 0\\ - \vdots &= \vdots \nonumber \\ -\frac{\partial L}{\partial x_n} &= \frac{\partial f}{\partial x_n} - \sum_{i = 1}^k\lambda_i\frac{\partial c_i}{\partial x_n} = 0\\ -\frac{\partial L}{\partial \lambda_1} &= c_1(x_i, \dots, x_n) - r_1 = 0\\ - \vdots &= \vdots \nonumber \\ -\frac{\partial L}{\partial \lambda_k} &= c_k(x_i, \dots, x_n) - r_k = 0 -\end{align*} - -We can then solve this system of equations, because there are \(n+k\) equations and \(n+k\) unknowns, to calculate the critical point \((x_1^*,\dots,x_n^*,\lambda_1^*,\dots,\lambda_k^*)\). - -\textbf{Second-order Conditions and Unconstrained Optimization:} There may be more than one critical point, i.e.~we need to verify that the critical point we find is a maximum/minimum. Similar to unconstrained optimization, we can do this by checking the second-order conditions. - -\begin{example}[Constrained optimization with two goods and a budget constraint] -\protect\hypertarget{exm:unnamed-chunk-60}{}{\label{exm:unnamed-chunk-60} \iffalse (Constrained optimization with two goods and a budget constraint) \fi{} } -Find the constrained optimization of -\[\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 = 4\] -\end{example} - -\begin{solution} -\iffalse{} {Solution. } \fi{} -1. Begin by writing the Lagrangian: -\[L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)\] -2. Take the partial derivatives and set equal to zero: - -\begin{align*} -\frac{\partial L}{\partial x_1} = -2x_1 - \lambda \quad \quad \quad &= 0\\ -\frac{\partial L}{\partial x_2} = -4x_2 - \lambda \quad \quad \quad &= 0\\ -\frac{\partial L}{\partial \lambda} = -(x_1 + x_2 - 4) \quad & = & 0\\ -\end{align*} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{2} -\item - Solve the system of equations: Using the first two partials, we see that \(\lambda = -2x_1\) and \(\lambda = -4x_2\) - Set these equal to see that \(x_1 = 2x_2\). - Using the third partial and the above equality, \(4 = 2x_2 + x_2\) from which we get \[x_2^* = 4/3, x_1^* = 8/3, \lambda = -16/3\] -\item - Therefore, the only critical point is \(x_1^* = \frac{8}{3}\) and \(x_2^* = \frac{4}{3}\) -\item - This gives \(f(\frac{8}{3}, \frac{4}{3}) = -\frac{96}{9}\), which is less than the unconstrained optimum \(f(0,0) = 0\) -\end{enumerate} -\end{solution} - -Notice that when we take the partial derivative of L with respect to the Lagrangian multiplier and set it equal to 0, we return exactly our constraint! This is why signs matter. - -\hypertarget{inequality-constraints}{% -\section{Inequality Constraints}\label{inequality-constraints}} - -Inequality constraints define the boundary of a region over which we seek to optimize the function. This makes inequality constraints more challenging because we do not know if the maximum/minimum lies along one of the constraints (the constraint binds) or in the interior of the region. - -We must introduce more variables in order to turn the problem into an unconstrained optimization. - -\textbf{Slack:} For each inequality constraint \(c_i(x_1, \dots, x_n) \leq a_i\), we define a slack variable \(s_i^2\) for which the expression \(c_i(x_1, \dots, x_n) \leq a_i - s_i^2\) would hold with equality. These slack variables capture how close the constraint comes to binding. We use \(s^2\) rather than \(s\) to ensure that the slack is positive. - -Slack is just a way to transform our constraints. - -Given a two-dimensional set-up and these edited constraints: -\[\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) \le a_1\] - -Adding in Slack: -\[\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2) \le a_1 - s_1^2\] - -We define the Lagrangian function \(L(x_1,x_2,\lambda_1,s_1)\) as follows: -\[L(x_1,x_2,\lambda_1,s_1) = f(x_1,x_2) - \lambda_1 ( c(x_1,x_2) + s_1^2 - a_1)\] - -More generally, in n dimensions: -\[ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k, s_1, \dots, s_k) = f(x_1, \dots, x_n) - \sum_{i = 1}^k \lambda_i(c_i(x_1,\dots, x_n) + s_i^2 - a_i)\] - -\textbf{Finding the Critical Points}: To find the critical points, we take the partial derivatives of the lagrangian function, \(L(x_1,\dots,x_n,\lambda_1,\dots,\lambda_k,s_1,\dots,s_k)\), with respect to each of its variables (all choice variables \(x\), all lagrangian multipliers \(\lambda\), and all slack variables \(s\)). At a critical point, \emph{each} of these partial derivatives must be equal to zero, so we obtain a system of \(n + 2k\) equations in \(n + 2k\) unknowns: - -\begin{align*} -\frac{\partial L}{\partial x_1} &= \frac{\partial f}{\partial x_1} - \sum_{i = 1}^k\lambda_i\frac{\partial c_i}{\partial x_1} = 0\\ - \vdots & = \vdots \\ -\frac{\partial L}{\partial x_n} &= \frac{\partial f}{\partial x_n} - \sum_{i = 1}^k\lambda_i\frac{\partial c_i}{\partial x_n} = 0\\ -\frac{\partial L}{\partial \lambda_1} &= c_1(x_i, \dots, x_n) + s_1^2 - b_1 = 0\\ - \vdots & = \vdots \\ -\frac{\partial L}{\partial \lambda_k} &= c_k(x_i, \dots, x_n) + s_k^2 - b_k = 0\\ -\frac{\partial L}{\partial s_1} &= 2s_1\lambda_1 = 0\\ - \vdots =\vdots \\ -\frac{\partial L}{\partial s_k} &= 2s_k\lambda_k = 0 -\end{align*} - -\textbf{Complementary slackness conditions}: The last set of first order conditions of the form \(2s_i\lambda_i = 0\) (the partials taken with respect to the slack variables) are known as complementary slackness conditions. These conditions can be satisfied one of three ways: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\lambda_i = 0\) and \(s_i \neq 0\): This implies that the slack is positive and thus \emph{the constraint does not bind}. -\item - \(\lambda_i \neq 0\) and \(s_i = 0\): This implies that there is no slack in the constraint and \emph{the constraint does bind}. -\item - \(\lambda_i = 0\) and \(s_i = 0\): In this case, there is no slack but the \emph{constraint binds trivially}, without changing the optimum. -\end{enumerate} - -Example: Find the critical points for the following constrained optimization: -\[\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4\] - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - Rewrite with the slack variables: - \[\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4 - s_1^2\] -\item - Write the Lagrangian: - \[L(x_1,x_2,\lambda_1,s_1) = -(x_1^2 + 2x_2^2) - \lambda_1 (x_1 + x_2 + s_1^2 - 4)\] -\item - Take the partial derivatives and set equal to 0: -\end{enumerate} - -\begin{align*} -\frac{\partial L}{\partial x_1} = -2x_1 - \lambda_1 &= 0\\ -\frac{\partial L}{\partial x_2} = -4x_2 - \lambda_1 &= 0\\ -\frac{\partial L}{\partial \lambda_1} = -(x_1 + x_2 + s_1^2 - 4)&= 0\\ -\frac{\partial L}{\partial s_1} = -2s_1\lambda_1 &= 0\\ -\end{align*} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{3} -\tightlist -\item - Consider all ways that the complementary slackness conditions are solved: - - \begin{center} - \begin{tabular}{|l|cccc|c|} - \hline - Hypothesis & $s_1$ & $\lambda_1$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ - \hline - $s_1 = 0$ $\lambda_1 = 0$ & \multicolumn{4}{l|}{No solution} & \\ - $s_1 \neq 0$ $\lambda_1 = 0$ & 2 & 0 & 0 & 0 & 0\\ - $s_1 = 0$ $\lambda_1 \neq 0$ & 0 & $\frac{-16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ - $s_1 \neq 0$ $\lambda_1 \neq 0$ & \multicolumn{4}{l|}{No solution} &\\ - \hline - \end{tabular} - \end{center} -\end{enumerate} - -This shows that there are two critical points: \((0,0)\) and \((\frac{8}{3},\frac{4}{3})\). - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{4} -\tightlist -\item - Find maximum: - Looking at the values of \(f(x_1,x_2)\) at the critical points, we see that \(f(x_1,x_2)\) is maximized at \(x_1^* = 0\) and \(x_2^*=0\). -\end{enumerate} - -\begin{exercise} -\protect\hypertarget{exr:unnamed-chunk-62}{}{\label{exr:unnamed-chunk-62} }Example: Find the critical points for the following constrained optimization: -\[\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } -\begin{array}{l} -x_1 + x_2 \le 4\\ -x_1 \ge 0\\ -x_2 \ge 0 -\end{array}\] -\end{exercise} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - Rewrite with the slack variables: - \[\phantom{max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } - \begin{array}{l} - x_1 + x_2 \le 4 - s_1^2\\ - -x_1 \le 0 - s_2^2\\ - -x_2 \le 0 - s_3^2 - \end{array}}\] -\item - Write the Lagrangian: - \[\phantom{L(x_1, x_2, \lambda_1, \lambda_2, \lambda_3, s_1, s_2, s_3) = -(x_1^2 + 2x_2^2) - \lambda_1(x_1 + x_2 + s_1^2 - 4) - \lambda_2(-x_1 + s_2^2) - \lambda_3(-x_2 + s_3^2)}\] -\item - Take the partial derivatives and set equal to zero: -\end{enumerate} - -\phantom{ -$\frac{\partial L}{\partial x_1} = -2x_1 - \lambda_1 + \lambda_2 = 0$\\ -$\frac{\partial L}{\partial x_2} = -4x_2 - \lambda_1 + \lambda_3 = 0$\\ -$\frac{\partial L}{\partial \lambda_1} = -(x_1 + x_2 + s_1^2 - 4) = 0$\\ -$\frac{\partial L}{\partial \lambda_2} = -(-x_1 + s_2^2) = 0$\\ -$\frac{\partial L}{\partial \lambda_3} = -(-x_2 + s_3^2) = 0$\\ -$\frac{\partial L}{\partial s_1} = 2s_1\lambda_1 = 0$\\ -$\frac{\partial L}{\partial s_2} = 2s_2\lambda_2 = 0$\\ -$\frac{\partial L}{\partial s_3} = 2s_3\lambda_3 = 0$} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{3} -\tightlist -\item - Consider all ways that the complementary slackness conditions are solved: - - \begin{center} - \begin{tabular}{|l|cccccccc|c|} - \hline - Hypothesis & $s_1$ & $s_2$ & $s_3$ & $\lambda_1$ & $\lambda_2$ & $\lambda_3$ & $x_1$ & $x_2$ & $f(x_1, x_2)$\\ - \hline - $s_1 = s_2 = s_3 = 0$ & \multicolumn{8}{l|}{\phantom{No solution}} & \\ - $s_1 \neq 0, s_2 = s_3 = 0$ & \phantom{2} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0} & \phantom{0}\\ - $s_2 \neq 0, s_1 = s_3 = 0$ & \phantom{0} & \phantom{2} & \phantom{0} & \phantom{-8} & \phantom{0} & \phantom{-8} & \phantom{4} & \phantom{0} & \phantom{-16}\\ - $s_3 \neq 0, s_1 = s_2 = 0$ & \phantom{0} & \phantom{0} & \phantom{2} & \phantom{-16} & \phantom{-16} & \phantom{0} & \phantom{0} & \phantom{4} & \phantom{-32}\\ - $s_1 \neq 0, s_2 \neq 0, s_3 = 0$ &\multicolumn{8}{l|}{\phantom{No solution}} & \\ - $s_1 \neq 0, s_3 \neq 0, s_2 = 0$ &\multicolumn{8}{l|}{\phantom{No solution}} & \\ - $s_2 \neq 0, s_3 \neq 0, s_1 = 0$ &\phantom{0} & \phantom{$\sqrt{\frac{8}{3}}$} & \phantom{$\sqrt{\frac{4}{3}}$} & \phantom{$-\frac{16}{3}$} & \phantom{0} & \phantom{0} & \phantom{$\frac{8}{3}$}& \phantom{$\frac{4}{3}$} & \phantom{$-\frac{32}{3}$}\\ - $s_1 \neq 0, s_2 \neq 0, s_3 \neq 0$ &\multicolumn{8}{l|}{\phantom{No solution}}& \\ - \hline - \end{tabular} - \end{center} -\end{enumerate} - -\phantom{This shows that there are four critical points: $(0,0)$, $(4,0)$, $(0,4)$, and $(\frac{8}{3},\frac{4}{3})$} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{4} -\tightlist -\item - Find maximum: \phantom{Looking at the values of $f(x_1,x_2)$ at the critical points, we see that the constrained maximum is located at $(x_1, x_2) = (0,0)$, which is the same as the unconstrained max. The constrained minimum is located at $(x_1, x_2) = (0,4)$, while there is no unconstrained minimum for this problem.} -\end{enumerate} - -\hypertarget{kuhn-tucker-conditions}{% -\section{Kuhn-Tucker Conditions}\label{kuhn-tucker-conditions}} - -As you can see, this can be a pain. When dealing explicitly with \emph{non-negativity constraints}, this process is simplified by using the Kuhn-Tucker method. - -Because the problem of maximizing a function subject to inequality and non-negativity constraints arises frequently in economics, the \textbf{Kuhn-Tucker conditions} provides a method that often makes it easier to both calculate the critical points and identify points that are (local) maxima. - -Given a \emph{two-dimensional set-up}: -\[\max_{x_1,x_2}/\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } -\begin{array}{l} -c(x_1,x_2) \le a_1\\ -x_1 \ge 0 \\ -gx_2 \ge 0 -\end{array}\] - -We define the Lagrangian function \(L(x_1,x_2,\lambda_1)\) the same as if we did not have the non-negativity constraints: -\[L(x_1,x_2,\lambda_2) = f(x_1,x_2) - \lambda_1(c(x_1,x_2) - a_1)\] - -More generally, in n dimensions: -\[ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda_k) = f(x_1, \dots, x_n) - \sum_{i=1}^k\lambda_i(c_i(x_1,\dots, x_n) - a_i)\] - -\textbf{Kuhn-Tucker and Complementary Slackness Conditions}: To find the critical points, we first calculate the Kuhn-Tucker conditions by taking the partial derivatives of the lagrangian function, \(L(x_1,\dots,x_n,\lambda_1,\dots,\lambda_k)\), with respect to each of its variables (all choice variable s\(x\) and all lagrangian multipliers \(\lambda\)) and we calculate the \emph{complementary slackness conditions} by multiplying each partial derivative by its respective variable \emph{and} include non-negativity conditions for all variables (choice variables \(x\) and lagrangian multipliers \(\lambda\)). - -\textbf{Kuhn-Tucker Conditions} - -\begin{align*} -\frac{\partial L}{\partial x_1} \leq 0, & \dots, \frac{\partial L}{\partial x_n} \leq 0\\ -\frac{\partial L}{\partial \lambda_1} \geq 0, & \dots, \frac{\partial L}{\partial \lambda_m} \geq 0 -\end{align*} - -\textbf{Complementary Slackness Conditions} - -\begin{align*} -x_1\frac{\partial L}{\partial x_1} = 0, & \dots, x_n\frac{\partial L}{\partial x_n} = 0\\ -\lambda_1\frac{\partial L}{\partial \lambda_1} = 0, & \dots, \lambda_m \frac{\partial L}{\partial \lambda_m} = 0 -\end{align*} - -\textbf{Non-negativity Conditions} -\begin{eqnarray*} -x_1 \geq 0 & \dots & x_n \geq 0\\ -\lambda_1 \geq 0 & \dots & \lambda_m \geq 0 -\end{eqnarray*} - -Note that some of these conditions are set equal to 0, while others are set as inequalities! - -Note also that to minimize the function \(f(x_1, \dots, x_n)\), the simplest thing to do is maximize the function \(-f(x_1, \dots, x_n)\); all of the conditions remain the same after reformulating as a maximization problem. - -There are additional assumptions (notably, f(x) is quasi-concave and the constraints are convex) that are sufficient to ensure that a point satisfying the Kuhn-Tucker conditions is a global max; if these assumptions do not hold, you may have to check more than one point. - -\textbf{Finding the Critical Points with Kuhn-Tucker Conditions}: Given the above conditions, to find the critical points we solve the above system of equations. To do so, we must check \textit{all} border and interior solutions to see if they satisfy the above conditions. - -In a two-dimensional set-up, this means we must check the following cases: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(x_1 = 0, x_2 = 0\) Border Solution -\item - \(x_1 = 0, x_2 \neq 0\) Border Solution -\item - \(x_1 \neq 0, x_2 = 0\) Border Solution -\item - \(x_1 \neq 0, x_2 \neq 0\) Interior Solution -\end{enumerate} - -\begin{example}[Kuhn-Tucker with two variables] -\protect\hypertarget{exm:unnamed-chunk-63}{}{\label{exm:unnamed-chunk-63} \iffalse (Kuhn-Tucker with two variables) \fi{} }Solve the following optimization problem with inequality constraints -\[\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2)\] - -\begin{align*} -\text{ s.t. } -\begin{cases} -&x_1 + x_2 *\le 4\\ -&x_1 *\ge 0\\ -&x_2 *\ge 0 -\end{cases} -\end{align*} -\end{example} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - Write the Lagrangian: - \[L(x_1, x_2, \lambda) = -(x_1^2 + 2x_2^2) - \lambda(x_1 + x_2 - 4)\] -\item - Find the First Order Conditions: -\end{enumerate} - -Kuhn-Tucker Conditions -\begin{align*} -\frac{\partial L}{\partial x_1} = -2x_1 - \lambda &\leq 0\\ -\frac{\partial L}{\partial x_2} = -4x_2 - \lambda & \leq 0\\ -\frac{\partial L}{\partial \lambda} = -(x_1 + x_2 - 4)& \geq 0 -\end{align*} - -Complementary Slackness Conditions -\begin{align*} -x_1\frac{\partial L}{\partial x_2} = x_1(-2x_1 - \lambda) &= 0\\ -x_2\frac{\partial L}{\partial x_2} = x_2(-4x_2 - \lambda) &= 0\\ -\lambda\frac{\partial L}{\partial \lambda} = -\lambda(x_1 + x_2 - 4)&= 0 -\end{align*} - -Non-negativity Conditions -\begin{align*} -x_1 & \geq 0\\ -x_2 & \geq 0\\ -\lambda & \geq 0 -\end{align*} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{2} -\tightlist -\item - Consider all border and interior cases: - - \begin{center} - \begin{tabular}{|l|ccc|c|} - \hline - Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ - \hline - $x_1 = 0, x_2 = 0$ &0 & 0 & 0 & 0\\ - $x_1 = 0, x_2 \neq 0$ &-16 & 0 & 4 & -32\\ - $x_1 \neq 0, x_2 = 0$ &-8 & 4 & 0 & -16\\ - $x_1 \neq 0, x_2 \neq 0$ & $-\frac{16}{3}$ & $\frac{8}{3}$ & $\frac{4}{3}$ & $-\frac{32}{3}$\\ - \hline - \end{tabular} - \end{center} -\item - Find Maximum: - Three of the critical points violate the requirement that \(\lambda \geq 0\), so the point \((0,0,0)\) is the maximum. -\end{enumerate} - -\begin{exercise}[Kuhn-Tucker with logs] -\protect\hypertarget{exr:unnamed-chunk-64}{}{\label{exr:unnamed-chunk-64} \iffalse (Kuhn-Tucker with logs) \fi{} }Solve the constrained optimization problem, - -\[\max_{x_1,x_2} f(x) = \frac{1}{3}\log (x_1 + 1) + \frac{2}{3}\log (x_2 + 1) \text{ s.t. } -\begin{array}{l} -x_1 + 2x_2 \leq 4\\ - x_1 \geq 0\\ - x_2 \geq 0 -\end{array}\] -\end{exercise} - -\begin{enumerate} -\item Write the Lagrangian: -$$\phantom{L(x_1, x_2, \lambda) = \frac{1}{3}\log(x_1+1) + \frac{2}{3}\log(x_2+1) - \lambda(x_1 + 2x_2 - 4)}$$ - -\item Find the First Order Conditions:\\ -Kuhn-Tucker Conditions\\ -\phantom{$\frac{\partial L}{\partial x_1} = \frac{1}{3(x_1+1)} - \lambda \leq 0$\\ -$\frac{\partial L}{\partial x_2} = \frac{2}{3(x_2+1)} - \lambda \leq 0$\\ -$\frac{\partial L}{\partial \lambda} = -(x_1 + 2x_2 - 4) \geq 0$}\\ -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] - -Complementary Slackness Conditions\\ -\phantom{$x_1\frac{\partial L}{\partial x_2} = x_1(\frac{1}{3(x_1+1)} - \lambda) = 0$\\ -$x_2\frac{\partial L}{\partial x_2} = x_2(\frac{2}{3(x_2+1)} - \lambda) = 0$\\ -$\lambda\frac{\partial L}{\partial \lambda} = -\lambda(x_1 + 2x_2 - 4) = 0$}\\ -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] -\item[] - -Non-negativity Conditions\\ -\phantom{$x_1 \geq 0$\\ -$x_2 \geq 0$\\ -$\lambda \geq $0}\\ -\item[] -\item[] -\item[] - -\item Consider all border and interior cases: -\begin{center} -\begin{tabular}{|l|ccc|c|} -\hline -Hypothesis & $\lambda$& $x_1$ & $x_2$ & $f(x_1, x_2)$\\ -\hline -$x_1 = 0, x_2 = 0$ &\multicolumn{3}{l|}{\phantom{No Solution}}& \\ -$x_1 = 0, x_2 \neq 0$ &\multicolumn{3}{l|}{\phantom{No Solution}}& \\ -$x_1 \neq 0, x_2 = 0$ & \multicolumn{3}{l|}{\phantom{No Solution}}& \\ -$x_1 \neq 0, x_2 \neq 0$ & & \phantom{$\frac{4}{3}$} & \phantom{$\frac{4}{3}$} & \phantom{$\log\frac{7}{3}$}\\ -\hline -\end{tabular} -\end{center} - -\item[] -\item[] -\item[] -\item[] - -\item Find Maximum:\\ -\phantom{Three of the critical points violate the constraints, so the point $(\frac{4}{3},\frac{4}{3})$ is the maximum.}\\ -\end{enumerate} - -\hypertarget{applications-of-quadratic-forms}{% -\section{Applications of Quadratic Forms}\label{applications-of-quadratic-forms}} - -\textbf{Curvature and The Taylor Polynomial as a Quadratic Form}: -The Hessian is used in a Taylor polynomial approximation to \(f(\mathbf{x})\) and provides information about the curvature of \(f({\bf x})\) at \(\mathbf{x}\) --- e.g., which tells us whether a critical point \(\mathbf{x}^*\) is a min, max, or saddle point. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - The second order Taylor polynomial about the critical point - \({\bf x}^*\) is - \[f({\bf x}^*+\bf h)=f({\bf x}^*)+\nabla f({\bf x}^*) \bf h +\frac{1}{2} \bf h^\top - {\bf H(x^*)} \bf h + R(\bf h)\] -\item - Since we're looking at a critical point, \(\nabla f({\bf x}^*)=0\); - and for small \(\bf h\), \(R(\bf h)\) is negligible. Rearranging, we get - \[f({\bf x}^*+\bf h)-f({\bf x}^*)\approx \frac{1}{2} \bf h^\top {\bf H(x^*)} - \bf h \] -\item - The Righthand side here is a quadratic form and we can determine the definiteness of \(\bf H(x^*)\). -\end{enumerate} - -\hypertarget{probability-theory}{% -\chapter{Probability Theory}\label{probability-theory}} - -Probability and Inferences are mirror images of each other, and both are integral to social science. Probability quantifies uncertainty, which is important because many things in the social world are at first uncertain. Inference is then the study of how to learn about facts you don't observe from facts you do observe. - -\hypertarget{counting-rules}{% -\section{Counting rules}\label{counting-rules}} - -\textbf{Fundamental Theorem of Counting}: If an object has \(j\) different characteristics that are independent of each other, and each characteristic \(i\) has \(n_i\) ways of being expressed, then there are \(\prod_{i = 1}^j n_i\) possible unique objects. - -Example: Cards can be either red or black and can take on any of 13 values. - -\(j =\) - -\(n_{\text{color}} =\) - -\(n_{\text{number}} =\) - -Number of Outcomes \(=\) - -We often need to count the number of ways to choose a subset from some set of possibilities. The number of outcomes depends on two characteristics of the process: does the order matter and is replacement allowed? - -\textbf{Sampling Table}: If there are \(n\) objects which are numbered 1 to \(n\) and we select \(k < n\) of them, how many different outcomes are possible? - -If the order in which a given object is selected matters, selecting 4 numbered objects in the following order (1, 3, 7, 2) and selecting the same four objects but in a different order such as (7, 2, 1, 3) will be counted as different outcomes. - -If replacement is allowed, there are always the same \(n\) objects to select from. However, if replacement is not allowed, there is always one less option than the previous round when making a selection. For example, if replacement is not allowed and I am selecting 3 elements from the following set \{1, 2, 3, 4, 5, 6\}, I will have 6 options at first, 5 options as I make my second selection, and 4 options as I make my third. - -So in counting how many different outcomes are possible, if \textbf{\emph{order matters}} AND we are sampling \textbf{\emph{with replacement}}, the number of different outcomes is \(n^k\). - -If \textbf{\emph{order matters}} AND we are sampling \textbf{\emph{without replacement}}, the number of different outcomes is \(n(n-1)(n-2)...(n-k+1)=\frac{n!}{(n-k)!}\). - -If \textbf{\emph{order doesn't matter}} AND we are sampling \textbf{\emph{without replacement}}, the number of different outcomes is \(\binom{n}{k} = \frac{n!}{(n-k)!k!}\). - -Expression \(\binom{n}{k}\) is read as ``n choose k'' and denotes \(\frac{n!}{(n-k)!k!}\). Also, note that \(0! = 1\). - -\begin{example}[Counting] -\protect\hypertarget{exm:counting}{}{\label{exm:counting} \iffalse (Counting) \fi{} } -There are five balls numbered from 1 through 5 in a jar. Three balls are chosen. How many possible choices are there? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - Ordered, with replacement \(=\) -\item - Ordered, without replacement \(=\) -\item - Unordered, without replacement \(=\) -\end{enumerate} -\end{example} - -\begin{exercise}[Counting] -\protect\hypertarget{exr:counting1}{}{\label{exr:counting1} \iffalse (Counting) \fi{} } -Four cards are selected from a deck of 52 cards. Once a card has been drawn, it is not reshuffled back into the deck. Moreover, we care only about the complete hand that we get (i.e.~we care about the set of selected cards, not the sequence in which it was drawn). How many possible outcomes are there? -\end{exercise} - -\hypertarget{setoper}{% -\section{Sets}\label{setoper}} - -\textbf{Set} : A set is any well defined collection of elements. If \(x\) is an element of \(S\), \(x \in S\). - -\textbf{Sample Space (S)}: A set or collection of all possible outcomes from some process. Outcomes in the set can be discrete elements (countable) or points along a continuous interval (uncountable). - -Examples: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Discrete: the numbers on a die, whether a vote cast is republican or democrat. -\item - Continuous: GNP, arms spending, age. -\end{enumerate} - -\textbf{Event}: Any collection of possible outcomes of an experiment. Any subset of the full set of possibilities, including the full set itself. Event A \(\subset\) S. - -\textbf{Empty Set}: a set with no elements. \(S = \{\}\). It is denoted by the symbol \(\emptyset\). - -Set operations: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \textbf{Union}: The union of two sets \(A\) and \(B\), \(A \cup B\), is the set containing all of the elements in \(A\) or \(B\). \[A_1 \cup A_2 \cup \cdots \cup A_n = \bigcup_{i=1}^n A_i\] -\item - \textbf{Intersection}: The intersection of sets \(A\) and \(B\), \(A \cap B\), is the set containing all of the elements in both \(A\) and \(B\). \[A_1 \cap A_2 \cap \cdots \cap A_n = \bigcap_{i=1}^n A_i\] -\item - \textbf{Complement}: If set \(A\) is a subset of \(S\), then the complement of \(A\), denoted \(A^C\), is the set containing all of the elements in \(S\) that are not in \(A\). -\end{enumerate} - -Properties of set operations: - -\begin{itemize} -\tightlist -\item - \textbf{Commutative}: \(A \cup B = B \cup A\); \(A \cap B = B \cap A\) -\item - \textbf{Associative}: \(A \cup (B \cup C) = (A \cup B) \cup C\); \(A \cap (B \cap C) = (A \cap B) \cap C\) -\item - \textbf{Distributive}: \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\); \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\) -\item - \textbf{de Morgan's laws}: \((A \cup B)^C = A^C \cap B^C\); \((A \cap B)^C = A^C \cup B^C\) -\item - \textbf{Disjointness}: Sets are disjoint when they do not intersect, such that \(A \cap B = \emptyset\). A collection of sets is pairwise disjoint (\textbf{mutually exclusive}) if, for all \(i \neq j\), \(A_i \cap A_j = \emptyset\). A collection of sets form a partition of set \(S\) if they are pairwise disjoint and they cover set \(S\), such that \(\bigcup_{i = 1}^k A_i = S\). -\end{itemize} - -\begin{example}[Sets] -\protect\hypertarget{exm:sets}{}{\label{exm:sets} \iffalse (Sets) \fi{} } -Let set \(A\) be \{1, 2, 3, 4\}, \(B\) be \{3, 4, 5, 6\}, and \(C\) be \{5, 6, 7, 8\}. Sets \(A\), \(B\), and \(C\) are all subsets of the sample space \(S\) which is \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} - -Write out the following sets: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(A \cup B\) -\item - \(C \cap B\) -\item - \(B^c\) -\item - \(A \cap (B \cup C)\) -\end{enumerate} -\end{example} - -\begin{exercise}[Sets] -\protect\hypertarget{exr:sets1}{}{\label{exr:sets1} \iffalse (Sets) \fi{} } - -Suppose you had a pair of four-sided dice. You sum the results from a single toss. - -What is the set of possible outcomes (i.e.~the sample space)? - -Consider subsets A \{2, 8\} and B \{2,3,7\} of the sample space you found. What is - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(A^c\) -\item - \((A \cup B)^c\) -\end{enumerate} -\end{exercise} - -\hypertarget{probdef}{% -\section{Probability}\label{probdef}} - -\begin{figure} -\centering -\includegraphics{images/probability.pdf} -\caption[\label{fig:prob-image}Probablity as a Measure]{\label{fig:prob-image}Probablity as a Measure\footnotemark{}} -\end{figure} -\footnotetext{Images of Probability and Random Variables drawn by Shiro Kuriwaki and inspired by Blitzstein and Morris} - -\hypertarget{probability-definitions-formal-and-informal}{% -\subsection*{Probability Definitions: Formal and Informal}\label{probability-definitions-formal-and-informal}} -\addcontentsline{toc}{subsection}{Probability Definitions: Formal and Informal} - -Many things in the world are uncertain. In everyday speech, we say that we are \emph{uncertain} about the outcome of random events. Probability is a formal model of uncertainty which provides a measure of uncertainty governed by a particular set of rules (Figure \ref{fig:prob-image}). A different model of uncertainty would, of course, have a set of rules different from anything we discuss here. Our focus on probability is justified because it has proven to be a particularly useful model of uncertainty. - -\textbf{Probability Distribution Function}: a mapping of each event in the sample space \(S\) to the real numbers that satisfy the following three axioms (also called Kolmogorov's Axioms). - -Formally, - -\begin{definition}[Probability] -\protect\hypertarget{def:unnamed-chunk-66}{}{\label{def:unnamed-chunk-66} \iffalse (Probability) \fi{} } -Probability is a function that maps events to a real number, obeying the axioms of probability. -\end{definition} - -The axioms of probability make sure that the separate events add up in terms of probability, and -- for standardization purposes -- that they add up to 1. - -\begin{definition}[Axioms of Probability] -\protect\hypertarget{def:unnamed-chunk-67}{}{\label{def:unnamed-chunk-67} \iffalse (Axioms of Probability) \fi{} } - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - For any event \(A\), \(P(A)\ge 0\). -\item - \(P(S)=1\) -\item - The Countable Additivity Axiom: For any sequence of \emph{disjoint} (mutually exclusive) events \(A_1,A_2,\ldots\) (of which there may be infinitely many), \[P\left( \bigcup\limits_{i=1}^k - A_i\right)=\sum\limits_{i=1}^k P(A_i)\] -\end{enumerate} - -The last axiom is an extension of a union to infinite sets. When there are only two events in the space, it boils down to: - -\begin{align*} -P(A_1 \cup A_2) = P(A_1) + P(A_2) \quad\text{for disjoint } A_1, A_2 -\end{align*} -\end{definition} - -\hypertarget{probability-operations}{% -\subsection*{Probability Operations}\label{probability-operations}} -\addcontentsline{toc}{subsection}{Probability Operations} - -Using these three axioms, we can define all of the common rules of probability. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(P(\emptyset)=0\) -\item - For any event \(A\), \(0\le P(A) \le 1\). -\item - \(P({A}^C)=1-P(A)\) -\item - If \(A\subset B\) (\(A\) is a subset of \(B\)), then \(P(A)\le P(B)\). -\item - For \emph{any} two events \(A\) and \(B\), \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) -\item - Boole's Inequality: For any sequence of \(n\) events (which need not be disjoint) \(A_1,A_2,\ldots,A_n\), then \(P\left( \bigcup\limits_{i=1}^n A_i\right) \leq \sum\limits_{i=1}^n P(A_i)\). -\end{enumerate} - -\begin{example}[Probability] -\protect\hypertarget{exm:prob}{}{\label{exm:prob} \iffalse (Probability) \fi{} } -Assume we have an evenly-balanced, six-sided die. - -Then, - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Sample space S = -\item - \(P(1)=\cdots=P(6)=\) -\item - \(P(\emptyset)=P(7)=\) -\item - \(P\left( \{ 1, 3, 5 \} \right)=\) -\item - \(P\left( \{ 1, 2 \}^C \right)= P\left( \{ 3, 4, 5, 6 \}\right)=\) -\item - Let \(A=\{ 1,2,3,4,5 \}\subset S\). Then \(P(A)=5/6 x) = 1 - P(X \le x)\). - -\begin{example} -\protect\hypertarget{exm:unnamed-chunk-77}{}{\label{exm:unnamed-chunk-77} }For a fair die with its value as \(Y\), What are the following? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(P(Y\le 1)\) -\item - \(P(Y\le 3)\) -\item - \(P(Y\le 6)\) -\end{enumerate} -\end{example} - -\hypertarget{continuous-random-variables}{% -\subsection*{Continuous Random Variables}\label{continuous-random-variables}} -\addcontentsline{toc}{subsection}{Continuous Random Variables} - -We also have a similar definition for \emph{continuous} random variables. - -\begin{definition}[Continuous Random Variable] -\protect\hypertarget{def:unnamed-chunk-78}{}{\label{def:unnamed-chunk-78} \iffalse (Continuous Random Variable) \fi{} } -\(X\) is a continuous random variable if there exists a nonnegative function \(f(x)\) defined for all real \(x\in (-\infty,\infty)\), such that for any interval \(A\), \(P(X\in A)=\int\limits_A f(x)dx\). Examples: age, income, GNP, temperature. -\end{definition} - -\begin{definition}[Probability Density Function] -\protect\hypertarget{def:unnamed-chunk-79}{}{\label{def:unnamed-chunk-79} \iffalse (Probability Density Function) \fi{} } -The function \(f\) above is called the probability density function (pdf) of \(X\) and must satisfy -\[f(x)\ge 0\] -\[\int\limits_{-\infty}^\infty f(x)dx=1\] - -Note also that \(P(X = x)=0\) --- i.e., the probability of any point \(y\) is zero. -\end{definition} - -\begin{comment} -\item[] \parbox[t]{4.5in}{Example: $f(y)=1, \quad 0\le y \le1$}\parbox{1.5in}{\hfill -\epsffile{contpdf.eps}} -\end{comment} - -For both discrete and continuous random variables, we have a unifying concept of another measure: the cumulative distribution: - -\begin{definition}[Cumulative Density Function] -\protect\hypertarget{def:unnamed-chunk-80}{}{\label{def:unnamed-chunk-80} \iffalse (Cumulative Density Function) \fi{} }Because the probability that a continuous random variable will assume any particular value is zero, we can only make statements about the probability of a continuous random variable being within an interval. The cumulative distribution gives the probability that \(Y\) lies on the interval \((-\infty,y)\) and is defined as \[F(x)=P(X\le x)=\int\limits_{-\infty}^x f(s)ds\] Note that \(F(x)\) has similar properties with continuous distributions as it does with discrete - non-decreasing, continuous (not just right-continuous), and \(\lim\limits_{x \to -\infty} F(x) = 0\) and \(\lim\limits_{x \to \infty} F(x) = 1\). -\end{definition} - -We can also make statements about the probability of \(Y\) falling in an interval \(a\le y\le b\). -\[P(a\le x\le b)=\int\limits_a^b f(x)dx\] - -The PDF and CDF are linked by the integral: The CDF of the integral of the PDF: \[f(x) = F'(x)=\frac{dF(x)}{dx}\] - -\begin{example} -\protect\hypertarget{exm:unnamed-chunk-81}{}{\label{exm:unnamed-chunk-81} } -For \(f(y)=1, \quad 0 0\) -\item - Continuous: \(f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)},\quad f_X(x) > 0\) -\end{itemize} - -\begin{exercise}[Discrete Outcomes] -\protect\hypertarget{exr:unnamed-chunk-82}{}{\label{exr:unnamed-chunk-82} \iffalse (Discrete Outcomes) \fi{} }Suppose we are interested in the outcomes of flipping a coin and rolling a 6-sided die at the same time. The sample space for this process contains 12 elements: \[\{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}\] We can define two random variables \(X\) and \(Y\) such that \(X = 1\) if heads and \(X = 0\) if tails, while \(Y\) equals the number on the die. - -We can then make statements about the joint distribution of \(X\) and \(Y\). What are the following? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(P(X=x)\) -\item - \(P(Y=y)\) -\item - \(P(X=x, Y=y)\) -\item - \(P(X=x|Y=y)\) -\item - Are X and Y independent? -\end{enumerate} -\end{exercise} - -\hypertarget{expectation}{% -\section{Expectation}\label{expectation}} - -We often want to summarize some characteristics of the distribution of a random variable. The most important summary is the expectation (or expected value, or mean), in which the possible values of a random variable are weighted by their probabilities. - -\begin{definition}[Expectation of a Discrete Random Variable] -\protect\hypertarget{def:unnamed-chunk-83}{}{\label{def:unnamed-chunk-83} \iffalse (Expectation of a Discrete Random Variable) \fi{} }The expected value of a discrete random variable \(Y\) is \[E(Y)=\sum\limits_{y} y P(Y=y)= \sum\limits_{y} y p(y)\]\\ -In words, it is the weighted average of all possible values of \(Y\), weighted by the probability that \(y\) occurs. It is not necessarily the number we would expect \(Y\) to take on, but the average value of \(Y\) after a large number of repetitions of an experiment. -\end{definition} - -\begin{example}[Expectation of a Discrete Random Variable] -\protect\hypertarget{exm:expectdiscrete}{}{\label{exm:expectdiscrete} \iffalse (Expectation of a Discrete Random Variable) \fi{} } - -What is the expectation of a fair, six-sided die? -\end{example} - -\textbf{Expectation of a Continuous Random Variable}: The expected -value of a continuous random variable is similar in concept to that of -the discrete random variable, except that instead of summing using -probabilities as weights, we integrate using the density to weight. -Hence, the expected value of the continuous variable \(Y\) is defined by -\[E(Y)=\int\limits_{y} y f(y) dy\] - -\begin{example}[Expectation of a Continuous Random Variable] -\protect\hypertarget{exm:expectconti}{}{\label{exm:expectconti} \iffalse (Expectation of a Continuous Random Variable) \fi{} } - -Find \(E(Y)\) for \(f(y)=\frac{1}{1.5}, \quad 00\] - -The Poisson has the unusual feature that its expectation equals its variance: \(E(Y)=\text{Var}(Y)=\lambda\). The Poisson distribution is often used to model rare event counts: counts of the number of events that occur during some unit of time. \(\lambda\) is often called the ``arrival rate.'' -\end{definition} - -\begin{example} -\protect\hypertarget{exm:unnamed-chunk-90}{}{\label{exm:unnamed-chunk-90} }Border disputes occur between two countries through a Poisson Distribution, at a rate of 2 per month. What is the probability of 0, 2, and less than 5 disputes occurring in a month? -\end{example} - -\begin{comment} - \parbox{1.5in}{\hfill \epsffile{poispmf.eps}} - \end{comment} - -Two \emph{continuous} distributions used often are: - -\begin{definition}[Uniform Distribution] -\protect\hypertarget{def:unnamed-chunk-91}{}{\label{def:unnamed-chunk-91} \iffalse (Uniform Distribution) \fi{} }A random variable \(Y\) has a continuous uniform distribution on the interval \((\alpha,\beta)\) if its density is given by \[f(y)=\frac{1}{\beta-\alpha}, \quad \alpha\le y\le \beta\] The mean and variance of \(Y\) are \(E(Y)=\frac{\alpha+\beta}{2}\) and \(\text{Var}(Y)=\frac{(\beta-\alpha)^2}{12}\). -\end{definition} - -\begin{example} -\protect\hypertarget{exm:unnamed-chunk-92}{}{\label{exm:unnamed-chunk-92} }For \(Y\) uniformly distributed over \((1,3)\), what are the following probabilities? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(P(Y=2)\) -\item - Its density evaluated at 2, or \(f(2)\) -\item - \(P(Y \le 2)\) -\item - \(P(Y > 2)\) -\end{enumerate} -\end{example} - -\begin{comment} -\parbox{1.5in}{\hfill \epsffile{unifpdf.eps}} -\end{comment} - -\begin{definition}[Normal Distribution] -\protect\hypertarget{def:unnamed-chunk-93}{}{\label{def:unnamed-chunk-93} \iffalse (Normal Distribution) \fi{} } -A random variable \(Y\) is normally distributed with mean \(E(Y)=\mu\) and variance \(\text{Var}(Y)=\sigma^2\) if its density is - -\[f(y)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(y-\mu)^2}{2\sigma^2}}\] -\end{definition} - -See Figure \ref{fig:normaldens} are various Normal Distributions with the same \(\mu = 1\) and two versions of the variance. - -\begin{figure} -\centering -\includegraphics{prefresher_files/figure-latex/normaldens-1.pdf} -\caption{\label{fig:normaldens}Normal Distribution Density} -\end{figure} - -\hypertarget{summarizing-observed-events-data}{% -\section{Summarizing Observed Events (Data)}\label{summarizing-observed-events-data}} - -So far, we've talked about distributions in a theoretical sense, looking at different properties of random variables. We don't observe random variables; we observe realizations of the random variable. These realizations of events are roughly equivalent to what we mean by ``data''. - -\textbf{Sample mean}: This is the most common measure of central tendency, calculated by summing across the observations and dividing by the number of observations. -\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i\] -The sample mean is an \emph{estimate} of the expected value of a distribution. - -\begin{framed} -Example: -\begin{center} -\begin{tabular}{|l|cccccccccc|} -\hline -X & 6 & 3 & 7 & 5 & 5 & 5 & 6 & 4 & 7 & 2\\ -\hline -Y & 1 & 2 & 1 & 2 & 2 & 1 & 2 & 0 & 2 & 0\\ -\hline -\end{tabular} -\end{center} - -\begin{enumerate} -\item $\bar{x} = $ \hspace{3.1cm} $\bar{y} = $ -\item median(x) $ = $ \hspace{1.5cm} median(y) $ = $ -\item $m_x = $ \hspace{2.75cm} $m_y =$\\ -\end{enumerate} -\end{framed} - -\textbf{Dispersion}: We also typically want to know how spread out the data are relative to the center of the observed distribution. There are several ways to measure dispersion. - -\textbf{Sample variance}: The sample variance is the sum of the squared deviations from the sample mean, divided by the number of observations minus 1. -\[ \hat{\text{Var}}(X) = \frac{1}{n-1}\sum_{i = 1}^n (x_i - \bar{x})^2\] - -Again, this is an \emph{estimate} of the variance of a random variable; we divide by \(n - 1\) instead of \(n\) in order to get an unbiased estimate. - -\textbf{Standard deviation}: The sample standard deviation is the square root of the sample variance. -\[ \hat{SD}(X) = \sqrt{\hat{\text{Var}}(X)} = \sqrt{\frac{1}{n-1}\sum_{i = 1}^n (x_i - \bar{x})^2}\] - -\begin{framed} -Example: Using table above, calculate: -\begin{enumerate} -\item $\text{Var}(X) = $ \hspace{1.5cm} $\text{Var}(Y) =$ -\item $\SD(X) = $ \hspace{1.65cm} $\SD(Y) =$ -\end{enumerate} -\end{framed} - -\textbf{Covariance and Correlation}: Both of these quantities measure the degree to which two variables vary together, and are estimates of the covariance and correlation of two random variables as defined above. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \textbf{Sample covariance}: \(\hat{\text{Cov}}(X,Y) = \frac{1}{n-1}\sum_{i = 1}^n(x_i - \bar{x})(y_i - \bar{y})\) -\item - \textbf{Sample correlation}: \(\hat{\text{Corr}} = \frac{\hat{\text{Cov}}(X,Y)}{\sqrt{\hat{\text{Var}}(X)\hat{\text{Var}}(Y)}}\) -\end{enumerate} - -\begin{example} -\protect\hypertarget{exm:unnamed-chunk-94}{}{\label{exm:unnamed-chunk-94} }Example: Using the above table, calculate the sample versions of: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\text{Cov}(X,Y)\) -\item - \(\text{Corr}(X, Y)\) -\end{enumerate} -\end{example} - -\hypertarget{asymptotic-theory}{% -\section{Asymptotic Theory}\label{asymptotic-theory}} - -In theoretical and applied research, asymptotic arguments are often made. In this section we briefly introduce some of this material. - -What are asymptotics? In probability theory, asymptotic analysis is the study of limiting behavior. By limiting behavior, we mean the behavior of some random process as the number of observations gets larger and larger. - -Why is this important? We rarely know the true process governing the events we see in the social world. It is helpful to understand how such unknown processes theoretically must behave and asymptotic theory helps us do this. - -\hypertarget{clt-and-lln}{% -\subsection{CLT and LLN}\label{clt-and-lln}} - -We are now finally ready to revisit, with a bit more precise terms, the two pillars of statistical theory we motivated Section \ref{limitsfun} with. - -\begin{theorem}[Central Limit Theorem (i.i.d. case)] -\protect\hypertarget{thm:clt}{}{\label{thm:clt} \iffalse (Central Limit Theorem (i.i.d. case)) \fi{} }Let \(\{X_n\} = \{X_1, X_2, \ldots\}\) be a sequence of i.i.d. random variables with finite mean (\(\mu\)) and variance (\(\sigma^2\)). Then, the sample mean \(\bar{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n}\) increasingly converges into a Normal distribution. - -\[\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \text{Normal}(0, 1),\] - -Another way to write this as a probability statement is that for all real numbers \(a\), - -\[P\left(\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \le a\right) \rightarrow \Phi(a)\] -as \(n\to \infty\), where \[\Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \, dx\] is the CDF of a Normal distribution with mean 0 and variance 1. - -This result means that, as \(n\) grows, the distribution of the sample mean \(\bar X_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n)\) is approximately normal with mean \(\mu\) and standard deviation \(\frac{\sigma}{\sqrt n}\), i.e., -\[\bar{X}_n \approx \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).\] The standard deviation of \(\bar X_n\) (which is roughly a measure of the precision of \(\bar X_n\) as an estimator of \(\mu\)) decreases at the rate \(1/\sqrt{n}\), so, for example, to increase its precision by \(10\) (i.e., to get one more digit right), one needs to collect \(10^2=100\) times more units of data. - -Intuitively, this result also justifies that whenever a lot of small, independent processes somehow combine together to form the realized observations, practitioners often feel comfortable assuming Normality. -\end{theorem} - -\begin{theorem}[Law of Large Numbers (LLN)] -\protect\hypertarget{thm:lln}{}{\label{thm:lln} \iffalse (Law of Large Numbers (LLN)) \fi{} }For any draw of independent random variables with the same mean \(\mu\), the sample average after \(n\) draws, \(\bar{X}_n = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)\), converges in probability to the expected value of \(X\), \(\mu\) as \(n \rightarrow \infty\): - -\[\lim\limits_{n\to \infty} P(|\bar{X}_n - \mu | > \varepsilon) = 0\] - -A shorthand of which is \(\bar{X}_n \xrightarrow{p} \mu\), where the arrow is read as ``converges in probability to''. -\end{theorem} - -as \(n\to \infty\). In other words, \(P( \lim_{n\to\infty}\bar{X}_n = \mu) = 1\). This is an important motivation for the widespread use of the sample mean, as well as the intuition link between averages and expected values. - -More precisely this version of the LLN is called the \emph{weak} law of large numbers because it leaves open the possibility that \(|\bar{X}_n - \mu | > \varepsilon\) occurs many times. The \emph{strong} law of large numbers states that, under a few more conditions, the probability that the limit of the sample average is the true mean is 1 (and other possibilities occur with probability 0), but the difference is rarely consequential in practice. - -The Strong Law of Large Numbers holds so long as the expected value exists; no other assumptions are needed. However, the rate of convergence will differ greatly depending on the distribution underlying the observed data. When extreme observations occur often (i.e.~kurtosis is large), the rate of convergence is much slower. Cf. The distribution of financial returns. - -\hypertarget{big-mathcalo-notation}{% -\subsection{\texorpdfstring{Big \(\mathcal{O}\) Notation}{Big \textbackslash mathcal\{O\} Notation}}\label{big-mathcalo-notation}} - -Some of you may encounter "big-OH'\,'-notation. If \(f, g\) are two functions, we say that \(f = \mathcal{O}(g)\) if there exists some constant, \(c\), such that \(f(n) \leq c \times g(n)\) for large enough \(n\). This notation is useful for simplifying complex problems in game theory, computer science, and statistics. - -Example. - -What is \(\mathcal{O}( 5\exp(0.5 n) + n^2 + n / 2)\)? Answer: \(\exp(n)\). Why? Because, for large \(n\), -\[ -\frac{ 5\exp(0.5 n) + n^2 + n / 2 }{ \exp(n)} \leq \frac{ c \exp(n) }{ \exp(n)} = c. -\] -whenever \(n > 4\) and where \(c = 1\). - -\begin{comment} -n_seq <- 1:100; numerator <- (5 * exp(0.5 * n_seq) + n_seq^2 + n_seq / 2) -denominator <- exp(n_seq) -numerator / denominator <= (1 * denominator) / denominator -\end{comment} - -\hypertarget{answers-to-examples-and-exercises-2}{% -\section*{Answers to Examples and Exercises}\label{answers-to-examples-and-exercises-2}} -\addcontentsline{toc}{section}{Answers to Examples and Exercises} - -Answer to Example \ref{exm:counting}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(5 \times 5 \times 5 = 125\) -\item - \(5 \times 4 \times 3 = 60\) -\item - \(\binom{5}{3} = \frac{5!}{(5-3)!3!} = \frac{5 \times 4}{2 \times 1} = 10\) -\end{enumerate} - -Answer to Exercise \ref{exr:counting1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\binom{52}{4} = \frac{52!}{(52-4)!4!} = 270725\) -\end{enumerate} - -Answer to Example \ref{exm:sets}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \{1, 2, 3, 4, 5, 6\} -\item - \{5, 6\} -\item - \{1, 2, 7, 8, 9, 10\} -\item - \{3, 4\} -\end{enumerate} - -Answer to Exercise \ref{exr:sets1}: - -Sample Space: \{2, 3, 4, 5, 6, 7, 8\} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \{3, 4, 5, 6, 7\} -\item - \{4, 5, 6\} -\end{enumerate} - -Answer to Example \ref{exm:prob}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \({1, 2, 3, 4, 5, 6}\) -\item - \(\frac{1}{6}\) -\item - \(0\) -\item - \(\frac{1}{2}\) -\item - \(\frac{4}{6} = \frac{2}{3}\) -\item - \(1\) -\item - \(A\cup B=\{1, 2, 3, 4, 6\}\), \(A\cap B=\{2\}\), \(\frac{5}{6}\) -\end{enumerate} - -Answer to Exercise \ref{exr:prob1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(P(X = 5) = \frac{4}{16}\), \(P(X = 3) = \frac{2}{16}\), \(P(X = 6) = \frac{3}{16}\) -\item - What is \(P(X=5 \cup X = 3)^C = \frac{10}{16}\)? -\end{enumerate} - -Answer to Example \ref{exm:condprobexm1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\frac{n_{ab} + n_{ab^c}}{N}\) -\item - \(\frac{n_{ab} + n_{a^cb}}{N}\) -\item - \(\frac{n_{ab}}{N}\) -\item - \(\frac{\frac{n_{ab}}{N}}{\frac{n_{ab} + n_{a^cb}}{N}} = \frac{n_{ab}}{n_{ab} + n_{a^cb}}\) -\item - \(\frac{\frac{n_{ab}}{N}}{\frac{n_{ab} + n_{ab^c}}{N}} = \frac{n_{ab}}{n_{ab} + n_{ab^c}}\) -\end{enumerate} - -Answer to Example \ref{exm:condprobexm2}: - -\(P(1|Odd) = \frac{P(1 \cap Odd)}{P(Odd)} = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{3}\) - -Answer to Example \ref{exm:bayesrule}: - -We are given that -\[P(D) = .4, P(D^c) = .6, P(S|D) = .5, P(S|D^c) = .9\] -Using this, Bayes' Law and the Law of Total Probability, we know: - -\[P(D|S) = \frac{P(D)P(S|D)}{P(D)P(S|D) + P(D^c)P(S|D^c)}\] -\[P(D|S) = \frac{.4 \times .5}{.4 \times .5 + .6 \times .9 } = .27\] - -Answer to Exercise \ref{exr:condprobexr}: - -We are given that - -\[P(M) = .02, P(C|M) = .95, P(C^c|M^c) = .97\] - -\[P(M|C) = \frac{P(C|M)P(M)}{P(C)}\] - -\[= \frac{P(C|M)P(M)}{P(C|M)P(M) + P(C|M^c)P(M^c)}\] - -\[= \frac{P(C|M)P(M)}{P(C|M)P(M) + [1-P(C^c|M^c)]P(M^c)}\] -\[ = \frac{.95 \times .02}{.95 \times .02 + .03 \times .98} = .38\] - -Answer to Example \ref{exm:expectdiscrete}: - -\(E(Y)=7/2\) - -We would never expect the result of a rolled die to be \(7/2\), but that would be the average over a large number of rolls of the die. - -Answer to Example \ref{exm:expectconti} - -0.75 - -Answer to Example \ref{exm:var}: - -\(E(X) = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 2 \times \frac{3}{8} + 3 \times \frac{1}{8} = \frac{3}{2}\) - -Since there is a 1 to 1 mapping from \(X\) to \(X^2:\) \(E(X^2) = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 4 \times \frac{3}{8} + 9 \times \frac{1}{8} = \frac{24}{8} = 3\) - -\begin{align*} -\text{Var}(x) &= E(X^2) - E(x)^2\\ -&= 3 - (\frac{3}{2})^2\\ -&= \frac{3}{4} -\end{align*} - -Answer to Exercise \ref{exr:expvar}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(E(X) = -2(\frac{1}{5}) + -1(\frac{1}{6}) + 0(\frac{1}{5}) + 1(\frac{1}{15}) + 2(\frac{11}{30}) = \frac{7}{30}\) -\item - \(E(Y) = 0(\frac{1}{5}) + 1(\frac{7}{30}) + 4(\frac{17}{30}) = \frac{5}{2}\) -\item -\end{enumerate} - -\begin{align*} -\text{Var}(X) &= E[X^2] - E[X]^2\\ -&= E(Y) - E(X)^2\\ -&= \frac{5}{2} - \frac{7}{30}^2 \approx 2.45 -\end{align*} - -Answer to Exercise \ref{exr:expvar2}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - expectation = \(\frac{6}{5}\), variance = \(\frac{6}{25}\) -\end{enumerate} - -Answer to Exercise \ref{exr:expvar3}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - mean = 2, standard deviation = \(\sqrt(\frac{2}{3})\) -\item - \(\frac{1}{8}(2 - \sqrt(\frac{2}{3}))^2\) -\end{enumerate} - -\hypertarget{linearalgebra}{% -\chapter{Linear Algebra}\label{linearalgebra}} - -Topics: -\(\bullet\) Working with Vectors -\(\bullet\) Linear Independence -\(\bullet\) Basics of Matrix Algebra -\(\bullet\) Square Matrices -\(\bullet\) Linear Equations -\(\bullet\) Systems of Linear Equations -\(\bullet\) Systems of Equations as Matrices -\(\bullet\) Solving Augmented Matrices and Systems of Equations -\(\bullet\) Rank -\(\bullet\) The Inverse of a Matrix -\(\bullet\) Inverse of Larger Matrices - -\hypertarget{vector-def}{% -\section{Working with Vectors}\label{vector-def}} - -\textbf{Vector}: A vector in \(n\)-space is an ordered list of \(n\) numbers. These numbers can be represented as either a row vector or a column vector: -\[ {\bf v} \begin{pmatrix} v_1 & v_2 & \dots & v_n\end{pmatrix} , {\bf v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}\] - -We can also think of a vector as defining a point in \(n\)-dimensional space, usually \({\bf R}^n\); each element of the vector defines the coordinate of the point in a particular direction. - -\textbf{Vector Addition and Subtraction}: If two vectors, \({\bf u}\) and \({\bf v}\), have the same length (i.e.~have the same number of elements), they can be added (subtracted) together: -\[ {\bf u} + {\bf v} = \begin{pmatrix} u_1 + v_1 & u_2 + v_2 & \cdots & u_k + v_n \end{pmatrix}\] -\[ {\bf u} - {\bf v} = \begin{pmatrix} u_1 - v_1 & u_2 - v_2 & \cdots & u_k - v_n \end{pmatrix}\] - -\textbf{Scalar Multiplication}: The product of a scalar \(c\) (i.e.~a constant) and vector \({\bf v}\) is:\\ -\[ c{\bf v} = \begin{pmatrix} cv_1 & cv_2 & \dots & cv_n \end{pmatrix} \] - -\textbf{Vector Inner Product}: The inner product (also called the dot product or scalar product) of two vectors \({\bf u}\) and \({\bf v}\) is again defined if and only if they have the same number of elements -\[ {\bf u} \cdot {\bf v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n = \sum_{i = 1}^n u_iv_i\] -If \({\bf u} \cdot {\bf v} = 0\), the two vectors are orthogonal (or perpendicular). - -\textbf{Vector Norm}: The norm of a vector is a measure of its length. There are many different ways to calculate the norm, but the most common is the Euclidean norm (which corresponds to our usual conception of distance in three-dimensional space): -\[ ||{\bf v}|| = \sqrt{{\bf v}\cdot{\bf v}} = \sqrt{ v_1v_1 + v_2v_2 + \cdots + v_nv_n}\] - -\begin{example}[Vector Algebra] -\protect\hypertarget{exm:vectors}{}{\label{exm:vectors} \iffalse (Vector Algebra) \fi{} } -Let \(a = \begin{pmatrix} 2&1&2\end{pmatrix}\), \(b = \begin{pmatrix} 3&4&5 \end{pmatrix}\). Calculate the following: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(a - b\) -\item - \(a \cdot b\) -\end{enumerate} -\end{example} - -\begin{exercise}[Vector Algebra] -\protect\hypertarget{exr:vectors1}{}{\label{exr:vectors1} \iffalse (Vector Algebra) \fi{} } -Let \(u = \begin{pmatrix} 7&1&-5&3\end{pmatrix}\), \(v = \begin{pmatrix} 9&-3&2&8 \end{pmatrix}\), \(w = \begin{pmatrix} 1&13& -7&2 &15 \end{pmatrix}\), and \(c = 2\). Calculate the following: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(u-v\) -\item - \(cw\) -\item - \(u \cdot v\) -\item - \(w \cdot v\) -\end{enumerate} -\end{exercise} - -\hypertarget{linearindependence}{% -\section{Linear Independence}\label{linearindependence}} - -\textbf{Linear combinations}: The vector \({\bf u}\) is a linear combination of the vectors \({\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k\) if -\[{\bf u} = c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k\] - -For example, \(\begin{pmatrix}9 & 13 & 17 \end{pmatrix}\) is a linear combination of the following three vectors: \(\begin{pmatrix}1 & 2 & 3 \end{pmatrix}\), \(\begin{pmatrix} 2 & 3& 4\end{pmatrix}\), and \(\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}\). This is because \(\begin{pmatrix}9 & 13 & 17 \end{pmatrix}\) = \((2)\begin{pmatrix}1 & 2 & 3 \end{pmatrix}\) \(+ (-1)\begin{pmatrix} 2 & 3& 4\end{pmatrix}\) + \(3\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}\) - -\textbf{Linear independence}: A set of vectors \({\bf v}_1, {\bf v}_2, \cdots , {\bf v}_k\) is linearly independent if the only solution to the equation -\[c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k = 0\] -is \(c_1 = c_2 = \cdots = c_k = 0\). If another solution exists, the set of vectors is linearly dependent. - -A set \(S\) of vectors is linearly dependent if and only if at least one of the vectors in \(S\) can be written as a linear combination of the other vectors in \(S\). - -Linear independence is only defined for sets of vectors with the same number of elements; any linearly independent set of vectors in \(n\)-space contains at most \(n\) vectors. - -Since \(\begin{pmatrix}9 & 13 & 17 \end{pmatrix}\) is a linear combination of \(\begin{pmatrix}1 & 2 & 3 \end{pmatrix}\), \(\begin{pmatrix} 2 & 3& 4\end{pmatrix}\), and \(\begin{pmatrix} 3 & 4 & 5 \end{pmatrix}\), these 4 vectors constitute a linearly dependent set. - -\begin{example}[Linear Independence] -\protect\hypertarget{exm:linearindep}{}{\label{exm:linearindep} \iffalse (Linear Independence) \fi{} } -Are the following sets of vectors linearly independent? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\begin{pmatrix}2 & 3 & 1 \end{pmatrix}\) and \(\begin{pmatrix}4 & 6 & 1 \end{pmatrix}\) -\item - \(\begin{pmatrix}1 & 0 & 0 \end{pmatrix}\), \(\begin{pmatrix}0 & 5 & 0 \end{pmatrix}\), and \(\begin{pmatrix}10 & 10 & 0 \end{pmatrix}\) -\end{enumerate} -\end{example} - -\begin{exercise}[Linear Independence] -\protect\hypertarget{exr:linearindep1}{}{\label{exr:linearindep1} \iffalse (Linear Independence) \fi{} } -Are the following sets of vectors linearly independent? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \[{\bf v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \] -\item - \[{\bf v}_1 = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} , {\bf v}_2 = \begin{pmatrix} -4 \\ 6 \\ 5 \end{pmatrix} , {\bf v}_3 = \begin{pmatrix} -2 \\ 8 \\ 6 \end{pmatrix} \] -\end{enumerate} -\end{exercise} - -\hypertarget{matrixbasics}{% -\section{Basics of Matrix Algebra}\label{matrixbasics}} - -\textbf{Matrix}: A matrix is an array of real numbers arranged in \(m\) rows by \(n\) columns. The dimensionality of the matrix is defined as the number of rows by the number of columns, \(m \times n\). - -\[{\bf A}=\begin{pmatrix} - a_{11} & a_{12} & \cdots & a_{1n} \\ - a_{21} & a_{22} & \cdots & a_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - a_{m1} & a_{m2} & \cdots & a_{mn} - \end{pmatrix}\] - -Note that you can think of vectors as special cases of matrices; a column vector of length \(k\) is a \(k \times 1\) matrix, while a row vector of the same length is a \(1 \times k\) matrix. - -It's also useful to think of matrices as being made up of a collection of row or column vectors. For example, -\[\bf A = \begin{pmatrix} {\bf a}_1 & {\bf a}_2 & \cdots & {\bf a}_m \end{pmatrix}\] - -\textbf{Matrix Addition}: Let \(\bf A\) and \(\bf B\) be two \(m\times n\) matrices. -\[{\bf A+B}=\begin{pmatrix} - a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ - a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{mn}+b_{mn} - \end{pmatrix}\] - -Note that matrices \({\bf A}\) and \({\bf B}\) must have the same dimensionality, in which case they are \textbf{conformable for addition}. - -\begin{example} -\protect\hypertarget{exm:matrixaddition}{}{\label{exm:matrixaddition} }\[{\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad - {\bf B}=\begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & 2 \end{pmatrix}\] -\({\bf A+B}=\) -\end{example} - -\textbf{Scalar Multiplication}: Given the scalar \(s\), the scalar multiplication of \(s {\bf A}\) is -\[ s {\bf A}= s \begin{pmatrix} - a_{11} & a_{12} & \cdots & a_{1n} \\ - a_{21} & a_{22} & \cdots & a_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - a_{m1} & a_{m2} & \cdots & a_{mn} - \end{pmatrix} - = \begin{pmatrix} - s a_{11} & s a_{12} & \cdots & s a_{1n} \\ - s a_{21} & s a_{22} & \cdots & s a_{2n} \\ - \vdots & \vdots & \ddots & \vdots \\ - s a_{m1} & s a_{m2} & \cdots & s a_{mn} - \end{pmatrix}\] - -\begin{example} -\protect\hypertarget{exm:scalarmulti}{}{\label{exm:scalarmulti} } - -\(s=2, \qquad {\bf A}=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\) - -\(s {\bf A} =\) -\end{example} - -\textbf{Matrix Multiplication}: If \({\bf A}\) is an \(m\times k\) matrix and \(\bf B\) is a \(k\times n\) matrix, then their product \(\bf C = A B\) is the \(m\times n\) matrix where -\[c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ik}b_{kj}\] - -\begin{example} -\protect\hypertarget{exm:matrixmulti}{}{\label{exm:matrixmulti} } - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\begin{pmatrix} a&b\\c&d\\e&f \end{pmatrix} \begin{pmatrix} A&B\\C&D \end{pmatrix} =\) -\item - \(\begin{pmatrix} 1&2&-1\\3&1&4 \end{pmatrix} \begin{pmatrix} -2&5\\4&-3\\2&1\end{pmatrix} =\) -\end{enumerate} -\end{example} - -Note that the number of columns of the first matrix must equal the number of rows of the second matrix, in which case they are \textbf{conformable for multiplication}. The sizes of the matrices (including the resulting product) must be \[(m\times k)(k\times n)=(m\times n)\] - -Also note that if \textbf{AB} exists, \textbf{BA} exists only if \(\dim({\bf A}) = m \times n\) and \(\dim({\bf B}) = n \times m\). - -This does not mean that \textbf{AB} = \textbf{BA}. \textbf{AB} = \textbf{BA} is true only in special circumstances, like when \({\bf A}\) or \({\bf B}\) is an identity matrix or \({\bf A} = {\bf B}^{-1}\). - -\textbf{Laws of Matrix Algebra}: - -\begin{enumerate} - \item \parbox[t]{1.5in}{Associative:} $\bf (A+B)+C = A+(B+C)$\\ - \parbox[t]{1.5in}{\quad} $\bf (AB)C = A(BC)$ - \item \parbox[t]{1.5in}{Commutative:} $\bf A+B=B+A$ - \item \parbox[t]{1.5in}{Distributive:} $\bf A(B+C)=AB+AC$\\ - \parbox[t]{1.5in}{\quad} $\bf (A+B)C=AC+BC$ -\end{enumerate} - -Commutative law for multiplication does not hold -- the order of multiplication matters: -\[\bf AB\ne BA\] - -For example, -\[{\bf A}=\begin{pmatrix} 1&2\\-1&3\end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 2&1\\0&1\end{pmatrix}\] -\[{\bf AB}=\begin{pmatrix} 2&3\\-2&2\end{pmatrix}, \qquad {\bf BA}=\begin{pmatrix} 1&7\\-1&3\end{pmatrix}\] - -\textbf{Transpose}: The transpose of the \(m\times n\) matrix \(\bf A\) is the \(n\times m\) matrix \({\bf A}^T\) (also written \({\bf A}'\)) obtained by interchanging the rows and columns of \(\bf A\). - -For example, - -\({\bf A}=\begin{pmatrix} 4&-2&3\\0&5&-1\end{pmatrix}, \qquad {\bf A}^T=\begin{pmatrix} 4&0\\-2&5\\3&-1 \end{pmatrix}\) - -\({\bf B}=\begin{pmatrix} 2\\-1\\3 \end{pmatrix}, \qquad {\bf B}^T=\begin{pmatrix} 2&-1&3\end{pmatrix}\) - -The following rules apply for transposed matrices: - -\begin{enumerate} - \item $({\bf A+B})^T = {\bf A}^T+{\bf B}^T$ - \item $({\bf A}^T)^T={\bf A}$ - \item $(s{\bf A})^T = s{\bf A}^T$ - \item $({\bf AB})^T = {\bf B}^T{\bf A}^T$; and by induction $({\bf ABC})^T = {\bf C}^T{\bf B}^T{\bf A}^T$ -\end{enumerate} - -Example of \(({\bf AB})^T = {\bf B}^T{\bf A}^T\): -\[{\bf A}=\begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix}\] -\[ ({\bf AB})^T = \left[ \begin{pmatrix} 1&3&2\\2&-1&3\end{pmatrix} \begin{pmatrix} 0&1\\2&2\\3&-1\end{pmatrix} \right]^T = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}\] -\[ {\bf B}^T{\bf A}^T= \begin{pmatrix} 0&2&3\\1&2&-1 \end{pmatrix} \begin{pmatrix} 1&2\\3&-1\\2&3 \end{pmatrix} = \begin{pmatrix} 12&7\\5&-3 \end{pmatrix}\] - -\begin{exercise}[Matrix Multiplication] -\protect\hypertarget{exr:matrixmulti1}{}{\label{exr:matrixmulti1} \iffalse (Matrix Multiplication) \fi{} } - -Let \[A = \begin{pmatrix} 2&0&-1&1\\1&2&0&1 \end{pmatrix}\] - -\[B = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix} \] - -\[C = \begin{pmatrix} 3&2&-1\\0&4&6 \end{pmatrix}\] - -Calculate the following: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \[AB\] -\item - \[BA\] -\item - \[(BC)^T\] -\item - \[BC^T\] -\end{enumerate} -\end{exercise} - -\hypertarget{systems-of-linear-equations}{% -\section{Systems of Linear Equations}\label{systems-of-linear-equations}} - -\textbf{Linear Equation}: \(a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b\) - -\(a_i\) are parameters or coefficients. \(x_i\) are variables or unknowns. - -Linear because only one variable per term and degree is at most 1. - -We are often interested in solving linear systems like - -\[\begin{matrix} - x & - & 3y & = & -3\\ - 2x & + & y & = & 8 - \end{matrix}\] - -\begin{comment} - \parbox[t]{1in}{\includegraphics[angle=270, width = 1in]{linsys.eps}} -\end{comment} - -More generally, we might have a system of \(m\) equations in \(n\) unknowns - -\[\begin{matrix} - a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ - a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ - \vdots & & & & \vdots & & & \vdots & \\ - a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m - \end{matrix}\] - -A \textbf{solution} to a linear system of \(m\) equations in \(n\) unknowns is a set of \(n\) numbers \(x_1, x_2, \cdots, x_n\) that satisfy each of the \(m\) equations. - -Example: \(x=3\) and \(y=2\) is the solution to the above \(2\times 2\) linear system. If you graph the two lines, you will find that they intersect at \((3,2)\). - -Does a linear system have one, no, or multiple solutions? For a system of 2 equations with 2 unknowns (i.e., two lines): -\_\\ -\textbf{One solution:} The lines intersect at exactly one point. - -\textbf{No solution:} The lines are parallel. - -\textbf{Infinite solutions:} The lines coincide. - -Methods to solve linear systems: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Substitution -\item - Elimination of variables -\item - Matrix methods -\end{enumerate} - -\begin{exercise}[Linear Equations] -\protect\hypertarget{exr:lineareq}{}{\label{exr:lineareq} \iffalse (Linear Equations) \fi{} } - -Provide a system of 2 equations with 2 unknowns that has - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - one solution -\item - no solution -\item - infinite solutions -\end{enumerate} -\end{exercise} - -\hypertarget{systems-of-equations-as-matrices}{% -\section{Systems of Equations as Matrices}\label{systems-of-equations-as-matrices}} - -Matrices provide an easy and efficient way to represent linear systems such as -\[\begin{matrix} - a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ - a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ - \vdots & & & & \vdots & & & \vdots & \\ - a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m - \end{matrix}\] - -as \[{\bf A x = b}\] where - -The \(m \times n\) \textbf{coefficient matrix} \({\bf A}\) is an array of \(m n\) real numbers arranged in \(m\) rows by \(n\) columns: -\[{\bf A}=\begin{pmatrix} - a_{11} & a_{12} & \cdots & a_{1n} \\ - a_{21} & a_{22} & \cdots & a_{2n} \\ - \vdots & & \ddots & \vdots \\ - a_{m1} & a_{m2} & \cdots & a_{mn} - \end{pmatrix}\] - -The unknown quantities are represented by the vector \({\bf x}=\begin{pmatrix} x_1\\x_2\\\vdots\\x_n \end{pmatrix}\). - -The right hand side of the linear system is represented by the vector \({\bf b}=\begin{pmatrix} b_1\\b_2\\\vdots\\b_m \end{pmatrix}\). - -\textbf{Augmented Matrix}: When we append \(\bf b\) to the coefficient matrix \(\bf A\), we get the augmented matrix \(\widehat{\bf A}=[\bf A | b]\) -\[\begin{pmatrix} - a_{11} & a_{12} & \cdots & a_{1n} & | & b_1\\ - a_{21} & a_{22} & \cdots & a_{2n} & | & b_2\\ - \vdots & & \ddots & \vdots & | & \vdots\\ - a_{m1} & a_{m2} & \cdots & a_{mn} & | & b_m - \end{pmatrix}\] - -\begin{exercise}[Augmented Matrix] -\protect\hypertarget{exr:augmatrix}{}{\label{exr:augmatrix} \iffalse (Augmented Matrix) \fi{} } - -Create an augmented matrix that represent the following system of equations: - -\[2x_1 -7x_2 + 9x_3 -4x_4 = 8\] -\[41x_2 + 9x_3 -5x_6 = 11\] -\[x_1 -15x_2 -11x_5 = 9\] -\end{exercise} - -\hypertarget{finding-solutions-to-augmented-matrices-and-systems-of-equations}{% -\section{Finding Solutions to Augmented Matrices and Systems of Equations}\label{finding-solutions-to-augmented-matrices-and-systems-of-equations}} - -\textbf{Row Echelon Form}: Our goal is to translate our augmented matrix or system of equations into row echelon form. This will provide us with the values of the vector \textbf{x} which solve the system. We use the row operations to change coefficients in the lower triangle of the augmented matrix to 0. An augmented matrix of the form - -\[\begin{pmatrix} - \fbox{$a'_{11}$}& a'_{12} & a'_{13}& \cdots & a'_{1n} & | & b'_1\\ - 0 & \fbox{$a'_{22}$} & a'_{23}& \cdots & a'_{2n} & | & b'_2\\ - 0 & 0 & \fbox{$a'_{33}$}& \cdots & a'_{3n} & | & b'_3\\ - 0 & 0 &0 & \ddots & \vdots & | & \vdots \\ - 0 & 0 &0 &0 & \fbox{$a'_{mn}$} & | & b'_m - \end{pmatrix}\] - -is said to be in row echelon form --- each row has more leading zeros than the row preceding it. - -\textbf{Reduced Row Echelon Form}: We can go one step further and put the matrix into reduced row echelon form. Reduced row echelon form makes the value of \textbf{x} which solves the system very obvious. For a system of \(m\) equations in \(m\) unknowns, with no all-zero rows, the reduced row echelon form would be - -\[\begin{pmatrix} - \fbox{$1$} & 0 & 0 & 0 & 0 & | & b^*_1\\ - 0 & \fbox{$1$} & 0 & 0 & 0 & | & b^*_2\\ - 0 & 0 & \fbox{$1$} & 0 & 0 & | & b^*_3\\ - 0 & 0 & 0 &\ddots & 0 & | &\vdots\\ - 0 & 0 & 0 & 0 & \fbox{$1$} & | & b^*_m - \end{pmatrix}\] - -\textbf{Gaussian and Gauss-Jordan elimination}: We can conduct elementary row operations to get our augmented matrix into row echelon or reduced row echelon form. The methods of transforming a matrix or system into row echelon and reduced row echelon form are referred to as Gaussian elimination and Gauss-Jordan elimination, respectively. - -\textbf{Elementary Row Operations}: To do Gaussian and Gauss-Jordan elimination, we use three basic operations to transform the augmented matrix into another augmented matrix that represents an equivalent linear system -- equivalent in the sense that the same values of \(x_j\) solve both the original and transformed matrix/system: - -\textbf{Interchanging Rows}: Suppose we have the augmented matrix -\[{\widehat{\bf A}}=\begin{pmatrix} a_{11} & a_{12} & | & b_1\\ - a_{21} & a_{22} & | & b_2 - \end{pmatrix}\] -If we interchange the two rows, we get the augmented matrix -\[\begin{pmatrix} - a_{21} & a_{22} & | & b_2\\ - a_{11} & a_{12} & | & b_1 - \end{pmatrix}\] -which represents a linear system equivalent to that represented by matrix \(\widehat{\bf A}\). - -\textbf{Multiplying by a Constant}: If we multiply the second row of matrix \(\widehat{\bf A}\) by a constant \(c\), we get the augmented matrix -\[\begin{pmatrix} - a_{11} & a_{12} & | & b_1\\ - c a_{21} & c a_{22} & | & c b_2 - \end{pmatrix}\] -which represents a linear system equivalent to that represented by matrix \(\widehat{\bf A}\). - -\textbf{Adding (subtracting) Rows}: If we add (subtract) the first row of matrix \(\widehat{\bf A}\) to the second, we obtain the augmented matrix -\[\begin{pmatrix} - a_{11} & a_{12} & | & b_1\\ - a_{11}+a_{21} & a_{12}+a_{22} & | & b_1+b_2 - \end{pmatrix}\] -which represents a linear system equivalent to that represented by matrix \(\widehat{\bf A}\). - -\begin{example} -\protect\hypertarget{exm:solvesys}{}{\label{exm:solvesys} } - -Solve the following system of equations by using elementary row operations: - -\(\begin{matrix} x & - & 3y & = & -3\\ 2x & + & y & = & 8 \end{matrix}\) -\end{example} - -\begin{exercise}[Solving Systems of Equations] -\protect\hypertarget{exr:solvesys1}{}{\label{exr:solvesys1} \iffalse (Solving Systems of Equations) \fi{} } - -Put the following system of equations into augmented matrix form. Then, using Gaussian or Gauss-Jordan elimination, solve the system of equations by putting the matrix into row echelon or reduced row echelon form. - -\[ - 1. \begin{cases} - x + y + 2z = 2\\ - 3x - 2y + z = 1\\ - y - z = 3 - \end{cases} - \] - -\[ - 2. \begin{cases} - 2x + 3y - z = -8\\ - x + 2y - z = 12\\ - -x -4y + z = -6 - \end{cases} - \] -\end{exercise} - -\hypertarget{rank-and-whether-a-system-has-one-infinite-or-no-solutions}{% -\section{Rank --- and Whether a System Has One, Infinite, or No Solutions}\label{rank-and-whether-a-system-has-one-infinite-or-no-solutions}} - -To determine how many solutions exist, we can use information about (1) the number of equations \(m\), (2) the number of unknowns \(n\), and (3) the \textbf{rank} of the matrix representing the linear system. - -\textbf{Rank}: The maximum number of linearly independent row or column vectors in the matrix. This is equivalent to the number of nonzero rows of a matrix in row echelon form. For any matrix \textbf{A}, the row rank always equals column rank, and we refer to this number as the rank of \textbf{A}. - -For example - -\(\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}\) - -Rank = 3 - -\(\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 0 \end{pmatrix}\) - -Rank = 2 - -\begin{exercise}[Rank of Matrices] -\protect\hypertarget{exr:rank}{}{\label{exr:rank} \iffalse (Rank of Matrices) \fi{} } - -Find the rank of each matrix below: - -(Hint: transform the matrices into row echelon form. Remember that the number of nonzero rows of a matrix in row echelon form is the rank of that matrix) - -1.\(\begin{pmatrix} 1 & 1 & 2 \\ 2 & 1 & 3 \\ 1 & 2 & 3 \end{pmatrix}\) - -\bigskip - -2.\(\begin{pmatrix} 1 & 3 & 3 & -3 & 3\\ 1 & 3 & 1 & 1 & 3 \\ 1 & 3 & 2 & -1 & -2 \\ 1 & 3 & 0 & 3 & -2 \end{pmatrix}\) -\end{exercise} - -Answer to Exercise \ref{exr:rank}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - rank is 2 -\item - rank is 3 -\end{enumerate} - -\hypertarget{the-inverse-of-a-matrix}{% -\section{The Inverse of a Matrix}\label{the-inverse-of-a-matrix}} - -\textbf{Identity Matrix}: The \(n\times n\) identity matrix \({\bf I}_n\) is the matrix whose diagonal elements are 1 and all off-diagonal elements are 0. Examples: -\[ {\bf I}_2=\begin{pmatrix} 1&0\\0&1 \end{pmatrix}, \qquad {\bf I}_3=\begin{pmatrix} 1&0&0\\ 0&1&0\\ - 0&0&1 \end{pmatrix}\] - -\textbf{Inverse Matrix}: An \(n\times n\) matrix \({\bf A}\) is \textbf{nonsingular} or \textbf{invertible} if there exists an \(n\times n\) matrix \({\bf A}^{-1}\) such that \[{\bf A} {\bf A}^{-1} = {\bf A}^{-1} {\bf A} = {\bf I}_n\] where \({\bf A}^{-1}\) is the inverse of \({\bf A}\). If there is no such \({\bf A}^{-1}\), then \({\bf A}\) is singular or not invertible. - -Example: Let -\[{\bf A} = \begin{pmatrix} 2&3\\2&2 \end{pmatrix}, \qquad {\bf B}=\begin{pmatrix} -1&\frac{3}{2}\\ 1&-1 - \end{pmatrix}\] -Since \[{\bf A} {\bf B} = {\bf B} {\bf A} = {\bf I}_n\] we conclude that \({\bf B}\) is the inverse, \({\bf A}^{-1}\), of \({\bf A}\) and that \({\bf A}\) is nonsingular. - -\textbf{Properties of the Inverse}: - -\begin{itemize} -\item - If the inverse exists, it is unique. -\item - If \({\bf A}\) is nonsingular, then \({\bf A}^{-1}\) is nonsingular. -\item - \(({\bf A}^{-1})^{-1} = {\bf A}\) -\item - If \({\bf A}\) and \({\bf B}\) are nonsingular, then \({\bf A}{\bf B}\) is nonsingular -\item - \(({\bf A}{\bf B})^{-1} = {\bf B}^{-1}{\bf A}^{-1}\) -\item - If \({\bf A}\) is nonsingular, then \(({\bf A}^T)^{-1}=({\bf A}^{-1})^T\) -\end{itemize} - -\textbf{Procedure to Find} \({\bf A}^{-1}\): We know that if \({\bf B}\) is the inverse of \({\bf A}\), then \[{\bf A} {\bf B} = {\bf B} {\bf A} = {\bf I}_n\] Looking only at the first and last parts of this \[{\bf A} {\bf B} = {\bf I}_n\] Solving for \({\bf B}\) is equivalent to solving for \(n\) linear systems, where each column of \({\bf B}\) is solved for the corresponding column in \({\bf I}_n\). We can solve the systems simultaneously by augmenting \({\bf A}\) with \({\bf I}_n\) and performing Gauss-Jordan elimination on \({\bf A}\). If Gauss-Jordan elimination on \([{\bf A} | {\bf I}_n]\) results in \([{\bf I}_n | {\bf B} ]\), then \({\bf B}\) is the inverse of \({\bf A}\). Otherwise, \({\bf A}\) is singular. - -To summarize: To calculate the inverse of \({\bf A}\) - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - Form the augmented matrix \([ {\bf A} | {\bf I}_n]\) -\item - Using elementary row operations, transform the augmented matrix to reduced row echelon form. -\item - The result of step 2 is an augmented matrix \([ {\bf C} | {\bf B} ]\). - - \begin{enumerate} - \def\labelenumii{\alph{enumii}.} - \item - If \({\bf C}={\bf I}_n\), then \({\bf B}={\bf A}^{-1}\). - \item - If \({\bf C}\ne{\bf I}_n\), then \(\bf C\) has a row of zeros. This means \({\bf A}\) is singular and \({\bf A}^{-1}\) does not exist. - \end{enumerate} -\end{enumerate} - -\begin{example} -\protect\hypertarget{exm:inverse}{}{\label{exm:inverse} } -Find the inverse of the following matricies: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \({\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}\) -\end{enumerate} -\end{example} - -\begin{exercise}[Finding the inverse of matrices] -\protect\hypertarget{exr:inverse1}{}{\label{exr:inverse1} \iffalse (Finding the inverse of matrices) \fi{} } - -Find the inverse of the following matrix: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \({\bf A}=\begin{pmatrix} 1&0&4\\0&2&0\\0&0&1 \end{pmatrix}\) -\end{enumerate} -\end{exercise} - -\hypertarget{linear-systems-and-inverses}{% -\section{Linear Systems and Inverses}\label{linear-systems-and-inverses}} - -Let's return to the matrix representation of a linear system - -\[\bf{Ax} = \bf{b}\] - -If \(\bf{A}\) is an \(n\times n\) matrix,then \(\bf{Ax}=\bf{b}\) is a system of \(n\) equations in \(n\) unknowns. Suppose \(\bf{A}\) is nonsingular. Then \(\bf{A}^{-1}\) exists. To solve this system, we can multiply each side by \(\bf{A}^{-1}\) and reduce it as follows: - -\begin{eqnarray*} -\bf{A}^{-1} (\bf{A} \bf{x}) & = & \bf{A}^{-1} \bf{b} \\ -(\bf{A}^{-1} \bf{A})\bf{x} & = & \bf{A}^{-1} \bf{b}\\ -\bf{I}_n \bf{x} & = & \bf{A}^{-1} \bf{b}\\ -\bf{x} & = & \bf{A}^{-1} \bf{b} -\end{eqnarray*} - -Hence, given \(\bf{A}\) and \(\bf{b}\) and given that \(\bf{A}\) is nonsingular, then \(\bf{x} = \bf{A}^{-1} \bf{b}\) is a unique solution to this system. - -\begin{exercise}[Solve linear system using inverses] -\protect\hypertarget{exr:invlinsys}{}{\label{exr:invlinsys} \iffalse (Solve linear system using inverses) \fi{} } -Use the inverse matrix to solve the following linear system: - -\begin{align*} --3x + 4y &= 5 \\ -2x - y &= -10 -\end{align*} - -\textbf{\emph{Hint: the linear system above can be written in the matrix form}} - -\(\textbf{A}\textbf{z} = \textbf{b}\) - -given \[\textbf{A} = \begin{pmatrix} -3&4\\2&-1 \end{pmatrix},\] - -\[\textbf{z} = \begin{pmatrix} x\\y \end{pmatrix},\] -and -\[\textbf{b} = \begin{pmatrix} 5\\-10 \end{pmatrix}\] -\end{exercise} - -\hypertarget{determinants}{% -\section{Determinants}\label{determinants}} - -\textbf{Singularity}: Determinants can be used to determine whether a square matrix is nonsingular. - -A square matrix is nonsingular if and only if its determinant is not zero. - -Determinant of a \(1 \times 1\) matrix, \textbf{A}, equals \(a_{11}\) - -Determinant of a \(2 \times 2\) matrix, \textbf{A}, -\(\begin{vmatrix} a_{11}&a_{12}\\ a_{21}&a_{22} \end{vmatrix}\): - -\begin{eqnarray*} -\det({\bf A}) &=& |{\bf A}|\\ - &=& a_{11}|a_{22}| - a_{12}|a_{21}|\\ - &=& a_{11}a_{22} - a_{12}a_{21} -\end{eqnarray*} - -We can extend the second to last equation above to get the definition of the determinant of a \(3 \times 3\) matrix: - -\begin{eqnarray*} - \begin{vmatrix} a_{11}&a_{12}&a_{13}\\ a_{21} & a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{vmatrix} - &=& - a_{11} \begin{vmatrix} a_{22}&a_{23}\\ a_{32}&a_{33} \end{vmatrix} - - a_{12} \begin{vmatrix} a_{21}&a_{23}\\ a_{31}&a_{33} \end{vmatrix} - + a_{13} \begin{vmatrix} a_{21}&a_{22}\\ a_{31}&a_{32} - \end{vmatrix}\\ - &=& a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) -\end{eqnarray*} - -Let's extend this now to any \(n\times n\) matrix. Let's define \({\bf A}_{ij}\) as the \((n-1)\times (n-1)\) submatrix of \({\bf A}\) obtained by deleting row \(i\) and column \(j\). Let the \((i,j)\)th \textbf{minor} of \({\bf A}\) be the determinant of \({\bf A}_{ij}\): -\[M_{ij}=|{\bf A}_{ij}|\] -Then for any \(n\times n\) matrix \({\bf A}\) -\[|{\bf A}|= a_{11}M_{11} - a_{12}M_{12} + \cdots + (-1)^{n+1} a_{1n} M_{1n}\] - -For example, in figuring out whether the following matrix has an inverse? -\[{\bf A}=\begin{pmatrix} 1&1&1\\0&2&3\\5&5&1 \end{pmatrix}\] -1. Calculate its determinant. -\begin{eqnarray} - &=& 1(2-15) - 1(0-15) + 1(0-10) \nonumber\\ - &=& -13+15-10 \nonumber\\ - &=& -8\nonumber -\end{eqnarray} -2. Since \(|{\bf A}|\ne 0\), we conclude that \({\bf A}\) has an inverse. - -\begin{exercise}[Determinants and Inverses] -\protect\hypertarget{exr:determinants}{}{\label{exr:determinants} \iffalse (Determinants and Inverses) \fi{} } - -Determine whether the following matrices are nonsingular: - -\[1. \begin{pmatrix} - 1 & 0 & 1\\ - 2 & 1 & 2\\ - 1 & 0 & -1 - \end{pmatrix}\] - -\[2. \begin{pmatrix} - 2 & 1 & 2\\ - 1 & 0 & 1\\ - 4 & 1 & 4 - \end{pmatrix}\] -\end{exercise} - -\hypertarget{getting-inverse-of-a-matrix-using-its-determinant}{% -\section{Getting Inverse of a Matrix using its Determinant}\label{getting-inverse-of-a-matrix-using-its-determinant}} - -Thus far, we have a number of algorithms to - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Find the solution of a linear system, -\item - Find the inverse of a matrix -\end{enumerate} - -but these remain just that --- algorithms. At this point, we have no way of telling how the solutions \(x_j\) change as the parameters \(a_{ij}\) and \(b_i\) change, except by changing the values and ``rerunning'' the algorithms. - -With determinants, we can provide an explicit formula for the inverse and -therefore provide an explicit formula for the solution of an \(n\times n\) linear system. - -Hence, we can examine how changes in the parameters and \(b_i\) affect the solutions \(x_j\). - -\textbf{Determinant Formula for the Inverse of a \(2 \times 2\)}: - -The determinant of a \(2 \times 2\) matrix \textbf{A} \(\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}\) is defined as: -\[\frac{1}{\det({\bf A})} \begin{pmatrix} - d & -b\\ - -c & a\\ - \end{pmatrix}\] - -For example, Let's calculate the inverse of matrix A from Exercise \ref{exr:invlinsys} using the determinant formula. - -Recall, - -\[A = \begin{pmatrix} - -3 & 4\\ - 2 & -1\\ - \end{pmatrix}\] - -\[\det({\bf A}) = (-3)(-1) - (4)(2) = 3 - 8 = -5\] - -\[\frac{1}{\det({\bf A})} \begin{pmatrix} - -1 & -4\\ - -2 & -3\\ - \end{pmatrix}\] - -\[\frac{1}{-5} \begin{pmatrix} - -1 & -4\\ - -2 & -3\\ - \end{pmatrix}\] - -\[ \begin{pmatrix} - \frac{1}{5} & \frac{4}{5}\\ - \frac{2}{5} & \frac{3}{5}\\ - \end{pmatrix}\] - -\begin{exercise}[Calculate Inverse using Determinant Formula] -\protect\hypertarget{exr:calcinverse}{}{\label{exr:calcinverse} \iffalse (Calculate Inverse using Determinant Formula) \fi{} } - -Caculate the inverse of A - -\[A = \begin{pmatrix} - 3 & 5\\ - -7 & 2\\ - \end{pmatrix}\] -\end{exercise} - -\hypertarget{answers-to-examples-and-exercises-3}{% -\section*{Answers to Examples and Exercises}\label{answers-to-examples-and-exercises-3}} -\addcontentsline{toc}{section}{Answers to Examples and Exercises} - -Answer to Example \ref{exm:vectors}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\begin{pmatrix} -1 &-3&-3 \end{pmatrix}\) -\item - 6 + 4 + 10 = 20 -\end{enumerate} - -Answer to Exercise \ref{exr:vectors1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\begin{pmatrix} -2 &4&-7&-5 \end{pmatrix}\) -\item - \(\begin{pmatrix} 2 &26&-14&4&30 \end{pmatrix}\) -\item - 63 -3 -10 + 24 = 74 -\item - undefined -\end{enumerate} - -Answer to Example \ref{exm:linearindep}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - yes -\item - no -\end{enumerate} - -Answer to Exercise \ref{exr:linearindep1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - yes -\item - no (\(-v_1 -v_2 + v_3 = 0\)) -\end{enumerate} - -Answer to Example \ref{exm:matrixaddition}: - -\({\bf A+B}=\begin{pmatrix} 2 & 4 & 4 \\ 6 & 6 & 8 \end{pmatrix}\) - -Answer to Example \ref{exm:scalarmulti}: - -\(s {\bf A} = \begin{pmatrix} 2 & 4 & 6 \\ 8 & 10 & 12 \end{pmatrix}\) - -Answer to Example \ref{exm:matrixmulti}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\begin{pmatrix} aA+bC&aB+bD\\cA+dC&cB+dD\\eA+fC&eB+fD \end{pmatrix}\) -\item - \(\begin{pmatrix} 1(-2)+2(4)-1(2)&1(5)+2(-3)-1(1)\\ 3(-2)+1(4)+4(2)&3(5)+1(-3)+4(1)\end{pmatrix} = \begin{pmatrix} 4&-2\\6&16\end{pmatrix}\) -\end{enumerate} - -Answer to Exercise \ref{exr:matrixmulti1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(AB = \begin{pmatrix} 4 & 11 & -15 \\ 5 & 7 & -7 \end{pmatrix}\) -\item - \(BA =\) undefined -\item - \((BC)^T =\) undefined -\item - \(BC^T = \begin{pmatrix} 1&5&-7\\1&1&0\\0&-1&1\\2&0&0\end{pmatrix}\begin{pmatrix} 3&0\\2&4\\-1&6 \end{pmatrix} =\begin{pmatrix}20 & -22 \\ 5 & 4 \\ -3 &2 \\6 & 0\end{pmatrix}\) -\end{enumerate} - -Answer to Exercise \ref{exr:lineareq}: - -There are many answers to this. Some possible simple ones are as follows: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - One solution: \[\begin{matrix} - -x & + & y & = & 0\\ - x & + & y & = & 2 - \end{matrix}\] -\item - No solution: \[\begin{matrix} - -x & + & y & = & 0\\ - x & - & y & = & 2 - \end{matrix}\] -\item - Infinite solutions: \[\begin{matrix} - -x & + & y & = & 0\\ - 2x & - & 2y & = & 0 - \end{matrix}\] -\end{enumerate} - -Answer to Exercise \ref{exr:augmatrix}: - -\(\begin{pmatrix} 2 & -7 & 9 & -4 & 0 & 0| & 8\\ 0 & 41 & 9 & 0 & 0 & 5 | & 11\\ 1 & -15 & 0 & 0 & -11 & 0 | & 9 \end{pmatrix}\) - -Answer to Example \ref{exm:solvesys}: - -\[\begin{matrix} - x & - & 3y & = & -3\\ - 2x & + & y & = & 8 - \end{matrix}\] - -\[\begin{matrix} - x & - & 3y & = & -3\\ - & & 7y & = & 14\\ - \end{matrix}\] - -\[\begin{matrix} - x & - & 3y & = & -3\\ - & & y & = & 2\\ - \end{matrix}\] - -\[\begin{matrix} - x & = & 3\\ - y & = & 2\\ - \end{matrix}\] - -Answer to Exercise \ref{exr:solvesys1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - x = 2, y = 2, z = -1 -\item - x = -17, y = -3, z = -35 -\end{enumerate} - -Answer to Exercise \ref{exr:rank}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - rank is 2 -\item - rank is 3 -\end{enumerate} - -Answer to Example \ref{exm:inverse}: - -\(\left(\begin{array}{ccc|ccc} 1&1&1&1&0&0\\ 0&2&3&0&1&0\\ 5&5&1&0&0&1 \end{array} \right)\) - -\(\left(\begin{array}{ccc|ccc} 1&1&1 &1 &0&0\\ 0&2&3 &0 &1&0\\ 0&0&-4&-5&0&1 \end{array} \right)\) - -\(\left(\begin{array}{ccc|ccc} 1&1&1&1 &0&0\\ 0&2&3&0 &1&0\\ 0&0&1&5/4&0&-1/4 \end{array} \right)\) - -\(\left(\begin{array}{ccc|ccc} 1&1&0&-1/4 &0&1/4\\ 0&2&0&-15/4&1&3/4\\ 0&0&1&5/4 &0&-1/4 \end{array} \right)\) - -\(\left(\begin{array}{ccc|ccc} 1&1&0&-1/4 &0 &1/4\\ 0&1&0&-15/8&1/2&3/8\\ 0&0&1&5/4 &0 &-1/4 \end{array} \right)\) - -\(\left(\begin{array}{ccc|ccc} 1&0&0&13/8 &-1/2&-1/8\\ 0&1&0&-15/8&1/2 &3/8\\ 0&0&1&5/4 &0 &-1/4 \end{array} \right)\) - -\({\bf A}^{-1} = \left(\begin{array}{ccc} 13/8 &-1/2&-1/8\\ -15/8&1/2 &3/8\\ 5/4 &0 &-1/4 \end{array} \right)\) - -Answer to Exercise \ref{exr:inverse1}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \({\bf A}^{-1}=\begin{pmatrix} 1&0&-4\\0&\frac{1}{2}&0\\0&0&1 \end{pmatrix}\) -\end{enumerate} - -Answer to Exercise \ref{exr:invlinsys}: - -\(\textbf{z} = \bf{A}^{-1} \bf{b} = \begin{pmatrix} 1/5 &4/5\\ 2/5&3/5 \end{pmatrix} \begin{pmatrix} 5 \\ -10 \end{pmatrix}= \begin{pmatrix} -7 \\ -4 \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\) - -Answer to Exercise \ref{exr:determinants}: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - nonsingular -\item - singular -\end{enumerate} - -Answer to Exercise \ref{exr:calcinverse}: - -\(\begin{pmatrix} \frac{2}{41} & \frac{-5}{41}\\ \frac{7}{41} & \frac{3}{41}\\ \end{pmatrix}\) - -\hypertarget{part-programming}{% -\part{Programming}\label{part-programming}} - -\hypertarget{dataimport}{% -\chapter[Orientation and Reading in Data]{\texorpdfstring{Orientation and Reading in Data\footnote{Module originally written by Shiro Kuriwaki}}{Orientation and Reading in Data}}\label{dataimport}} - -Welcome to the first in-class session for programming. Up till this point, you should have already: - -\begin{itemize} -\tightlist -\item - Completed the R Visualization and Programming primers (under ``The Basics'') on your own at \url{https://rstudio.cloud/learn/primers/}, -\item - Made an account at RStudio Cloud and join the Math Prefresher 2019 Space, and -\item - Successfully signed up for the University wi-fi: \url{https://getonline.harvard.edu/} (Access Harvard Secure with your HarvardKey. Try to get a HarvardKey as soon as possible.) -\end{itemize} - -\hypertarget{motivation-data-and-you}{% -\section*{Motivation: Data and You}\label{motivation-data-and-you}} -\addcontentsline{toc}{section}{Motivation: Data and You} - -The modal social science project starts by importing existing datasets. Datasets come in all shapes and sizes. As you search for new data you may encounter dozens of file extensions -- csv, xlsx, dta, sav, por, Rdata, Rds, txt, xml, json, shp \ldots{} the list continues. Although these files can often be cumbersome, its a good to be able to find a way to encounter any file that your research may call for. - -Reviewing data import will allow us to get on the same page on how computer systems work. - -\hypertarget{where-are-we-where-are-we-headed}{% -\subsection*{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed}} -\addcontentsline{toc}{subsection}{Where are we? Where are we headed?} - -Today we'll cover: - -\begin{itemize} -\tightlist -\item - What's what in RStudio -\item - What R is, at a high level -\item - How to read in data -\item - Comment on coding style on the way -\end{itemize} - -\hypertarget{check-your-understanding}{% -\subsection*{Check your understanding}\label{check-your-understanding}} -\addcontentsline{toc}{subsection}{Check your understanding} - -\begin{itemize} -\tightlist -\item - What is the difference between a file and a folder? -\item - In the RStudio windows, what is the difference between the ``Source'' Pane and the ``Console''? What is a ``code chunk''? -\item - How do you read a R help page? What is the \texttt{Usage} section, the \texttt{Values} section, and the \texttt{Examples} section? -\item - What use is the ``Environment'' Pane? -\item - How would you read in a spreadsheet in R? -\item - How would you figure out what variables are in the data? size of the data? -\item - How would you read in a \texttt{csv} file, a \texttt{dta} file, a \texttt{sav} file? -\end{itemize} - -\hypertarget{orienting}{% -\section{Orienting}\label{orienting}} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - We will be using a cloud version of RStudio at \url{https://rstudio.cloud}. You should join the Math Prefresher Space 2019 from the link that was emailed to you. Each day, click on the project with the day's date on it. - - Although most of you will probably doing your work on RStudio local rather than cloud, we are trying to use cloud because it makes it easier to standardize people's settings. -\item - RStudio (either cloud or desktop) is a \textbf{GUI} and an IDE for the programming language R. A Graphical User Interface allows users to interface with the software (in this case R) using graphical aids like buttons and tabs. Often we don't think of GUIs because to most computer users, everything is a GUI (like Microsoft Word or your ``Control Panel''), but it's always there! A Integrated Development Environment just says that the software to interface with R comes with useful useful bells and whistles to give you shortcuts. - - The \textbf{Console} is kind of a the core window through which you see your GUI actually operating through R. It's not graphical so might not be as intuitive. But all your results, commands, errors, warnings.. you see them in here. A console tells you what's going on now. -\end{enumerate} - -\begin{figure} -\centering -\includegraphics{images/11_1_rstudio-startup.png} -\caption{A Typical RStudio Window at Startup} -\end{figure} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{2} -\item - via the GUI, you the analyst needs to sends instructions, or \textbf{commands}, to the R application. The verb for this is ``run'' or ``execute'' the command. Computer programs ask users to provide instructions in very specific formats. While a English-speaking human can understand a sentence with a few typos in it by filling in the blanks, the same typo or misplaced character would halt a computer program. Each program has its own requirements for how commands should be typed; after all, each of these is its own language. We refer to the way a program needs its commands to be formatted as its \textbf{syntax}. -\item - Theoretically, one could do all their work by typing in commands into the Console. But that would be a lot of work, because you'd have to give instructions each time you start your data analysis. Moreover, you'll have no record of what you did. That's why you need a \textbf{script}. This is a type of \textbf{code}. It can be referred to as a \textbf{source} because that is the source of your commands. Source is also used as a verb; ``source the script'' just means execute it. RStudio doesn't start out with a script, so you can make one from ``File \textgreater{} New'' or the New file icon. -\end{enumerate} - -\begin{figure} -\centering -\includegraphics{images/11_2_rstudio-script.png} -\caption{Opening New Script (as opposed to the Console)} -\end{figure} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{3} -\item - You can also open scripts that are in folders in your computer. A script is a type of File. Find your Files in the bottom-right ``Files'' pane. - - To load a dataset, you need to specify where that file is. Computer files (data, documents, programs) are organized hiearchically, like a branching tree. Folders can contain files, and also other folders. The GUI toolbar makes this lineaer and hiearchical relationship apparent. When we turn to locate the file in our commands, we need another set of syntax. Importantly, denote the hierarchy of a folder by the \texttt{/} (slash) symbol. \texttt{data/input/2018-08} indicates the \texttt{2018-08} folder, which is included in the \texttt{input} folder, which is in turn included in the \texttt{data} folder. - - Files (but not folders) have ``file extensions'' which you are probably familiar with already: \texttt{.docx}, \texttt{.pdf}, and \texttt{.pdf}. The file extensions you will see in a stats or quantitative social science class are: - - \begin{itemize} - \item - \texttt{.pdf}: PDF, a convenient format to view documents and slides in, regardless of Mac/Windows. - \item - \texttt{.csv}: A comma separated values file - \item - \texttt{.xlsx}: Microsoft Excel file - \item - \texttt{.dta}: Stata data - \item - \texttt{.sav}: SPSS data - \item - \texttt{.R}: R code (script) - \item - \texttt{.Rmd}: Rmarkdown code (text + code) - \item - \texttt{.do}: Stata code (script) - \end{itemize} -\end{enumerate} - -\begin{figure} -\centering -\includegraphics{images/11_3_rstudio-files.png} -\caption{Opening an Existing Script from Files} -\end{figure} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{4} -\tightlist -\item - In R, there are two main types of scripts. A classic \texttt{.R} file and a \texttt{.Rmd} file (for Rmarkdown). A .R file is just lines and lines of R code that is meant to be inserted right into the Console. A .Rmd tries to weave code and English together, to make it easier for users to create reports that interact with data and intersperse R code with explanation. For example, we built this book in Rmds. -\end{enumerate} - -\begin{verbatim} -The Rmarkdown facilitates is the use of __code chunks__, which are used here. These start and end with three back-ticks. In the beginning, we can add options in curly braces (`{}`). Specifying `r` in the beginning tells to render it as R code. Options like `echo = TRUE` switch between showing the code that was executed or not; `eval = TRUE` switch between evaluating the code. More about Rmarkdown in Section \\ref{nonwysiwyg}. For example, this code chunk would evaluate `1 + 1` and show its output when compiled, but not display the code that was executed. -\end{verbatim} - -\begin{figure} -\centering -\includegraphics{images/11_4_codechunk.png} -\caption{A code chunk in Rmarkdown (before rendering)} -\end{figure} - -\newpage - -\hypertarget{but-what-is-r}{% -\section{But what is R?}\label{but-what-is-r}} - -R is an object oriented programming language primarily used for statistical computing. An object oriented language is a programming language built around manipulating objects. - -\begin{itemize} -\tightlist -\item - In R, objects can be matrices, vectors, scalars, strings, and data frames, for example -\item - Objects contain different types of information -\item - Different objects have different allowable procedures: -\end{itemize} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Adding a string and a string does not work because the \textquotesingle{}+\textquotesingle{} operator } -\CommentTok{\# does not work for strings:} - -\CommentTok{\# \textquotesingle{}Harvard\textquotesingle{} + \textquotesingle{}Gov\textquotesingle{}} - -\CommentTok{\# The \textquotesingle{}+\textquotesingle{} operator can add numbers just fine} -\DecValTok{9} \OperatorTok{+}\StringTok{ }\DecValTok{13} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 22 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Reminder: to figure out the type of an object use:} -\NormalTok{x <{-}}\StringTok{ }\DecValTok{9} -\KeywordTok{class}\NormalTok{(}\DecValTok{9}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "numeric" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(x)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "numeric" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(}\StringTok{\textquotesingle{}Harvard\textquotesingle{}}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "character" -\end{verbatim} - -Object oriented programming makes languages flexible and powerful: - -\begin{itemize} -\tightlist -\item - You can create custom functions and objects for your needs -\item - Other people can create great packages for everyone to use -\item - Many errors come from using the wrong data type and a lot of programming in R is getting data into the right format and type to work with. -\end{itemize} - -It is helpful to think in terms of object manipulation at a high level while programming in R, particularly at the beginning of tackling a new problem. Think about what objects you want to manipulate, what types they are, and how they fit together. Once you have the logic of your solution ready then you can write it in R. - -\hypertarget{the-computer-and-you-giving-instructions}{% -\section{The Computer and You: Giving Instructions}\label{the-computer-and-you-giving-instructions}} - -We'll do the Peanut Butter and Jelly Exercise in class as an introduction to programming for those who are new.\footnote{This Exercise is taken from Harvard's Introductory Undergraduate Class, CS50 (\url{https://www.youtube.com/watch?v=kcbT3hrEi9s}), and many other writeups.} - -Assignment: Take 5 minutes to write down on a piece of paper, how to make a peanut butter and jelly sandwich. Be as concise and unambiguous as possible so that a robot (who doesn't know what a PBJ is) would understand. You can assume that there will be loaf of sliced bread, a jar of jelly, a jar of peanut butter, and a knife. - -Simpler assignment: Say we just want a robot to be able to tell us if we have enough ingredients to make a peanut butter and jelly sandwich. Write down instructions so that if told how many slices of bread, servings of peanut butter, and servings of jelly you have, the robot can tell you if you can make a PBJ. - -Now, translate the simpler assignment into R code using the code below as a starting point: - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{n\_bread <{-}}\StringTok{ }\DecValTok{8} -\NormalTok{n\_pb <{-}}\StringTok{ }\DecValTok{3} -\NormalTok{n\_jelly <{-}}\StringTok{ }\DecValTok{9} - -\CommentTok{\# write instructions in R here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{base-r-vs.-tidyverse}{% -\section{Base-R vs.~tidyverse}\label{base-r-vs.-tidyverse}} - -One last thing before we jump into data. Many things in R and other open source packages have competing standards. A lecture on a technique inevitably biases one standard over another. Right now among R users in this area, there are two families of functions: base-R and tidyverse. R instructors thus face a dilemma about which to teach primarily.\footnote{See for example this community discussion: \url{https://community.rstudio.com/t/base-r-and-the-tidyverse/2965/17}} - -In this prefresher, we try our best to choose the one that is most useful to the modal task of social science researchers, and make use of the tidyverse functions in most applications. but feel free to suggest changes to us or to the booklet. - -Although you do not need to choose one over the other, for beginners it is confusing what is a tidyverse function and what is not. Many of the tidyverse \emph{packages} are covered in this 2017 graphic below, and the cheat-sheets that other programmers have written: \url{https://www.rstudio.com/resources/cheatsheets/} - -\begin{figure} -\centering -\includegraphics{images/tidyverse-packages.png} -\caption{Names of Packages in the tidyverse Family} -\end{figure} - -The following side-by-side comparison of commands for a particular function compares some tidyverse and non-tidyverse functions (which we refer to loosely as base-R). This list is not meant to be comprehensive and more to give you a quick rule of thumb. - -\hypertarget{dataframe-subsetting}{% -\subsection*{Dataframe subsetting}\label{dataframe-subsetting}} -\addcontentsline{toc}{subsection}{Dataframe subsetting} - -\begin{longtable}[]{@{}lll@{}} -\toprule -\begin{minipage}[b]{0.29\columnwidth}\raggedright -In order to \ldots{}\strut -\end{minipage} & \begin{minipage}[b]{0.33\columnwidth}\raggedright -in tidyverse:\strut -\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright -in base-R:\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Count each category\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{count(df,\ var)}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{table(df\$var)}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Filter rows by condition\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{filter(df,\ var\ ==\ "Female")}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{df{[}df\$var\ ==\ "Female",\ {]}} or \texttt{subset(df,\ var\ ==\ "Female")}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Extract columns\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{select(df,\ var1,\ var2)}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{df{[},\ c("var1",\ "var2"){]}}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Extract a single column as a vector\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{pull(df,\ var)}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{df{[}{[}"var"{]}{]}} or \texttt{df{[},\ "var"{]}}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Combine rows\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{bind\_rows()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{rbind()}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Combine columns\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{bind\_cols()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{cbind()}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Create a dataframe\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{tibble(x\ =\ vec1,\ y\ =\ vec2)}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{data.frame(x\ =\ vec1,\ y\ =\ vec2)}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Turn a dataframe into a tidyverse dataframe\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{tbl\_df(df)}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Remember that tidyverse applies to \emph{dataframes} only, not vectors. For subsetting vectors, use the base-R functions with the square brackets. - -\hypertarget{read-data}{% -\subsection*{Read data}\label{read-data}} -\addcontentsline{toc}{subsection}{Read data} - -Some non-tidyverse functions are not quite ``base-R'' but have similar relationships to tidyverse. For these, we recommend using the \emph{tidyverse} functions as a general rule due to their common format, simplicity, and scalability. - -\begin{longtable}[]{@{}lll@{}} -\toprule -\begin{minipage}[b]{0.29\columnwidth}\raggedright -In order to \ldots{}\strut -\end{minipage} & \begin{minipage}[b]{0.33\columnwidth}\raggedright -in tidyverse:\strut -\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright -in base-R:\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Read a Excel file\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{read\_excel()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{read.xlsx()}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Read a csv\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{read\_csv()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{read.csv()}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Read a Stata file\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{read\_dta()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{read.dta()}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Substitute strings\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{str\_replace()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{gsub()}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Return matching strings\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{str\_subset()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{grep(.,\ value\ =\ TRUE)}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Merge \texttt{data1} and \texttt{data2} on variables \texttt{x1} and \texttt{x2}\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{left\_join(data1,\ data2,\ by\ =\ c("x1",\ "x2"))}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{merge(data1,\ data2,\ by.x\ =\ "x1",\ by.y\ =\ "x2",\ all.x\ =\ TRUE)}\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -\hypertarget{visualization}{% -\subsection*{Visualization}\label{visualization}} -\addcontentsline{toc}{subsection}{Visualization} - -Plotting by ggplot2 (from your tutorials) is also a tidyverse family. - -\begin{longtable}[]{@{}lll@{}} -\toprule -\begin{minipage}[b]{0.29\columnwidth}\raggedright -In order to \ldots{}\strut -\end{minipage} & \begin{minipage}[b]{0.33\columnwidth}\raggedright -in tidyverse:\strut -\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright -in base-R:\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Make a scatter plot\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{ggplot(data,\ aes(x,\ y))\ +\ geom\_point()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{plot(data\$x,\ data\$y)}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Make a line plot\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{ggplot(data,\ aes(x,\ y))\ +\ geom\_line()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{plot(data\$x,\ data\$y,\ type\ =\ "l")}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Make a histogram\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -\texttt{ggplot(data,\ aes(x,\ y))\ +\ geom\_histogram()}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -\texttt{hist(data\$x,\ data\$y)}\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.29\columnwidth}\raggedright -Make a barplot\strut -\end{minipage} & \begin{minipage}[t]{0.33\columnwidth}\raggedright -See Section \ref{dataviz}\strut -\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright -See Section \ref{dataviz}\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -\hypertarget{a-is-for-athens}{% -\section{A is for Athens}\label{a-is-for-athens}} - -For our first dataset, let's try reading in a dataset on the Ancient Greek world. Political Theorists and Political Historians study the domestic systems, international wars, cultures and writing of this era to understand the first instance of democracy, the rise and overturning of tyranny, and the legacies of political institutions. - -This POLIS dataset was generously provided by Professor Josiah Ober of Stanford University. This dataset includes information on city states in the Ancient Greek world, parts of it collected by careful work by historians and archaeologists. It is part of his recent books on Greece (Ober 2015), ``The Rise and Fall of Classical Greece''\footnote{\href{https://press.princeton.edu/titles/10423.html}{Ober, Josiah (2015). \emph{The Rise and Fall of Classical Greece}. Princeton University Press.}} and Institutions in Ancient Athens (Ober 2010) , ``Democracy and Knowledge: Innovation and Learning in Classical Athens.''\footnote{\href{https://press.princeton.edu/titles/8742.html}{Ober, Josiah (2010). \emph{Democracy and Knowledge: Innovation and Learning in Classical Athens}. Princeton University Press.}} - -\hypertarget{locating-the-data}{% -\subsection{Locating the Data}\label{locating-the-data}} - -What files do we have in the \texttt{data/input} folder? - -\begin{verbatim} -## data/input/Nunn_Wantchekon_AER_2011.dta data/input/Nunn_Wantchekon_sample.dta -## data/input/acs2015_1percent.csv data/input/gapminder_wide.Rds -## data/input/gapminder_wide.tab data/input/german_credit.sav -## data/input/justices_court-median.csv data/input/ober_2018.xlsx -## data/input/sample_mid.csv data/input/sample_polity.csv -## data/input/upshot-siena-polls.csv data/input/usc2010_001percent.Rds -## data/input/usc2010_001percent.csv -\end{verbatim} - -A typical file format is Microsoft Excel. Although this is not usually the best format for R because of its highly formatted structure as opposed to plain text (more on this in Section \ref@(sec:wysiwyg)), recent packages have made this fairly easy. - -\hypertarget{reading-in-data}{% -\subsection{Reading in Data}\label{reading-in-data}} - -In Rstudio, a good way to start is to use the GUI and the Import tool. Once you click a file, an option to ``Import Dataset'' comes up. RStudio picks the right function for you, and you can copy that code, but it's important to eventually be able to write that code yourself. - -For the first time using an outside package, you first need to install it. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{install.packages}\NormalTok{(}\StringTok{"readxl"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -After that, you don't need to install it again. But you \textbf{do} need to load it each time. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(readxl)} -\end{Highlighting} -\end{Shaded} - -The package \texttt{readxl} has a website: \url{https://readxl.tidyverse.org/}. Other packages are not as user-friendly, but they have a help page with a table of contents of all their functions. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{help}\NormalTok{(}\DataTypeTok{package =}\NormalTok{ readxl)} -\end{Highlighting} -\end{Shaded} - -From the help page, we see that \texttt{read\_excel()} is the function that we want to use. - -Let's try it. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(readxl)} -\NormalTok{ober <{-}}\StringTok{ }\KeywordTok{read\_excel}\NormalTok{(}\StringTok{"data/input/ober\_2018.xlsx"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -Review: what does the \texttt{/} mean? Why do we need the \texttt{data} term first? Does the argument need to be in quotes? - -\hypertarget{inspecting}{% -\subsection{Inspecting}\label{inspecting}} - -For almost any dataset, you usually want to do a couple of standard checks first to understand what you loaded. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{ober} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 1,035 x 10 -## polis_number Name Latitude Longitude Hellenicity Fame Size Colonies Regime -## -## 1 1 Alal~ 42.1 9.51 most Greek 1.12 100-~ 0 -## 2 2 Empo~ 42.1 3.11 most barba~ 2.12 25-1~ 0 -## 3 3 Mass~ 43.3 5.38 most Greek 4 25-1~ 2 no ev~ -## 4 4 Rhode 42.3 3.17 most Greek 0.87 0 -## 5 5 Abak~ 38.1 15.1 most barba~ 1 0 -## 6 6 Adra~ 37.7 14.8 most Greek 1 0 -## 7 7 Agyr~ 37.7 14.5 most Greek 1.25 0 no ev~ -## 8 8 Aitna 38.2 15.6 most Greek 3.25 200-~ 1 no ev~ -## 9 9 Akra~ 37.3 13.6 most Greek 6.37 500 ~ 0 evide~ -## 10 10 Akrai 37.1 14.9 most Greek 1.25 0 -## # ... with 1,025 more rows, and 1 more variable: Delian -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{dim}\NormalTok{(ober)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 1035 10 -\end{verbatim} - -From your tutorials, you also know how to do graphics! Graphics are useful for grasping your data, but we will cover them more deeply in Chapter \ref{dataviz}. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ggplot}\NormalTok{(ober, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ Fame)) }\OperatorTok{+}\StringTok{ }\KeywordTok{geom\_histogram}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`. -\end{verbatim} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-105-1.pdf} - -What about the distribution of fame by regime? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ggplot}\NormalTok{(ober, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{y =}\NormalTok{ Fame, }\DataTypeTok{x =}\NormalTok{ Regime, }\DataTypeTok{group =}\NormalTok{ Regime)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_boxplot}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-106-1.pdf} - -What do the 1's, 2's, and 3's stand for? - -\hypertarget{finding-observations}{% -\subsection{Finding observations}\label{finding-observations}} - -These \texttt{tidyverse} commands from the \texttt{dplyr} package are newer and not built-in, but they are one of the increasingly more popular ways to wrangle data. - -\begin{itemize} -\tightlist -\item - 80 percent of your data wrangling needs might be doable with these basic \texttt{dplyr} functions: \texttt{select}, \texttt{mutate}, \texttt{group\_by}, \texttt{summarize}, and \texttt{arrange}. -\item - These verbs roughly correspond to the same commands in SQL, another important language in data science. -\item - The \texttt{\%\textgreater{}\%} symbol is a pipe. It takes the thing on the left side and pipes it down to the function on the right side. We could have done \texttt{count(cen10,\ race)} as \texttt{cen10\ \%\textgreater{}\%\ count(race)}. That means take \texttt{cen10} and pass it on to the function \texttt{count}, which will count observations by race and return a collapsed dataset with the categories in its own variable and their respective counts in \texttt{n}. -\end{itemize} - -\hypertarget{extra-a-sneak-peak-at-obers-data}{% -\subsection{Extra: A sneak peak at Ober's data}\label{extra-a-sneak-peak-at-obers-data}} - -Although this is a bit beyond our current stage, it's hard to resist the temptation to see what you can do with data like this. For example, you can map it.\footnote{In mid-2018, changes in Google's services made it no longer possible to render maps on the fly. Therefore, the map is not currently rendered automatically (but can be rendered once the user registers their API). Instead, you now need to register with Google. See the \href{https://github.com/dkahle/ggmap/blob/e55c0b22b0d16a010b4b45dd2fce800ff0ef19b8/NEWS\#L6-L12}{change} to the pacakge ggmap.} - -Using the \texttt{ggmap} package - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(ggmap)} -\end{Highlighting} -\end{Shaded} - -First get a map of the Greek world. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{greece <{-}}\StringTok{ }\KeywordTok{get\_map}\NormalTok{(}\DataTypeTok{location =} \KeywordTok{c}\NormalTok{(}\DataTypeTok{lon =} \FloatTok{22.6382849}\NormalTok{, }\DataTypeTok{lat =} \FloatTok{39.543287}\NormalTok{),} - \DataTypeTok{zoom =} \DecValTok{5}\NormalTok{, } - \DataTypeTok{source =} \StringTok{"stamen"}\NormalTok{,} - \DataTypeTok{maptype =} \StringTok{"toner"}\NormalTok{)} -\KeywordTok{ggmap}\NormalTok{(greece)} -\end{Highlighting} -\end{Shaded} - -\includegraphics{images/ober_ggmap_default.png} - -I chose the specifications for arguments \texttt{zoom} and \texttt{maptype} by looking at the webpage and Googling some examples. - -Ober's data has the latitude and longitude of each polis. Because the map of Greece has the same coordinates, we can add the polei on the same map. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{gg\_ober <{-}}\StringTok{ }\KeywordTok{ggmap}\NormalTok{(greece) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_point}\NormalTok{(}\DataTypeTok{data =}\NormalTok{ ober, } - \KeywordTok{aes}\NormalTok{(}\DataTypeTok{y =}\NormalTok{ Latitude, }\DataTypeTok{x =}\NormalTok{ Longitude), } - \DataTypeTok{size =} \FloatTok{0.5}\NormalTok{,} - \DataTypeTok{color =} \StringTok{"orange"}\NormalTok{)} -\NormalTok{gg\_ober }\OperatorTok{+}\StringTok{ } -\StringTok{ }\KeywordTok{scale\_x\_continuous}\NormalTok{(}\DataTypeTok{limits =} \KeywordTok{c}\NormalTok{(}\DecValTok{10}\NormalTok{, }\DecValTok{35}\NormalTok{)) }\OperatorTok{+}\StringTok{ } -\StringTok{ }\KeywordTok{scale\_y\_continuous}\NormalTok{(}\DataTypeTok{limits =} \KeywordTok{c}\NormalTok{(}\DecValTok{32}\NormalTok{, }\DecValTok{44}\NormalTok{)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{theme\_void}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\includegraphics{images/ober_ggmap_polis.png} - -\hypertarget{exercises}{% -\section*{Exercises}\label{exercises}} -\addcontentsline{toc}{section}{Exercises} - -\hypertarget{section}{% -\subsection*{1}\label{section}} -\addcontentsline{toc}{subsection}{1} - -What is the Fame value of Delphoi? - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-1}{% -\subsection*{2}\label{section-1}} -\addcontentsline{toc}{subsection}{2} - -Find the polis with the top 10 Fame values. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-2}{% -\subsection*{3}\label{section-2}} -\addcontentsline{toc}{subsection}{3} - -Make a scatterplot with the number of colonies on the x-axis and Fame on the y-axis. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-3}{% -\subsection*{4}\label{section-3}} -\addcontentsline{toc}{subsection}{4} - -Find the correct function to read the following datasets (available in your rstudio.cloud session) into your R window. - -\begin{itemize} -\tightlist -\item - \texttt{data/input/acs2015\_1percent.csv}: A one percent sample of the American Community Survey -\item - \texttt{data/input/gapminder\_wide.tab}: Country-level wealth and health from Gapminder\footnote{Formatted and taken from \url{https://doi.org/10.7910/DVN/GJQNEQ}} -\item - \texttt{data/input/gapminder\_wide.Rds}: A Rds version of the Gapminder (What is a Rds file? What's the difference?) -\item - \texttt{data/input/Nunn\_Wantchekon\_sample.dta}: A sample from the Afrobarometer survey (which we'll explore tomorrow). \texttt{.dta} is a Stata format. -\item - \texttt{data/input/german\_credit.sav}: A hypothetical dataset on consumer credit. \texttt{.sav} is a SPSS format. -\end{itemize} - -Our Recommendations: Look at the packages \texttt{haven} and \texttt{readr} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here, perhaps making a chunk for each file.} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-4}{% -\subsection*{5}\label{section-4}} -\addcontentsline{toc}{subsection}{5} - -Read Ober's codebook and find a variable that you think is interesting. Check the distribution of that variable in your data, get a couple of statistics, and summarize it in English. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-5}{% -\subsection*{6}\label{section-5}} -\addcontentsline{toc}{subsection}{6} - -This is day 1 and we covered a lot of material. Some of you might have found this completely new; others not so. Please click through this survey before you leave so we can adjust accordingly on the next few days. - -\url{https://harvard.az1.qualtrics.com/jfe/form/SV_8As7Y7C83iBiQzH} - -\hypertarget{rmatrices}{% -\chapter[Manipulating Vectors and Matrices]{\texorpdfstring{Manipulating Vectors and Matrices\footnote{Module originally written by Shiro Kuriwaki and Yon Soo Park}}{Manipulating Vectors and Matrices}}\label{rmatrices}} - -\hypertarget{motivation}{% -\subsection*{Motivation}\label{motivation}} -\addcontentsline{toc}{subsection}{Motivation} - -\href{https://dash.harvard.edu/bitstream/handle/1/11986331/nunn-slave-trade.pdf}{Nunn and Wantchekon (2011)} -- ``The Slave Trade and the Origins of Mistrust in Africa''\footnote{\href{https://dash.harvard.edu/bitstream/handle/1/11986331/nunn-slave-trade.pdf}{Nunn, Nathan, and Leonard Wantchekon. 2011. ``The Slave Trade and the Origins of Mistrust in Africa.'' American Economic Review 101(7): 3221--52.}} -- argues that across African countries, the distrust of co-ethnics fueled by the slave trade has had long-lasting effects on modern day trust in these territories. They argued that the slave trade created distrust in these societies in part because as some African groups were employed by European traders to capture their neighbors and bring them to the slave ships. - -Nunn and Wantchekon use a variety of statistical tools to make their case (adding controls, ordered logit, instrumental variables, falsification tests, causal mechanisms), many of which will be covered in future courses. In this module we will only touch on their first set of analysis that use Ordinary Least Squares (OLS). OLS is likely the most common application of linear algebra in the social sciences. We will cover some linear algebra, matrix manipulation, and vector manipulation from this data. - -\hypertarget{where-are-we-where-are-we-headed-1}{% -\subsection*{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed-1}} -\addcontentsline{toc}{subsection}{Where are we? Where are we headed?} - -Up till now, you should have covered: - -\begin{itemize} -\tightlist -\item - R basic programming -\item - Data Import -\item - Statistical Summaries. -\end{itemize} - -Today we'll cover - -\begin{itemize} -\tightlist -\item - Matrices \& Dataframes in R -\item - Manipulating variables -\item - And other \texttt{R} tips -\end{itemize} - -\hypertarget{read-data-1}{% -\section{Read Data}\label{read-data-1}} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(haven)} -\NormalTok{nunn\_full <{-}}\StringTok{ }\KeywordTok{read\_dta}\NormalTok{(}\StringTok{"data/input/Nunn\_Wantchekon\_AER\_2011.dta"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -Nunn and Wantchekon's main dataset has more than 20,000 observations. Each observation is a respondent from the Afrobarometer survey. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{head}\NormalTok{(nunn\_full)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 6 x 59 -## respno ethnicity murdock_name isocode region district townvill location_id -## -## 1 BEN00~ fon FON BEN atlna~ KPOMASSE TOKPA-D~ 30 -## 2 BEN00~ fon FON BEN atlna~ KPOMASSE TOKPA-D~ 30 -## 3 BEN00~ fon FON BEN atlna~ OUIDAH 3ARROND 31 -## 4 BEN00~ fon FON BEN atlna~ OUIDAH 3ARROND 31 -## 5 BEN00~ fon FON BEN atlna~ OUIDAH PAHOU 32 -## 6 BEN00~ fon FON BEN atlna~ OUIDAH PAHOU 32 -## # ... with 51 more variables: trust_relatives , trust_neighbors , -## # intra_group_trust , inter_group_trust , -## # trust_local_council , ln_export_area , export_area , -## # export_pop , ln_export_pop , age , age2 , male , -## # urban_dum , occupation , religion , living_conditions , -## # education , near_dist , distsea , loc_murdock_name , -## # loc_ln_export_area , local_council_performance , -## # council_listen , corrupt_local_council , school_present , -## # electricity_present , piped_water_present , sewage_present , -## # health_clinic_present , district_ethnic_frac , -## # frac_ethnicity_in_district , townvill_nonethnic_mean_exports , -## # district_nonethnic_mean_exports , region_nonethnic_mean_exports , -## # country_nonethnic_mean_exports , murdock_centr_dist_coast , -## # centroid_lat , centroid_long , explorer_contact , -## # railway_contact , dist_Saharan_node , dist_Saharan_line , -## # malaria_ecology , v30 , v33 , fishing , -## # exports , ln_exports , total_missions_area , -## # ln_init_pop_density , cities_1400_dum -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{colnames}\NormalTok{(nunn\_full)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "respno" "ethnicity" -## [3] "murdock_name" "isocode" -## [5] "region" "district" -## [7] "townvill" "location_id" -## [9] "trust_relatives" "trust_neighbors" -## [11] "intra_group_trust" "inter_group_trust" -## [13] "trust_local_council" "ln_export_area" -## [15] "export_area" "export_pop" -## [17] "ln_export_pop" "age" -## [19] "age2" "male" -## [21] "urban_dum" "occupation" -## [23] "religion" "living_conditions" -## [25] "education" "near_dist" -## [27] "distsea" "loc_murdock_name" -## [29] "loc_ln_export_area" "local_council_performance" -## [31] "council_listen" "corrupt_local_council" -## [33] "school_present" "electricity_present" -## [35] "piped_water_present" "sewage_present" -## [37] "health_clinic_present" "district_ethnic_frac" -## [39] "frac_ethnicity_in_district" "townvill_nonethnic_mean_exports" -## [41] "district_nonethnic_mean_exports" "region_nonethnic_mean_exports" -## [43] "country_nonethnic_mean_exports" "murdock_centr_dist_coast" -## [45] "centroid_lat" "centroid_long" -## [47] "explorer_contact" "railway_contact" -## [49] "dist_Saharan_node" "dist_Saharan_line" -## [51] "malaria_ecology" "v30" -## [53] "v33" "fishing" -## [55] "exports" "ln_exports" -## [57] "total_missions_area" "ln_init_pop_density" -## [59] "cities_1400_dum" -\end{verbatim} - -First, let's consider a small subset of this dataset. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{nunn <{-}}\StringTok{ }\KeywordTok{read\_dta}\NormalTok{(}\StringTok{"data/input/Nunn\_Wantchekon\_sample.dta"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{nunn} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 10 x 5 -## trust_neighbors exports ln_exports export_area ln_export_area -## -## 1 3 0.388 0.328 0.00407 0.00406 -## 2 3 0.631 0.489 0.0971 0.0926 -## 3 3 0.994 0.690 0.0125 0.0124 -## 4 0 183. 5.21 1.82 1.04 -## 5 3 0 0 0 0 -## 6 2 0 0 0 0 -## 7 2 666. 6.50 14.0 2.71 -## 8 0 0.348 0.298 0.00608 0.00606 -## 9 3 0.435 0.361 0.0383 0.0376 -## 10 3 0 0 0 0 -\end{verbatim} - -\hypertarget{data.frame-vs.-matricies}{% -\section{data.frame vs.~matricies}\label{data.frame-vs.-matricies}} - -This is a \texttt{data.frame} object. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(nunn)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "tbl_df" "tbl" "data.frame" -\end{verbatim} - -But it can be also consider a matrix in the linear algebra sense. What are the dimensions of this matrix? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{nrow}\NormalTok{(nunn)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 10 -\end{verbatim} - -\texttt{data.frame}s and matrices have much overlap in \texttt{R}, but to explicitly treat an object as a matrix, you'd need to coerce its class. Let's call this matrix \texttt{X}. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X <{-}}\StringTok{ }\KeywordTok{as.matrix}\NormalTok{(nunn)} -\end{Highlighting} -\end{Shaded} - -What is the difference between a \texttt{data.frame} and a matrix? A \texttt{data.frame} can have columns that are of different types, whereas --- in a matrix --- all columns must be of the same type (usually either ``numeric'' or ``character''). - -You can think of data frames maybe as matrices-plus, because a column can take on characters as well as numbers. As we just saw, this is often useful for real data analyses. - -Another way to think about data frames is that it is a type of list. Try the \texttt{str()} code below and notice how it is organized in slots. Each slot is a vector. They can be vectors of numbers or characters. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# enter this on your console} -\KeywordTok{str}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\hypertarget{handling-matricies-in-r}{% -\section{\texorpdfstring{Handling matricies in \texttt{R}}{Handling matricies in R}}\label{handling-matricies-in-r}} - -You can easily transpose a matrix - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## trust_neighbors exports ln_exports export_area ln_export_area -## [1,] 3 0.3883497 0.3281158 0.004067405 0.004059155 -## [2,] 3 0.6311236 0.4892691 0.097059444 0.092633367 -## [3,] 3 0.9941893 0.6902376 0.012524694 0.012446908 -## [4,] 0 182.5891266 5.2127004 1.824284434 1.038255095 -## [5,] 3 0.0000000 0.0000000 0.000000000 0.000000000 -## [6,] 2 0.0000000 0.0000000 0.000000000 0.000000000 -## [7,] 2 665.9652100 6.5027380 13.975566864 2.706419945 -## [8,] 0 0.3476418 0.2983562 0.006082553 0.006064130 -## [9,] 3 0.4349871 0.3611559 0.038332380 0.037615947 -## [10,] 3 0.0000000 0.0000000 0.000000000 0.000000000 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{t}\NormalTok{(X)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [,1] [,2] [,3] [,4] [,5] [,6] -## trust_neighbors 3.000000000 3.00000000 3.00000000 0.000000 3 2 -## exports 0.388349682 0.63112360 0.99418926 182.589127 0 0 -## ln_exports 0.328115761 0.48926911 0.69023758 5.212700 0 0 -## export_area 0.004067405 0.09705944 0.01252469 1.824284 0 0 -## ln_export_area 0.004059155 0.09263337 0.01244691 1.038255 0 0 -## [,7] [,8] [,9] [,10] -## trust_neighbors 2.000000 0.000000000 3.00000000 3 -## exports 665.965210 0.347641766 0.43498713 0 -## ln_exports 6.502738 0.298356235 0.36115587 0 -## export_area 13.975567 0.006082553 0.03833238 0 -## ln_export_area 2.706420 0.006064130 0.03761595 0 -\end{verbatim} - -What are the values of all rows in the first column? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X[, }\DecValTok{1}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 3 3 3 0 3 2 2 0 3 3 -\end{verbatim} - -What are all the values of ``exports''? (i.e.~return the whole ``exports'' column) - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X[, }\StringTok{"exports"}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.3883497 0.6311236 0.9941893 182.5891266 0.0000000 0.0000000 -## [7] 665.9652100 0.3476418 0.4349871 0.0000000 -\end{verbatim} - -What is the first observation (i.e.~first row)? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X[}\DecValTok{1}\NormalTok{, ]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## trust_neighbors exports ln_exports export_area ln_export_area -## 3.000000000 0.388349682 0.328115761 0.004067405 0.004059155 -\end{verbatim} - -What is the value of the first variable of the first observation? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X[}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## trust_neighbors -## 3 -\end{verbatim} - -Pause and consider the following problem on your own. What is the following code doing? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X[X[, }\StringTok{"trust\_neighbors"}\NormalTok{] }\OperatorTok{==}\StringTok{ }\DecValTok{0}\NormalTok{, }\StringTok{"export\_area"}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 1.824284434 0.006082553 -\end{verbatim} - -Why does it give the same output as the following? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X[}\KeywordTok{which}\NormalTok{(X[, }\StringTok{"trust\_neighbors"}\NormalTok{] }\OperatorTok{==}\StringTok{ }\DecValTok{0}\NormalTok{), }\StringTok{"export\_area"}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 1.824284434 0.006082553 -\end{verbatim} - -Some more manipulation - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X }\OperatorTok{+}\StringTok{ }\NormalTok{X} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## trust_neighbors exports ln_exports export_area ln_export_area -## [1,] 6 0.7766994 0.6562315 0.008134809 0.00811831 -## [2,] 6 1.2622472 0.9785382 0.194118887 0.18526673 -## [3,] 6 1.9883785 1.3804752 0.025049388 0.02489382 -## [4,] 0 365.1782532 10.4254007 3.648568869 2.07651019 -## [5,] 6 0.0000000 0.0000000 0.000000000 0.00000000 -## [6,] 4 0.0000000 0.0000000 0.000000000 0.00000000 -## [7,] 4 1331.9304199 13.0054760 27.951133728 5.41283989 -## [8,] 0 0.6952835 0.5967125 0.012165107 0.01212826 -## [9,] 6 0.8699743 0.7223117 0.076664761 0.07523189 -## [10,] 6 0.0000000 0.0000000 0.000000000 0.00000000 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X }\OperatorTok{{-}}\StringTok{ }\NormalTok{X} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## trust_neighbors exports ln_exports export_area ln_export_area -## [1,] 0 0 0 0 0 -## [2,] 0 0 0 0 0 -## [3,] 0 0 0 0 0 -## [4,] 0 0 0 0 0 -## [5,] 0 0 0 0 0 -## [6,] 0 0 0 0 0 -## [7,] 0 0 0 0 0 -## [8,] 0 0 0 0 0 -## [9,] 0 0 0 0 0 -## [10,] 0 0 0 0 0 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{t}\NormalTok{(X) }\OperatorTok{\%*\%}\StringTok{ }\NormalTok{X} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## trust_neighbors exports ln_exports export_area -## trust_neighbors 62.000000 1339.276 18.61181 28.40709 -## exports 1339.276369 476850.298 5283.76294 9640.42990 -## ln_exports 18.611811 5283.763 70.50077 100.46202 -## export_area 28.407085 9640.430 100.46202 198.65558 -## ln_export_area 5.853106 1992.047 23.08189 39.72847 -## ln_export_area -## trust_neighbors 5.853106 -## exports 1992.046502 -## ln_exports 23.081893 -## export_area 39.728468 -## ln_export_area 8.412887 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{cbind}\NormalTok{(X, }\DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## trust_neighbors exports ln_exports export_area ln_export_area -## [1,] 3 0.3883497 0.3281158 0.004067405 0.004059155 1 -## [2,] 3 0.6311236 0.4892691 0.097059444 0.092633367 2 -## [3,] 3 0.9941893 0.6902376 0.012524694 0.012446908 3 -## [4,] 0 182.5891266 5.2127004 1.824284434 1.038255095 4 -## [5,] 3 0.0000000 0.0000000 0.000000000 0.000000000 5 -## [6,] 2 0.0000000 0.0000000 0.000000000 0.000000000 6 -## [7,] 2 665.9652100 6.5027380 13.975566864 2.706419945 7 -## [8,] 0 0.3476418 0.2983562 0.006082553 0.006064130 8 -## [9,] 3 0.4349871 0.3611559 0.038332380 0.037615947 9 -## [10,] 3 0.0000000 0.0000000 0.000000000 0.000000000 10 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{cbind}\NormalTok{(X, }\DecValTok{1}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## trust_neighbors exports ln_exports export_area ln_export_area -## [1,] 3 0.3883497 0.3281158 0.004067405 0.004059155 1 -## [2,] 3 0.6311236 0.4892691 0.097059444 0.092633367 1 -## [3,] 3 0.9941893 0.6902376 0.012524694 0.012446908 1 -## [4,] 0 182.5891266 5.2127004 1.824284434 1.038255095 1 -## [5,] 3 0.0000000 0.0000000 0.000000000 0.000000000 1 -## [6,] 2 0.0000000 0.0000000 0.000000000 0.000000000 1 -## [7,] 2 665.9652100 6.5027380 13.975566864 2.706419945 1 -## [8,] 0 0.3476418 0.2983562 0.006082553 0.006064130 1 -## [9,] 3 0.4349871 0.3611559 0.038332380 0.037615947 1 -## [10,] 3 0.0000000 0.0000000 0.000000000 0.000000000 1 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{colnames}\NormalTok{(X)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "trust_neighbors" "exports" "ln_exports" "export_area" -## [5] "ln_export_area" -\end{verbatim} - -\hypertarget{variable-transformations}{% -\section{Variable Transformations}\label{variable-transformations}} - -\texttt{exports} is the total number of slaves that were taken from the individual's ethnic group between Africa's four slave trades between 1400-1900. - -What is \texttt{ln\_exports}? The article describes this as the natural log of one plus the \texttt{exports}. This is a transformation of one column by a particular function - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{log}\NormalTok{(}\DecValTok{1} \OperatorTok{+}\StringTok{ }\NormalTok{X[, }\StringTok{"exports"}\NormalTok{])} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.3281158 0.4892691 0.6902376 5.2127003 0.0000000 0.0000000 6.5027379 -## [8] 0.2983562 0.3611559 0.0000000 -\end{verbatim} - -Question for you: why add the 1? - -Verify that this is the same as \texttt{X{[},\ "ln\_exports"{]}} - -\hypertarget{linear-combinations}{% -\section{Linear Combinations}\label{linear-combinations}} - -In Table 1 we see ``OLS Estimates''. These are estimates of OLS coefficients and standard errors. You do not need to know what these are for now, but it doesn't hurt to getting used to seeing them. - -\includegraphics{images/nunn_wantchekon_table1.png} - -A very crude way to describe regression is through linear combinations. The simplest linear combination is a one-to-one transformation. - -Take the first number in Table 1, which is -0.00068. Now, multiply this by \texttt{exports} - -\begin{Shaded} -\begin{Highlighting}[] -\FloatTok{{-}0.00068} \OperatorTok{*}\StringTok{ }\NormalTok{X[, }\StringTok{"exports"}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] -0.0002640778 -0.0004291640 -0.0006760487 -0.1241606061 0.0000000000 -## [6] 0.0000000000 -0.4528563428 -0.0002363964 -0.0002957912 0.0000000000 -\end{verbatim} - -Now, just one more step. Make a new matrix with just exports and the value 1 - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X2 <{-}}\StringTok{ }\KeywordTok{cbind}\NormalTok{(}\DecValTok{1}\NormalTok{, X[, }\StringTok{"exports"}\NormalTok{])} -\end{Highlighting} -\end{Shaded} - -name this new column ``intercept'' - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{colnames}\NormalTok{(X2)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## NULL -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{colnames}\NormalTok{(X2) <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"intercept"}\NormalTok{, }\StringTok{"exports"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -What are the dimensions of the matrix \texttt{X2}? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{dim}\NormalTok{(X2)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 10 2 -\end{verbatim} - -Now consider a new matrix, called \texttt{B}. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{B <{-}}\StringTok{ }\KeywordTok{matrix}\NormalTok{(}\KeywordTok{c}\NormalTok{(}\FloatTok{1.62}\NormalTok{, }\FloatTok{{-}0.00068}\NormalTok{))} -\end{Highlighting} -\end{Shaded} - -What are the dimensions of \texttt{B}? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{dim}\NormalTok{(B)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 2 1 -\end{verbatim} - -What is the product of \texttt{X2} and \texttt{B}? From the dimensions, can you tell if it will be conformable? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{X2 }\OperatorTok{\%*\%}\StringTok{ }\NormalTok{B} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [,1] -## [1,] 1.619736 -## [2,] 1.619571 -## [3,] 1.619324 -## [4,] 1.495839 -## [5,] 1.620000 -## [6,] 1.620000 -## [7,] 1.167144 -## [8,] 1.619764 -## [9,] 1.619704 -## [10,] 1.620000 -\end{verbatim} - -What is this multiplication doing in terms of equations? - -\hypertarget{matrix-basics}{% -\section{Matrix Basics}\label{matrix-basics}} - -Let's take a look at Matrices in the context of R - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/usc2010\_001percent.csv"}\NormalTok{)} -\KeywordTok{head}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 6 x 4 -## state sex age race -## -## 1 New York Female 8 White -## 2 Ohio Male 24 White -## 3 Nevada Male 37 White -## 4 Michigan Female 12 White -## 5 Maryland Female 18 Black/Negro -## 6 New Hampshire Male 50 White -\end{verbatim} - -What is the dimension of this dataframe? What does the number of rows represent? What does the number of columns represent? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{dim}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 30871 4 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{nrow}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 30871 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ncol}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 4 -\end{verbatim} - -What variables does this dataset hold? What kind of information does it have? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{colnames}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "state" "sex" "age" "race" -\end{verbatim} - -We can access column vectors, or vectors that contain values of variables by using the \$ sign - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{head}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{state)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "New York" "Ohio" "Nevada" "Michigan" -## [5] "Maryland" "New Hampshire" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{head}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "White" "White" "White" "White" "Black/Negro" -## [6] "White" -\end{verbatim} - -We can look at a unique set of variable values by calling the unique function - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{unique}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{state)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "New York" "Ohio" "Nevada" -## [4] "Michigan" "Maryland" "New Hampshire" -## [7] "Iowa" "Missouri" "New Jersey" -## [10] "California" "Texas" "Pennsylvania" -## [13] "Washington" "West Virginia" "Idaho" -## [16] "North Carolina" "Massachusetts" "Connecticut" -## [19] "Arkansas" "Indiana" "Wisconsin" -## [22] "Maine" "Tennessee" "Minnesota" -## [25] "Florida" "Oklahoma" "Montana" -## [28] "Georgia" "Arizona" "Colorado" -## [31] "Virginia" "Illinois" "Oregon" -## [34] "Kentucky" "South Carolina" "Kansas" -## [37] "Louisiana" "Alabama" "District of Columbia" -## [40] "Mississippi" "Utah" "Delaware" -## [43] "Nebraska" "Alaska" "New Mexico" -## [46] "South Dakota" "Hawaii" "Vermont" -## [49] "Rhode Island" "Wyoming" "North Dakota" -\end{verbatim} - -How many different states are represented (this dataset includes DC as a state)? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{length}\NormalTok{(}\KeywordTok{unique}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{state))} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 51 -\end{verbatim} - -Matrices are rectangular structures of numbers (they have to be numbers, and they can't be characters). - -A cross-tab can be considered a matrix: - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{table}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race, cen10}\OperatorTok{$}\NormalTok{sex)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## -## Female Male -## American Indian or Alaska Native 142 153 -## Black/Negro 2070 1943 -## Chinese 192 162 -## Japanese 51 26 -## Other Asian or Pacific Islander 587 542 -## Other race, nec 877 962 -## Three or more major races 37 51 -## Two major races 443 426 -## White 11252 10955 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cross\_tab <{-}}\StringTok{ }\KeywordTok{table}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race, cen10}\OperatorTok{$}\NormalTok{sex)} -\KeywordTok{dim}\NormalTok{(cross\_tab)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 9 2 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cross\_tab[}\DecValTok{6}\NormalTok{, }\DecValTok{2}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 962 -\end{verbatim} - -But a subset of your data -- individual values-- can be considered a matrix too. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# First 20 rows of the entire data} -\CommentTok{\# Below two lines of code do the same thing} -\NormalTok{cen10[}\DecValTok{1}\OperatorTok{:}\DecValTok{20}\NormalTok{, ]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 20 x 4 -## state sex age race -## -## 1 New York Female 8 White -## 2 Ohio Male 24 White -## 3 Nevada Male 37 White -## 4 Michigan Female 12 White -## 5 Maryland Female 18 Black/Negro -## 6 New Hampshire Male 50 White -## 7 Iowa Female 51 White -## 8 Missouri Female 41 White -## 9 New Jersey Male 62 White -## 10 California Male 25 White -## 11 Texas Female 23 White -## 12 Pennsylvania Female 66 White -## 13 California Female 57 White -## 14 Texas Female 73 Other race, nec -## 15 California Male 43 White -## 16 Washington Male 29 White -## 17 Texas Male 8 White -## 18 Missouri Male 78 White -## 19 West Virginia Male 10 White -## 20 Idaho Female 9 White -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{slice}\NormalTok{(}\DecValTok{1}\OperatorTok{:}\DecValTok{20}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 20 x 4 -## state sex age race -## -## 1 New York Female 8 White -## 2 Ohio Male 24 White -## 3 Nevada Male 37 White -## 4 Michigan Female 12 White -## 5 Maryland Female 18 Black/Negro -## 6 New Hampshire Male 50 White -## 7 Iowa Female 51 White -## 8 Missouri Female 41 White -## 9 New Jersey Male 62 White -## 10 California Male 25 White -## 11 Texas Female 23 White -## 12 Pennsylvania Female 66 White -## 13 California Female 57 White -## 14 Texas Female 73 Other race, nec -## 15 California Male 43 White -## 16 Washington Male 29 White -## 17 Texas Male 8 White -## 18 Missouri Male 78 White -## 19 West Virginia Male 10 White -## 20 Idaho Female 9 White -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Of the first 20 rows of the entire data, look at values of just race and age} -\CommentTok{\# Below two lines of code do the same thing} -\NormalTok{cen10[}\DecValTok{1}\OperatorTok{:}\DecValTok{20}\NormalTok{, }\KeywordTok{c}\NormalTok{(}\StringTok{"race"}\NormalTok{, }\StringTok{"age"}\NormalTok{)]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 20 x 2 -## race age -## -## 1 White 8 -## 2 White 24 -## 3 White 37 -## 4 White 12 -## 5 Black/Negro 18 -## 6 White 50 -## 7 White 51 -## 8 White 41 -## 9 White 62 -## 10 White 25 -## 11 White 23 -## 12 White 66 -## 13 White 57 -## 14 Other race, nec 73 -## 15 White 43 -## 16 White 29 -## 17 White 8 -## 18 White 78 -## 19 White 10 -## 20 White 9 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{slice}\NormalTok{(}\DecValTok{1}\OperatorTok{:}\DecValTok{20}\NormalTok{) }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{select}\NormalTok{(race, age)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 20 x 2 -## race age -## -## 1 White 8 -## 2 White 24 -## 3 White 37 -## 4 White 12 -## 5 Black/Negro 18 -## 6 White 50 -## 7 White 51 -## 8 White 41 -## 9 White 62 -## 10 White 25 -## 11 White 23 -## 12 White 66 -## 13 White 57 -## 14 Other race, nec 73 -## 15 White 43 -## 16 White 29 -## 17 White 8 -## 18 White 78 -## 19 White 10 -## 20 White 9 -\end{verbatim} - -A vector is a special type of matrix with only one column or only one row - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# One column} -\NormalTok{cen10[}\DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{, }\KeywordTok{c}\NormalTok{(}\StringTok{"age"}\NormalTok{)]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 10 x 1 -## age -## -## 1 8 -## 2 24 -## 3 37 -## 4 12 -## 5 18 -## 6 50 -## 7 51 -## 8 41 -## 9 62 -## 10 25 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{slice}\NormalTok{(}\DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{) }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{select}\NormalTok{(}\KeywordTok{c}\NormalTok{(}\StringTok{"age"}\NormalTok{))} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 10 x 1 -## age -## -## 1 8 -## 2 24 -## 3 37 -## 4 12 -## 5 18 -## 6 50 -## 7 51 -## 8 41 -## 9 62 -## 10 25 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# One row} -\NormalTok{cen10[}\DecValTok{2}\NormalTok{, ]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 1 x 4 -## state sex age race -## -## 1 Ohio Male 24 White -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{slice}\NormalTok{(}\DecValTok{2}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 1 x 4 -## state sex age race -## -## 1 Ohio Male 24 White -\end{verbatim} - -What if we want a special subset of the data? For example, what if I only want the records of individuals in California? What if I just want the age and race of individuals in California? - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# subset for CA rows} -\NormalTok{ca\_subset <{-}}\StringTok{ }\NormalTok{cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ "California"}\NormalTok{, ]} - -\NormalTok{ca\_subset\_tidy <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(state }\OperatorTok{==}\StringTok{ "California"}\NormalTok{)} - -\KeywordTok{all\_equal}\NormalTok{(ca\_subset, ca\_subset\_tidy)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] TRUE -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# subset for CA rows and select age and race} -\NormalTok{ca\_subset\_age\_race <{-}}\StringTok{ }\NormalTok{cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ "California"}\NormalTok{, }\KeywordTok{c}\NormalTok{(}\StringTok{"age"}\NormalTok{, }\StringTok{"race"}\NormalTok{)]} - -\NormalTok{ca\_subset\_age\_race\_tidy <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(state }\OperatorTok{==}\StringTok{ "California"}\NormalTok{) }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{select}\NormalTok{(age, race)} - -\KeywordTok{all\_equal}\NormalTok{(ca\_subset\_age\_race, ca\_subset\_age\_race\_tidy)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] TRUE -\end{verbatim} - -Some common operators that can be used to filter or to use as a condition. Remember, you can use the unique function to look at the set of all values a variable holds in the dataset. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# all individuals older than 30 and younger than 70} -\NormalTok{s1 <{-}}\StringTok{ }\NormalTok{cen10[cen10}\OperatorTok{$}\NormalTok{age }\OperatorTok{>}\StringTok{ }\DecValTok{30} \OperatorTok{\&}\StringTok{ }\NormalTok{cen10}\OperatorTok{$}\NormalTok{age }\OperatorTok{<}\StringTok{ }\DecValTok{70}\NormalTok{, ]} -\NormalTok{s2 <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(age }\OperatorTok{>}\StringTok{ }\DecValTok{30} \OperatorTok{\&}\StringTok{ }\NormalTok{age }\OperatorTok{<}\StringTok{ }\DecValTok{70}\NormalTok{)} -\KeywordTok{all\_equal}\NormalTok{(s1, s2)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] TRUE -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# all individuals in either New York or California} -\NormalTok{s3 <{-}}\StringTok{ }\NormalTok{cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ "New York"} \OperatorTok{|}\StringTok{ }\NormalTok{cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ "California"}\NormalTok{, ]} -\NormalTok{s4 <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(state }\OperatorTok{==}\StringTok{ "New York"} \OperatorTok{|}\StringTok{ }\NormalTok{state }\OperatorTok{==}\StringTok{ "California"}\NormalTok{)} -\KeywordTok{all\_equal}\NormalTok{(s3, s4)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] TRUE -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# all individuals in any of the following states: California, Ohio, Nevada, Michigan} -\NormalTok{s5 <{-}}\StringTok{ }\NormalTok{cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{\%in\%}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Ohio"}\NormalTok{, }\StringTok{"Nevada"}\NormalTok{, }\StringTok{"Michigan"}\NormalTok{), ]} -\NormalTok{s6 <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(state }\OperatorTok{\%in\%}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Ohio"}\NormalTok{, }\StringTok{"Nevada"}\NormalTok{, }\StringTok{"Michigan"}\NormalTok{))} -\KeywordTok{all\_equal}\NormalTok{(s5, s6)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] TRUE -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# all individuals NOT in any of the following states: California, Ohio, Nevada, Michigan} -\NormalTok{s7 <{-}}\StringTok{ }\NormalTok{cen10[}\OperatorTok{!}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{\%in\%}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Ohio"}\NormalTok{, }\StringTok{"Nevada"}\NormalTok{, }\StringTok{"Michigan"}\NormalTok{)), ]} -\NormalTok{s8 <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(}\OperatorTok{!}\NormalTok{state }\OperatorTok{\%in\%}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Ohio"}\NormalTok{, }\StringTok{"Nevada"}\NormalTok{, }\StringTok{"Michigan"}\NormalTok{))} -\KeywordTok{all\_equal}\NormalTok{(s7, s8)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] TRUE -\end{verbatim} - -\hypertarget{checkpoint}{% -\section*{Checkpoint}\label{checkpoint}} -\addcontentsline{toc}{section}{Checkpoint} - -\hypertarget{section-6}{% -\subsection*{1}\label{section-6}} -\addcontentsline{toc}{subsection}{1} - -Get the subset of cen10 for non-white individuals (Hint: look at the set of values for the race variable by using the unique function) - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-7}{% -\subsection*{2}\label{section-7}} -\addcontentsline{toc}{subsection}{2} - -Get the subset of cen10 for females over the age of 40 - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-8}{% -\subsection*{3}\label{section-8}} -\addcontentsline{toc}{subsection}{3} - -Get all the serial numbers for black, male individuals who don't live in Ohio or Nevada. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{exercises-1}{% -\section*{Exercises}\label{exercises-1}} -\addcontentsline{toc}{section}{Exercises} - -\hypertarget{section-9}{% -\subsection*{1}\label{section-9}} -\addcontentsline{toc}{subsection}{1} - -Let -\[\mathbf{A} = \left[\begin{array} -{rrr} -0.6 & 0.2\\ -0.4 & 0.8\\ -\end{array}\right] -\] - -Use R to write code that will create the matrix \(A\), and then consecutively multiply \(A\) to itself 4 times. What is the value of \(A^{4}\)? - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -Note that R notation of matrices is different from the math notation. Simply trying \texttt{X\^{}n} where \texttt{X} is a matrix will only take the power of each element to \texttt{n}. Instead, this problem asks you to perform matrix multiplication. - -\hypertarget{section-10}{% -\subsection*{2}\label{section-10}} -\addcontentsline{toc}{subsection}{2} - -Let's apply what we learned about subsetting or filtering/selecting. Use the \texttt{nunn\_full} dataset you have already loaded - -\begin{enumerate} -\def\labelenumi{\alph{enumi})} -\tightlist -\item - First, show all observations (rows) that have a \texttt{"male"} variable higher than 0.5 -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -\begin{enumerate} -\def\labelenumi{\alph{enumi})} -\setcounter{enumi}{1} -\tightlist -\item - Next, create a matrix / dataframe with only two columns: \texttt{"trust\_neighbors"} and \texttt{"age"} -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -\begin{enumerate} -\def\labelenumi{\alph{enumi})} -\setcounter{enumi}{2} -\tightlist -\item - Lastly, show all values of \texttt{"trust\_neighbors"} and \texttt{"age"} for observations (rows) that have the ``male'' variable value that is higher than 0.5 -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-11}{% -\subsection*{3}\label{section-11}} -\addcontentsline{toc}{subsection}{3} - -Find a way to generate a vector of ``column averages'' of the matrix \texttt{X} from the Nunn and Wantchekon data in one line of code. Each entry in the vector should contain the sample average of the values in the column. So a 100 by 4 matrix should generate a length-4 matrix. - -\hypertarget{section-12}{% -\subsection*{4}\label{section-12}} -\addcontentsline{toc}{subsection}{4} - -Similarly, generate a vector of ``column medians''. - -\hypertarget{section-13}{% -\subsection*{5}\label{section-13}} -\addcontentsline{toc}{subsection}{5} - -Consider the regression that was run to generate Table 1: - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{form <{-}}\StringTok{ "trust\_neighbors \textasciitilde{} exports + age + age2 + male + urban\_dum + factor(education) + factor(occupation) + factor(religion) + factor(living\_conditions) + district\_ethnic\_frac + frac\_ethnicity\_in\_district + isocode"} -\NormalTok{lm\_}\DecValTok{1}\NormalTok{\_}\DecValTok{1}\NormalTok{ <{-}}\StringTok{ }\KeywordTok{lm}\NormalTok{(}\KeywordTok{as.formula}\NormalTok{(form), nunn\_full)} - -\CommentTok{\# The below coef function returns a vector of OLS coefficiants} -\KeywordTok{coef}\NormalTok{(lm\_}\DecValTok{1}\NormalTok{\_}\DecValTok{1}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## (Intercept) exports -## 1.619913e+00 -6.791360e-04 -## age age2 -## 8.395936e-03 -5.473436e-05 -## male urban_dum -## 4.550246e-02 -1.404551e-01 -## factor(education)1 factor(education)2 -## 1.709816e-02 -5.224591e-02 -## factor(education)3 factor(education)4 -## -1.373770e-01 -1.889619e-01 -## factor(education)5 factor(education)6 -## -1.893494e-01 -2.400767e-01 -## factor(education)7 factor(education)8 -## -2.850748e-01 -1.232085e-01 -## factor(education)9 factor(occupation)1 -## -2.406437e-01 6.185655e-02 -## factor(occupation)2 factor(occupation)3 -## 7.392168e-02 3.356158e-02 -## factor(occupation)4 factor(occupation)5 -## 7.942048e-03 6.661126e-02 -## factor(occupation)6 factor(occupation)7 -## -7.563297e-02 1.699699e-02 -## factor(occupation)8 factor(occupation)9 -## -9.428177e-02 -9.981440e-02 -## factor(occupation)10 factor(occupation)11 -## -3.307068e-02 -2.300045e-02 -## factor(occupation)12 factor(occupation)13 -## -1.564540e-01 -1.441370e-02 -## factor(occupation)14 factor(occupation)15 -## -5.566414e-02 -2.343762e-01 -## factor(occupation)16 factor(occupation)18 -## -1.306947e-02 -1.729589e-01 -## factor(occupation)19 factor(occupation)20 -## -1.770261e-01 -2.457800e-02 -## factor(occupation)21 factor(occupation)22 -## -4.936813e-02 -1.068511e-01 -## factor(occupation)23 factor(occupation)24 -## -9.712205e-02 1.292371e-02 -## factor(occupation)25 factor(occupation)995 -## 2.623186e-02 -1.195063e-03 -## factor(religion)2 factor(religion)3 -## 5.395953e-02 7.887878e-02 -## factor(religion)4 factor(religion)5 -## 4.749150e-02 4.318455e-02 -## factor(religion)6 factor(religion)7 -## -1.787694e-02 -3.616542e-02 -## factor(religion)10 factor(religion)11 -## 6.015041e-02 2.237845e-01 -## factor(religion)12 factor(religion)13 -## 2.627086e-01 -6.812813e-02 -## factor(religion)14 factor(religion)15 -## 4.673681e-02 3.844555e-01 -## factor(religion)360 factor(religion)361 -## 3.656843e-01 3.416413e-01 -## factor(religion)362 factor(religion)363 -## 8.230393e-01 3.856565e-01 -## factor(religion)995 factor(living_conditions)2 -## 4.161301e-02 4.395862e-02 -## factor(living_conditions)3 factor(living_conditions)4 -## 8.627372e-02 1.197428e-01 -## factor(living_conditions)5 district_ethnic_frac -## 1.203606e-01 -1.553648e-02 -## frac_ethnicity_in_district isocodeBWA -## 1.011222e-01 -4.258953e-01 -## isocodeGHA isocodeKEN -## 1.135307e-02 -1.819556e-01 -## isocodeLSO isocodeMDG -## -5.511200e-01 -3.315727e-01 -## isocodeMLI isocodeMOZ -## 7.528101e-02 8.223730e-02 -## isocodeMWI isocodeNAM -## 3.062497e-01 -1.397541e-01 -## isocodeNGA isocodeSEN -## -2.381525e-01 3.867371e-01 -## isocodeTZA isocodeUGA -## 2.079366e-01 -6.443732e-02 -## isocodeZAF isocodeZMB -## -2.179153e-01 -2.172868e-01 -\end{verbatim} - -First, get a small subset of the nunn\_full dataset. This time, sample 20 rows and select for variables \texttt{exports}, \texttt{age}, \texttt{age2}, \texttt{male}, and \texttt{urban\_dum}. To this small subset, add (\texttt{bind\_cols()} in tidyverse or \texttt{cbind()} in base R) a column of 1's; this represents the intercept. If you need some guidance, look at how we sampled 10 rows selected for a different set of variables above in the lecture portion. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter here} -\end{Highlighting} -\end{Shaded} - -Next let's try calculating predicted values of levels of trust in neighbors by multiplying coefficients for the intercept, \texttt{exports}, \texttt{age}, \texttt{age2}, \texttt{male}, and \texttt{urban\_dum} to the actual observed values for those variables in the small subset you've just created. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Hint: You can get just selected elements from the vector returned by coef(lm\_1\_1)} - -\CommentTok{\# For example, the below code gives you the first 3 elements of the original vector} -\KeywordTok{coef}\NormalTok{(lm\_}\DecValTok{1}\NormalTok{\_}\DecValTok{1}\NormalTok{)[}\DecValTok{1}\OperatorTok{:}\DecValTok{3}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## (Intercept) exports age -## 1.619913146 -0.000679136 0.008395936 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Also, the below code gives you the coefficient elements for intercept and male} -\KeywordTok{coef}\NormalTok{(lm\_}\DecValTok{1}\NormalTok{\_}\DecValTok{1}\NormalTok{)[}\KeywordTok{c}\NormalTok{(}\StringTok{"(Intercept)"}\NormalTok{, }\StringTok{"male"}\NormalTok{)]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## (Intercept) male -## 1.61991315 0.04550246 -\end{verbatim} - -\hypertarget{robjloops}{% -\chapter{Objects, Functions, Loops}\label{robjloops}} - -\hypertarget{where-are-we-where-are-we-headed-2}{% -\subsection*{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed-2}} -\addcontentsline{toc}{subsection}{Where are we? Where are we headed?} - -Up till now, you should have covered: - -\begin{itemize} -\tightlist -\item - R basic programming -\item - Data Import -\item - Statistical Summaries -\item - Visualization -\end{itemize} - -Today we'll cover - -\begin{itemize} -\tightlist -\item - Objects -\item - Functions -\item - Loops -\end{itemize} - -\hypertarget{what-is-an-object}{% -\section{What is an object?}\label{what-is-an-object}} - -Now that we have covered some hands-on ways to use graphics, let's go into some fundamentals of the R language. - -Let's first set up - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(dplyr)} -\KeywordTok{library}\NormalTok{(readr)} -\KeywordTok{library}\NormalTok{(haven)} -\KeywordTok{library}\NormalTok{(ggplot2)} -\end{Highlighting} -\end{Shaded} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/usc2010\_001percent.csv"}\NormalTok{, }\DataTypeTok{col\_types =} \KeywordTok{cols}\NormalTok{())} -\end{Highlighting} -\end{Shaded} - -Objects are abstract symbols in which you store data. Here we will create an object from \texttt{copy}, and assign \texttt{cen10} to it. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{copy <{-}}\StringTok{ }\NormalTok{cen10 } -\end{Highlighting} -\end{Shaded} - -This looks the same as the original dataset: - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{copy} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 30,871 x 4 -## state sex age race -## -## 1 New York Female 8 White -## 2 Ohio Male 24 White -## 3 Nevada Male 37 White -## 4 Michigan Female 12 White -## 5 Maryland Female 18 Black/Negro -## 6 New Hampshire Male 50 White -## 7 Iowa Female 51 White -## 8 Missouri Female 41 White -## 9 New Jersey Male 62 White -## 10 California Male 25 White -## # ... with 30,861 more rows -\end{verbatim} - -What happens if you do this next? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{copy <{-}}\StringTok{ ""} -\end{Highlighting} -\end{Shaded} - -It got reassigned: - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{copy} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "" -\end{verbatim} - -\hypertarget{lists}{% -\subsection{lists}\label{lists}} - -Lists are one of the most generic and flexible type of object. You can make an empty list by the function \texttt{list()} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_list <{-}}\StringTok{ }\KeywordTok{list}\NormalTok{()} -\NormalTok{my\_list} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## list() -\end{verbatim} - -And start filling it in. Slots on the list are invoked by double square brackets \texttt{{[}{[}{]}{]}} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_list[[}\DecValTok{1}\NormalTok{]] <{-}}\StringTok{ "contents of the first slot {-}{-} this is a string"} -\NormalTok{my\_list[[}\StringTok{"slot 2"}\NormalTok{]] <{-}}\StringTok{ "contents of slot named slot 2"} -\NormalTok{my\_list} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [[1]] -## [1] "contents of the first slot -- this is a string" -## -## $`slot 2` -## [1] "contents of slot named slot 2" -\end{verbatim} - -each slot can be anything. What are we doing here? We are defining the 1st slot of the list \texttt{my\_list} to be a vector \texttt{c(1,\ 2,\ 3,\ 4,\ 5)} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_list[[}\DecValTok{1}\NormalTok{]] <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\DecValTok{1}\NormalTok{, }\DecValTok{2}\NormalTok{, }\DecValTok{3}\NormalTok{, }\DecValTok{4}\NormalTok{, }\DecValTok{5}\NormalTok{)} -\NormalTok{my\_list} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [[1]] -## [1] 1 2 3 4 5 -## -## $`slot 2` -## [1] "contents of slot named slot 2" -\end{verbatim} - -You can even make nested lists. Let's say we want the 1st slot of the list to be another list of three elements. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_list[[}\DecValTok{1}\NormalTok{]][[}\DecValTok{1}\NormalTok{]] <{-}}\StringTok{ "subitem 1 in slot 1 of my\_list"} -\NormalTok{my\_list[[}\DecValTok{1}\NormalTok{]][[}\DecValTok{2}\NormalTok{]] <{-}}\StringTok{ "subitem 1 in slot 2 of my\_list"} -\NormalTok{my\_list[[}\DecValTok{1}\NormalTok{]][[}\DecValTok{3}\NormalTok{]] <{-}}\StringTok{ "subitem 1 in slot 3 of my\_list"} - -\NormalTok{my\_list} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [[1]] -## [1] "subitem 1 in slot 1 of my_list" "subitem 1 in slot 2 of my_list" -## [3] "subitem 1 in slot 3 of my_list" "4" -## [5] "5" -## -## $`slot 2` -## [1] "contents of slot named slot 2" -\end{verbatim} - -\hypertarget{making-your-own-objects}{% -\section{Making your own objects}\label{making-your-own-objects}} - -We've covered one type of object, which is a list. You saw it was quite flexible. How many types of objects are there? - -There are an infinite number of objects, because people make their own class of object. You can detect the type of the object (the class) by the function \texttt{class} - -Object can be said to be an instance of a class. - -\textbf{\emph{Analogies}}: - -\textbf{class} - Pokemon, \textbf{object} - Pikachu - -\textbf{class} - Book, \textbf{object} - To Kill a Mockingbird - -\textbf{class} - DataFrame, \textbf{object} - 2010 census data - -\textbf{class} - Character, \textbf{object} - ``Programming is Fun'' - -What is type (class) of object is \texttt{cen10}? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "spec_tbl_df" "tbl_df" "tbl" "data.frame" -\end{verbatim} - -What about this text? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(}\StringTok{"some random text"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "character" -\end{verbatim} - -To change or create the class of any object, you can \emph{assign} it. To do this, assign the name of your class to character to an object's \texttt{class()}. - -We can start from a simple list. For example, say we wanted to store data about pokemon. Because there is no pre-made package for this, we decide to make our own class. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{pikachu <{-}}\StringTok{ }\KeywordTok{list}\NormalTok{(}\DataTypeTok{name =} \StringTok{"Pikachu"}\NormalTok{,} - \DataTypeTok{number =} \DecValTok{25}\NormalTok{,} - \DataTypeTok{type =} \StringTok{"Electric"}\NormalTok{,} - \DataTypeTok{color =} \StringTok{"Yellow"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -and we can give it any class name we want. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(pikachu) <{-}}\StringTok{ "Pokemon"} -\KeywordTok{str}\NormalTok{(pikachu)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## List of 4 -## $ name : chr "Pikachu" -## $ number: num 25 -## $ type : chr "Electric" -## $ color : chr "Yellow" -## - attr(*, "class")= chr "Pokemon" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{pikachu}\OperatorTok{$}\NormalTok{type} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Electric" -\end{verbatim} - -\hypertarget{seeing-r-through-objects}{% -\subsection{Seeing R through objects}\label{seeing-r-through-objects}} - -Most of the R objects that you will see as you advance are their own objects. For example, here's a linear regression object (which you will learn more about in Gov 2000): - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{ols <{-}}\StringTok{ }\KeywordTok{lm}\NormalTok{(mpg }\OperatorTok{\textasciitilde{}}\StringTok{ }\NormalTok{wt }\OperatorTok{+}\StringTok{ }\NormalTok{vs }\OperatorTok{+}\StringTok{ }\NormalTok{gear }\OperatorTok{+}\StringTok{ }\NormalTok{carb, mtcars)} -\KeywordTok{class}\NormalTok{(ols)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "lm" -\end{verbatim} - -Anything can be an object! Even graphs (in \texttt{ggplot}) can be assigned, re-assigned, and edited. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{grp\_race <{-}}\StringTok{ }\KeywordTok{group\_by}\NormalTok{(cen10, race)}\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{summarize}\NormalTok{(}\DataTypeTok{count =} \KeywordTok{n}\NormalTok{())} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## `summarise()` ungrouping output (override with `.groups` argument) -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{grp\_race\_ordered <{-}}\StringTok{ }\KeywordTok{arrange}\NormalTok{(grp\_race, count) }\OperatorTok{\%>\%}\StringTok{ } -\StringTok{ }\KeywordTok{mutate}\NormalTok{(}\DataTypeTok{race =}\NormalTok{ forcats}\OperatorTok{::}\KeywordTok{as\_factor}\NormalTok{(race))} - -\NormalTok{gg\_tab <{-}}\StringTok{ }\KeywordTok{ggplot}\NormalTok{(}\DataTypeTok{data =}\NormalTok{ grp\_race\_ordered) }\OperatorTok{+} -\StringTok{ }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ race, }\DataTypeTok{y =}\NormalTok{ count) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_col}\NormalTok{() }\OperatorTok{+} -\StringTok{ }\KeywordTok{labs}\NormalTok{(}\DataTypeTok{caption =} \StringTok{"Source: U.S. Census 2010"}\NormalTok{)} - -\NormalTok{gg\_tab} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-185-1.pdf} - -You can change the orientation - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{gg\_tab<{-}}\StringTok{ }\NormalTok{gg\_tab }\OperatorTok{+}\StringTok{ }\KeywordTok{coord\_flip}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\hypertarget{parsing-an-object-by-strs}{% -\subsection{\texorpdfstring{Parsing an object by \texttt{str()s}}{Parsing an object by str()s}}\label{parsing-an-object-by-strs}} - -It can be hard to understand an \texttt{R} object because it's contents are unknown. The function \texttt{str}, short for structure, is a quick way to look into the innards of an object - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{str}\NormalTok{(my\_list)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## List of 2 -## $ : chr [1:5] "subitem 1 in slot 1 of my_list" "subitem 1 in slot 2 of my_list" "subitem 1 in slot 3 of my_list" "4" ... -## $ slot 2: chr "contents of slot named slot 2" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(my\_list)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "list" -\end{verbatim} - -Same for the object we just made - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{str}\NormalTok{(pikachu)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## List of 4 -## $ name : chr "Pikachu" -## $ number: num 25 -## $ type : chr "Electric" -## $ color : chr "Yellow" -## - attr(*, "class")= chr "Pokemon" -\end{verbatim} - -What does a \texttt{ggplot} object look like? Very complicated, but at least you can see it: - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# enter this on your console} -\KeywordTok{str}\NormalTok{(gg\_tab)} -\end{Highlighting} -\end{Shaded} - -\hypertarget{types-of-variables}{% -\section{Types of variables}\label{types-of-variables}} - -In the social science we often analyze variables. As you saw in the tutorial, different types of variables require different care. - -A key link with what we just learned is that variables are also types of R objects. - -\hypertarget{scalars}{% -\subsection{scalars}\label{scalars}} - -One number. How many people did we count in our Census sample? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{nrow}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 30871 -\end{verbatim} - -Question: What proportion of our census sample is Native American? This number is also a scalar - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter yourself} -\KeywordTok{unique}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "White" "Black/Negro" -## [3] "Other race, nec" "American Indian or Alaska Native" -## [5] "Chinese" "Other Asian or Pacific Islander" -## [7] "Two major races" "Three or more major races" -## [9] "Japanese" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{mean}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race }\OperatorTok{==}\StringTok{ "American Indian or Alaska Native"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.009555894 -\end{verbatim} - -Hint: you can use the function \texttt{mean()} to calcualte the sample mean. The sample proportion is the mean of a sequence of number, where your event of interest is a 1 (or \texttt{TRUE}) and others are 0 (or \texttt{FALSE}). - -\hypertarget{numeric-vectors}{% -\subsection{numeric vectors}\label{numeric-vectors}} - -A sequence of numbers. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{grp\_race\_ordered}\OperatorTok{$}\NormalTok{count} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 77 88 295 354 869 1129 1839 4013 22207 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(grp\_race\_ordered}\OperatorTok{$}\NormalTok{count)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "integer" -\end{verbatim} - -Or even, all the ages of the millions of people in our Census. Here are just the first few numbers of the list. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{head}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{age)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 8 24 37 12 18 50 -\end{verbatim} - -\hypertarget{characters-aka-strings}{% -\subsection{characters (aka strings)}\label{characters-aka-strings}} - -This can be just one stretch of characters - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_name <{-}}\StringTok{ "Meg"} -\NormalTok{my\_name} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Meg" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(my\_name)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "character" -\end{verbatim} - -or more characters. Notice here that there's a difference between a vector of individual characters and a length-one object of characters. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_name\_letters <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"M"}\NormalTok{,}\StringTok{"e"}\NormalTok{,}\StringTok{"g"}\NormalTok{)} -\NormalTok{my\_name\_letters} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "M" "e" "g" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{class}\NormalTok{(my\_name\_letters)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "character" -\end{verbatim} - -Finally, remember that lower vs.~upper case matters in R! - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_name2 <{-}}\StringTok{ "shiro"} -\NormalTok{my\_name }\OperatorTok{==}\StringTok{ }\NormalTok{my\_name2} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] FALSE -\end{verbatim} - -\hypertarget{what-is-a-function}{% -\section{What is a function?}\label{what-is-a-function}} - -Most of what we do in R is executing a function. \texttt{read\_csv()}, \texttt{nrow()}, \texttt{ggplot()} .. pretty much anything with a parentheses is a function. And even things like \texttt{\textless{}-} and \texttt{{[}} are functions as well. - -A function is a set of instructions with specified ingredients. It takes an \textbf{input}, then \textbf{manipulates} it -- changes it in some way -- and then returns the manipulated product. - -One way to see what a function actually does is to enter it without parentheses. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# enter this on your console} -\NormalTok{table} -\end{Highlighting} -\end{Shaded} - -You'll see below that the most basic functions are quite complicated internally. - -You'll notice that functions contain other functions. \emph{wrapper} functions are functions that ``wrap around'' existing functions. This sounds redundant, but it's an important feature of programming. If you find yourself repeating a command more than two times, you should make your own function, rather than writing the same type of code. - -\hypertarget{write-your-own-function}{% -\subsection{Write your own function}\label{write-your-own-function}} - -It's worth remembering the basic structure of a function. You create a new function, call it \texttt{my\_fun} by this: - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_fun <{-}}\StringTok{ }\ControlFlowTok{function}\NormalTok{() \{} - -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -If we wanted to generate a function that computed the number of men in your data, what would that look like? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{count\_men <{-}}\StringTok{ }\ControlFlowTok{function}\NormalTok{(data) \{} - -\NormalTok{ nmen <{-}}\StringTok{ }\KeywordTok{sum}\NormalTok{(data}\OperatorTok{$}\NormalTok{sex }\OperatorTok{==}\StringTok{ "Male"}\NormalTok{)} - - \KeywordTok{return}\NormalTok{(nmen)} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -Then all we need to do is feed this function a dataset - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{count\_men}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 15220 -\end{verbatim} - -The point of a function is that you can use it again and again without typing up the set of constituent manipulations. So, what if we wanted to figure out the number of men in California? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{count\_men}\NormalTok{(cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ "California"}\NormalTok{,])} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 1876 -\end{verbatim} - -Let's go one step further. What if we want to know the proportion of non-whites in a state, just by entering the name of the state? There's multiple ways to do it, but it could look something like this - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{nw\_in\_state <{-}}\StringTok{ }\ControlFlowTok{function}\NormalTok{(data, state) \{} - -\NormalTok{ s.subset <{-}}\StringTok{ }\NormalTok{data[data}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state,]} -\NormalTok{ total.s <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(s.subset)} -\NormalTok{ nw.s <{-}}\StringTok{ }\KeywordTok{sum}\NormalTok{(s.subset}\OperatorTok{$}\NormalTok{race }\OperatorTok{!=}\StringTok{ "White"}\NormalTok{)} - -\NormalTok{ nw.s }\OperatorTok{/}\StringTok{ }\NormalTok{total.s} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -The last line is what gets generated from the function. To be more explicit you can wrap the last line around \texttt{return()}. (as in \texttt{return(nw.s/total.s}). \texttt{return()} is used when you want to break out of a function in the middle of it and not wait till the last line. - -Try it on your favorite state! - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{nw\_in\_state}\NormalTok{(cen10, }\StringTok{"Massachusetts"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.2040185 -\end{verbatim} - -\hypertarget{checkpoint-1}{% -\section*{Checkpoint}\label{checkpoint-1}} -\addcontentsline{toc}{section}{Checkpoint} - -\hypertarget{section-14}{% -\subsection*{1}\label{section-14}} -\addcontentsline{toc}{subsection}{1} - -Try making your own function, \texttt{average\_age\_in\_state}, that will give you the average age of people in a given state. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter on your own} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-15}{% -\subsection*{2}\label{section-15}} -\addcontentsline{toc}{subsection}{2} - -Try making your own function, \texttt{asians\_in\_state}, that will give you the number of \texttt{Chinese}, \texttt{Japanese}, and \texttt{Other\ Asian\ or\ Pacific\ Islander} people in a given state. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter on your own} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-16}{% -\subsection*{3}\label{section-16}} -\addcontentsline{toc}{subsection}{3} - -Try making your own function, `top\_10\_oldest\_cities', that will give you the names of cities whose population's average age is top 10 oldest. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter on your own} -\end{Highlighting} -\end{Shaded} - -\hypertarget{what-is-a-package}{% -\section{What is a package?}\label{what-is-a-package}} - -You can think of a package as a suite of functions that other people have already built for you to make your life easier. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{help}\NormalTok{(}\DataTypeTok{package =} \StringTok{"ggplot2"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -To use a package, you need to do two things: (1) install it, and then (2) load it. - -Installing is a one-time thing - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{install.packages}\NormalTok{(}\StringTok{"ggplot2"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -But you need to load each time you start a R instance. So always keep these commands on a script. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(ggplot2)} -\end{Highlighting} -\end{Shaded} - -In \texttt{rstudio.cloud}, we already installed a set of packages for you. But when you start your own R instance, you need to have installed the package at some point. - -\hypertarget{conditionals}{% -\section{Conditionals}\label{conditionals}} - -Sometimes, you want to execute a command only under certain conditions. This is done through the almost universal function, \texttt{if()}. Inside the \texttt{if} function we enter a logical statement. The line that is adjacent to, or follows, the \texttt{if()} statement only gets executed if the statement returns \texttt{TRUE}. - -For example, - -For example, - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{x <{-}}\StringTok{ }\DecValTok{5} -\ControlFlowTok{if}\NormalTok{ (x }\OperatorTok{>}\DecValTok{0}\NormalTok{) \{} - \KeywordTok{print}\NormalTok{(}\StringTok{"positive number"}\NormalTok{)} -\NormalTok{\} }\ControlFlowTok{else} \ControlFlowTok{if}\NormalTok{ (x }\OperatorTok{==}\StringTok{ }\DecValTok{0}\NormalTok{) \{} - \KeywordTok{print}\NormalTok{ (}\StringTok{"zero"}\NormalTok{)} -\NormalTok{\} }\ControlFlowTok{else}\NormalTok{ \{} - \KeywordTok{print}\NormalTok{(}\StringTok{"negative number"}\NormalTok{)} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "positive number" -\end{verbatim} - -You can wrap that whole things in a function - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{is\_positive <{-}}\StringTok{ }\ControlFlowTok{function}\NormalTok{(number) \{} - \ControlFlowTok{if}\NormalTok{ (number }\OperatorTok{>}\DecValTok{0}\NormalTok{) \{} - \KeywordTok{print}\NormalTok{(}\StringTok{"positive number"}\NormalTok{)} -\NormalTok{ \} }\ControlFlowTok{else} \ControlFlowTok{if}\NormalTok{ (number }\OperatorTok{==}\StringTok{ }\DecValTok{0}\NormalTok{) \{} - \KeywordTok{print}\NormalTok{ (}\StringTok{"zero"}\NormalTok{)} -\NormalTok{ \} }\ControlFlowTok{else}\NormalTok{ \{} - \KeywordTok{print}\NormalTok{(}\StringTok{"negative number"}\NormalTok{)} -\NormalTok{ \}} -\NormalTok{\}} - -\KeywordTok{is\_positive}\NormalTok{(}\DecValTok{5}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "positive number" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{is\_positive}\NormalTok{(}\OperatorTok{{-}}\DecValTok{3}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "negative number" -\end{verbatim} - -\hypertarget{for-loops}{% -\section{For-loops}\label{for-loops}} - -Loops repeat the same statement, although the statement can be ``the same'' only in an abstract sense. Use the \texttt{for(x\ in\ X)} syntax to repeat the subsequent command as many times as there are elements in the right-hand object \texttt{X}. Each of these elements will be referred to the left-hand index \texttt{x} - -First, come up with a vector. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{fruits <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"apples"}\NormalTok{, }\StringTok{"oranges"}\NormalTok{, }\StringTok{"grapes"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -Now we use the \texttt{fruits} vector in a \texttt{for} loop. - -\begin{Shaded} -\begin{Highlighting}[] -\ControlFlowTok{for}\NormalTok{ (fruit }\ControlFlowTok{in}\NormalTok{ fruits) \{} - \KeywordTok{print}\NormalTok{(}\KeywordTok{paste}\NormalTok{(}\StringTok{"I love"}\NormalTok{, fruit))} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "I love apples" -## [1] "I love oranges" -## [1] "I love grapes" -\end{verbatim} - -Here \texttt{for()} and \texttt{in} must be part of any for loop. The right hand side \texttt{fruits} must be a thing that exists. Finally the \texttt{left-hand} side object is ``Pick your favor name.'' It is analogous to how we can index a sum with any letter. \(\sum_{i=1}^{10}i\) and \texttt{sum\_\{j\ =\ 1\}\^{}\{10\}j} are in fact the same thing. - -\begin{Shaded} -\begin{Highlighting}[] -\ControlFlowTok{for}\NormalTok{ (i }\ControlFlowTok{in} \DecValTok{1}\OperatorTok{:}\KeywordTok{length}\NormalTok{(fruits)) \{} - \KeywordTok{print}\NormalTok{(}\KeywordTok{paste}\NormalTok{(}\StringTok{"I love"}\NormalTok{, fruits[i]))} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "I love apples" -## [1] "I love oranges" -## [1] "I love grapes" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{states\_of\_interest <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Massachusetts"}\NormalTok{, }\StringTok{"New Hampshire"}\NormalTok{, }\StringTok{"Washington"}\NormalTok{)} - -\ControlFlowTok{for}\NormalTok{( state }\ControlFlowTok{in}\NormalTok{ states\_of\_interest)\{} -\NormalTok{ state\_data <{-}}\StringTok{ }\NormalTok{cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state,]} -\NormalTok{ nmen <{-}}\StringTok{ }\KeywordTok{sum}\NormalTok{(state\_data}\OperatorTok{$}\NormalTok{sex }\OperatorTok{==}\StringTok{ "Male"}\NormalTok{)} - -\NormalTok{ n <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(state\_data)} -\NormalTok{ men\_perc <{-}}\StringTok{ }\KeywordTok{round}\NormalTok{(}\DecValTok{100}\OperatorTok{*}\NormalTok{(nmen}\OperatorTok{/}\NormalTok{n), }\DataTypeTok{digits=}\DecValTok{2}\NormalTok{)} - \KeywordTok{print}\NormalTok{(}\KeywordTok{paste}\NormalTok{(}\StringTok{"Percentage of men in"}\NormalTok{,state, }\StringTok{"is"}\NormalTok{, men\_perc))} - -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Percentage of men in California is 49.85" -## [1] "Percentage of men in Massachusetts is 47.6" -## [1] "Percentage of men in New Hampshire is 48.55" -## [1] "Percentage of men in Washington is 48.19" -\end{verbatim} - -Instead of printing, you can store the information in a vector - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{states\_of\_interest <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Massachusetts"}\NormalTok{, }\StringTok{"New Hampshire"}\NormalTok{, }\StringTok{"Washington"}\NormalTok{)} -\NormalTok{male\_percentages <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{()} -\NormalTok{iter <{-}}\DecValTok{1} - -\ControlFlowTok{for}\NormalTok{( state }\ControlFlowTok{in}\NormalTok{ states\_of\_interest)\{} -\NormalTok{ state\_data <{-}}\StringTok{ }\NormalTok{cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state,]} -\NormalTok{ nmen <{-}}\StringTok{ }\KeywordTok{sum}\NormalTok{(state\_data}\OperatorTok{$}\NormalTok{sex }\OperatorTok{==}\StringTok{ "Male"}\NormalTok{)} -\NormalTok{ n <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(state\_data)} -\NormalTok{ men\_perc <{-}}\StringTok{ }\KeywordTok{round}\NormalTok{(}\DecValTok{100}\OperatorTok{*}\NormalTok{(nmen}\OperatorTok{/}\NormalTok{n), }\DataTypeTok{digits=}\DecValTok{2}\NormalTok{)} - -\NormalTok{ male\_percentages <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(male\_percentages, men\_perc)} - \KeywordTok{names}\NormalTok{(male\_percentages)[iter] <{-}}\StringTok{ }\NormalTok{state} -\NormalTok{ iter <{-}}\StringTok{ }\NormalTok{iter }\OperatorTok{+}\StringTok{ }\DecValTok{1} -\NormalTok{\}} - -\NormalTok{male\_percentages} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## California Massachusetts New Hampshire Washington -## 49.85 47.60 48.55 48.19 -\end{verbatim} - -\hypertarget{nested-loops}{% -\section{Nested Loops}\label{nested-loops}} - -What if I want to calculate the population percentage of a race group for all race groups in states of interest? -You could probably use tidyverse functions to do this, but let's try using loops! - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{states\_of\_interest <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Massachusetts"}\NormalTok{, }\StringTok{"New Hampshire"}\NormalTok{, }\StringTok{"Washington"}\NormalTok{)} -\ControlFlowTok{for}\NormalTok{ (state }\ControlFlowTok{in}\NormalTok{ states\_of\_interest) \{} - \ControlFlowTok{for}\NormalTok{ (race }\ControlFlowTok{in} \KeywordTok{unique}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race)) \{} -\NormalTok{ race\_state\_num <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(cen10[cen10}\OperatorTok{$}\NormalTok{race }\OperatorTok{==}\StringTok{ }\NormalTok{race }\OperatorTok{\&}\StringTok{ }\NormalTok{cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state, ])} -\NormalTok{ state\_pop <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state, ])} -\NormalTok{ race\_perc <{-}}\StringTok{ }\KeywordTok{round}\NormalTok{(}\DecValTok{100}\OperatorTok{*}\NormalTok{(race\_state\_num}\OperatorTok{/}\NormalTok{(state\_pop)), }\DataTypeTok{digits=}\DecValTok{2}\NormalTok{)} - \KeywordTok{print}\NormalTok{(}\KeywordTok{paste}\NormalTok{(}\StringTok{"Percentage of "}\NormalTok{, race , }\StringTok{"in"}\NormalTok{, state, }\StringTok{"is"}\NormalTok{, race\_perc))} -\NormalTok{ \}} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Percentage of White in California is 57.61" -## [1] "Percentage of Black/Negro in California is 6.72" -## [1] "Percentage of Other race, nec in California is 15.55" -## [1] "Percentage of American Indian or Alaska Native in California is 1.12" -## [1] "Percentage of Chinese in California is 3.75" -## [1] "Percentage of Other Asian or Pacific Islander in California is 9.54" -## [1] "Percentage of Two major races in California is 4.62" -## [1] "Percentage of Three or more major races in California is 0.37" -## [1] "Percentage of Japanese in California is 0.72" -## [1] "Percentage of White in Massachusetts is 79.6" -## [1] "Percentage of Black/Negro in Massachusetts is 5.87" -## [1] "Percentage of Other race, nec in Massachusetts is 4.02" -## [1] "Percentage of American Indian or Alaska Native in Massachusetts is 0.77" -## [1] "Percentage of Chinese in Massachusetts is 2.32" -## [1] "Percentage of Other Asian or Pacific Islander in Massachusetts is 4.33" -## [1] "Percentage of Two major races in Massachusetts is 2.78" -## [1] "Percentage of Three or more major races in Massachusetts is 0" -## [1] "Percentage of Japanese in Massachusetts is 0.31" -## [1] "Percentage of White in New Hampshire is 93.48" -## [1] "Percentage of Black/Negro in New Hampshire is 0.72" -## [1] "Percentage of Other race, nec in New Hampshire is 0.72" -## [1] "Percentage of American Indian or Alaska Native in New Hampshire is 0.72" -## [1] "Percentage of Chinese in New Hampshire is 0.72" -## [1] "Percentage of Other Asian or Pacific Islander in New Hampshire is 2.17" -## [1] "Percentage of Two major races in New Hampshire is 0.72" -## [1] "Percentage of Three or more major races in New Hampshire is 0" -## [1] "Percentage of Japanese in New Hampshire is 0.72" -## [1] "Percentage of White in Washington is 76.05" -## [1] "Percentage of Black/Negro in Washington is 2.9" -## [1] "Percentage of Other race, nec in Washington is 5.37" -## [1] "Percentage of American Indian or Alaska Native in Washington is 2.03" -## [1] "Percentage of Chinese in Washington is 1.31" -## [1] "Percentage of Other Asian or Pacific Islander in Washington is 6.68" -## [1] "Percentage of Two major races in Washington is 4.79" -## [1] "Percentage of Three or more major races in Washington is 0.29" -## [1] "Percentage of Japanese in Washington is 0.58" -\end{verbatim} - -\hypertarget{exercises-2}{% -\section*{Exercises}\label{exercises-2}} -\addcontentsline{toc}{section}{Exercises} - -\hypertarget{exercise-1-write-your-own-function}{% -\subsection*{Exercise 1: Write your own function}\label{exercise-1-write-your-own-function}} -\addcontentsline{toc}{subsection}{Exercise 1: Write your own function} - -Write your own function that makes some task of data analysis simpler. Ideally, it would be a function that helps you do either of the previous tasks in fewer lines of code. You can use the three lines of code that was provided in exercise 1 to wrap that into another function too! - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -\hypertarget{exercise-2-using-loops}{% -\subsection*{Exercise 2: Using Loops}\label{exercise-2-using-loops}} -\addcontentsline{toc}{subsection}{Exercise 2: Using Loops} - -Using a loop, create a crosstab of sex and race for each state in the set ``states\_of\_interest'' - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{states\_of\_interest <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Massachusetts"}\NormalTok{, }\StringTok{"New Hampshire"}\NormalTok{, }\StringTok{"Washington"}\NormalTok{)} -\CommentTok{\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -\hypertarget{exercise-3-storing-information-derived-within-loops-in-a-global-dataframe}{% -\subsection*{Exercise 3: Storing information derived within loops in a global dataframe}\label{exercise-3-storing-information-derived-within-loops-in-a-global-dataframe}} -\addcontentsline{toc}{subsection}{Exercise 3: Storing information derived within loops in a global dataframe} - -Recall the following nested loop - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{states\_of\_interest <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Massachusetts"}\NormalTok{, }\StringTok{"New Hampshire"}\NormalTok{, }\StringTok{"Washington"}\NormalTok{)} -\ControlFlowTok{for}\NormalTok{ (state }\ControlFlowTok{in}\NormalTok{ states\_of\_interest) \{} - \ControlFlowTok{for}\NormalTok{ (race }\ControlFlowTok{in} \KeywordTok{unique}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race)) \{} -\NormalTok{ race\_state\_num <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(cen10[cen10}\OperatorTok{$}\NormalTok{race }\OperatorTok{==}\StringTok{ }\NormalTok{race }\OperatorTok{\&}\StringTok{ }\NormalTok{cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state, ])} -\NormalTok{ state\_pop <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state, ])} -\NormalTok{ race\_perc <{-}}\StringTok{ }\KeywordTok{round}\NormalTok{(}\DecValTok{100}\OperatorTok{*}\NormalTok{(race\_state\_num}\OperatorTok{/}\NormalTok{(state\_pop)), }\DataTypeTok{digits=}\DecValTok{2}\NormalTok{)} - \KeywordTok{print}\NormalTok{(}\KeywordTok{paste}\NormalTok{(}\StringTok{"Percentage of "}\NormalTok{, race , }\StringTok{"in"}\NormalTok{, state, }\StringTok{"is"}\NormalTok{, race\_perc))} -\NormalTok{ \}} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Percentage of White in California is 57.61" -## [1] "Percentage of Black/Negro in California is 6.72" -## [1] "Percentage of Other race, nec in California is 15.55" -## [1] "Percentage of American Indian or Alaska Native in California is 1.12" -## [1] "Percentage of Chinese in California is 3.75" -## [1] "Percentage of Other Asian or Pacific Islander in California is 9.54" -## [1] "Percentage of Two major races in California is 4.62" -## [1] "Percentage of Three or more major races in California is 0.37" -## [1] "Percentage of Japanese in California is 0.72" -## [1] "Percentage of White in Massachusetts is 79.6" -## [1] "Percentage of Black/Negro in Massachusetts is 5.87" -## [1] "Percentage of Other race, nec in Massachusetts is 4.02" -## [1] "Percentage of American Indian or Alaska Native in Massachusetts is 0.77" -## [1] "Percentage of Chinese in Massachusetts is 2.32" -## [1] "Percentage of Other Asian or Pacific Islander in Massachusetts is 4.33" -## [1] "Percentage of Two major races in Massachusetts is 2.78" -## [1] "Percentage of Three or more major races in Massachusetts is 0" -## [1] "Percentage of Japanese in Massachusetts is 0.31" -## [1] "Percentage of White in New Hampshire is 93.48" -## [1] "Percentage of Black/Negro in New Hampshire is 0.72" -## [1] "Percentage of Other race, nec in New Hampshire is 0.72" -## [1] "Percentage of American Indian or Alaska Native in New Hampshire is 0.72" -## [1] "Percentage of Chinese in New Hampshire is 0.72" -## [1] "Percentage of Other Asian or Pacific Islander in New Hampshire is 2.17" -## [1] "Percentage of Two major races in New Hampshire is 0.72" -## [1] "Percentage of Three or more major races in New Hampshire is 0" -## [1] "Percentage of Japanese in New Hampshire is 0.72" -## [1] "Percentage of White in Washington is 76.05" -## [1] "Percentage of Black/Negro in Washington is 2.9" -## [1] "Percentage of Other race, nec in Washington is 5.37" -## [1] "Percentage of American Indian or Alaska Native in Washington is 2.03" -## [1] "Percentage of Chinese in Washington is 1.31" -## [1] "Percentage of Other Asian or Pacific Islander in Washington is 6.68" -## [1] "Percentage of Two major races in Washington is 4.79" -## [1] "Percentage of Three or more major races in Washington is 0.29" -## [1] "Percentage of Japanese in Washington is 0.58" -\end{verbatim} - -Instead of printing the percentage of each race in each state, create a dataframe, and store all that information in that dataframe. (Hint: look at how I stored information about male percentage in each state of interest in a vector.) - -\hypertarget{dataviz}{% -\chapter[Visualization]{\texorpdfstring{Visualization\footnote{Module originally written by Shiro Kuriwaki}}{Visualization}}\label{dataviz}} - -\hypertarget{motivation-the-law-of-the-census}{% -\subsection*{Motivation: The Law of the Census}\label{motivation-the-law-of-the-census}} -\addcontentsline{toc}{subsection}{Motivation: The Law of the Census} - -In this module, let's visualize some cross-sectional stats with an actual Census. Then, we'll do an example on time trends with Supreme Court ideal points. - -Why care about the Census? The Census is one of the fundamental acts of a government. See the Law Review article by \href{http://cardozolawreview.com/Joomla1.5/content/32-3/Persily.32-3.pdf}{Persily (2011)}, ``The Law of the Census.''\footnote{\href{http://cardozolawreview.com/Joomla1.5/content/32-3/Persily.32-3.pdf}{Persily, Nathaniel. 2011. ``The Law of the Census: How to Count, What to Count, Whom to Count, and Where to Count Them.''}. \emph{Cardozo Law Review} 32(3): 755--91.} The Census is government's primary tool for apportionment (allocating seats to districts), appropriations (allocating federal funding), and tracking demographic change. See for example \href{https://dash.harvard.edu/bitstream/handle/1/3153295/hoschschild_racialreorganization.pdf?sequence=2}{Hochschild and Powell (2008)} on how the categorizations of race in the Census during 1850-1930.\footnote{\href{https://dash.harvard.edu/bitstream/handle/1/3153295/hoschschild_racialreorganization.pdf?sequence=2}{Hochschild, Jennifer L., and Brenna Marea Powell. 2008. ``Racial Reorganization and the United States Census 1850--1930: Mulattoes, Half-Breeds, Mixed Parentage, Hindoos, and the Mexican Race.''}. \emph{Studies in American Political Development} 22(1): 59--96.} Notice also that both of these pieces are not inherently ``quantitative'' --- the Persily article is a Law Review and the Hochschild and Powell article is on American Historical Development --- but data analysis would be certainly relevant. - -Time series data is a common form of data in social science data, and there is growing methodological work on making causal inferences with time series.\footnote{\href{https://doi.org/10.1017/S0003055418000357}{Blackwell, Matthew, and Adam Glynn. 2018. ``How to Make Causal Inferences with Time-Series Cross-Sectional Data under Selection on Observables.''} \emph{American Political Science Review}} We will use the the ideological estimates of the Supreme court. - -\hypertarget{where-are-we-where-are-we-headed-3}{% -\subsection*{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed-3}} -\addcontentsline{toc}{subsection}{Where are we? Where are we headed?} - -Up till now, you should have covered: - -\begin{itemize} -\tightlist -\item - The R Visualization and Programming primers at \url{https://rstudio.cloud/primers/} -\item - Reading and handling data -\item - Matrices and Vectors - - \begin{itemize} - \tightlist - \item - What does \texttt{:} mean in R? What about \texttt{==}? \texttt{,}?, \texttt{!=} , \texttt{\&}, \texttt{\textbar{}}, \texttt{\%in\%} - \item - What does \texttt{\%\textgreater{}\%} do? - \end{itemize} -\end{itemize} - -Today we'll cover: - -\begin{itemize} -\tightlist -\item - Visualization -\item - A bit of data wrangling -\end{itemize} - -\hypertarget{check-your-understanding-1}{% -\subsection*{Check your understanding}\label{check-your-understanding-1}} -\addcontentsline{toc}{subsection}{Check your understanding} - -\begin{itemize} -\tightlist -\item - How do you make a barplot, in base-R and in ggplot? -\item - How do you add layers to a ggplot? -\item - How do you change the axes of a ggplot? -\item - How do you make a histogram? -\item - How do you make a graph that looks like this? -\end{itemize} - -\begin{figure} -\centering -\includegraphics{images/Martin-Quinn_Wikipedia.png} -\caption{By Randy Schutt - Own work, CC BY-SA 3.0, \href{https://commons.wikimedia.org/w/index.php?curid=29585342}{Wikimedia.}} -\end{figure} - -\hypertarget{read-data-2}{% -\section{Read data}\label{read-data-2}} - -First, the census. Read in a subset of the 2010 Census that we looked at earlier. This time, it is in Rds form. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 <{-}}\StringTok{ }\KeywordTok{readRDS}\NormalTok{(}\StringTok{"data/input/usc2010\_001percent.Rds"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -The data comes from IPUMS\footnote{\href{http://doi.org/10.18128/D010.V6.0}{Ruggles, Steven, Katie Genadek, Ronald Goeken, Josiah Grover, and Matthew Sobek. 2015. Integrated Public Use Microdata Series: Version 6.0 dataset}}, a great source to extract and analyze Census and Census-conducted survey (ACS, CPS) data. - -\hypertarget{counting}{% -\section{Counting}\label{counting}} - -How many people are in your sample? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{nrow}\NormalTok{(cen10)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 30871 -\end{verbatim} - -This and all subsequent tasks involve manipulating and summarizing data, sometimes called ``wrangling''. As per last time, there are both ``base-R'' and ``tidyverse'' approaches. - -We have already seen several functions from the tidyverse: - -\begin{itemize} -\tightlist -\item - \texttt{select} selects columns -\item - \texttt{filter} selects rows based on a logical (boolean) statement -\item - \texttt{slice} selects rows based on the row number -\item - \texttt{arrange} reordered the rows in descending order. -\end{itemize} - -In this visualization section, we'll make use of the pair of functions \texttt{group\_by()} and \texttt{summarize()}. - -\hypertarget{tabulating}{% -\section{Tabulating}\label{tabulating}} - -Summarizing data is the key part of communication; good data viz gets the point across.\footnote{\href{http://www.princeton.edu/~jkastell/Tables2Graphs/graphs.pdf}{Kastellec, Jonathan P., and Eduardo L. Leoni. 2007. ``Using Graphs Instead of Tables in Political Science.''}. \emph{Perspectives on Politics} 5 (4): 755--71.} Summaries of data come in two forms: tables and figures. - -Here are two ways to count by group, or to tabulate. - -In base-R Use the \texttt{table} function, that provides how many rows exist for an unique value of the vector (remember \texttt{unique} from yesterday?) - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{table}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## -## American Indian or Alaska Native Black/Negro -## 295 4013 -## Chinese Japanese -## 354 77 -## Other Asian or Pacific Islander Other race, nec -## 1129 1839 -## Three or more major races Two major races -## 88 869 -## White -## 22207 -\end{verbatim} - -With tidyverse, a quick convenience function is \texttt{count}, with the variable to count on included. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{count}\NormalTok{(cen10, race)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 9 x 2 -## race n -## -## 1 American Indian or Alaska Native 295 -## 2 Black/Negro 4013 -## 3 Chinese 354 -## 4 Japanese 77 -## 5 Other Asian or Pacific Islander 1129 -## 6 Other race, nec 1839 -## 7 Three or more major races 88 -## 8 Two major races 869 -## 9 White 22207 -\end{verbatim} - -We can check out the arguments of \texttt{count} and see that there is a \texttt{sort} option. What does this do? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{count}\NormalTok{(cen10, race, }\DataTypeTok{sort =} \OtherTok{TRUE}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 9 x 2 -## race n -## -## 1 White 22207 -## 2 Black/Negro 4013 -## 3 Other race, nec 1839 -## 4 Other Asian or Pacific Islander 1129 -## 5 Two major races 869 -## 6 Chinese 354 -## 7 American Indian or Alaska Native 295 -## 8 Three or more major races 88 -## 9 Japanese 77 -\end{verbatim} - -\texttt{count} is a kind of shorthand for \texttt{group\_by()} and \texttt{summarize}. This code would have done the same. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ } -\StringTok{ }\KeywordTok{group\_by}\NormalTok{(race) }\OperatorTok{\%>\%}\StringTok{ } -\StringTok{ }\KeywordTok{summarize}\NormalTok{(}\DataTypeTok{n =} \KeywordTok{n}\NormalTok{())} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## `summarise()` ungrouping output (override with `.groups` argument) -\end{verbatim} - -\begin{verbatim} -## # A tibble: 9 x 2 -## race n -## -## 1 American Indian or Alaska Native 295 -## 2 Black/Negro 4013 -## 3 Chinese 354 -## 4 Japanese 77 -## 5 Other Asian or Pacific Islander 1129 -## 6 Other race, nec 1839 -## 7 Three or more major races 88 -## 8 Two major races 869 -## 9 White 22207 -\end{verbatim} - -If you are new to tidyverse, what would you \emph{think} each row did? Reading the function help page, verify if your intuition was correct. - -where \texttt{n()} is a function that counts rows. - -\hypertarget{base-r-graphics-and-ggplot}{% -\section{base R graphics and ggplot}\label{base-r-graphics-and-ggplot}} - -Two prevalent ways of making graphing are referred to as ``base-R'' and ``ggplot''. - -\hypertarget{base-r}{% -\subsection{base R}\label{base-r}} - -``Base-R'' graphics are graphics that are made with R's default graphics commands. First, let's assign our tabulation to an object, -then put it in the \texttt{barplot()} function. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{barplot}\NormalTok{(}\KeywordTok{table}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race))} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-229-1.pdf} - -\hypertarget{ggplot}{% -\subsection{ggplot}\label{ggplot}} - -A popular alternative a \texttt{ggplot} graphics, that you were introduced to in the tutorial. \texttt{gg} stands for grammar of graphics by Hadley Wickham, and it has a new semantics of explaining graphics in R. Again, first let's set up the data. - -Although the tutorial covered making scatter plots as the first cut, often data requires summaries before they made into graphs. - -For this example, let's group and count first like we just did. But assign it to a new object. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{grp\_race <{-}}\StringTok{ }\KeywordTok{count}\NormalTok{(cen10, race)} -\end{Highlighting} -\end{Shaded} - -We will now plot this grouped set of numbers. Recall that the \texttt{ggplot()} function takes two main arguments, \texttt{data} and \texttt{aes}. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - First enter a single dataframe from which you will draw a plot. -\item - Then enter the \texttt{aes}, or aesthetics. This defines which variable in the data the plotting functions should take for pre-set dimensions in graphics. The dimensions \texttt{x} and \texttt{y} are the most important. We will assign \texttt{race} and \texttt{count} to them, respectively, -\item - After you close \texttt{ggplot()} .. add \textbf{layers} by the plus sign. A \texttt{geom} is a layer of graphical representation, for example \texttt{geom\_histogram} renders a histogram, \texttt{geom\_point} renders a scatter plot. For a barplot, we can use \texttt{geom\_col()} -\end{enumerate} - -What is the right geometry layer to make a barplot? Turns out: - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ggplot}\NormalTok{(}\DataTypeTok{data =}\NormalTok{ grp\_race, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ race, }\DataTypeTok{y =}\NormalTok{ n)) }\OperatorTok{+}\StringTok{ }\KeywordTok{geom\_col}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-231-1.pdf} - -\hypertarget{improving-your-graphics}{% -\section{Improving your graphics}\label{improving-your-graphics}} - -Adjusting your graphics to make the point clear is an important skill. Here is a base-R example of showing the same numbers but with a different design, in a way that aims to maximize the ``data-to-ink ratio''. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{par}\NormalTok{(}\DataTypeTok{oma =} \KeywordTok{c}\NormalTok{(}\DecValTok{1}\NormalTok{, }\DecValTok{11}\NormalTok{, }\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{))} -\KeywordTok{barplot}\NormalTok{(}\KeywordTok{sort}\NormalTok{(}\KeywordTok{table}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race)), }\CommentTok{\# sort numbers} - \DataTypeTok{horiz =} \OtherTok{TRUE}\NormalTok{, }\CommentTok{\# flip} - \DataTypeTok{border =} \OtherTok{NA}\NormalTok{, }\CommentTok{\# border is extraneous} - \DataTypeTok{xlab =} \StringTok{"Number in Race Category"}\NormalTok{, } - \DataTypeTok{bty =} \StringTok{"n"}\NormalTok{, }\CommentTok{\# no box} - \DataTypeTok{las =} \DecValTok{1}\NormalTok{) }\CommentTok{\# alignment of axis labels is horizontal} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-232-1.pdf} - -Notice that we applied the \texttt{sort()} function to order the bars in terms of their counts. The default ordering of a categorical variable / factor is alphabetical. Alphabetical ordering is uninformative and almost never the way you should order variables. - -In ggplot you might do this by: - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(forcats)} - -\NormalTok{grp\_race\_ordered <{-}}\StringTok{ }\KeywordTok{arrange}\NormalTok{(grp\_race, n) }\OperatorTok{\%>\%}\StringTok{ } -\StringTok{ }\KeywordTok{mutate}\NormalTok{(}\DataTypeTok{race =} \KeywordTok{as\_factor}\NormalTok{(race))} - -\KeywordTok{ggplot}\NormalTok{(}\DataTypeTok{data =}\NormalTok{ grp\_race\_ordered, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ race, }\DataTypeTok{y =}\NormalTok{ n)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_col}\NormalTok{() }\OperatorTok{+} -\StringTok{ }\KeywordTok{coord\_flip}\NormalTok{() }\OperatorTok{+} -\StringTok{ }\KeywordTok{labs}\NormalTok{(}\DataTypeTok{y =} \StringTok{"Number in Race Category"}\NormalTok{,} - \DataTypeTok{x =} \StringTok{""}\NormalTok{,} - \DataTypeTok{caption =} \StringTok{"Source: 2010 U.S. Census sample"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-233-1.pdf} - -The data ink ratio was popularized by Ed Tufte (originally a political economy scholar who has recently become well known for his data visualization work). See Tufte (2001), \emph{The Visual Display of Quantitative Information} and his website \url{https://www.edwardtufte.com/tufte/}. For a R and ggplot focused example using social science examples, check out Healy (2018), \emph{Data Visualization: A Practical Introduction} with a draft at \url{https://socviz.co/}\footnote{Healy, Kieran. forthcoming. \emph{Data Visualization: A Practical Introduction}. Princeton University Press}. There are a growing number of excellent books on data visualization. - -\hypertarget{cross-tabs}{% -\section{Cross-tabs}\label{cross-tabs}} - -Visualizations and Tables each have their strengths. A rule of thumb is that more than a dozen numbers on a table is too much to digest, but less than a dozen is too few for a figure to be worth it. Let's look at a table first. - -A cross-tab is counting with two types of variables, and is a simple and powerful tool to show the relationship between multiple variables. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{xtab\_race\_state <{-}}\StringTok{ }\KeywordTok{table}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{state, cen10}\OperatorTok{$}\NormalTok{race)} -\NormalTok{xtab\_race\_state} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## -## American Indian or Alaska Native Black/Negro Chinese -## Alabama 2 128 1 -## Alaska 11 6 0 -## Arizona 28 23 1 -## Arkansas 1 45 0 -## California 42 253 141 -## Colorado 7 26 3 -## Connecticut 1 39 7 -## Delaware 3 28 1 -## District of Columbia 0 35 0 -## Florida 9 304 4 -## Georgia 2 304 5 -## Hawaii 0 0 2 -## Idaho 2 0 0 -## Illinois 5 194 6 -## Indiana 2 66 3 -## Iowa 0 9 1 -## Kansas 2 24 2 -## Kentucky 2 35 2 -## Louisiana 3 161 1 -## Maine 0 4 1 -## Maryland 2 177 4 -## Massachusetts 5 38 15 -## Michigan 5 147 8 -## Minnesota 6 25 5 -## Mississippi 1 116 0 -## Missouri 4 74 2 -## Montana 8 0 0 -## Nebraska 2 11 0 -## Nevada 6 15 6 -## New Hampshire 1 1 1 -## New Jersey 0 130 19 -## New Mexico 21 3 1 -## New York 13 305 55 -## North Carolina 12 220 4 -## North Dakota 4 1 0 -## Ohio 1 122 5 -## Oklahoma 21 20 0 -## Oregon 5 5 4 -## Pennsylvania 2 156 10 -## Rhode Island 2 3 0 -## South Carolina 2 120 1 -## South Dakota 7 1 0 -## Tennessee 0 97 0 -## Texas 14 316 15 -## Utah 8 0 1 -## Vermont 0 2 0 -## Virginia 0 171 8 -## Washington 14 20 9 -## West Virginia 0 5 0 -## Wisconsin 6 27 0 -## Wyoming 1 1 0 -## -## Japanese Other Asian or Pacific Islander Other race, nec -## Alabama 0 3 8 -## Alaska 0 5 2 -## Arizona 0 12 74 -## Arkansas 0 1 11 -## California 27 359 585 -## Colorado 0 10 28 -## Connecticut 0 16 20 -## Delaware 0 4 5 -## District of Columbia 0 1 1 -## Florida 1 24 72 -## Georgia 0 35 35 -## Hawaii 16 35 0 -## Idaho 1 0 8 -## Illinois 3 53 75 -## Indiana 0 8 20 -## Iowa 0 4 10 -## Kansas 0 8 6 -## Kentucky 0 4 5 -## Louisiana 0 5 7 -## Maine 0 1 0 -## Maryland 1 12 28 -## Massachusetts 2 28 26 -## Michigan 1 23 8 -## Minnesota 1 28 13 -## Mississippi 0 3 2 -## Missouri 0 9 6 -## Montana 0 0 1 -## Nebraska 0 5 6 -## Nevada 2 15 41 -## New Hampshire 1 3 1 -## New Jersey 2 65 69 -## New Mexico 1 1 23 -## New York 3 68 154 -## North Carolina 1 12 40 -## North Dakota 0 0 0 -## Ohio 2 17 7 -## Oklahoma 0 5 15 -## Oregon 0 11 21 -## Pennsylvania 1 28 30 -## Rhode Island 0 4 6 -## South Carolina 0 4 6 -## South Dakota 1 1 2 -## Tennessee 0 13 13 -## Texas 2 92 253 -## Utah 1 6 14 -## Vermont 0 0 1 -## Virginia 2 29 29 -## Washington 4 46 37 -## West Virginia 0 0 0 -## Wisconsin 1 11 13 -## Wyoming 0 2 2 -## -## Three or more major races Two major races White -## Alabama 1 8 344 -## Alaska 0 15 37 -## Arizona 2 11 485 -## Arkansas 1 2 247 -## California 14 174 2168 -## Colorado 1 22 401 -## Connecticut 1 7 284 -## Delaware 1 1 66 -## District of Columbia 0 2 21 -## Florida 2 42 1435 -## Georgia 1 21 587 -## Hawaii 14 27 39 -## Idaho 1 6 129 -## Illinois 2 35 856 -## Indiana 1 6 514 -## Iowa 0 8 287 -## Kansas 0 8 237 -## Kentucky 1 9 357 -## Louisiana 0 6 273 -## Maine 0 1 117 -## Maryland 1 13 302 -## Massachusetts 0 18 515 -## Michigan 2 23 792 -## Minnesota 1 10 483 -## Mississippi 2 1 167 -## Missouri 2 14 516 -## Montana 0 0 88 -## Nebraska 0 5 155 -## Nevada 1 16 171 -## New Hampshire 0 1 129 -## New Jersey 3 25 589 -## New Mexico 1 6 146 -## New York 8 51 1220 -## North Carolina 2 20 648 -## North Dakota 0 1 46 -## Ohio 3 20 931 -## Oklahoma 3 24 266 -## Oregon 4 9 279 -## Pennsylvania 1 27 1045 -## Rhode Island 1 4 74 -## South Carolina 1 6 325 -## South Dakota 0 2 72 -## Tennessee 0 9 474 -## Texas 2 71 1792 -## Utah 0 8 255 -## Vermont 0 4 59 -## Virginia 4 24 548 -## Washington 2 33 524 -## West Virginia 0 3 168 -## Wisconsin 1 8 497 -## Wyoming 0 2 47 -\end{verbatim} - -Another function to make a cross-tab is the \texttt{xtabs} command, which uses formula notation. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{xtabs}\NormalTok{(}\OperatorTok{\textasciitilde{}}\StringTok{ }\NormalTok{state }\OperatorTok{+}\StringTok{ }\NormalTok{race, cen10)} -\end{Highlighting} -\end{Shaded} - -What if we care about proportions within states, rather than counts? Say we'd like to compare the racial composition of a small state (like Delaware) and a large state (like California). In fact, most tasks of inference is about the unobserved population, not the observed data --- and proportions are estimates of a quantity in the population. - -One way to transform a table of counts to a table of proportions is the function \texttt{prop.table}. Be careful what you want to take proportions of -- this is set by the \texttt{margin} argument. In R, the first margin (\texttt{margin\ =\ 1}) is \emph{rows} and the second (\texttt{margin\ =\ 2}) is \emph{columns}. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{ptab\_race\_state <{-}}\StringTok{ }\KeywordTok{prop.table}\NormalTok{(xtab\_race\_state, }\DataTypeTok{margin =} \DecValTok{2}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -Check out each of these table objects in your console and familiarize yourself with the difference. - -\hypertarget{composition-plots}{% -\section{Composition Plots}\label{composition-plots}} - -How would you make the same figure with \texttt{ggplot()}? First, we want a count for each state \(\times\) race combination. So group by those two factors and count how many observations are in each two-way categorization. \texttt{group\_by()} can take any number of variables, separated by commas. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{grp\_race\_state <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ } -\StringTok{ }\KeywordTok{count}\NormalTok{(race, state)} -\end{Highlighting} -\end{Shaded} - -Can you tell from the code what \texttt{grp\_race\_state} will look like? - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# run on your own} -\NormalTok{grp\_race\_state} -\end{Highlighting} -\end{Shaded} - -Now, we want to tell \texttt{ggplot2} something like the following: I want bars by state, where heights indicate racial groups. Each bar should be colored by the race. With some googling, you will get something like this: - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ggplot}\NormalTok{(}\DataTypeTok{data =}\NormalTok{ grp\_race\_state, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ state, }\DataTypeTok{y =}\NormalTok{ n, }\DataTypeTok{fill =}\NormalTok{ race)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_col}\NormalTok{(}\DataTypeTok{position =} \StringTok{"fill"}\NormalTok{) }\OperatorTok{+}\StringTok{ }\CommentTok{\# the position is determined by the fill ae} -\StringTok{ }\KeywordTok{scale\_fill\_brewer}\NormalTok{(}\DataTypeTok{name =} \StringTok{"Census Race"}\NormalTok{, }\DataTypeTok{palette =} \StringTok{"OrRd"}\NormalTok{, }\DataTypeTok{direction =} \DecValTok{{-}1}\NormalTok{) }\OperatorTok{+}\StringTok{ }\CommentTok{\# choose palette} -\StringTok{ }\KeywordTok{coord\_flip}\NormalTok{() }\OperatorTok{+}\StringTok{ }\CommentTok{\# flip axes} -\StringTok{ }\KeywordTok{scale\_y\_continuous}\NormalTok{(}\DataTypeTok{labels =}\NormalTok{ percent) }\OperatorTok{+}\StringTok{ }\CommentTok{\# label numbers as percentage} -\StringTok{ }\KeywordTok{labs}\NormalTok{(}\DataTypeTok{y =} \StringTok{"Proportion of Racial Group within State"}\NormalTok{,} - \DataTypeTok{x =} \StringTok{""}\NormalTok{,} - \DataTypeTok{source =} \StringTok{"Source: 2010 Census sample"}\NormalTok{) }\OperatorTok{+} -\StringTok{ }\KeywordTok{theme\_minimal}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-239-1.pdf} - -\hypertarget{line-graphs}{% -\section{Line graphs}\label{line-graphs}} - -Line graphs are useful for plotting time trends. - -The Census does not track individuals over time. So let's take up another example: The U.S. Supreme Court. Take the dataset \texttt{justices\_court-median.csv}. - -This data is adapted from the estimates of Martin and Quinn on their website \url{http://mqscores.lsa.umich.edu/}.\footnote{This exercise inspired from Princeton's R Camp Assignment.} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{justice <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/justices\_court{-}median.csv"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -What does the data look like? How do you think it is organized? What does each row represent? - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{justice} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 746 x 7 -## term justice_id justice idealpt idealpt_sd median_idealpt median_justice -## -## 1 1937 67 McReynolds 3.44 0.54 -0.568 Brandeis -## 2 1937 68 Brandeis -0.612 0.271 -0.568 Brandeis -## 3 1937 71 Sutherland 1.59 0.549 -0.568 Brandeis -## 4 1937 72 Butler 2.06 0.426 -0.568 Brandeis -## 5 1937 74 Stone -0.774 0.259 -0.568 Brandeis -## 6 1937 75 Hughes2 -0.368 0.232 -0.568 Brandeis -## 7 1937 76 O. Roberts 0.008 0.228 -0.568 Brandeis -## 8 1937 77 Cardozo -1.59 0.634 -0.568 Brandeis -## 9 1937 78 Black -2.90 0.334 -0.568 Brandeis -## 10 1937 79 Reed -1.06 0.342 -0.568 Brandeis -## # ... with 736 more rows -\end{verbatim} - -As you might have guessed, these data can be shown in a time trend from the range of the \texttt{term} variable. As there are only nine justices at any given time and justices have life tenure, there times on the court are staggered. With a common measure of ``preference'', we can plot time trends of these justices ideal points on the same y-axis scale. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ggplot}\NormalTok{(justice, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ term, }\DataTypeTok{y =}\NormalTok{ idealpt)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_line}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-242-1.pdf} -Why does the above graph not look like the the put in the beginning? Fix it by adding just one aesthetic to the graph. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# enter a correction that draws separate lines by group.} -\end{Highlighting} -\end{Shaded} - -If you got the right aesthetic, this seems to ``work'' off the shelf. But take a moment to see why the code was written as it is and how that maps on to the graphics. What is the \texttt{group} aesthetic doing for you? - -Now, this graphic already indicates a lot, but let's improve the graphics so people can actually read it. This is left for a Exercise. - -As social scientists, we should also not forget to ask ourselves whether these numerical measures are fit for what we care about, or actually succeeds in measuring what we'd like to measure. The estimation of these ``ideal points'' is a subfield of political methodology beyond this prefresher. For more reading, skim through the original paper by Martin and Quinn (2002).\footnote{\href{http://mqscores.lsa.umich.edu/media/pa02.pdf}{Martin, Andrew D. and Kevin M. Quinn. 2002. ``Dynamic Ideal Point Estimation via Markov Chain Monte Carlo for the U.S. Supreme Court, 1953-1999''}. \emph{Political Analysis.} 10(2): 134-153.} Also for a methodological discussion on the difficulty of measuring time series of preferences, check out Bailey (2013).\footnote{\href{https://michaelbailey.georgetown.domains/wp-content/uploads/2018/05/JOP_proofs_June2013.pdf}{Bailey, Michael A. 2013. ``Is Today's Court the Most Conservative in Sixty Years? Challenges and Opportunities in Measuring Judicial Preferences.''}. \emph{Journal of Politics} 75(3): 821-834} - -\hypertarget{exercises-3}{% -\section*{Exercises}\label{exercises-3}} -\addcontentsline{toc}{section}{Exercises} - -In the time remaining, try the following exercises. Order doesn't matter. - -\hypertarget{rural-states}{% -\subsection*{1: Rural states}\label{rural-states}} -\addcontentsline{toc}{subsection}{1: Rural states} - -Make a well-labelled figure that plots the proportion of the state's population (as per the census) that is 65 years or older. Each state should be visualized as a point, rather than a bar, and there should be 51 points, ordered by their value. All labels should be readable. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -\hypertarget{the-swing-justice}{% -\subsection*{2: The swing justice}\label{the-swing-justice}} -\addcontentsline{toc}{subsection}{2: The swing justice} - -Using the \texttt{justices\_court-median.csv} dataset and building off of the plot that was given, make an improved plot by implementing as many of the following changes (which hopefully improves the graph): - -\begin{itemize} -\tightlist -\item - Label axes -\item - Use a black-white background. -\item - Change the breaks of the x-axis to print numbers for every decade, not just every two decades. -\item - Plots each line in translucent gray, so the overlapping lines can be visualized clearly. (Hint: in ggplot the \texttt{alpha} argument controls the degree of transparency) -\item - Limit the scale of the y-axis to {[}-5, 5{]} so that the outlier justice in the 60s is trimmed and the rest of the data can be seen more easily (also, who is that justice?) -\item - Plot the ideal point of the justice who holds the ``median'' ideal point in a given term. To distinguish this with the others, plot this line separately in a very light red \emph{below} the individual justice's lines. -\item - Highlight the trend-line of only the nine justices who are \emph{currently} sitting on SCOTUS. Make sure this is clearer than the other past justices. -\item - Add the current nine justice's names to the right of the endpoint of the 2016 figure, alongside their ideal point. -\item - Make sure the text labels do not overlap with each other for readability using the \texttt{ggrepel} package. -\item - Extend the x-axis label to about 2020 so the text labels of justices are to the right of the trend-lines. -\item - Add a caption to your text describing the data briefly, as well as any features relevant for the reader (such as the median line and the trimming of the y-axis) -\end{itemize} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -\hypertarget{dont-sort-by-the-alphabet}{% -\subsection*{3: Don't sort by the alphabet}\label{dont-sort-by-the-alphabet}} -\addcontentsline{toc}{subsection}{3: Don't sort by the alphabet} - -The Figure we made that shows racial composition by state has one notable shortcoming: it orders the states alphabetically, which is not particularly useful if you want see an overall pattern, without having particular states in mind. - -Find a way to modify the figures so that the states are ordered by the \emph{proportion} of White residents in the sample. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Enter yourself} -\end{Highlighting} -\end{Shaded} - -\hypertarget{what-to-show-and-how-to-show-it}{% -\subsection*{4 What to show and how to show it}\label{what-to-show-and-how-to-show-it}} -\addcontentsline{toc}{subsection}{4 What to show and how to show it} - -As a student of politics our goal is not necessarily to make pretty pictures, but rather make pictures that tell us something about politics, government, or society. If you could augment either the census dataset or the justices dataset in some way, what would be an substantively significant thing to show as a graphic? - -\hypertarget{dempeace}{% -\chapter[Joins and Merges, Wide and Long]{\texorpdfstring{Joins and Merges, Wide and Long\footnote{Module originally written by Shiro Kuriwaki, Connor Jerzak, and Yon Soo Park}}{Joins and Merges, Wide and Long}}\label{dempeace}} - -\hypertarget{motivation-1}{% -\subsection*{Motivation}\label{motivation-1}} -\addcontentsline{toc}{subsection}{Motivation} - -The ``Democratic Peace'' is one of the most widely discussed propositions in political science, covering the fields of International Relations and Comparative Politics, with insights to domestic politics of democracies (e.g.~American Politics). The one-sentence idea is that democracies do not fight with each other. There have been much theoretical debate -- for example in earlier work, \href{https://blackboard.angelo.edu/bbcswebdav/institution/LFA/CSS/Course\%20Material/SEC6302/Readings/Lesson_3/Oneal-Russett.pdf}{Oneal and Russet (1999)} argue that the democratic peace is not due to the hegemony of strong democracies like the U.S. and attempt to distinguish between realist and what they call Kantian propositions (e.g.~democratic governance, international organizations)\footnote{\href{https://blackboard.angelo.edu/bbcswebdav/institution/LFA/CSS/Course\%20Material/SEC6302/Readings/Lesson_3/Oneal-Russett.pdf}{The Kantian Peace: The Pacific Benefits of Democracy, Interdependence, and International Organizations, 1885-1992. \emph{World Politics} 52(1):1-37}}. - -An empirical demonstration of the democratic peace is also a good example of a \textbf{Time Series Cross Sectional} (or panel) dataset, where the same units (in this case countries) are observed repeatedly for multiple time periods. Experience in assembling and analyzing a TSCS dataset will prepare you for any future research in this area. - -\hypertarget{where-are-we-where-are-we-headed-4}{% -\section*{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed-4}} -\addcontentsline{toc}{section}{Where are we? Where are we headed?} - -Up till now, you should have covered: - -\begin{itemize} -\tightlist -\item - R basic programming -\item - Counting. -\item - Visualization. -\item - Objects and Classes. -\item - Matrix algebra in R -\item - Functions. -\end{itemize} - -Today you will work on your own, but feel free to ask a fellow classmate nearby or the instructor. The objective for this session is to get more experience using R, but in the process (a) test a prominent theory in the political science literature and (b) explore related ideas of interest to you. - -\hypertarget{setting-up}{% -\section{Setting up}\label{setting-up}} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(dplyr)} -\KeywordTok{library}\NormalTok{(tidyr)} -\KeywordTok{library}\NormalTok{(readr)} -\KeywordTok{library}\NormalTok{(ggplot2)} -\end{Highlighting} -\end{Shaded} - -\hypertarget{create-a-project-directory}{% -\section{Create a project directory}\label{create-a-project-directory}} - -First start a directory for this project. This can be done manually or through RStudio's Project feature(\texttt{File\ \textgreater{}\ New\ Project...}) - -Directories is the computer science / programming name for folders. While advice about how to structure your working directories might strike you as petty, we believe that starting from some well-tested guides will go a long way in improving the quality and efficiency of your work. - -Chapter 4 of Gentzkow and Shapiro's memo, \href{https://web.stanford.edu/~gentzkow/research/CodeAndData.pdf}{Code and Data for the Social Scientist}{]} provides a good template. - -\hypertarget{data-sources}{% -\section{Data Sources}\label{data-sources}} - -Most projects you do will start with downloading data from elsewhere. For this task, you'll probably want to track down and download the following: - -\begin{itemize} -\tightlist -\item - \textbf{Correlates of war dataset (COW):} Find and download the Militarized Interstate Disputes (MIDs) data from the Correlates of War website: \url{http://www.correlatesofwar.org/data-sets}. Or a dyad-version on dataverse: \url{https://dataverse.harvard.edu/dataset.xhtml?persistentId=hdl:1902.1/11489} -\item - \textbf{PRIO Data on Armed Conflict:} Find and download the Uppsala Conflict Data Program (UCDP) and PRIO dyad-year data on armed conflict(\url{https://www.prio.org}) or this link to to the flat csv file (\url{http://ucdp.uu.se/downloads/dyadic/ucdp-dyadic-171.csv}). -\item - \textbf{Polity:} The Polity data can be downloaded from their website (\url{http://www.systemicpeace.org/inscrdata.html}). Look for the newest version of the time series that has the widest coverage. -\end{itemize} - -\hypertarget{example-with-2-datasets}{% -\section{Example with 2 Datasets}\label{example-with-2-datasets}} - -Let's read in a sample dataset. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{polity <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/sample\_polity.csv"}\NormalTok{)} -\NormalTok{mid <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/sample\_mid.csv"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -What does \texttt{polity} look like? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{unique}\NormalTok{(polity}\OperatorTok{$}\NormalTok{country)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "France" "Prussia" "Germany" "United States" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ggplot}\NormalTok{(polity, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ year, }\DataTypeTok{y =}\NormalTok{ polity2)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{facet\_wrap}\NormalTok{(}\OperatorTok{\textasciitilde{}}\StringTok{ }\NormalTok{country) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_line}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-249-1.pdf} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{head}\NormalTok{(polity)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 6 x 5 -## scode ccode country year polity2 -## -## 1 FRN 220 France 1800 -8 -## 2 FRN 220 France 1801 -8 -## 3 FRN 220 France 1802 -8 -## 4 FRN 220 France 1803 -8 -## 5 FRN 220 France 1804 -8 -## 6 FRN 220 France 1805 -8 -\end{verbatim} - -MID is a dataset that captures a \texttt{dispute} for a given country and year. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{mid} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 6,132 x 5 -## ccode polity_code dispute StYear EndYear -## -## 1 200 UKG 1 1902 1903 -## 2 2 USA 1 1902 1903 -## 3 345 YGS 1 1913 1913 -## 4 300 1 1913 1913 -## 5 339 ALB 1 1946 1946 -## 6 200 UKG 1 1946 1946 -## 7 200 UKG 1 1951 1952 -## 8 651 EGY 1 1951 1952 -## 9 630 IRN 1 1856 1857 -## 10 200 UKG 1 1856 1857 -## # ... with 6,122 more rows -\end{verbatim} - -\hypertarget{loops}{% -\section{Loops}\label{loops}} - -Notice that in the \texttt{mid} data, we have a start of a dispute vs.~an end of a dispute.In order to combine this into the \texttt{polity} data, we want a way to give each of the interval years a row. - -There are many ways to do this, but one is a loop. We go through one row at a time, and then for each we make a new dataset. that has \texttt{year} as a sequence of each year. A lengthy loop like this is typically slow, and you'd want to recast the task so you can do things with functions. But, a loop is a good place to start. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{mid\_year\_by\_year <{-}}\StringTok{ }\KeywordTok{data\_frame}\NormalTok{(}\DataTypeTok{ccode =} \KeywordTok{numeric}\NormalTok{(),} - \DataTypeTok{year =} \KeywordTok{numeric}\NormalTok{(),} - \DataTypeTok{dispute =} \KeywordTok{numeric}\NormalTok{())} - -\ControlFlowTok{for}\NormalTok{(i }\ControlFlowTok{in} \DecValTok{1}\OperatorTok{:}\KeywordTok{nrow}\NormalTok{(mid)) \{} -\NormalTok{ x <{-}}\StringTok{ }\KeywordTok{data\_frame}\NormalTok{(}\DataTypeTok{ccode =}\NormalTok{ mid}\OperatorTok{$}\NormalTok{ccode[i], }\CommentTok{\#\# row i\textquotesingle{}s country} - \DataTypeTok{year =}\NormalTok{ mid}\OperatorTok{$}\NormalTok{StYear[i]}\OperatorTok{:}\NormalTok{mid}\OperatorTok{$}\NormalTok{EndYear[i], }\CommentTok{\#\# sequence of years for dispute in row i} - \DataTypeTok{dispute =} \DecValTok{1}\NormalTok{) } -\NormalTok{ mid\_year\_by\_year <{-}}\StringTok{ }\KeywordTok{rbind}\NormalTok{(mid\_year\_by\_year, x)} -\NormalTok{\}} - -\KeywordTok{head}\NormalTok{(mid\_year\_by\_year)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 6 x 3 -## ccode year dispute -## -## 1 200 1902 1 -## 2 200 1903 1 -## 3 2 1902 1 -## 4 2 1903 1 -## 5 345 1913 1 -## 6 300 1913 1 -\end{verbatim} - -\hypertarget{merging}{% -\section{Merging}\label{merging}} - -We want to combine these two datasets by merging. Base-R has a function called \texttt{merge}. \texttt{dplyr} has several types of \texttt{joins} (the same thing). Those names are based on SQL syntax. - -\includegraphics{images/dplyr-joins.png} - -Here we can do a \texttt{left\_join} matching rows from \texttt{mid} to \texttt{polity}. We want to keep the rows in \texttt{polity} that do not match in \texttt{mid}, and label them as non-disputes. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{p\_m <{-}}\StringTok{ }\KeywordTok{left\_join}\NormalTok{(polity,} - \KeywordTok{distinct}\NormalTok{(mid\_year\_by\_year),} - \DataTypeTok{by =} \KeywordTok{c}\NormalTok{(}\StringTok{"ccode"}\NormalTok{, }\StringTok{"year"}\NormalTok{))} - -\KeywordTok{head}\NormalTok{(p\_m)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 6 x 6 -## scode ccode country year polity2 dispute -## -## 1 FRN 220 France 1800 -8 NA -## 2 FRN 220 France 1801 -8 NA -## 3 FRN 220 France 1802 -8 NA -## 4 FRN 220 France 1803 -8 NA -## 5 FRN 220 France 1804 -8 NA -## 6 FRN 220 France 1805 -8 NA -\end{verbatim} - -Replace \texttt{dispute} = \texttt{NA} rows with a zero. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{p\_m}\OperatorTok{$}\NormalTok{dispute[}\KeywordTok{is.na}\NormalTok{(p\_m}\OperatorTok{$}\NormalTok{dispute)] <{-}}\StringTok{ }\DecValTok{0} -\end{Highlighting} -\end{Shaded} - -Reshape the dataset long to wide - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{p\_m\_wide <{-}}\StringTok{ }\KeywordTok{pivot\_wider}\NormalTok{(p\_m, } - \DataTypeTok{id\_cols =} \KeywordTok{c}\NormalTok{(scode, ccode, country),} - \DataTypeTok{names\_from =}\NormalTok{ year,} - \DataTypeTok{values\_from =}\NormalTok{ polity2)} - -\KeywordTok{select}\NormalTok{(p\_m\_wide, }\DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## # A tibble: 4 x 10 -## scode ccode country `1800` `1801` `1802` `1803` `1804` `1805` `1806` -## -## 1 FRN 220 France -8 -8 -8 -8 -8 -8 -8 -## 2 GMY 255 Prussia -10 -10 -10 -10 -10 -10 NA -## 3 GMY 255 Germany NA NA NA NA NA NA NA -## 4 USA 2 United States 4 4 4 4 4 4 4 -\end{verbatim} - -\hypertarget{main-project}{% -\section{Main Project}\label{main-project}} - -Try building a panel that would be useful in answering the Democratic Peace Question, perhaps in these steps. - -\hypertarget{task-1-data-input-and-standardization}{% -\subsection*{Task 1: Data Input and Standardization}\label{task-1-data-input-and-standardization}} -\addcontentsline{toc}{subsection}{Task 1: Data Input and Standardization} - -Often, files we need are saved in the \texttt{.xls} or \texttt{xlsx} format. It is possible to read these files directly into \texttt{R}, but experience suggests that this process is slower than converting them first to \texttt{.csv} format and reading them in as \texttt{.csv} files. - -\texttt{readxl}/\texttt{readr}/\texttt{haven} packages(\url{https://github.com/tidyverse/tidyverse}) is constantly expanding to capture more file types. In day 1, we used the package \texttt{readxl}, using the \texttt{read\_excel()} function. - -\hypertarget{task-2-data-merging}{% -\subsection*{Task 2: Data Merging}\label{task-2-data-merging}} -\addcontentsline{toc}{subsection}{Task 2: Data Merging} - -We will use data to test a version of the Democratic Peace Thesis (DPS). Democracies are said to go to war less because the leaders who wage wars are accountable to voters who have to bear the costs of war. Are democracies less likely to engage in militarized interstate disputes? - -To start, let's download and merge some data. - -\begin{itemize} -\tightlist -\item - Load in the Militarized Interstate Dispute (MID) files. Militarized interstate disputes are hostile action between two formally recognized states. Examples of this would be threats to use force, threats to declare war, beginning war, fortifying a border with troops, and so on. -\item - Find a way to \textbf{merge} the Polity IV dataset and the MID data. This process can be a bit tricky. -\item - An \emph{advanced} version of this task would be to download the dyadic form of the data and try merging that with polity. -\end{itemize} - -\hypertarget{task-3-tabulations-and-visualization}{% -\subsection*{Task 3: Tabulations and Visualization}\label{task-3-tabulations-and-visualization}} -\addcontentsline{toc}{subsection}{Task 3: Tabulations and Visualization} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Calculate the mean Polity2 score by year. Plot the result. Use graphical indicators of your choosing to show where key events fall in this timeline (such as 1914, 1929, 1939, 1989, 2008). Speculate on why the behavior from 1800 to 1920 seems to be qualitatively different than behavior afterwards. -\item - Do the same but only among state-years that were invovled in a MID. Plot this line together with your results from 1. -\item - Do the same but only among state years that were \emph{not} involved in a MID. -\item - Arrive at a tentative conclusion for how well the Democratic Peace argument seems to hold up in this dataset. Visualize this conclusion. -\end{enumerate} - -\hypertarget{simulation}{% -\chapter[Simulation]{\texorpdfstring{Simulation\footnote{Module originally written by Connor Jerzak and Shiro Kuriwaki}}{Simulation}}\label{simulation}} - -\hypertarget{motivation-simulation-as-an-analytical-tool}{% -\subsection*{Motivation: Simulation as an Analytical Tool}\label{motivation-simulation-as-an-analytical-tool}} -\addcontentsline{toc}{subsection}{Motivation: Simulation as an Analytical Tool} - -An increasing amount of political science contributions now include a simulation. - -\begin{itemize} -\tightlist -\item - \href{http://www-personal.umich.edu/~axe/research/Dissemination.pdf}{Axelrod (1977)} demonstrated via simulation how atomized individuals evolve to be grouped in similar clusters or countries, a model of culture.\footnote{\href{http://www-personal.umich.edu/~axe/research/Dissemination.pdf}{Axelrod, Robert. 1997. ``The Dissemination of Culture.'' \emph{Journal of Conflict Resolution} 41(2): 203--26.}} -\item - \href{http://www-personal.umich.edu/~jowei/florida.pdf}{Chen and Rodden (2013)} argued in a 2013 article that the vote-seat inequality in U.S. elections that is often attributed to intentional partisan gerrymandering can actually attributed to simply the reality of ``human geography'' -- Democratic voters tend to be concentrated in smaller area. Put another way, no feasible form of gerrymandering could spread out Democratic voters in such a way to equalize their vote-seat translation effectiveness. After demonstrating the empirical pattern of human geography, they advance their key claim by simulating thousands of redistricting plans and record the vote-seat ratio.\footnote{\href{http://www-personal.umich.edu/~jowei/florida.pdf}{Chen, Jowei, and Jonathan Rodden. ``Unintentional Gerrymandering: Political Geography and Electoral Bias in Legislatures. \emph{Quarterly Journal of Political Science}, 8:239-269''}} -\item - \href{https://gking.harvard.edu/files/abs/evil-abs.shtml}{Gary King, James Honaker, and multiple other authors} propose a way to analyze missing data with a method of multiple imputation, which uses a lot of simulation from a researcher's observed dataset.\footnote{\href{https://gking.harvard.edu/files/abs/evil-abs.shtml}{King, Gary, et al.~``Analyzing Incomplete Political Science Data: An Alternative Algorithm for Multiple Imputation''. \emph{American Political Science Review}, 95: 49-69.}} (Software: Amelia\footnote{\href{http://www.jstatsoft.org/v45/i07/}{James Honaker, Gary King, Matthew Blackwell (2011). Amelia II: A Program for Missing Data. Journal of - Statistical Software, 45(7), 1-47.}}) -\end{itemize} - -Statistical methods also incorporate simulation: - -\begin{itemize} -\tightlist -\item - The bootstrap: a statistical method for estimating uncertainty around some parameter by re-sampling observations. -\item - Bagging: a method for improving machine learning predictions by re-sampling observations, storing the estimate across many re-samples, and averaging these estimates to form the final estimate. A variance reduction technique. -\item - Statistical reasoning: if you are trying to understand a quantitative problem, a wonderful first-step to understand the problem better is to simulate it! The analytical solution is often very hard (or impossible), but the simulation is often much easier :-) -\end{itemize} - -\hypertarget{where-are-we-where-are-we-headed-5}{% -\subsection*{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed-5}} -\addcontentsline{toc}{subsection}{Where are we? Where are we headed?} - -Up till now, you should have covered: - -\begin{itemize} -\tightlist -\item - \texttt{R} basics -\item - Visualization -\item - Matrices and vectors -\item - Functions, objects, loops -\item - Joining real data -\end{itemize} - -In this module, we will start to work with generating data within R, from thin air, as it were. Doing simulation also strengthens your understanding of Probability (Section @ref\{probability\}). - -\hypertarget{check-your-understanding-2}{% -\subsection*{Check your Understanding}\label{check-your-understanding-2}} -\addcontentsline{toc}{subsection}{Check your Understanding} - -\begin{itemize} -\tightlist -\item - What does the \texttt{sample()} function do? -\item - What does \texttt{runif()} stand for? -\item - What is a \texttt{seed}? -\item - What is a Monte Carlo? -\end{itemize} - -Check if you have an idea of how you might code the following tasks: - -\begin{itemize} -\tightlist -\item - Simulate 100 rolls of a die -\item - Simulate one random ordering of 25 numbers -\item - Simulate 100 values of white noise (uniform random variables) -\item - Generate a ``bootstrap'' sample of an existing dataset -\end{itemize} - -We're going to learn about this today! - -\hypertarget{pick-a-sample-any-sample}{% -\section{Pick a sample, any sample}\label{pick-a-sample-any-sample}} - -\hypertarget{the-sample-function}{% -\section{\texorpdfstring{The \texttt{sample()} function}{The sample() function}}\label{the-sample-function}} - -The core functions for coding up stochastic data revolves around several key functions, so we will simply review them here. - -Suppose you have a vector of values \texttt{x} and from it you want to randomly sample a sample of length \texttt{size}. For this, use the \texttt{sample} function - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sample}\NormalTok{(}\DataTypeTok{x =} \DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{, }\DataTypeTok{size =} \DecValTok{5}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 8 7 9 10 6 -\end{verbatim} - -There are two subtypes of sampling -- with and without replacement. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Sampling without replacement (\texttt{replace\ =\ FALSE}) means once an element of \texttt{x} is chosen, it will not be considered again: -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sample}\NormalTok{(}\DataTypeTok{x =} \DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{, }\DataTypeTok{size =} \DecValTok{10}\NormalTok{, }\DataTypeTok{replace =} \OtherTok{FALSE}\NormalTok{) }\CommentTok{\#\# no number appears more than once} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 4 3 10 7 6 8 2 9 1 5 -\end{verbatim} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{1} -\tightlist -\item - Sampling with replacement (\texttt{replace\ =\ TRUE}) means that even if an element of \texttt{x} is chosen, it is put back in the pool and may be chosen again. -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sample}\NormalTok{(}\DataTypeTok{x =} \DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{, }\DataTypeTok{size =} \DecValTok{10}\NormalTok{, }\DataTypeTok{replace =} \OtherTok{TRUE}\NormalTok{) }\CommentTok{\#\# any number can appear more than once} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 9 6 6 3 1 1 4 1 7 3 -\end{verbatim} - -It follows then that you cannot sample without replacement a sample that is larger than the pool. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sample}\NormalTok{(}\DataTypeTok{x =} \DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{, }\DataTypeTok{size =} \DecValTok{100}\NormalTok{, }\DataTypeTok{replace =} \OtherTok{FALSE}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## Error in sample.int(length(x), size, replace, prob): cannot take a sample larger than the population when 'replace = FALSE' -\end{verbatim} - -So far, every element in \texttt{x} has had an equal probability of being chosen. In some application, we want a sampling scheme where some elements are more likely to be chosen than others. The argument \texttt{prob} handles this. - -For example, this simulates 20 fair coin tosses (each outcome is equally likely to happen) - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sample}\NormalTok{(}\KeywordTok{c}\NormalTok{(}\StringTok{"Head"}\NormalTok{, }\StringTok{"Tail"}\NormalTok{), }\DataTypeTok{size =} \DecValTok{20}\NormalTok{, }\DataTypeTok{prob =} \KeywordTok{c}\NormalTok{(}\FloatTok{0.5}\NormalTok{, }\FloatTok{0.5}\NormalTok{), }\DataTypeTok{replace =} \OtherTok{TRUE}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Head" "Head" "Tail" "Head" "Head" "Tail" "Head" "Head" "Tail" "Tail" -## [11] "Tail" "Head" "Tail" "Tail" "Tail" "Head" "Head" "Head" "Tail" "Tail" -\end{verbatim} - -But this simulates 20 biased coin tosses, where say the probability of Tails is 4 times more likely than the number of Heads - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sample}\NormalTok{(}\KeywordTok{c}\NormalTok{(}\StringTok{"Head"}\NormalTok{, }\StringTok{"Tail"}\NormalTok{), }\DataTypeTok{size =} \DecValTok{20}\NormalTok{, }\DataTypeTok{prob =} \KeywordTok{c}\NormalTok{(}\FloatTok{0.2}\NormalTok{, }\FloatTok{0.8}\NormalTok{), }\DataTypeTok{replace =} \OtherTok{TRUE}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Tail" "Tail" "Tail" "Head" "Tail" "Tail" "Tail" "Tail" "Tail" "Tail" -## [11] "Head" "Tail" "Tail" "Tail" "Tail" "Tail" "Tail" "Tail" "Tail" "Tail" -\end{verbatim} - -\hypertarget{sampling-rows-from-a-dataframe}{% -\subsection{Sampling rows from a dataframe}\label{sampling-rows-from-a-dataframe}} - -In tidyverse, there is a convenience function to sample rows randomly: \texttt{sample\_n()} and \texttt{sample\_frac()}. - -For example, load the dataset on cars, \texttt{mtcars}, which has 32 observations. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{mtcars} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## mpg cyl disp hp drat wt qsec vs am gear carb -## Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4 -## Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4 -## Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1 -## Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1 -## Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2 -## Valiant 18.1 6 225.0 105 2.76 3.460 20.22 1 0 3 1 -## Duster 360 14.3 8 360.0 245 3.21 3.570 15.84 0 0 3 4 -## Merc 240D 24.4 4 146.7 62 3.69 3.190 20.00 1 0 4 2 -## Merc 230 22.8 4 140.8 95 3.92 3.150 22.90 1 0 4 2 -## Merc 280 19.2 6 167.6 123 3.92 3.440 18.30 1 0 4 4 -## Merc 280C 17.8 6 167.6 123 3.92 3.440 18.90 1 0 4 4 -## Merc 450SE 16.4 8 275.8 180 3.07 4.070 17.40 0 0 3 3 -## Merc 450SL 17.3 8 275.8 180 3.07 3.730 17.60 0 0 3 3 -## Merc 450SLC 15.2 8 275.8 180 3.07 3.780 18.00 0 0 3 3 -## Cadillac Fleetwood 10.4 8 472.0 205 2.93 5.250 17.98 0 0 3 4 -## Lincoln Continental 10.4 8 460.0 215 3.00 5.424 17.82 0 0 3 4 -## Chrysler Imperial 14.7 8 440.0 230 3.23 5.345 17.42 0 0 3 4 -## Fiat 128 32.4 4 78.7 66 4.08 2.200 19.47 1 1 4 1 -## Honda Civic 30.4 4 75.7 52 4.93 1.615 18.52 1 1 4 2 -## Toyota Corolla 33.9 4 71.1 65 4.22 1.835 19.90 1 1 4 1 -## Toyota Corona 21.5 4 120.1 97 3.70 2.465 20.01 1 0 3 1 -## Dodge Challenger 15.5 8 318.0 150 2.76 3.520 16.87 0 0 3 2 -## AMC Javelin 15.2 8 304.0 150 3.15 3.435 17.30 0 0 3 2 -## Camaro Z28 13.3 8 350.0 245 3.73 3.840 15.41 0 0 3 4 -## Pontiac Firebird 19.2 8 400.0 175 3.08 3.845 17.05 0 0 3 2 -## Fiat X1-9 27.3 4 79.0 66 4.08 1.935 18.90 1 1 4 1 -## Porsche 914-2 26.0 4 120.3 91 4.43 2.140 16.70 0 1 5 2 -## Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.90 1 1 5 2 -## Ford Pantera L 15.8 8 351.0 264 4.22 3.170 14.50 0 1 5 4 -## Ferrari Dino 19.7 6 145.0 175 3.62 2.770 15.50 0 1 5 6 -## Maserati Bora 15.0 8 301.0 335 3.54 3.570 14.60 0 1 5 8 -## Volvo 142E 21.4 4 121.0 109 4.11 2.780 18.60 1 1 4 2 -\end{verbatim} - -sample\_n picks a user-specified number of rows from the dataset: - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sample\_n}\NormalTok{(mtcars, }\DecValTok{3}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## mpg cyl disp hp drat wt qsec vs am gear carb -## Merc 450SE 16.4 8 275.8 180 3.07 4.07 17.4 0 0 3 3 -## Porsche 914-2 26.0 4 120.3 91 4.43 2.14 16.7 0 1 5 2 -## Merc 240D 24.4 4 146.7 62 3.69 3.19 20.0 1 0 4 2 -\end{verbatim} - -Sometimes you want a X percent sample of your dataset. In this case use \texttt{sample\_frac()} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sample\_frac}\NormalTok{(mtcars, }\FloatTok{0.10}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## mpg cyl disp hp drat wt qsec vs am gear carb -## Cadillac Fleetwood 10.4 8 472.0 205 2.93 5.250 17.98 0 0 3 4 -## Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.90 1 1 5 2 -## Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4 -\end{verbatim} - -As a side-note, these functions have very practical uses for any type of data analysis: - -\begin{itemize} -\tightlist -\item - Inspecting your dataset: using \texttt{head()} all the same time and looking over the first few rows might lead you to ignore any issues that end up in the bottom for whatever reason. -\item - Testing your analysis with a small sample: If running analyses on a dataset takes more than a handful of seconds, change your dataset upstream to a fraction of the size so the rest of the code runs in less than a second. Once verifying your analysis code runs, then re-do it with your full dataset (by simply removing the \texttt{sample\_n} / \texttt{sample\_frac} line of code in the beginning). While three seconds may not sound like much, they accumulate and eat up time. -\end{itemize} - -\hypertarget{random-numbers-from-specific-distributions}{% -\section{Random numbers from specific distributions}\label{random-numbers-from-specific-distributions}} - -\hypertarget{rbinom}{% -\subsection*{\texorpdfstring{\texttt{rbinom()}}{rbinom()}}\label{rbinom}} -\addcontentsline{toc}{subsection}{\texttt{rbinom()}} - -\texttt{rbinom} builds upon \texttt{sample} as a tool to help you answer the question -- what is the \emph{total number of successes} I would get if I sampled a binary (Bernoulli) result from a test with \texttt{size} number of trials each, with a event-wise probability of \texttt{prob}. The first argument \texttt{n} asks me how many such numbers I want. - -For example, I want to know how many Heads I would get if I flipped a fair coin 100 times. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{rbinom}\NormalTok{(}\DataTypeTok{n =} \DecValTok{1}\NormalTok{, }\DataTypeTok{size =} \DecValTok{100}\NormalTok{, }\DataTypeTok{prob =} \FloatTok{0.5}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 50 -\end{verbatim} - -Now imagine this I wanted to do this experiment 10 times, which would require I flip the coin 10 x 100 = 1000 times! Helpfully, we can do this in one line - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{rbinom}\NormalTok{(}\DataTypeTok{n =} \DecValTok{10}\NormalTok{, }\DataTypeTok{size =} \DecValTok{100}\NormalTok{, }\DataTypeTok{prob =} \FloatTok{0.5}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 48 51 59 54 51 59 49 52 50 44 -\end{verbatim} - -\hypertarget{runif}{% -\subsection*{\texorpdfstring{\texttt{runif()}}{runif()}}\label{runif}} -\addcontentsline{toc}{subsection}{\texttt{runif()}} - -\texttt{runif} also simulates a stochastic scheme where each event has equal probability of getting chosen like \texttt{sample}, but is a continuous rather than discrete system. We will cover this more in the next math module. - -The intuition to emphasize here is that one can generate potentially infinite amounts (size \texttt{n}) of noise that is a essentially random - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{runif}\NormalTok{(}\DataTypeTok{n =} \DecValTok{5}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.198848957 0.333367717 0.009064493 0.503594777 0.744089354 -\end{verbatim} - -\hypertarget{rnorm}{% -\subsection*{\texorpdfstring{\texttt{rnorm()}}{rnorm()}}\label{rnorm}} -\addcontentsline{toc}{subsection}{\texttt{rnorm()}} - -\texttt{rnorm} is also a continuous distribution, but draws from a Normal distribution -- perhaps the most important distribution in statistics. It runs the same way as \texttt{runif} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{rnorm}\NormalTok{(}\DataTypeTok{n =} \DecValTok{5}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] -0.9168349 -0.5256658 0.2999925 -2.2437300 -0.4587616 -\end{verbatim} - -To better visualize the difference between the output of \texttt{runif} and \texttt{rnorm}, let's generate lots of each and plot a histogram. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{from\_runif <{-}}\StringTok{ }\KeywordTok{runif}\NormalTok{(}\DataTypeTok{n =} \DecValTok{1000}\NormalTok{)} -\NormalTok{from\_rnorm <{-}}\StringTok{ }\KeywordTok{rnorm}\NormalTok{(}\DataTypeTok{n =} \DecValTok{1000}\NormalTok{)} - -\KeywordTok{par}\NormalTok{(}\DataTypeTok{mfrow =} \KeywordTok{c}\NormalTok{(}\DecValTok{1}\NormalTok{, }\DecValTok{2}\NormalTok{)) }\CommentTok{\#\# base{-}R parameter for two plots at once} -\KeywordTok{hist}\NormalTok{(from\_runif)} -\KeywordTok{hist}\NormalTok{(from\_rnorm)} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-272-1.pdf} - -\hypertarget{r-p-and-d}{% -\section{r, p, and d}\label{r-p-and-d}} - -Each distribution can do more than generate random numbers (the prefix \texttt{r}). We can compute the cumulative probability by the function \texttt{pbinom()}, \texttt{punif()}, and \texttt{pnorm()}. Also the density -- the value of the PDF -- by \texttt{dbinom()}, \texttt{dunif()} and \texttt{dnorm()}. - -\hypertarget{set.seed}{% -\section{\texorpdfstring{\texttt{set.seed()}}{set.seed()}}\label{set.seed}} - -\texttt{R} doesn't have the ability to generate truly random numbers! Random numbers are actually very hard to generate. (Think: flipping a coin --\textgreater{} can be perfectly predicted if I know wind speed, the angle the coin is flipped, etc.). Some people use random noise in the atmosphere or random behavior in quantum systems to generate ``truly'' (?) random numbers. Conversely, R uses deterministic algorithms which take as an input a ``seed'' and which then perform a series of operations to generate a sequence of random-seeming numbers (that is, numbers whose sequence is sufficiently hard to predict). - -Let's think about this another way. Sampling is a stochastic process, so every time you run \texttt{sample()} or \texttt{runif()} you are bound to get a different output (because different random seeds are used). This is intentional in some cases but you might want to avoid it in others. For example, you might want to diagnose a coding discrepancy by setting the random number generator to give the same number each time. To do this, use the function \texttt{set.seed()}. - -In the function goes any number. When you run a sample function in the same command as a preceding \texttt{set.seed()}, the sampling function will always give you the same sequence of numbers. In a sense, the sampler is no longer random (in the sense of unpredictable to use; remember: it never was ``truly'' random in the first place) - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{set.seed}\NormalTok{(}\DecValTok{02138}\NormalTok{)} -\KeywordTok{runif}\NormalTok{(}\DataTypeTok{n =} \DecValTok{10}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.51236144 0.61530551 0.37451441 0.43541258 0.21166530 0.17812129 -## [7] 0.04420775 0.45567854 0.88718264 0.06970056 -\end{verbatim} - -The random number generator should give you the exact same sequence of numbers if you precede the function by the same seed, - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{set.seed}\NormalTok{(}\DecValTok{02138}\NormalTok{)} -\KeywordTok{runif}\NormalTok{(}\DataTypeTok{n =} \DecValTok{10}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.51236144 0.61530551 0.37451441 0.43541258 0.21166530 0.17812129 -## [7] 0.04420775 0.45567854 0.88718264 0.06970056 -\end{verbatim} - -\hypertarget{exercises-4}{% -\section*{Exercises}\label{exercises-4}} -\addcontentsline{toc}{section}{Exercises} - -\hypertarget{census-sampling}{% -\subsection*{Census Sampling}\label{census-sampling}} -\addcontentsline{toc}{subsection}{Census Sampling} - -What can we learn from surveys of populations, and how wrong do we get if our sampling is biased?\footnote{This example is inspired from \href{https://statistics.fas.harvard.edu/files/statistics-2/files/statistical_paradises_and_paradoxes.pdf}{Meng, Xiao-Li (2018). Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election. \emph{Annals of Applied Statistics} 12:2, 685--726. doi:10.1214/18-AOAS1161SF.}} Suppose we want to estimate the proportion of U.S. residents who are non-white (\texttt{race\ !=\ "White"}). In reality, we do not have any population dataset to utilize and so we \emph{only see the sample survey}. Here, however, to understand how sampling works, let's conveniently use the Census extract in some cases and pretend we didn't in others. - -\begin{enumerate} -\def\labelenumi{(\alph{enumi})} -\item - First, load \texttt{usc2010\_001percent.csv} into your R session. After loading the \texttt{library(tidyverse)}, browse it. Although this is only a 0.01 percent extract, treat this as your population for pedagogical purposes. What is the population proportion of non-White residents? -\item - Setting a seed to \texttt{1669482}, sample 100 respondents from this sample. What is the proportion of non-White residents in this \emph{particular} sample? By how many percentage points are you off from (what we labelled as) the true proportion? -\item - Now imagine what you did above was one survey. What would we get if we did 20 surveys? -\end{enumerate} - -To simulate this, write a loop that does the same exercise 20 times, each time computing a sample proportion. Use the same seed at the top, but be careful to position the \texttt{set.seed} function such that it generates the same sequence of 20 samples, rather than 20 of the same sample. - -Try doing this with a \texttt{for} loop and storing your sample proportions in a new length-20 vector. (Suggestion: make an empty vector first as a container). After running the loop, show a histogram of the 20 values. Also what is the average of the 20 sample estimates? - -\begin{enumerate} -\def\labelenumi{(\alph{enumi})} -\setcounter{enumi}{3} -\tightlist -\item - Now, to make things more real, let's introduce some response bias. The goal here is not to correct response bias but to induce it and see how it affects our estimates. Suppose that non-White residents are 10 percent less likely to respond to enter your survey than White respondents. This is plausible if you think that the Census is from 2010 but you are polling in 2018, and racial minorities are more geographically mobile than Whites. Repeat the same exercise in (c) by modeling this behavior. -\end{enumerate} - -You can do this by creating a variable, e.g.~\texttt{propensity}, that is 0.9 for non-Whites and 1 otherwise. Then, you can refer to it in the propensity argument. - -\begin{enumerate} -\def\labelenumi{(\alph{enumi})} -\setcounter{enumi}{4} -\item - Finally, we want to see if more data (``Big Data'') will improve our estimates. Using the same unequal response rates framework as (d), repeat the same exercise but instead of each poll collecting 100 responses, we collect 10,000. -\item - Optional - visualize your 2 pairs of 20 estimates, with a bar showing the ``correct'' population average. -\end{enumerate} - -\hypertarget{conditional-proportions}{% -\subsection*{Conditional Proportions}\label{conditional-proportions}} -\addcontentsline{toc}{subsection}{Conditional Proportions} - -This example is not on simulation, but is meant to reinforce some of the probability discussion from math lecture. - -Read in the Upshot Siena poll from Fall 2016, \texttt{data/input/upshot-siena-polls.csv}. - -In addition to some standard demographic questions, we will focus on one called \texttt{vt\_pres\_2} in the csv. This is a two-way presidential vote question, asking respondents who they plan to vote for President if the election were held today -- Donald Trump, the Republican, or Hilary Clinton, the Democrat, with options for Other candidates as well. For this problem, use the two-way vote question rather than the 4-way vote question. - -\begin{enumerate} -\def\labelenumi{(\alph{enumi})} -\item - Drop the the respondents who answered the November poll (i.e.~those for which \texttt{poll\ ==\ "November"}). We do this in order to ignore this November population in all subsequent parts of this question because they were not asked the Presidential vote question. -\item - Using the dataset after the procedure in (a), find the proportion of \emph{poll respondents} (those who are in the sample) who support Donald Trump. -\item - Among those who supported Donald Trump, what proportion of them has a Bachelor's degree or higher (i.e.~have a Bachelor's, Graduate, or other Professional Degree)? -\item - Among those who did not support Donald Trump (i.e.~including supporters of Hilary Clinton, another candidate, or those who refused to answer the question), what proportion of them has a Bachelor's degree or higher? -\item - Express the numbers in the previous parts as probabilities of specified events. Define your own symbols: For example, we can let \(T\) be the event that a randomly selected respondent in the poll supports Donald Trump, then the proportion in part (b) is the probability \(P(T).\) -\item - Suppose we randomly sampled a person who participated in the survey and found that he/she had a Bachelor's degree or higher. Given this evidence, what is the probability that the same person supports Donald Trump? Use Bayes Rule and show your work -- that is, do not use data or R to compute the quantity directly. Then, verify this is the case via R. -\end{enumerate} - -\hypertarget{the-birthday-problem}{% -\subsection*{The Birthday problem}\label{the-birthday-problem}} -\addcontentsline{toc}{subsection}{The Birthday problem} - -Write code that will answer the well-known birthday problem via simulation.\footnote{This exercise draws from Imai (2017)} - -The problem is fairly simple: Suppose \(k\) people gather together in a room. What is the probability at least two people share the same birthday? - -To simplify reality a bit, assume that (1) there are no leap years, and so there are always 365 days in a year, and (2) a given individual's birthday is randomly assigned and independent from each other. - -\emph{Step 1}: Set \texttt{k} to a concrete number. Pick a number from 1 to 365 randomly, \texttt{k} times to simulate birthdays (would this be with replacement or without?). - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Your code} -\end{Highlighting} -\end{Shaded} - -\emph{Step 2}: Write a line (or two) of code that gives a \texttt{TRUE} or \texttt{FALSE} statement of whether or not at least two people share the same birth date. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Your code} -\end{Highlighting} -\end{Shaded} - -\emph{Step 3}: The above steps will generate a \texttt{TRUE} or \texttt{FALSE} answer for your event of interest, but only for one realization of an event in the sample space. In order to estimate the \emph{probability} of your event happening, we need a ``stochastic'', as opposed to ``deterministic'', method. To do this, write a loop that does Steps 1 and 2 repeatedly for many times, call that number of times \texttt{sims}. For each of \texttt{sims} iteration, your code should give you a \texttt{TRUE} or \texttt{FALSE} answer. Code up a way to store these estimates. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Your code} -\end{Highlighting} -\end{Shaded} - -\emph{Step 4}: Finally, generalize the function further by letting \texttt{k} be a user-defined number. You have now created a \emph{Monte Carlo simulation}! - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Your code} -\end{Highlighting} -\end{Shaded} - -\emph{Step 5}: Generate a table or plot that shows how the probability of sharing a birthday changes by \texttt{k} (fixing \texttt{sims} at a large number like \texttt{1000}). Also generate a similar plot that shows how the probability of sharing a birthday changes by \texttt{sims} (fixing \texttt{k} at some arbitrary number like \texttt{10}). - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Your code} -\end{Highlighting} -\end{Shaded} - -\emph{Extra credit}: Give an ``analytical'' answer to this problem, that is an answer through deriving the mathematical expressions of the probability. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Your equations} -\end{Highlighting} -\end{Shaded} - -\hypertarget{nonwysiwyg}{% -\chapter[LaTeX and markdown]{\texorpdfstring{LaTeX and markdown\footnote{Module originally written by Shiro Kuriwaki}}{LaTeX and markdown}}\label{nonwysiwyg}} - -\hypertarget{where-are-we-where-are-we-headed-6}{% -\subsection*{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed-6}} -\addcontentsline{toc}{subsection}{Where are we? Where are we headed?} - -Up till now, you should have covered: - -\begin{itemize} -\tightlist -\item - Statistical Programming in \texttt{R} -\end{itemize} - -This is only the beginning of \texttt{R} -- programming is like learning a language, so learn more as we use it. And yet \texttt{R} is of likely not the only programming language you will want to use. While we cannot introduce everything, we'll pick out a few that we think are particularly helpful. - -Here will cover - -\begin{itemize} -\tightlist -\item - Markdown -\item - LaTeX (and BibTeX) -\end{itemize} - -as examples of a non-WYSIWYG editor - -and the next chapter (you can read it without reading this LaTeX chapter) covers - -\begin{itemize} -\tightlist -\item - command-line -\item - git -\end{itemize} - -command-line are a basic set of tools that you may have to use from time to time. It also clarifies what more complicated programs are doing. Markdown is an example of compiling a plain text file. LaTeX is a typesetting program and git is a version control program -- both are useful for non-quantitative work as well. - -\hypertarget{check-your-understanding-3}{% -\subsection*{Check your understanding}\label{check-your-understanding-3}} -\addcontentsline{toc}{subsection}{Check your understanding} - -Check if you have an idea of how you might code the following tasks: - -\begin{itemize} -\tightlist -\item - What does ``WYSIWYG'' stand for? How would a non-WYSIWYG format text? -\item - How do you start a header in markdown? -\item - What are some ``plain text'' editors? -\item - How do you start a document in \texttt{.tex}? -\item - How do you start a environment in \texttt{.tex}? -\item - How do you insert a figure in \texttt{.tex}? -\item - How do you reference a figure in \texttt{.tex}? -\item - What is a \texttt{.bib} file? -\item - Say you came across a interesting journal article. How would you want to maintain this reference so that you can refer to its citation in all your subsequent papers? -\end{itemize} - -\hypertarget{motivation-2}{% -\section{Motivation}\label{motivation-2}} - -Statistical programming is a fast-moving field. The beta version of \texttt{R} was released in 2000, \texttt{ggplot2} was released on 2005, and \texttt{RStudio} started around 2010. Of course, some programming technologies are quite ``old'': (\texttt{C} in 1969, \texttt{C++} around 1989, \texttt{TeX} in 1978, \texttt{Linux} in 1991, Mac OS in 1984). But it is easy to feel you are falling behind in the recent developments of programming. Today we will do a \textbf{brief} and rough overview of some fundamental and new tools other than \texttt{R}, with the general aim of having you break out of your comfort zone so you won't be shut out from learning these tools in the future. - -\hypertarget{markdown}{% -\section{Markdown}\label{markdown}} - -Markdown is the text we have been using throughout this course! At its core markdown is just plain text. Plain text does not have any formatting embedded in it. Instead, the formatting is coded up as text. Markdown is \emph{not} a WYSIWYG (What you see is what you get) text editor like Microsoft Word or Google Docs. This will mean that you need to explicitly code for \texttt{bold\{text\}} rather than hitting Command+B and making your text look \textbf{bold} on your own computer. - -Markdown is known as a ``light-weight'' editor, which means that it is relatively easy to write code that will compile. It is quick and easy and satisfies most presentation purposes; you might want to try \texttt{LaTeX} for more involved papers. - -\hypertarget{markdown-commands}{% -\subsection{markdown commands}\label{markdown-commands}} - -For italic and bold, use either the asterisks or the underlines, - -\begin{verbatim} -*italic* **bold** -_italic_ __bold__ -\end{verbatim} - -And for headers use the hash symbols, - -\begin{verbatim} -# Main Header -## Sub-headers -\end{verbatim} - -\hypertarget{your-own-markdown}{% -\subsection{your own markdown}\label{your-own-markdown}} - -RStudio makes it easy to compile your very first markdown file by giving you templates. Got to \texttt{New\ \textgreater{}\ R\ Markdown}, pick a document and click Ok. This will give you a skeleton of a document you can compile -- or ``knit''. - -Rmd is actually a slight modification of real markdown. It is a type of file that R reads and turns into a proper \texttt{md} file. Then, it uses a document-conversion called pandoc to compile your \texttt{md} into documents like PDF or HTML. - -\begin{figure} -\centering -\includegraphics{images/RMarkdownFlow.png} -\caption{How Rmds become PDFs or HTMLs} -\end{figure} - -\hypertarget{a-note-on-plain-text-editors}{% -\subsection{A note on plain-text editors}\label{a-note-on-plain-text-editors}} - -Multiple software exist where you can edit plain-text (roughly speaking, text that is not WYSIWYG). - -\begin{itemize} -\tightlist -\item - RStudio (especially for R-related links) -\item - TeXMaker, TeXShop (especially for TeX) -\item - \href{https://www.gnu.org/software/emacs/}{emacs}, aquamacs (general) -\item - \href{http://www.vim.org/download.php}{vim} (general) -\item - \href{https://www.sublimetext.com}{Sublime Text} (general) -\end{itemize} - -Each has their own keyboard shortcuts and special features. You can browse a couple and see which one(s) you like. - -\hypertarget{latex}{% -\section{LaTeX}\label{latex}} - -LaTeX is a typesetting program. You'd engage with LaTeX much like you engage with your \texttt{R} code. You will interact with LaTeX in a text editor, and will writing code which will be interpreted by the LaTeX compiler and which will finally be parsed to form your final PDF. - -\hypertarget{compile-online}{% -\subsection{compile online}\label{compile-online}} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Go to \url{https://www.overleaf.com} -\item - Scroll down and go to ``CREATE A NEW PAPER'' if you don't have an account. -\item - Let's discuss the default template. -\item - Make a new document, and set it as your main document. Then type in the Minimal Working Example (MWE): -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{\textbackslash{}}\ExtensionTok{documentclass}\DataTypeTok{\{article\}} -\NormalTok{\textbackslash{}}\ExtensionTok{begin}\DataTypeTok{\{document\}} -\ExtensionTok{Hello}\NormalTok{ World} -\NormalTok{\textbackslash{}}\ExtensionTok{end}\DataTypeTok{\{document\}} -\end{Highlighting} -\end{Shaded} - -\hypertarget{compile-your-first-latex-document-locally}{% -\subsection{compile your first LaTeX document locally}\label{compile-your-first-latex-document-locally}} - -LaTeX is a very stable system, and few changes to it have been made since the 1990s. The main benefit: better control over how your papers will look; better methods for writing equations or making tables; overall pleasing aesthetic. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Open a plain text editor. Then type in the MWE -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{\textbackslash{}}\ExtensionTok{documentclass}\DataTypeTok{\{article\}} -\NormalTok{\textbackslash{}}\ExtensionTok{begin}\DataTypeTok{\{document\}} -\ExtensionTok{Hello}\NormalTok{ World} -\NormalTok{\textbackslash{}}\ExtensionTok{end}\DataTypeTok{\{document\}} -\end{Highlighting} -\end{Shaded} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{1} -\tightlist -\item - Save this as \texttt{hello\_world.tex}. Make sure you get the file extension right. -\item - Open this in your ``LaTeX'' editor. This can be \texttt{TeXMaker}, \texttt{Aqumacs}, etc.. -\item - Go through the click/dropdown interface and click compile. -\end{enumerate} - -\hypertarget{main-latex-commands}{% -\subsection{main LaTeX commands}\label{main-latex-commands}} - -LaTeX can cover most of your typesetting needs, to clean equations and intricate diagrams. - -Some main commands you'll be using are below, and a very concise cheat sheet here: \url{https://wch.github.io/latexsheet/latexsheet.pdf} - -Most involved features require that you begin a specific ``environment'' for that feature, clearly demarcating them by the notation \texttt{\textbackslash{}begin\{figure\}} and then \texttt{\textbackslash{}end\{figure\}}, e.g.~in the case of figures. - -\begin{verbatim} -\begin{figure} -\includegraphics{histogram.pdf} -\end{figure} -\end{verbatim} - -where \texttt{histogram.pdf} is a path to one of your files. - -Notice that each line starts with a backslash \texttt{\textbackslash{}} -- in LaTeX this is the symbol to run a command. - -The following syntax at the endpoints are shorthand for math equations. - -\begin{verbatim} -\[\int x^2 dx\] -\end{verbatim} - -these compile math symbols: \(\displaystyle \int x^2 dx.\)\footnote{Enclosing with \texttt{\$\$} instead of \texttt{\textbackslash{}{[}} also has the same effect, so you may see it too. But this is now discouraged due to its inflexibility.} - -The \texttt{align} environment is useful to align your multi-line math, for example. - -\begin{verbatim} -\begin{align} -P(A \mid B) &= \frac{P(A \cap B)}{P(B)}\\ -&= \frac{P(B \mid A)P(A)}{P(B)} -\end{align} -\end{verbatim} - -\begin{align} -P(A \mid B) &= \frac{P(A \cap B)}{P(B)}\\ -&= \frac{P(B \mid A)P(A)}{P(B)} -\end{align} - -Regression tables should be outputted as \texttt{.tex} files with packages like \texttt{xtable} and \texttt{stargazer}, and then called into LaTeX by \texttt{\textbackslash{}input\{regression\_table.tex\}} where \texttt{regression\_table.tex} is the path to your regression output. - -Figures and equations should be labelled with the tag (e.g.~\texttt{label\{tab:regression\}} so that you can refer to them later with their tag \texttt{Table\ \textbackslash{}ref\{tab:regression\}}, instead of hard-coding \texttt{Table\ 2}). - -For some LaTeX commands you might need to load a separate package that someone else has written. Do this in your preamble (i.e.~before \texttt{\textbackslash{}begin\{document\}}): - -\begin{verbatim} -\usepackage[options]{package} -\end{verbatim} - -where \texttt{package} is the name of the package and \texttt{options} are options specific to the package. - -\hypertarget{further-guides}{% -\subsection*{Further Guides}\label{further-guides}} -\addcontentsline{toc}{subsection}{Further Guides} - -For a more comprehensive listing of LaTeX commands, Mayya Komisarchik has a great tutorial set of folders: \url{https://scholar.harvard.edu/mkomisarchik/tutorials-0} - -There is a version of LaTeX called Beamer, which is a popular way of making a slideshow. Slides in markdown is also a competitor. The language of Beamer is the same as LaTeX but has some special functions for slides. - -\hypertarget{bibtex}{% -\section{BibTeX}\label{bibtex}} - -BibTeX is a reference system for bibliographical tests. We have a \texttt{.bib} file separately on our computer. This is also a plain text file, but it encodes bibliographical resources with special syntax so that a program can rearrange parts accordingly for different citation systems. - -\hypertarget{what-is-a-.bib-file}{% -\subsection{\texorpdfstring{what is a \texttt{.bib} file?}{what is a .bib file?}}\label{what-is-a-.bib-file}} - -For example, here is the Nunn and Wantchekon article entry in \texttt{.bib} form. - -\begin{verbatim} -@article{nunn2011slave, - title={The Slave Trade and the Origins of Mistrust in Africa}, - author={Nunn, Nathan and Wantchekon, Leonard}, - journal={American Economic Review}, - volume={101}, - number={7}, - pages={3221--3252}, - year={2011} -} -\end{verbatim} - -The first entry, \texttt{nunn2011slave}, is ``pick your favorite'' -- pick your own name for your reference system. The other slots in this \texttt{@article} entry are entries that refer to specific bibliographical text. - -\hypertarget{what-does-latex-do-with-.bib-files}{% -\subsection{what does LaTeX do with .bib files?}\label{what-does-latex-do-with-.bib-files}} - -Now, in LaTeX, if you type - -\begin{verbatim} - \textcite{nunn2011slave} argue that current variation in the trust among citizens of African countries has historical roots in the European slave trade in the 1600s. - -\end{verbatim} - -as part of your text, then when the \texttt{.tex} file is compiled the PDF shows something like - -\includegraphics{images/biblatex_inline.png} - -in whatever citation style (APSA, APA, Chicago) you pre-specified! - -Also at the end of your paper you will have a bibliography with entries ordered and formatted in the appropriate citation. - -\includegraphics{images/biblatex_bibliography.png} - -This is a much less frustrating way of keeping track of your references -- no need to hand-edit formatting the bibliography to conform to citation rules (which biblatex already knows) and no need to update your bibliography as you add and drop references (biblatex will only show entries that are used in the main text). - -\hypertarget{stocking-up-on-your-.bib-files}{% -\subsection{stocking up on your .bib files}\label{stocking-up-on-your-.bib-files}} - -You should keep your own \texttt{.bib} file that has all your bibliographical resources. Storing entries is cheap (does not take much memory), so it is fine to keep all your references in one place (but you'll want to make a new one for collaborative projects where multiple people will compile a \texttt{.tex} file). - -For example, Gary's BibTeX file is here: \url{https://github.com/iqss-research/gkbibtex/blob/master/gk.bib} - -Citation management software (Mendeley or Zotero) automatically generates .bib entries from your library of PDFs for you, provided you have the bibliography attributes right. - -\hypertarget{exercise}{% -\section*{Exercise}\label{exercise}} -\addcontentsline{toc}{section}{Exercise} - -Create a LaTeX document for a hypothetical research paper on your laptop and, once you've verified it compiles into a PDF, come show it to either one of the instructors. - -You can also use overleaf if you have preference for a cloud-based system. But don't swallow the built-in templates without understanding or testing them. - -Each student will have slightly different substantive interests, so we won't impose much of a standard. But at a minimum, the LaTeX document should have: - -\begin{itemize} -\tightlist -\item - A title, author, date, and abstract -\item - Sections -\item - Italics and boldface -\item - A figure with a caption and in-text reference to it. -\end{itemize} - -Depending on your subfield or interests, try to implement some of the following: - -\begin{itemize} -\tightlist -\item - A bibliographical reference drawing from a separate \texttt{.bib} file -\item - A table -\item - A math expression -\item - A different font -\item - Different page margins -\item - Different line spacing -\end{itemize} - -\hypertarget{concluding-the-prefresher}{% -\section*{Concluding the Prefresher}\label{concluding-the-prefresher}} -\addcontentsline{toc}{section}{Concluding the Prefresher} - -Math may not be the perfect tool for every aspiring political scientist, but hopefully it was useful background to have at the least: - -Historians think this totally meaningless and nonsensical statistic is the product of an early-modern epistemological shift in which numbers and quantifiable data became revered above other kinds of knowledge as the most useful and credible form of truth https://t.co/wVFyAQGxEv - ---- Gina Anne Tam 譚吉娜 (\citet{DGTam86}) May 29, 2018 - -But we should be aware that too much slant towards math and programming can miss the point: - -To be clear, PhD training in Econ (first year) is often a disaster-- like how to prove the Central Limit Theorem (the LeBron James of Statistics) with polar-cooardinates. This is mostly a way to demoralize actual economists and select a bunch of unimaginative math jocks. - ---- Amitabh Chandra (\citet{amitabhchandra2}) August 14, 2018 - -Keep on learning, trying new techniques to improve your work, and learn from others! - -What \#rstats tricks did it take you way too long to learn? One of mine is using readRDS and saveRDS instead of repeatedly loading from CSV - ---- Emily Riederer (\citet{EmilyRiederer}) August 19, 2017 - -\hypertarget{your-feedback-matters}{% -\subsection*{Your Feedback Matters}\label{your-feedback-matters}} -\addcontentsline{toc}{subsection}{Your Feedback Matters} - -\emph{Please tell us how we can improve the Prefresher}: The Prefresher is a work in progress, with material mainly driven by graduate students. Please tell us how we should change (or not change) each of its elements: - -\url{https://harvard.az1.qualtrics.com/jfe/form/SV_esbzN8ZFAOPTqiV} - -\hypertarget{rtext}{% -\chapter[Text]{\texorpdfstring{Text\footnote{Module originally written by Connor Jerzak}}{Text}}\label{rtext}} - -\hypertarget{where-are-we-where-are-we-headed-7}{% -\section*{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed-7}} -\addcontentsline{toc}{section}{Where are we? Where are we headed?} - -Up till now, you should have covered: - -\begin{itemize} -\tightlist -\item - Loading in data; -\item - \texttt{R} notation; -\item - Matrix algebra. -\end{itemize} - -\hypertarget{review}{% -\section{Review}\label{review}} - -\begin{itemize} -\tightlist -\item - \texttt{"} and \texttt{\textquotesingle{}} are usually equivalent. -\item - \texttt{\textless{}-} and \texttt{=} are usually interchangeable\footnote{Only equal signs are allowed to define the values of a functions' argument}. (\texttt{x\ \textless{}-\ 3} is equivalent to \texttt{x\ =\ 3}, although the former is more preferred because it explicitly states the assignment). -\item - Use \texttt{(} \texttt{)} when you are giving input to a function: -\end{itemize} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# my\_results <{-} FunctionName(FunctionInputs)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -note `c(1,2,3)` is inputting three numbers in the function `c` -\end{verbatim} - -\begin{itemize} -\tightlist -\item - Use \texttt{\{} \texttt{\}} when you are defining a function or writing a \texttt{for} loop: -\end{itemize} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#function } -\NormalTok{MyFunction <{-}}\StringTok{ }\ControlFlowTok{function}\NormalTok{(InputMatrix)\{ } -\NormalTok{ TempMat <{-}}\StringTok{ }\NormalTok{InputMatrix} - \ControlFlowTok{for}\NormalTok{(i }\ControlFlowTok{in} \DecValTok{1}\OperatorTok{:}\DecValTok{5}\NormalTok{)\{} -\NormalTok{ TempMat <{-}}\StringTok{ }\KeywordTok{t}\NormalTok{(TempMat) }\OperatorTok{\%*\%}\StringTok{ }\NormalTok{TempMat }\OperatorTok{/}\StringTok{ }\DecValTok{10} -\NormalTok{ \} } - \KeywordTok{return}\NormalTok{( TempMat )} -\NormalTok{\}} -\NormalTok{myMat <{-}}\StringTok{ }\KeywordTok{matrix}\NormalTok{(}\KeywordTok{rnorm}\NormalTok{(}\DecValTok{100}\OperatorTok{*}\DecValTok{5}\NormalTok{), }\DataTypeTok{nrow =} \DecValTok{100}\NormalTok{, }\DataTypeTok{ncol =} \DecValTok{5}\NormalTok{)} -\KeywordTok{print}\NormalTok{( }\KeywordTok{MyFunction}\NormalTok{(myMat) ) } -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [,1] [,2] [,3] [,4] [,5] -## [1,] 342.3602 196.1668 856.7638 -732.7517 173.1954 -## [2,] 196.1668 515.3176 762.8554 -277.1625 299.6710 -## [3,] 856.7638 762.8554 2697.1230 -1868.8323 461.6741 -## [4,] -732.7517 -277.1625 -1868.8323 1678.3580 -264.6936 -## [5,] 173.1954 299.6710 461.6741 -264.6936 219.0823 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# loop } -\NormalTok{x <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{() } -\ControlFlowTok{for}\NormalTok{(i }\ControlFlowTok{in} \DecValTok{1}\OperatorTok{:}\DecValTok{20}\NormalTok{)\{} -\NormalTok{ x[i] <{-}}\StringTok{ }\NormalTok{i } -\NormalTok{\}} -\KeywordTok{print}\NormalTok{(x) } -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -\end{verbatim} - -\hypertarget{goals-for-today}{% -\section{Goals for today}\label{goals-for-today}} - -Today, we will learn more about using text data. Our objectives are: - -\begin{itemize} -\tightlist -\item - Reading and writing in text in \texttt{R}. -\item - To learn how to use paste and sprintf; -\item - To learn how to use regular expressions; -\item - To learn about other tools for representing + analyzing text in \texttt{R}. -\end{itemize} - -\hypertarget{reading-and-writing-text-in-r}{% -\section{Reading and writing text in R}\label{reading-and-writing-text-in-r}} - -\begin{itemize} -\tightlist -\item - To read in a text file, use readLines -\end{itemize} - -\begin{verbatim} -readLines("~/Downloads/Carboxylic acid - Wikipedia.html") -\end{verbatim} - -\begin{itemize} -\tightlist -\item - To write a text file, use: -\end{itemize} - -\begin{verbatim} -write.table(my_string_vector, "~/mydata.txt", sep="\t") -\end{verbatim} - -\hypertarget{paste-and-sprintf}{% -\section{\texorpdfstring{\texttt{paste()} and \texttt{sprintf()}}{paste() and sprintf()}}\label{paste-and-sprintf}} - -paste and sprintf are useful commands in text processing, such as for automatically naming files or automatically performing a series of command over a subset of your data. Table making also will often need these commands. - -Paste concatenates vectors together. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#use collapse for inputs of length > 1 } -\NormalTok{my\_string <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"Not"}\NormalTok{, }\StringTok{"one"}\NormalTok{, }\StringTok{"could"}\NormalTok{, }\StringTok{"equal"}\NormalTok{)} -\KeywordTok{paste}\NormalTok{(my\_string, }\DataTypeTok{collapse =} \StringTok{" "}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Not one could equal" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#use sep for inputs of length == 1 } -\KeywordTok{paste}\NormalTok{(}\StringTok{"Not"}\NormalTok{, }\StringTok{"one"}\NormalTok{, }\StringTok{"could"}\NormalTok{, }\StringTok{"equal"}\NormalTok{, }\DataTypeTok{sep =} \StringTok{" "}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Not one could equal" -\end{verbatim} - -For more sophisticated concatenation, use sprintf. This is very useful for automatically making tables. - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sprintf}\NormalTok{(}\StringTok{"Coefficient for \%s: \%.3f (\%.2f)"}\NormalTok{, }\StringTok{"Gender"}\NormalTok{, }\FloatTok{1.52324}\NormalTok{, }\FloatTok{0.03143}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Coefficient for Gender: 1.523 (0.03)" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#\%s is replaced by a character string} -\CommentTok{\#\%.3f is replaced by a floating point digit with 3 decimal places} -\CommentTok{\#\%.2f is replaced by a floating point digit with 2 decimal places} -\end{Highlighting} -\end{Shaded} - -\hypertarget{regular-expressions}{% -\section{Regular expressions}\label{regular-expressions}} - -A regular expression is a special text string for describing a search pattern. -They are most often used in functions for detecting, locating, and replacing desired text in a corpus. - -Use cases: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - TEXT PARSING. E.g. I have 10000 congressional speaches. Find all those which mention Iran. -\item - WEB SCRAPING. E.g. Parse html code in order to extract research information from an online table. -\item - CLEANING DATA. E.g. After loading in a dataset, we might need to remove mistakes from the dataset, orsubset the data using regular expression tools. -\end{enumerate} - -Example in \texttt{R}. Extract the tweet mentioning Indonesia. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{s1 <{-}}\StringTok{ "If only Bradley\textquotesingle{}s arm was longer. RT"} -\NormalTok{s2 <{-}}\StringTok{ "Share our love in Indonesia and in the World. RT if you agree."} -\NormalTok{my\_string <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(s1, s2)} -\KeywordTok{grepl}\NormalTok{(my\_string, }\DataTypeTok{pattern =} \StringTok{"Indonesia"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] FALSE TRUE -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_string[ }\KeywordTok{grepl}\NormalTok{(my\_string, }\DataTypeTok{pattern =} \StringTok{"Indonesia"}\NormalTok{)]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Share our love in Indonesia and in the World. RT if you agree." -\end{verbatim} - -Key point: Many R commands use regular expressions. See \texttt{?grepl}. Assume that \texttt{x} is a character vector and that \texttt{pattern} is the target pattern. In the earlier example, \texttt{x} could have been something like \texttt{my\_string} and \texttt{pattern} would have been ``\texttt{Indonesia}''. Here are other key uses: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - DETECT PATTERNS. \texttt{grepl(pattern,\ x)} goes through all the entries of \texttt{x} and returns a string of TRUE and FALSE values of the same size as \texttt{x}. It will return a \texttt{TRUE} whenever that string entry has the target pattern, and \texttt{FALSE} whenever it doesn't. -\item - REPLACE PATTERNS. \texttt{gsub(pattern,\ x,\ replacement)} goes through all the entries of \texttt{x} replaces the \texttt{pattern} with \texttt{replacement}. -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{gsub}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ my\_string,} - \DataTypeTok{pattern =} \StringTok{"o"}\NormalTok{, } - \DataTypeTok{replacement =} \StringTok{"AAAA"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "If AAAAnly Bradley's arm was lAAAAnger. RT" -## [2] "Share AAAAur lAAAAve in IndAAAAnesia and in the WAAAArld. RT if yAAAAu agree." -\end{verbatim} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{2} -\tightlist -\item - LOCATE PATTERNS. \texttt{regexpr(pattern,\ text)} goes through each element of the character string. It returns a vector of the same length, with the entries of the vector corresponding to the location of the first pattern match, or a -1 if no match was obtained. -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{regex\_object <{-}}\StringTok{ }\KeywordTok{regexpr}\NormalTok{(}\DataTypeTok{pattern =} \StringTok{"was"}\NormalTok{, }\DataTypeTok{text =}\NormalTok{ my\_string)} -\KeywordTok{attr}\NormalTok{(regex\_object, }\StringTok{"match.length"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 3 -1 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{attr}\NormalTok{(regex\_object, }\StringTok{"useBytes"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] TRUE -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{regexpr}\NormalTok{(}\DataTypeTok{pattern =} \StringTok{"was"}\NormalTok{, }\DataTypeTok{text =}\NormalTok{ my\_string)[}\DecValTok{1}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 23 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{regexpr}\NormalTok{(}\DataTypeTok{pattern =} \StringTok{"was"}\NormalTok{, }\DataTypeTok{text =}\NormalTok{ my\_string)[}\DecValTok{2}\NormalTok{]} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] -1 -\end{verbatim} - -Seems simple? The problem: the patterns can get pretty complex! - -\hypertarget{character-classes}{% -\subsection{Character classes}\label{character-classes}} - -Some types of symbols are stand in for some more complex thing, rather than taken literally. - -\texttt{{[}{[}:digit:{]}{]}} Matches with all digits. - -\texttt{{[}{[}:lower:{]}{]}} Matches with lower case letters. - -\texttt{{[}{[}:alpha:{]}{]}} Matches with all alphabetic characters. - -\texttt{{[}{[}:punct:{]}{]}} Matches with all punctuation characters. - -\texttt{{[}{[}:cntrl:{]}{]}} Matches with ``control'' characters such as \texttt{\textbackslash{}n}, \texttt{\textbackslash{}r}, etc. - -Example in \texttt{R}: - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_string <{-}}\StringTok{ "Do you think that 34\% of apples are red?"} -\KeywordTok{gsub}\NormalTok{(my\_string, }\DataTypeTok{pattern =} \StringTok{"[[:digit:]]"}\NormalTok{, }\DataTypeTok{replace =}\StringTok{"DIGIT"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Do you think that DIGITDIGIT% of apples are red?" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{gsub}\NormalTok{(my\_string, }\DataTypeTok{pattern =} \StringTok{"[[:alpha:]]"}\NormalTok{, }\DataTypeTok{replace =}\StringTok{""}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] " 34% ?" -\end{verbatim} - -\hypertarget{special-characters.}{% -\subsection{Special Characters.}\label{special-characters.}} - -Certain characters (such as \texttt{.,\ *,\ \textbackslash{}}) have special meaning in the regular expressions framework (they are used to form conditional patterns as discussed below). Thus, when we want our pattern to explicitly include those characters as characters, we must ``escape'' them by using \textbackslash{} or encoding them in \textbackslash Q\ldots\textbackslash E. - -Example in \texttt{R}: - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_string <{-}}\StringTok{ "Do *really* think he will win?"} -\KeywordTok{gsub}\NormalTok{(my\_string, }\DataTypeTok{pattern =} \StringTok{"}\CharTok{\textbackslash{}\textbackslash{}}\StringTok{*"}\NormalTok{, }\DataTypeTok{replace =}\StringTok{""}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Do really think he will win?" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_string <{-}}\StringTok{ "Now be brave! }\CharTok{\textbackslash{}n}\StringTok{ Dread what comrades say of you here in combat! "} -\KeywordTok{gsub}\NormalTok{(my\_string, }\DataTypeTok{pattern =} \StringTok{"}\CharTok{\textbackslash{}\textbackslash{}\textbackslash{}n}\StringTok{"}\NormalTok{, }\DataTypeTok{replace =}\StringTok{""}\NormalTok{) } -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Now be brave! Dread what comrades say of you here in combat! " -\end{verbatim} - -\hypertarget{conditional-patterns}{% -\subsection{Conditional patterns}\label{conditional-patterns}} - -\texttt{{[}{]}} The target characters to match are located between the brackets. For example, \texttt{{[}aAbB{]}} will match with the characters \texttt{a,\ A,\ b,\ B}. - -\texttt{{[}\^{}...{]}} Matches with everything except the material between the brackets. For example, \texttt{{[}\^{}aAbB{]}} will match with everything but the characters \texttt{a,\ A,\ b,\ B}. - -\texttt{(?=)} Lookahead -- match something that IS followed by the pattern. - -\texttt{(?!)} Negative lookahead --- match something that is NOT followed by the pattern. - -\texttt{(?\textless{}=)} Lookbehind -- match with something that follows the pattern. - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_string <{-}}\StringTok{ "Do you think that 34\%of the 23\%of apples are red?"} -\KeywordTok{gsub}\NormalTok{(my\_string, }\DataTypeTok{pattern =} \StringTok{"(?<=\%)"}\NormalTok{, }\DataTypeTok{replace =} \StringTok{" "}\NormalTok{, }\DataTypeTok{perl =} \OtherTok{TRUE}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "Do you think that 34% of the 23% of apples are red?" -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_string <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"legislative1\_term1.png"}\NormalTok{, } - \StringTok{"legislative1\_term1.pdf"}\NormalTok{,} - \StringTok{"legislative1\_term2.png"}\NormalTok{,} - \StringTok{"legislative1\_term2.pdf"}\NormalTok{,} - \StringTok{"term2\_presidential1.png"}\NormalTok{, } - \StringTok{"presidential1.png"}\NormalTok{, } - \StringTok{"presidential1\_term2.png"}\NormalTok{,} - \StringTok{"presidential1\_term1.pdf"}\NormalTok{, } - \StringTok{"presidential1\_term2.pdf"}\NormalTok{)} - -\KeywordTok{grepl}\NormalTok{(my\_string, }\DataTypeTok{pattern =} \StringTok{"\^{}(?!presidential1).*}\CharTok{\textbackslash{}\textbackslash{}}\StringTok{.png"}\NormalTok{, }\DataTypeTok{perl =} \OtherTok{TRUE}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] TRUE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE -\end{verbatim} - -\begin{itemize} -\tightlist -\item - Indicates which file names don't start with \texttt{presidential1} but do end in \texttt{.png} -\item - \texttt{\^{}} indicates that the pattern should start at the beginning of the string. -\item - \texttt{?!} indicates negative lookahead -- we're looking for any pattern NOT following presidential1 which meets the subsequent conditions. (see below) -\item - The first \texttt{.} indicates that, following the negative lookahead, there can be any characters and the * says that it doesn't matter how many. Note that we have to escape the . in \texttt{.png}. (by writing \texttt{\textbackslash{}\textbackslash{}.} instead of just \texttt{.}) -\end{itemize} - -You will have the chance to try out some regular expressions for yourself at the end! - -\hypertarget{representing-text}{% -\section{Representing Text}\label{representing-text}} - -In courses and research, we often want to analyze text, to extract meaning out of it. -One of the key decisions we need to make is how to represent the text as numbers. -Once the text is represented numerically, we can then apply a host of statistical -and machine learning methods to it. Those methods are discussed more in the Gov methods sequence (Gov 2000-2003). Here's a summary of the decisions you must make: - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - WHICH TEXT TO USE? Which text do I want to analyze? What is my universe of documents? -\item - HOW TO REPRESENT THE TEXT NUMERICALLY? How do I use numbers to represent different things about the text? -\item - HOW TO ANALYZE THE NUMERICAL REPRESENTATION? How do I extract meaning out of the numerical representation? -\end{enumerate} - -Representing text numerically. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Document term matrix. The document term matrix (DTM) is a common method for representing text. The DTM is a matrix. Each row of this matrix corresponds to a document; each column corresponds to a word. It is often useful to look at summary statistics such as the percentage of speaches in which a Democratic lawmaker used the word ``inequality'' compared to a Republican; the DTM would be very helpful for this and other tasks. -\end{enumerate} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{doc1 <{-}}\StringTok{ "Rage{-}{-}{-}Goddess, sing the rage of Peleus’ son Achilles,} -\StringTok{ murderous, doomed, that cost the Achaeans countless losses,} -\StringTok{ hurling down to the House of Death so many sturdy souls,} -\StringTok{ great fighters’ souls."} -\NormalTok{doc2 <{-}}\StringTok{ "And fate? No one alive has ever escaped it,} -\StringTok{ neither brave man nor coward, I tell you, } -\StringTok{ it\textquotesingle{}s born with us the day that we are born."} -\NormalTok{doc3 <{-}}\StringTok{ "Many cities of men he saw and learned their minds,} -\StringTok{ many pains he suffered, heartsick on the open sea,} -\StringTok{ fighting to save his life and bring his comrades home."} -\end{Highlighting} -\end{Shaded} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{DocVec <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(doc1, doc2, doc3)} -\end{Highlighting} -\end{Shaded} - -Now we can use utility functions in the \texttt{tm} package: - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(tm)} -\NormalTok{DocCorpus <{-}}\StringTok{ }\KeywordTok{Corpus}\NormalTok{(}\KeywordTok{VectorSource}\NormalTok{(DocVec) ) } -\NormalTok{DTM1 <{-}}\StringTok{ }\KeywordTok{inspect}\NormalTok{( }\KeywordTok{DocumentTermMatrix}\NormalTok{(DocCorpus) ) } -\end{Highlighting} -\end{Shaded} - -Consider the effect of different ``pre-processing'' choices on the resulting DTM! - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{DocVec <{-}}\StringTok{ }\KeywordTok{tolower}\NormalTok{(DocVec)} -\NormalTok{DocVec <{-}}\StringTok{ }\KeywordTok{gsub}\NormalTok{(DocVec, }\DataTypeTok{pattern =}\StringTok{"[[:punct:]]"}\NormalTok{, }\DataTypeTok{replace =} \StringTok{" "}\NormalTok{)} -\NormalTok{DocVec <{-}}\StringTok{ }\KeywordTok{gsub}\NormalTok{(DocVec, }\DataTypeTok{pattern =}\StringTok{"[[:cntrl:]]"}\NormalTok{, }\DataTypeTok{replace =} \StringTok{" "}\NormalTok{)} -\NormalTok{DocCorpus <{-}}\StringTok{ }\KeywordTok{Corpus}\NormalTok{(}\KeywordTok{VectorSource}\NormalTok{(DocVec) ) } -\NormalTok{DTM2 <{-}}\StringTok{ }\KeywordTok{inspect}\NormalTok{(}\KeywordTok{DocumentTermMatrix}\NormalTok{(DocCorpus, } - \DataTypeTok{control =} \KeywordTok{list}\NormalTok{(}\DataTypeTok{stopwords =} \OtherTok{TRUE}\NormalTok{, }\DataTypeTok{stemming =} \OtherTok{TRUE}\NormalTok{))) } -\end{Highlighting} -\end{Shaded} - -Stemming is the process of reducing inflected/derived words to their word stem or base (e.g.~stemming, stemmed, stemmer --\textgreater{} stem*) - -\hypertarget{important-packages-for-parsing-text}{% -\section{Important packages for parsing text}\label{important-packages-for-parsing-text}} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - rvest -- Useful for downloading and manipulating HTML and XM. -\item - tm -- Useful for converting text into a numerical representation (forming DTMs). -\item - stringr -- Useful for string parsing. -\end{enumerate} - -\hypertarget{exercises-5}{% -\section*{Exercises}\label{exercises-5}} -\addcontentsline{toc}{section}{Exercises} - -\hypertarget{section-17}{% -\subsection*{1}\label{section-17}} -\addcontentsline{toc}{subsection}{1} - -Figure out why this command does what it does: - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{sprintf}\NormalTok{(}\StringTok{"\%s of spontaneous events are \%s in the mind. } -\StringTok{ Really, \%.2f?"}\NormalTok{, } - \StringTok{"15.03322123"}\NormalTok{, }\StringTok{"puzzles"}\NormalTok{, }\FloatTok{15.03322123}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "15.03322123 of spontaneous events are puzzles in the mind. \n Really, 15.03?" -\end{verbatim} - -\hypertarget{section-18}{% -\subsection*{2}\label{section-18}} -\addcontentsline{toc}{subsection}{2} - -Why does this command not work? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{try}\NormalTok{(}\KeywordTok{sprintf}\NormalTok{(}\StringTok{"\%s of spontaneous events are \%s in the mind. Really, \%.2f?"}\NormalTok{,} - \StringTok{"15.03322123"}\NormalTok{, }\StringTok{"puzzles"}\NormalTok{, }\StringTok{"15.03322123"}\NormalTok{ ), }\OtherTok{TRUE}\NormalTok{) } -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-19}{% -\subsection*{3}\label{section-19}} -\addcontentsline{toc}{subsection}{3} - -Using \texttt{grepl}, these materials, Google, and your friends, describe what the following command does. What changes when \texttt{value\ =\ FALSE}? - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{grep}\NormalTok{(}\StringTok{\textquotesingle{}}\CharTok{\textbackslash{}\textquotesingle{}}\StringTok{\textquotesingle{}}\NormalTok{, } - \KeywordTok{c}\NormalTok{(}\StringTok{"To dare is to lose one\textquotesingle{}s footing momentarily."}\NormalTok{, }\StringTok{"To not dare is to lose oneself."}\NormalTok{), }\DataTypeTok{value =} \OtherTok{TRUE}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "To dare is to lose one's footing momentarily." -\end{verbatim} - -\hypertarget{section-20}{% -\subsection*{4}\label{section-20}} -\addcontentsline{toc}{subsection}{4} - -Write code to automatically extract the file names that DO end start with presidential and DO end in .pdf - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{my\_string <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"legislative1\_term1.png"}\NormalTok{, } - \StringTok{"legislative1\_term1.pdf"}\NormalTok{,} - \StringTok{"legislative1\_term2.png"}\NormalTok{, } - \StringTok{"legislative1\_term2.pdf"}\NormalTok{,} - \StringTok{"term2\_presidential1.png"}\NormalTok{, } - \StringTok{"presidential1.png"}\NormalTok{, } - \StringTok{"presidential1\_term2.png"}\NormalTok{,} - \StringTok{"presidential1\_term1.pdf"}\NormalTok{,} - \StringTok{"presidential1\_term2.pdf"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-21}{% -\subsection*{5}\label{section-21}} -\addcontentsline{toc}{subsection}{5} - -Using the same string as in the above, write code to automatically extract the file names that end in .pdf and that contain the text \texttt{term2}. - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Your code here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-22}{% -\subsection*{6}\label{section-22}} -\addcontentsline{toc}{subsection}{6} - -Combine these two strings into a single string separated by a ``-''. Desired output: ``The carbonyl group in aldehydes and ketones is an oxygen analog of the carbon--carbon double bond.'' - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{string1 <{-}}\StringTok{ "The carbonyl group in aldehydes and ketones } -\StringTok{ is an oxygen analog of the carbon"} -\NormalTok{string2 <{-}}\StringTok{ "–carbon double bond."} -\end{Highlighting} -\end{Shaded} - -\hypertarget{section-23}{% -\subsection*{7}\label{section-23}} -\addcontentsline{toc}{subsection}{7} - -Challenge problem! Download this webpage \url{https://en.wikipedia.org/wiki/Odyssey} - -\begin{itemize} -\tightlist -\item - Read the html file into your R workspace. -\item - Remove all of the htlm tags (you may need Google to help with this one). -\item - Remove all punctuation. -\item - Make all the characters lower case. -\item - Do this same process with this webpage (\url{https://en.wikipedia.org/wiki/Iliad}). -\item - Form a document term matrix from the two resulting text strings. -\end{itemize} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\# Your code here} -\end{Highlighting} -\end{Shaded} - -\hypertarget{commandline-git}{% -\chapter[Command-line, git]{\texorpdfstring{Command-line, git\footnote{Module originally written by Shiro Kuriwaki}}{Command-line, git}}\label{commandline-git}} - -\hypertarget{where-are-we-where-are-we-headed-8}{% -\section{Where are we? Where are we headed?}\label{where-are-we-where-are-we-headed-8}} - -Up till now, you should have covered: - -\begin{itemize} -\tightlist -\item - Statistical Programming in \texttt{R} -\end{itemize} - -In conjunction with the markdown/LaTeX chapter, which is mostly used for typesetting and presentation, here we'll introduce the command-line and git, more used for software extensions and version control - -\hypertarget{check-your-understanding-4}{% -\section{Check your understanding}\label{check-your-understanding-4}} - -Check if you have an idea of how you might code the following tasks: - -\begin{itemize} -\tightlist -\item - What is a GUI? -\item - What do the following commands stand for in shell: \texttt{ls} (or \texttt{dir} in Windows), \texttt{cd}, \texttt{rm}, \texttt{mv} (or \texttt{move} in windows), \texttt{cp} (or \texttt{copy} in Windows). -\item - What is the difference between a relative path and an absolute path? -\item - What paths do these refer to in shell/terminal: \texttt{\textasciitilde{}/}, \texttt{.}, \texttt{..} -\item - What is a \emph{repository} in github? -\item - What does it mean to ``clone'' a repository? -\end{itemize} - -\hypertarget{command-line}{% -\section{command-line}\label{command-line}} - -Elementary programming operations are done on the command-line, or by entering commands into your computer. This is different from a UI or GUI -- graphical user-interface -- which are interfaces that allow you to click buttons and enter commands in more readable form. Although there are good enough GUIs for most of your needs, you still might need to go under the hood sometimes and run a command. - -\hypertarget{command-line-commands}{% -\subsection{command-line commands}\label{command-line-commands}} - -Open up \texttt{Terminal} in a Mac. (\texttt{Command\ Prompt} in Windows) - -Running this command in a Mac (\texttt{dir} in Windows) should show you a list of all files in the directory that you are currently in. - -\begin{Shaded} -\begin{Highlighting}[] -\FunctionTok{ls} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## 01_warmup.Rmd -## 02_functions.Rmd -## 03_limits.Rmd -## 04_calculus.Rmd -## 05_optimization.Rmd -## 06_probability.Rmd -## 07_linear-algebra.Rmd -## 11_data-handling_counting.Rmd -## 12_matricies-manipulation.Rmd -## 13_functions_obj_loops.Rmd -## 14_visualization.Rmd -## 15_project-dempeace.Rmd -## 16_simulation.Rmd -## 17_non-wysiwyg.Rmd -## 18_text.Rmd -## 19_command-line_git.Rmd -## 21_solutions-warmup.Rmd -## 23_solution_programming.Rmd -## CODE_OF_CONDUCT.md -## CONTRIBUTING.md -## DESCRIPTION -## LICENSE -## README.md -## R_exercises -## _book -## _bookdown.yml -## _bookdown_files -## _build.sh -## _deploy.sh -## _output.yml -## data -## images -## index.Rmd -## preamble.tex -## prefresher.Rmd -## prefresher.Rproj -## prefresher.log -## prefresher_files -## rsconnect -## style.css -\end{verbatim} - -\texttt{pwd} stands for present working directory (\texttt{cd} in Windows) - -\begin{Shaded} -\begin{Highlighting}[] -\BuiltInTok{pwd} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## /Users/shirokuriwaki/Dropbox/prefresher -\end{verbatim} - -\texttt{cd} means change directory. You need to give it what to change your current directory \emph{to}. You can specify a name of another directory in your directory. - -Or you can go up to your parent directory. The syntax for that are two periods, \texttt{..} . One period \texttt{.} refers to the current directory. - -\begin{Shaded} -\begin{Highlighting}[] -\BuiltInTok{cd}\NormalTok{ ..} -\BuiltInTok{pwd} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## /Users/shirokuriwaki/Dropbox -\end{verbatim} - -\texttt{\textasciitilde{}/} stands for your home directory defined by your computer. - -\begin{Shaded} -\begin{Highlighting}[] -\BuiltInTok{cd}\NormalTok{ \textasciitilde{}/} -\FunctionTok{ls} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## Applications -## Box -## Desktop -## Documents -## Downloads -## Dropbox -## Google Drive File Stream -## Library -## Movies -## Music -## Pictures -## Projects -## Public -## go -## opt -\end{verbatim} - -Using \texttt{..} and \texttt{.} are ``relative'' to where you are currently at. So are things like \texttt{figures/figure1.pdf}, which is implicitly writing \texttt{./figures/figure1.pdf}. These are called relative paths. In contrast, \texttt{/Users/shirokuriwaki/project1/figures/figure1.pdf} is an ``absolute'' path because it does not start from your current directory. - -Relative paths are nice if you have a shared Dropbox, for example, and I had \texttt{/Users/shirokuriwaki/mathcamp} but Connor's path to the same folder is \texttt{/Users/connorjerzak/mathcamp}. To run the same code in \texttt{mathcamp}, we should be using relative paths that start from ``\texttt{mathcamp}''. Relative paths are also shorter, and they are invariant to higher-level changes in your computer. - -\hypertarget{running-things-via-command-line}{% -\subsection{running things via command-line}\label{running-things-via-command-line}} - -Suppose you have a simple Rscript, call it \texttt{hello\_world.R}. This is simply a plain text file that contains - -\begin{verbatim} -cat("Hello World") -\end{verbatim} - -Then in command-line, go to the directory that contains \texttt{hello\_world.R} and enter - -\begin{Shaded} -\begin{Highlighting}[] -\ExtensionTok{Rscript}\NormalTok{ hello\_world.R} -\end{Highlighting} -\end{Shaded} - -This should give you the output \texttt{Hello\ World}, which verifies that you ``executed'' the file with R via the command-line. - -\hypertarget{why-do-command-line}{% -\subsection{why do command-line?}\label{why-do-command-line}} - -If you know exactly what you want to do your files and the changes are local, then command-line might be faster and be more sensible than navigating yourself through a GUI. For example, what if you wanted a single command that will run 10 R scripts successively at once (as Gentzkow and Shapiro suggest you should do in your research)? It is tedious to run each of your scripts on Rstudio, especially if running some take more than a few minutes. Instead you could write a ``batch'' script that you can run on the terminal, - -\begin{Shaded} -\begin{Highlighting}[] -\ExtensionTok{Rscript}\NormalTok{ 01\_read\_data.R} -\ExtensionTok{Rscript}\NormalTok{ 02\_merge\_data.R} -\ExtensionTok{Rscript}\NormalTok{ 03\_run\_regressions.R} -\ExtensionTok{Rscript}\NormalTok{ 04\_make\_graphs.R} -\ExtensionTok{Rscript}\NormalTok{ 05\_maketable.R} -\end{Highlighting} -\end{Shaded} - -Then run this single file, call it \texttt{run\_all\_Rscripts.sh}, on your terminal as - -\begin{Shaded} -\begin{Highlighting}[] -\FunctionTok{sh}\NormalTok{ run\_all\_Rscripts.sh} -\end{Highlighting} -\end{Shaded} - -On the other hand, command-line prompts may require more keystrokes, and is also less intuitive than a good GUI. It can also be dangerous for beginners, because it can allow you to make large irreversible changes inadvertently. For example, removing a file (\texttt{rm}) has no ``Undo'' feature. - -\hypertarget{git}{% -\section{git}\label{git}} - -Git is a tool for version control. It comes pre-installed on Macs, you will probably need to install it yourself on Windows. - -\hypertarget{why-version-control}{% -\subsection{why version control?}\label{why-version-control}} - -All version control software should be built to - -\begin{itemize} -\tightlist -\item - preserve all snapshots of your work -\item - and catalog them in such a way that you can refer back or even revert back your files to the past snapshot. -\item - makes it easy to see exactly which parts of your files you changed between directories. -\end{itemize} - -Further, git is most commonly used for collaborative work. - -\begin{itemize} -\tightlist -\item - maintains ``branches'', or parallel universes of your files that people can switch back and forth on, doing version control on each one -\item - makes it easy to ``merge'' a sub-branch to a master branch when it is ready. -\end{itemize} - -Note that Dropbox is useful for collaborative work too. But the added value of git's branches is that people can make different changes simultaneously on their computers and merge them to the master branch later. In Dropbox, there is only one copy of each thing so simultaneous editing is not possible. - -\hypertarget{open-source-code-at-your-fingertips}{% -\subsection{open-source code at your fingertips}\label{open-source-code-at-your-fingertips}} - -Some links to check out: - -\begin{itemize} -\tightlist -\item - \url{https://github.com/tidyverse/dplyr} -\item - \url{https://github.com/apple/swift} -\item - \url{https://github.com/kosukeimai/qss} -\end{itemize} - -GitHub \url{https://github.com} is the GUI to git. Making an account there is free. Making an account will allow you to be a part of the collaborative programming community. It will also allow you to ``fork'' other people's ``repositories''. ``Forking'' is making your own copy of the project that forks off from the master project at a point in time. A ``repository'' is simply the name of your main project directory. - -``cloning'' someone else's repository is similar to forking -- it gives you your own copy. - -\hypertarget{commands-in-git}{% -\subsection{commands in git}\label{commands-in-git}} - -As you might have noticed from all the quoted terms, git uses a lot of its own terms that are not intuitive and hard to remember at first. The nuts and bolts of maintaining your version control further requires ``adding'', ``committing'', and ``push''ing, sometimes ``pull''ing. - -The tutorial \url{https://try.github.io/} is quite good. You'd want to have familiarity with command-line to fully understand this and use it in your work. - -RStudio Projects has a great git GUI as well. - -\hypertarget{is-git-worth-it}{% -\subsection{is git worth it?}\label{is-git-worth-it}} - -While git is a powerful tool, you may choose to not use it for everything because - -\begin{itemize} -\tightlist -\item - git is mainly for code, not data. It has a fairly strict limit on the size of your dataset that you cover. -\item - your collaborators might want to work with Dropbox -\item - unless you get a paid account, all your repositories will be public. -\end{itemize} - -\hypertarget{part-solutions}{% -\part{Solutions}\label{part-solutions}} - -\hypertarget{solutions-to-warmup-questions}{% -\chapter*{Solutions to Warmup Questions}\label{solutions-to-warmup-questions}} -\addcontentsline{toc}{chapter}{Solutions to Warmup Questions} - -\hypertarget{linear-algebra-1}{% -\section*{Linear Algebra}\label{linear-algebra-1}} -\addcontentsline{toc}{section}{Linear Algebra} - -\hypertarget{vectors-1}{% -\subsection*{Vectors}\label{vectors-1}} -\addcontentsline{toc}{subsection}{Vectors} - -Define the vectors \(u = \begin{pmatrix} 1 \\2 \\3 \end{pmatrix}\), \(v = \begin{pmatrix} 4\\5\\6 \end{pmatrix}\), and the scalar \(c = 2\). - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(u + v = \begin{pmatrix}5\\7\\9\end{pmatrix}\) -\item - \(cv = \begin{pmatrix}8\\10\\12\end{pmatrix}\) -\item - \(u \cdot v = 1(4) + 2(5) + 3(6) = 32\) -\end{enumerate} - -If you are having trouble with these problems, please review Section \ref{vector-def} ``Working with Vectors'' in Chapter \ref{linearalgebra}. - -Are the following sets of vectors linearly independent? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(u = \begin{pmatrix} 1\\ 2\end{pmatrix}\), \(v = \begin{pmatrix} 2\\4\end{pmatrix}\) -\end{enumerate} - -\(\leadsto\) No: \[2u = \begin{pmatrix} 2\\ 4\end{pmatrix}, v = \begin{pmatrix} 2\\ 4\end{pmatrix}\] -so infinitely many linear combinations of \(u\) and \(v\) that amount to 0 exist. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{1} -\tightlist -\item - \(u = \begin{pmatrix} 1\\ 2\\ 5 \end{pmatrix}\), \(v = \begin{pmatrix} 3\\ 7\\ 9 \end{pmatrix}\) -\end{enumerate} - -\(\leadsto\) Yes: we cannot find linear combination of these two vectors that would amount to zero. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\setcounter{enumi}{2} -\tightlist -\item - \(a = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}\), \(b = \begin{pmatrix} 3\\ -4\\ -2 \end{pmatrix}\), \(c = \begin{pmatrix} 5\\ -10\\ -8 \end{pmatrix}\) -\end{enumerate} - -\(\leadsto\) No: After playing around with some numbers, we can find that -\[-2a = \begin{pmatrix} -4\\ 2\\ -2 \end{pmatrix}, 3b = \begin{pmatrix} 9\\ -12\\ -6 \end{pmatrix}, -1c = \begin{pmatrix} -5\\ 10\\ 8 \end{pmatrix}\] - -So -\[-2a + 3b - c = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\] - -i.e., a linear combination of these three vectors that would amount to zero exists. - -If you are having trouble with these problems, please review Section \ref{linearindependence}. - -\hypertarget{matrices-1}{% -\subsection*{Matrices}\label{matrices-1}} -\addcontentsline{toc}{subsection}{Matrices} - -\[{\bf A}=\begin{pmatrix} - 7 & 5 & 1 \\ - 11 & 9 & 3 \\ - 2 & 14 & 21 \\ - 4 & 1 & 5 - \end{pmatrix}\] - -What is the dimensionality of matrix \({\bf A}\)? 4 \(\times\) 3 - -What is the element \(a_{23}\) of \({\bf A}\)? 3 - -Given that - -\[{\bf B}=\begin{pmatrix} - 1 & 2 & 8 \\ - 3 & 9 & 11 \\ - 4 & 7 & 5 \\ - 5 & 1 & 9 - \end{pmatrix}\] - -\[\mathbf{A} + \mathbf{B} = \begin{pmatrix} - 8 & 7 & 9 \\ - 14 & 18 & 14 \\ - 6 & 21 & 26 \\ - 9 & 2 & 14 - \end{pmatrix}\] - -Given that - -\[{\bf C}=\begin{pmatrix} - 1 & 2 & 8 \\ - 3 & 9 & 11 \\ - 4 & 7 & 5 \\ - \end{pmatrix}\] - -\[\mathbf{A} + \mathbf{C} = \text{No solution, matrices non-conformable}\] - -Given that - -\[c = 2\] - -\[c\textbf{A} = \begin{pmatrix} - 14 & 10 & 2 \\ - 22 & 18 & 6 \\ - 4 & 28 & 42 \\ - 8 & 2 & 10 - \end{pmatrix}\] - -If you are having trouble with these problems, please review Section \ref{matrixbasics}. - -\hypertarget{operations-1}{% -\section*{Operations}\label{operations-1}} -\addcontentsline{toc}{section}{Operations} - -\hypertarget{summation-1}{% -\subsection*{Summation}\label{summation-1}} -\addcontentsline{toc}{subsection}{Summation} - -Simplify the following - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\sum\limits_{i = 1}^3 i = 1 + 2+ 3 = 6\) -\item - \(\sum\limits_{k = 1}^3(3k + 2) = 3\sum\limits_{k=1}^3k + \sum\limits_{k=1}^3 2= 3\times 6 + 3\times 2 = 24\) -\item - \(\sum\limits_{i= 1}^4 (3k + i + 2) = 3\sum\limits_{i= 1}^4k + \sum\limits_{i= 1}^4i + \sum\limits_{i= 1}^42 = 12k + 10 + 8 = 12k + 18\) -\end{enumerate} - -\hypertarget{products-1}{% -\subsection*{Products}\label{products-1}} -\addcontentsline{toc}{subsection}{Products} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \(\prod\limits_{i= 1}^3 i = 1\cdot 2\cdot 3 = 6\) -\item - \(\prod\limits_{k=1}^3(3k + 2) = (3 + 2)\cdot (6 + 2)\cdot (9 + 2) = 440\) -\end{enumerate} - -To review this material, please see Section \ref{sum-notation}. - -\hypertarget{logs-and-exponents-1}{% -\subsection*{Logs and exponents}\label{logs-and-exponents-1}} -\addcontentsline{toc}{subsection}{Logs and exponents} - -Simplify the following - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(4^2 = 16\) -\item - \(4^2 2^3 = 2^{2\cdot 2}2^{3} = 2^{4 + 3} = 128\) -\item - \(\log_{10}100 = \log_{10}10^2 = 2\) -\item - \(\log_{2}4 = \log_{2}2^2 = 2\) -\item - when \(\log\) is the natural log, \(\log e = \log_{e} e^1 = 1\) -\item - when \(a, b, c\) are each constants, \(e^a e^b e^c = e^{a + b + c}\), -\item - \(\log 0 = \text{undefined}\) -- no exponentiation of anything will result in a 0. -\item - \(e^0 = 1\) -- any number raised to the 0 is always 1. -\item - \(e^1 = e\) -- any number raised to the 1 is always itself -\item - \(\log e^2 = \log_e e^2 = 2\) -\end{enumerate} - -To review this material, please see Section \ref{logexponents} - -\hypertarget{limits-1}{% -\section*{Limits}\label{limits-1}} -\addcontentsline{toc}{section}{Limits} - -Find the limit of the following. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(\lim\limits_{x \to 2} (x - 1) = 1\) -\item - \(\lim\limits_{x \to 2} \frac{(x - 2) (x - 1)}{(x - 2)} = 1\), though note that the original function \(\frac{(x - 2) (x - 1)}{(x - 2)}\) would have been undefined at \(x = 2\) because of a divide by zero problem; otherwise it would have been equal to \(x - 1\). -\item - \(\lim\limits_{x \to 2}\frac{x^2 - 3x + 2}{x- 2} = 1\), same as above. -\end{enumerate} - -To review this material please see Section \ref{limitsfun} - -\hypertarget{calculus-1}{% -\section*{Calculus}\label{calculus-1}} -\addcontentsline{toc}{section}{Calculus} - -For each of the following functions \(f(x)\), find the derivative \(f'(x)\) or \(\frac{d}{dx}f(x)\) - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x)=c\), \(f'(x) = 0\) -\item - \(f(x)=x\), \(f'(x) = 1\) -\item - \(f(x)=x^2\), \(f'(x) = 2x\) -\item - \(f(x)=x^3\), \(f'(x) = 3x^2\) -\item - \(f(x)=3x^2+2x^{1/3}\), \(f'(x) = 6x + \frac{2}{3}x^{-2/3}\) -\item - \(f(x)=(x^3)(2x^4)\), \(f'(x) = \frac{d}{dx}2x^7 = 14x^6\) -\end{enumerate} - -For a review, please see Section \ref{derivintro} - \ref{derivpoly} - -\hypertarget{optimization-1}{% -\section*{Optimization}\label{optimization-1}} -\addcontentsline{toc}{section}{Optimization} - -For each of the followng functions \(f(x)\), does a maximum and minimum exist in the domain \(x \in \mathbf{R}\)? If so, for what are those values and for which values of \(x\)? - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(f(x) = x\) \(\leadsto\) neither exists. -\item - \(f(x) = x^2\) \(\leadsto\) a minimum \(f(x) = 0\) exists at \(x = 0\), but not a maximum. -\item - \(f(x) = -(x - 2)^2\) \(\leadsto\) a maximum \(f(x) = 0\) exists at \(x = 2\), but not a minimum. -\end{enumerate} - -If you are stuck, please try sketching out a picture of each of the functions. - -\hypertarget{probability-1}{% -\section*{Probability}\label{probability-1}} -\addcontentsline{toc}{section}{Probability} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - If there are 12 cards, numbered 1 to 12, and 4 cards are chosen, \(\binom{12}{4} = \frac{12\cdot 11\cdot 10\cdot 9}{4!} = 495\) possible hands exist (unordered, without replacement) . -\item - Let \(A = \{1,3,5,7,8\}\) and \(B = \{2,4,7,8,12,13\}\). Then \(A \cup B = \{1, 2, 3, 4, 5, 7, 8, 12, 13\}\), \(A \cap B = \{7, 8\}\)? If \(A\) is a subset of the Sample Space \(S = \{1,2,3,4,5,6,7,8,9,10\}\), then the complement \(A^C = \{2, 4, 6, 9, 10\}\) -\item - If we roll two fair dice, what is the probability that their sum would be 11? \(\leadsto \frac{1}{18}\) -\item - If we roll two fair dice, what is the probability that their sum would be 12? \(\leadsto \frac{1}{36}\). There are two independent dice, so \(6^2 = 36\) options in total. While the previous question had two possibilities for a sum of 11 (5,6 and 6,5), there is only one possibility out of 36 for a sum of 12 (6,6). -\end{enumerate} - -For a review, please see Sections \ref{setoper} - \ref{probdef} - -\hypertarget{suggested-programming-solutions}{% -\chapter*{Suggested Programming Solutions}\label{suggested-programming-solutions}} -\addcontentsline{toc}{chapter}{Suggested Programming Solutions} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{library}\NormalTok{(dplyr)} -\KeywordTok{library}\NormalTok{(readr)} -\KeywordTok{library}\NormalTok{(ggplot2)} -\KeywordTok{library}\NormalTok{(ggrepel)} -\KeywordTok{library}\NormalTok{(forcats)} -\KeywordTok{library}\NormalTok{(scales)} -\end{Highlighting} -\end{Shaded} - -\hypertarget{chapter-refdataviz-visualization}{% -\section{Chapter \ref{dataviz}: Visualization}\label{chapter-refdataviz-visualization}} - -\hypertarget{state-proportions}{% -\subsection*{1 State Proportions}\label{state-proportions}} -\addcontentsline{toc}{subsection}{1 State Proportions} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 <{-}}\StringTok{ }\KeywordTok{readRDS}\NormalTok{(}\StringTok{"data/input/usc2010\_001percent.Rds"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -Group by state, noting that the mean of a set of logicals is a mean of 1s (\texttt{TRUE}) and 0s (\texttt{FALSE}). - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{grp\_st <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{group\_by}\NormalTok{(state) }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{summarize}\NormalTok{(}\DataTypeTok{prop =} \KeywordTok{mean}\NormalTok{(age }\OperatorTok{>=}\StringTok{ }\DecValTok{65}\NormalTok{)) }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{arrange}\NormalTok{(prop) }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{mutate}\NormalTok{(}\DataTypeTok{state =} \KeywordTok{as\_factor}\NormalTok{(state))} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## `summarise()` ungrouping output (override with `.groups` argument) -\end{verbatim} - -Plot points - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ggplot}\NormalTok{(grp\_st, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ state, }\DataTypeTok{y =}\NormalTok{ prop)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_point}\NormalTok{() }\OperatorTok{+} -\StringTok{ }\KeywordTok{coord\_flip}\NormalTok{() }\OperatorTok{+} -\StringTok{ }\KeywordTok{scale\_y\_continuous}\NormalTok{(}\DataTypeTok{labels =} \KeywordTok{percent\_format}\NormalTok{(}\DataTypeTok{accuracy =} \DecValTok{1}\NormalTok{)) }\OperatorTok{+}\StringTok{ }\CommentTok{\# use the scales package to format percentages} -\StringTok{ }\KeywordTok{labs}\NormalTok{(} - \DataTypeTok{y =} \StringTok{"Proportion Senior"}\NormalTok{,} - \DataTypeTok{x =} \StringTok{""}\NormalTok{,} - \DataTypeTok{caption =} \StringTok{"Source: 2010 Census sample"} -\NormalTok{ )} -\end{Highlighting} -\end{Shaded} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-325-1.pdf} - -\hypertarget{swing-justice}{% -\subsection*{2 Swing Justice}\label{swing-justice}} -\addcontentsline{toc}{subsection}{2 Swing Justice} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{justices <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/justices\_court{-}median.csv"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -Keep justices who are in the dataset in 2016, - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{in\_}\DecValTok{2017}\NormalTok{ <{-}}\StringTok{ }\NormalTok{justices }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{filter}\NormalTok{(term }\OperatorTok{>=}\StringTok{ }\DecValTok{2016}\NormalTok{) }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{distinct}\NormalTok{(justice) }\OperatorTok{\%>\%}\StringTok{ }\CommentTok{\# unique values} -\StringTok{ }\KeywordTok{mutate}\NormalTok{(}\DataTypeTok{present\_2016 =} \DecValTok{1}\NormalTok{) }\CommentTok{\# keep an indicator to distinguish from rest after merge} - -\NormalTok{df\_indicator <{-}}\StringTok{ }\NormalTok{justices }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{left\_join}\NormalTok{(in\_}\DecValTok{2017}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## Joining, by = "justice" -\end{verbatim} - -All together - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{ggplot}\NormalTok{(df\_indicator, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ term, }\DataTypeTok{y =}\NormalTok{ idealpt, }\DataTypeTok{group =}\NormalTok{ justice\_id)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_line}\NormalTok{(}\KeywordTok{aes}\NormalTok{(}\DataTypeTok{y =}\NormalTok{ median\_idealpt), }\DataTypeTok{color =} \StringTok{"red"}\NormalTok{, }\DataTypeTok{size =} \DecValTok{2}\NormalTok{, }\DataTypeTok{alpha =} \FloatTok{0.1}\NormalTok{) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_line}\NormalTok{(}\DataTypeTok{alpha =} \FloatTok{0.5}\NormalTok{) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_line}\NormalTok{(}\DataTypeTok{data =} \KeywordTok{filter}\NormalTok{(df\_indicator, }\OperatorTok{!}\KeywordTok{is.na}\NormalTok{(present\_}\DecValTok{2016}\NormalTok{))) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_point}\NormalTok{(}\DataTypeTok{data =} \KeywordTok{filter}\NormalTok{(df\_indicator, }\OperatorTok{!}\KeywordTok{is.na}\NormalTok{(present\_}\DecValTok{2016}\NormalTok{), term }\OperatorTok{==}\StringTok{ }\DecValTok{2018}\NormalTok{)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_text\_repel}\NormalTok{(} - \DataTypeTok{data =} \KeywordTok{filter}\NormalTok{(df\_indicator, term }\OperatorTok{==}\StringTok{ }\DecValTok{2016}\NormalTok{), }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{label =}\NormalTok{ justice),} - \DataTypeTok{nudge\_x =} \DecValTok{10}\NormalTok{,} - \DataTypeTok{direction =} \StringTok{"y"} -\NormalTok{ ) }\OperatorTok{+}\StringTok{ }\CommentTok{\# labels nudged and vertical} -\StringTok{ }\KeywordTok{scale\_x\_continuous}\NormalTok{(}\DataTypeTok{breaks =} \KeywordTok{seq}\NormalTok{(}\DecValTok{1940}\NormalTok{, }\DecValTok{2020}\NormalTok{, }\DecValTok{10}\NormalTok{), }\DataTypeTok{limits =} \KeywordTok{c}\NormalTok{(}\DecValTok{1937}\NormalTok{, }\DecValTok{2020}\NormalTok{)) }\OperatorTok{+}\StringTok{ }\CommentTok{\# axis breaks} -\StringTok{ }\KeywordTok{scale\_y\_continuous}\NormalTok{(}\DataTypeTok{limits =} \KeywordTok{c}\NormalTok{(}\OperatorTok{{-}}\DecValTok{5}\NormalTok{, }\DecValTok{5}\NormalTok{)) }\OperatorTok{+}\StringTok{ }\CommentTok{\# axis limits} -\StringTok{ }\KeywordTok{labs}\NormalTok{(} - \DataTypeTok{x =} \StringTok{"SCOTUS Term"}\NormalTok{,} - \DataTypeTok{y =} \StringTok{"Estimated Martin{-}Quinn Ideal Point"}\NormalTok{,} - \DataTypeTok{caption =} \StringTok{"Outliers capped at {-}5 to 5. Red lines indicate median justice. Current justices of the 2017 Court in black."} -\NormalTok{ ) }\OperatorTok{+} -\StringTok{ }\KeywordTok{theme\_bw}\NormalTok{()} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## Warning: Removed 19 row(s) containing missing values (geom_path). -\end{verbatim} - -\includegraphics{prefresher_files/figure-latex/unnamed-chunk-328-1.pdf} - -\hypertarget{chapter-refrobjloops-objects-and-loops}{% -\section{Chapter \ref{robjloops}: Objects and Loops}\label{chapter-refrobjloops-objects-and-loops}} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/usc2010\_001percent.csv"}\NormalTok{)} -\NormalTok{sample\_acs <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/acs2015\_1percent.csv"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\hypertarget{checkpoint-3}{% -\subsection*{Checkpoint \#3}\label{checkpoint-3}} -\addcontentsline{toc}{subsection}{Checkpoint \#3} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{cen10 }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{group\_by}\NormalTok{(state) }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{summarise}\NormalTok{(}\DataTypeTok{avg\_age =} \KeywordTok{mean}\NormalTok{(age)) }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{arrange}\NormalTok{(}\KeywordTok{desc}\NormalTok{(avg\_age)) }\OperatorTok{\%>\%} -\StringTok{ }\KeywordTok{slice}\NormalTok{(}\DecValTok{1}\OperatorTok{:}\DecValTok{10}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## `summarise()` ungrouping output (override with `.groups` argument) -\end{verbatim} - -\begin{verbatim} -## # A tibble: 10 x 2 -## state avg_age -## -## 1 West Virginia 44.1 -## 2 Maine 42.1 -## 3 Florida 41.3 -## 4 New Hampshire 41.2 -## 5 North Dakota 41.1 -## 6 Montana 40.6 -## 7 Vermont 40.3 -## 8 Connecticut 40.1 -## 9 Wisconsin 39.9 -## 10 New Mexico 39.3 -\end{verbatim} - -\hypertarget{exercise-2}{% -\subsection*{Exercise 2}\label{exercise-2}} -\addcontentsline{toc}{subsection}{Exercise 2} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{states\_of\_interest <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{(}\StringTok{"California"}\NormalTok{, }\StringTok{"Massachusetts"}\NormalTok{, }\StringTok{"New Hampshire"}\NormalTok{, }\StringTok{"Washington"}\NormalTok{)} - -\ControlFlowTok{for}\NormalTok{ (state\_i }\ControlFlowTok{in}\NormalTok{ states\_of\_interest) \{} -\NormalTok{ state\_subset <{-}}\StringTok{ }\NormalTok{cen10 }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(state }\OperatorTok{==}\StringTok{ }\NormalTok{state\_i)} - - \KeywordTok{print}\NormalTok{(state\_i)} - - \KeywordTok{print}\NormalTok{(}\KeywordTok{table}\NormalTok{(state\_subset}\OperatorTok{$}\NormalTok{race, state\_subset}\OperatorTok{$}\NormalTok{sex))} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] "California" -## -## Female Male -## American Indian or Alaska Native 21 21 -## Black/Negro 127 126 -## Chinese 76 65 -## Japanese 15 12 -## Other Asian or Pacific Islander 182 177 -## Other race, nec 283 302 -## Three or more major races 7 7 -## Two major races 91 83 -## White 1085 1083 -## [1] "Massachusetts" -## -## Female Male -## American Indian or Alaska Native 4 1 -## Black/Negro 21 17 -## Chinese 8 7 -## Japanese 1 1 -## Other Asian or Pacific Islander 14 14 -## Other race, nec 9 17 -## Two major races 10 8 -## White 272 243 -## [1] "New Hampshire" -## -## Female Male -## American Indian or Alaska Native 1 0 -## Black/Negro 0 1 -## Chinese 0 1 -## Japanese 1 0 -## Other Asian or Pacific Islander 2 1 -## Other race, nec 1 0 -## Two major races 0 1 -## White 66 63 -## [1] "Washington" -## -## Female Male -## American Indian or Alaska Native 9 5 -## Black/Negro 11 9 -## Chinese 2 7 -## Japanese 4 0 -## Other Asian or Pacific Islander 28 18 -## Other race, nec 19 18 -## Three or more major races 0 2 -## Two major races 17 16 -## White 267 257 -\end{verbatim} - -\hypertarget{exercise-3}{% -\subsection*{Exercise 3}\label{exercise-3}} -\addcontentsline{toc}{subsection}{Exercise 3} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{race\_d <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{()} -\NormalTok{state\_d <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{()} -\NormalTok{proportion\_d <{-}}\StringTok{ }\KeywordTok{c}\NormalTok{()} -\NormalTok{answer <{-}}\StringTok{ }\KeywordTok{data.frame}\NormalTok{(race\_d, state\_d, proportion\_d)} -\end{Highlighting} -\end{Shaded} - -Then - -\begin{Shaded} -\begin{Highlighting}[] -\ControlFlowTok{for}\NormalTok{ (state }\ControlFlowTok{in}\NormalTok{ states\_of\_interest) \{} - \ControlFlowTok{for}\NormalTok{ (race }\ControlFlowTok{in} \KeywordTok{unique}\NormalTok{(cen10}\OperatorTok{$}\NormalTok{race)) \{} -\NormalTok{ race\_state\_num <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(cen10[cen10}\OperatorTok{$}\NormalTok{race }\OperatorTok{==}\StringTok{ }\NormalTok{race }\OperatorTok{\&}\StringTok{ }\NormalTok{cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state, ])} -\NormalTok{ state\_pop <{-}}\StringTok{ }\KeywordTok{nrow}\NormalTok{(cen10[cen10}\OperatorTok{$}\NormalTok{state }\OperatorTok{==}\StringTok{ }\NormalTok{state, ])} -\NormalTok{ race\_perc <{-}}\StringTok{ }\KeywordTok{round}\NormalTok{(}\DecValTok{100} \OperatorTok{*}\StringTok{ }\NormalTok{(race\_state\_num }\OperatorTok{/}\StringTok{ }\NormalTok{(state\_pop)), }\DataTypeTok{digits =} \DecValTok{2}\NormalTok{)} -\NormalTok{ line <{-}}\StringTok{ }\KeywordTok{data.frame}\NormalTok{(}\DataTypeTok{race\_d =}\NormalTok{ race, }\DataTypeTok{state\_d =}\NormalTok{ state, }\DataTypeTok{proportion\_d =}\NormalTok{ race\_perc)} -\NormalTok{ answer <{-}}\StringTok{ }\KeywordTok{rbind}\NormalTok{(answer, line)} -\NormalTok{ \}} -\NormalTok{\}} -\end{Highlighting} -\end{Shaded} - -\hypertarget{chapter-refdempeace-demoratic-peace-project}{% -\section{Chapter \ref{dempeace}: Demoratic Peace Project}\label{chapter-refdempeace-demoratic-peace-project}} - -\hypertarget{task-1-data-input-and-standardization-1}{% -\subsection*{Task 1: Data Input and Standardization}\label{task-1-data-input-and-standardization-1}} -\addcontentsline{toc}{subsection}{Task 1: Data Input and Standardization} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{mid\_b <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/MIDB\_4.2.csv"}\NormalTok{)} -\NormalTok{polity <{-}}\StringTok{ }\KeywordTok{read\_excel}\NormalTok{(}\StringTok{"data/input/p4v2017.xls"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\hypertarget{task-2-data-merging-1}{% -\subsection*{Task 2: Data Merging}\label{task-2-data-merging-1}} -\addcontentsline{toc}{subsection}{Task 2: Data Merging} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{mid\_y\_by\_y <{-}}\StringTok{ }\KeywordTok{data\_frame}\NormalTok{(}\DataTypeTok{ccode =} \KeywordTok{numeric}\NormalTok{(),} - \DataTypeTok{year =} \KeywordTok{numeric}\NormalTok{(),} - \DataTypeTok{dispute =} \KeywordTok{numeric}\NormalTok{())} -\KeywordTok{colnames}\NormalTok{(mid\_b)} -\ControlFlowTok{for}\NormalTok{(i }\ControlFlowTok{in} \DecValTok{1}\OperatorTok{:}\KeywordTok{nrow}\NormalTok{(mid\_b)) \{} -\NormalTok{ x <{-}}\StringTok{ }\KeywordTok{data\_frame}\NormalTok{(}\DataTypeTok{ccode =}\NormalTok{ mid\_b}\OperatorTok{$}\NormalTok{ccode[i], }\CommentTok{\#\# row i\textquotesingle{}s country} - \DataTypeTok{year =}\NormalTok{ mid\_b}\OperatorTok{$}\NormalTok{styear[i]}\OperatorTok{:}\NormalTok{mid\_b}\OperatorTok{$}\NormalTok{endyear[i], }\CommentTok{\#\# sequence of years for dispute in row i} - \DataTypeTok{dispute =} \DecValTok{1}\NormalTok{)}\CommentTok{\#\# there was a dispute} -\NormalTok{ mid\_y\_by\_y <{-}}\StringTok{ }\KeywordTok{rbind}\NormalTok{(mid\_y\_by\_y, x)} -\NormalTok{\}} - -\NormalTok{merged\_mid\_polity <{-}}\StringTok{ }\KeywordTok{left\_join}\NormalTok{(polity,} - \KeywordTok{distinct}\NormalTok{(mid\_y\_by\_y),} - \DataTypeTok{by =} \KeywordTok{c}\NormalTok{(}\StringTok{"ccode"}\NormalTok{, }\StringTok{"year"}\NormalTok{))} -\end{Highlighting} -\end{Shaded} - -\hypertarget{task-3-tabulations-and-visualization-1}{% -\subsection*{Task 3: Tabulations and Visualization}\label{task-3-tabulations-and-visualization-1}} -\addcontentsline{toc}{subsection}{Task 3: Tabulations and Visualization} - -\begin{Shaded} -\begin{Highlighting}[] -\CommentTok{\#don\textquotesingle{}t include the {-}88, {-}77, {-}66 values in calculating the mean of polity} -\NormalTok{mean\_polity\_by\_year <{-}}\StringTok{ }\NormalTok{merged\_mid\_polity }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{group\_by}\NormalTok{(year) }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{summarise}\NormalTok{(}\DataTypeTok{mean\_polity =} \KeywordTok{mean}\NormalTok{(polity[}\KeywordTok{which}\NormalTok{(polity }\OperatorTok{<}\DecValTok{11} \OperatorTok{\&}\StringTok{ }\NormalTok{polity }\OperatorTok{>}\StringTok{ }\DecValTok{{-}11}\NormalTok{)]))} - -\NormalTok{mean\_polity\_by\_year\_ordered <{-}}\StringTok{ }\KeywordTok{arrange}\NormalTok{(mean\_polity\_by\_year, year) } - -\NormalTok{mean\_polity\_by\_year\_mid <{-}}\StringTok{ }\NormalTok{merged\_mid\_polity }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{group\_by}\NormalTok{(year, dispute) }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{summarise}\NormalTok{(}\DataTypeTok{mean\_polity\_mid =} \KeywordTok{mean}\NormalTok{(polity[}\KeywordTok{which}\NormalTok{(polity }\OperatorTok{<}\DecValTok{11} \OperatorTok{\&}\StringTok{ }\NormalTok{polity }\OperatorTok{>}\StringTok{ }\DecValTok{{-}11}\NormalTok{)]))} - -\NormalTok{mean\_polity\_by\_year\_mid\_ordered <{-}}\StringTok{ }\KeywordTok{arrange}\NormalTok{(mean\_polity\_by\_year\_mid, year) } - -\NormalTok{mean\_polity\_no\_mid <{-}}\StringTok{ }\NormalTok{mean\_polity\_by\_year\_mid\_ordered }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(dispute }\OperatorTok{==}\StringTok{ }\DecValTok{0}\NormalTok{)} -\NormalTok{mean\_polity\_yes\_mid <{-}}\StringTok{ }\NormalTok{mean\_polity\_by\_year\_mid\_ordered }\OperatorTok{\%>\%}\StringTok{ }\KeywordTok{filter}\NormalTok{(dispute }\OperatorTok{==}\StringTok{ }\DecValTok{1}\NormalTok{)} - - -\NormalTok{answer <{-}}\StringTok{ }\KeywordTok{ggplot}\NormalTok{(}\DataTypeTok{data =}\NormalTok{ mean\_polity\_by\_year\_ordered, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ year, }\DataTypeTok{y =}\NormalTok{ mean\_polity)) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_line}\NormalTok{() }\OperatorTok{+} -\StringTok{ }\KeywordTok{labs}\NormalTok{(}\DataTypeTok{y =} \StringTok{"Mean Polity Score"}\NormalTok{,} - \DataTypeTok{x =} \StringTok{""}\NormalTok{) }\OperatorTok{+} -\StringTok{ }\KeywordTok{geom\_vline}\NormalTok{(}\DataTypeTok{xintercept =} \KeywordTok{c}\NormalTok{(}\DecValTok{1914}\NormalTok{, }\DecValTok{1929}\NormalTok{, }\DecValTok{1939}\NormalTok{, }\DecValTok{1989}\NormalTok{, }\DecValTok{2008}\NormalTok{), }\DataTypeTok{linetype =} \StringTok{"dashed"}\NormalTok{)} - -\NormalTok{answer }\OperatorTok{+}\StringTok{ }\KeywordTok{geom\_line}\NormalTok{(}\DataTypeTok{data =}\NormalTok{mean\_polity\_no\_mid, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ year, }\DataTypeTok{y =}\NormalTok{ mean\_polity\_mid), }\DataTypeTok{col =} \StringTok{"indianred"}\NormalTok{) }\OperatorTok{+}\StringTok{ }\KeywordTok{geom\_line}\NormalTok{(}\DataTypeTok{data =}\NormalTok{mean\_polity\_yes\_mid, }\KeywordTok{aes}\NormalTok{(}\DataTypeTok{x =}\NormalTok{ year, }\DataTypeTok{y =}\NormalTok{ mean\_polity\_mid), }\DataTypeTok{col =} \StringTok{"dodgerblue"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\hypertarget{chapter-refsimulation-simulation}{% -\section{Chapter \ref{simulation}: Simulation}\label{chapter-refsimulation-simulation}} - -\hypertarget{census-sampling-1}{% -\subsection{Census Sampling}\label{census-sampling-1}} - -\begin{Shaded} -\begin{Highlighting}[] -\NormalTok{pop <{-}}\StringTok{ }\KeywordTok{read\_csv}\NormalTok{(}\StringTok{"data/input/usc2010\_001percent.csv"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## Parsed with column specification: -## cols( -## state = col_character(), -## sex = col_character(), -## age = col_double(), -## race = col_character() -## ) -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{mean}\NormalTok{(pop}\OperatorTok{$}\NormalTok{race }\OperatorTok{!=}\StringTok{ "White"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - -\begin{verbatim} -## [1] 0.2806517 -\end{verbatim} - -\begin{Shaded} -\begin{Highlighting}[] -\KeywordTok{set.seed}\NormalTok{(}\DecValTok{1669482}\NormalTok{)} -\NormalTok{samp <{-}}\StringTok{ }\KeywordTok{sample\_n}\NormalTok{(pop, }\DecValTok{100}\NormalTok{)} -\KeywordTok{mean}\NormalTok{(samp}\OperatorTok{$}\NormalTok{race }\OperatorTok{!=}\StringTok{ "White"}\NormalTok{)} -\end{Highlighting} -\end{Shaded} - 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-\KeywordTok{mean}\NormalTok{(ests)} -\end{Highlighting} -\end{Shaded} - - -\end{document} diff --git a/_book/style.css b/_book/style.css deleted file mode 100644 index f317b43..0000000 --- a/_book/style.css +++ /dev/null @@ -1,14 +0,0 @@ -p.caption { - color: #777; - margin-top: 10px; -} -p code { - white-space: inherit; -} -pre { - word-break: normal; - word-wrap: normal; -} -pre code { - white-space: inherit; -} From f986ee573e486a666c2a4cca04d35f3e8ab9ec94 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Thu, 4 Jul 2024 23:36:16 -0400 Subject: [PATCH 26/34] all refs fixed --- 02_functions.qmd | 2 +- 03_limits.qmd | 2 +- 04_calculus.qmd | 4 ++-- 06_probability.qmd | 8 ++++---- 16_simulation.qmd | 2 +- 5 files changed, 9 insertions(+), 9 deletions(-) diff --git a/02_functions.qmd b/02_functions.qmd index ac98124..33eab42 100644 --- a/02_functions.qmd +++ b/02_functions.qmd @@ -138,7 +138,7 @@ The degree of a polynomial is the highest degree of its monomial terms. Also, it **Exponential Functions**: Example: $y=2^x$ -## $\log$ and $\exp$ {#logexponents} +## $\log$ and $\exp$ {#sec-logexponents} **Relationship of logarithmic and exponential functions**: $$y=\log_a(x) \iff a^y=x$$ diff --git a/03_limits.qmd b/03_limits.qmd index f0c0d75..3108661 100644 --- a/03_limits.qmd +++ b/03_limits.qmd @@ -155,7 +155,7 @@ Find the following limits of sequences, then explain in English the intuition fo 2. $\lim\limits_{n\to\infty} (n^3 - 100n^2)$ ::: -## Limits of a Function {#limitsfun} +## Limits of a Function {#sec-limitsfun} We've now covered functions and just covered limits of sequences, so now is the time to combine the two. diff --git a/04_calculus.qmd b/04_calculus.qmd index eb9b28a..8d6c790 100644 --- a/04_calculus.qmd +++ b/04_calculus.qmd @@ -28,7 +28,7 @@ even more concretely, if the potential values of $X$ are finite, then we can wri $$E(X) = \large \sum_{x} \quad\left( \underbrace{x}_{\text{value}}\cdot \underbrace{P(X = x)}_{\text{weight, or PMF}}\right)$$ -## Derivatives {#derivintro} +## Derivatives {#sec-derivintro} The derivative of $f$ at $x$ is its rate of change at $x$: how much $f(x)$ changes with a change in $x$. The rate of change is a fraction --- rise over run --- but because not all lines are straight and the rise over run formula will give us different values depending on the range we examine, we need to take a limit (Section @sec-limits-precalc). @@ -148,7 +148,7 @@ For each of the following functions, find the first-order derivative $f^\prime(x 10. $f(x)=\frac{x^2+1}{x^2-1}$ ::: -## Higher-Order Derivatives (Derivatives of Derivatives of Derivatives) {#derivpoly} +## Higher-Order Derivatives (Derivatives of Derivatives of Derivatives) {#sec-derivpoly} The first derivative is applying the definition of derivatives on the function, and it can be expressed as diff --git a/06_probability.qmd b/06_probability.qmd index 89104eb..e1c1ed1 100644 --- a/06_probability.qmd +++ b/06_probability.qmd @@ -62,7 +62,7 @@ There are five balls numbered from 1 through 5 in a jar. Three balls are chosen. Four cards are selected from a deck of 52 cards. Once a card has been drawn, it is not reshuffled back into the deck. Moreover, we care only about the complete hand that we get (i.e. we care about the set of selected cards, not the sequence in which it was drawn). How many possible outcomes are there? ::: -## Sets {#setoper} +## Sets {#sec-setoper} Probability is about quantifying the uncertainty of events. *Sets* (set theory) are the mathematical way we choose to formalize those events. Events are not inherently numerical: the onset of war or the stock market crashing is not inherently a number. Sets can define such events, and we wrap math around so that we have a transparent language to communicate about those events. *Measure theory* might sound mysterious or hard, but it is also just a mathematical way to quantify things like length, volume, and mass. Probability can be thought of as a particular application of measure theory where we want to quantify the measure of a set. @@ -119,10 +119,10 @@ Consider subsets A {2, 8} and B {2,3,7} of the sample space you found. What is 2. $(A \cup B)^c$ ::: -## Probability {#probdef} +## Probability {#sec-probdef} ```{r} -#| label: prob-image +#| label: fig-prob-image #| echo: false #| fig-cap: Probablity as a Measure^[Images of Probability and Random Variables drawn #| by Shiro Kuriwaki and inspired by Blitzstein and Morris] @@ -321,7 +321,7 @@ Perhaps more counter-intuitively: If two events are already independent, then it Most questions in the social sciences involve events, rather than numbers per se. To analyze and reason about events quantitatively, we need a way of mapping events to numbers. A random variable does exactly that. ```{r} -#| label: rv-image +#| label: fig-rv-image #| echo: false #| fig-cap: The Random Variable as a Real-Valued Function knitr::include_graphics('images/rv.png') diff --git a/16_simulation.qmd b/16_simulation.qmd index 283fd1f..984602b 100644 --- a/16_simulation.qmd +++ b/16_simulation.qmd @@ -41,7 +41,7 @@ Up till now, you should have covered: - Functions, objects, loops - Joining real data -In this module, we will start to work with generating data within R, from thin air, as it were. Doing simulation also strengthens your understanding of Probability (Section @sec-probability). +In this module, we will start to work with generating data within R, from thin air, as it were. Doing simulation also strengthens your understanding of Probability (Section @sec-probability-theory). ### Check your Understanding {.unnumbered} From 889b1eb425edd9f59b4e928ad895e698d19ba0c5 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Fri, 5 Jul 2024 00:13:11 -0400 Subject: [PATCH 27/34] get pdf rendering --- 05_optimization.qmd | 28 +++++++++++++++++++++------- 13_functions_obj_loops.qmd | 11 ++++------- _quarto.yml | 20 ++++++++++---------- preamble.tex | 2 ++ 4 files changed, 37 insertions(+), 24 deletions(-) diff --git a/05_optimization.qmd b/05_optimization.qmd index 1144bcd..cd464d7 100644 --- a/05_optimization.qmd +++ b/05_optimization.qmd @@ -503,9 +503,17 @@ For example, the budget constraint binds when you spend your entire budget. This Any number of constraints can be placed on an optimization problem. When working with multiple constraints, always make sure that the set of constraints are not pathological; it must be possible for all of the constraints to be satisfied simultaneously. -\textbf{Set-up for Constrained Optimization:} $$\max_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ $$\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ This tells us to maximize/minimize our function, $f(x_1,x_2)$, with respect to the choice variables, $x_1,x_2$, subject to the constraint. +\textbf{Set-up for Constrained Optimization:} -Example: $$\max_{x_1,x_2} f(x_1, x_2) = -(x_1^2 + 2x_2^2) \text{ s.t. }x_1 + x_2 = 4$$ It is easy to see that the \textit{unconstrained} maximum occurs at $(x_1, x_2) = (0,0)$, but that does not satisfy the constraint. How should we proceed? +$$\max_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ $$\min_{x_1,x_2} f(x_1,x_2) \text{ s.t. } c(x_1,x_2)$$ + +This tells us to maximize/minimize our function, $f(x_1,x_2)$, with respect to the choice variables, $x_1,x_2$, subject to the constraint. + +Example: + +$$\max_{x_1,x_2} f(x_1, x_2) = -(x_1^2 + 2x_2^2) \text{ s.t. }x_1 + x_2 = 4$$ + +It is easy to see that the \textit{unconstrained} maximum occurs at $(x_1, x_2) = (0,0)$, but that does not satisfy the constraint. How should we proceed? ### Equality Constraints {.unnumbered} @@ -606,11 +614,17 @@ More generally, in n dimensions: $$ L(x_1, \dots, x_n, \lambda_1, \dots, \lambda 2. $\lambda_i \neq 0$ and $s_i = 0$: This implies that there is no slack in the constraint and *the constraint does bind*. 3. $\lambda_i = 0$ and $s_i = 0$: In this case, there is no slack but the *constraint binds trivially*, without changing the optimum. -Example: Find the critical points for the following constrained optimization: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4$$ +Example: Find the critical points for the following constrained optimization: + +$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4$$ -1. Rewrite with the slack variables: $$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4 - s_1^2$$ +1. Rewrite with the slack variables: + +$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } x_1 + x_2 \le 4 - s_1^2$$ -2. Write the Lagrangian: $$L(x_1,x_2,\lambda_1,s_1) = -(x_1^2 + 2x_2^2) - \lambda_1 (x_1 + x_2 + s_1^2 - 4)$$ +2. Write the Lagrangian: + +$$L(x_1,x_2,\lambda_1,s_1) = -(x_1^2 + 2x_2^2) - \lambda_1 (x_1 + x_2 + s_1^2 - 4)$$ 3. Take the partial derivatives and set equal to 0: @@ -654,12 +668,12 @@ x_2 \ge 0 1. Rewrite with the slack variables: ```{=latex} -$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. +$$\max_{x_1,x_2} f(x) = -(x_1^2 + 2x_2^2) \text{ s.t. } \begin{array}{l} x_1 + x_2 \le 4 - s_1^2\\ -x_1 \le 0 - s_2^2\\ -x_2 \le 0 - s_3^2 -\end{array}}$$ +\end{array}$$ ``` 2. Write the Lagrangian: diff --git a/13_functions_obj_loops.qmd b/13_functions_obj_loops.qmd index 0637113..be15aee 100644 --- a/13_functions_obj_loops.qmd +++ b/13_functions_obj_loops.qmd @@ -271,8 +271,7 @@ A function is a set of instructions with specified ingredients. It takes an **in One way to see what a function actually does is to enter it without parentheses. -```{r} -#| eval: false +``` r # enter this on your console table ``` @@ -285,8 +284,7 @@ You'll notice that functions contain other functions. *wrapper* functions are fu It's worth remembering the basic structure of a function. You create a new function, call it `my_fun` by this: -```{r} -#| eval: !expr F +``` r my_fun <- function() { } @@ -368,7 +366,7 @@ Try making your own function, 'top_10_oldest_cities', that will give you the nam You can think of a package as a suite of functions that other people have already built for you to make your life easier. -```{r} +``` r help(package = "ggplot2") ``` @@ -376,8 +374,7 @@ To use a package, you need to do two things: (1) install it, and then (2) load i Installing is a one-time thing -```{r} -#| eval: false +``` r install.packages("ggplot2") ``` diff --git a/_quarto.yml b/_quarto.yml index 9875d46..6f0550b 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -2,11 +2,12 @@ project: type: book output-dir: _book - book: title: "Math Prefresher for Political Scientists" description: "Text for Harvard Department of Government Math Prefresher" site-url: https://iqss.github.io/prefresher/ + cover-image: "images/logo.png" + repo-url: "https://github.com/IQSS/prefresher" chapters: - index.qmd - 01_warmup.qmd @@ -36,16 +37,15 @@ book: format: html: - theme: + theme: - cosmo - -delete_merged_file: true + biblio-style: apalike + pdf: + include-in-header: preamble.tex + biblio-style: apalike + geometry: "margin=1.5in" + link-citations: true + language: ui: chapter_name: "Chapter " - -github-repo: "IQSS/prefresher" -geometry: "margin=1.5in" -biblio-style: apalike -link-citations: true -cover-image: "./images/logo.png" diff --git a/preamble.tex b/preamble.tex index 872b20b..fb14767 100644 --- a/preamble.tex +++ b/preamble.tex @@ -12,6 +12,8 @@ \usepackage{xcolor} \usepackage{verbatim} +\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} + \usepackage{bm} \setcounter{MaxMatrixCols}{20} \newcommand{\Var}{\mathrm{Var}} From ccca4063f7579bb2676482b46cd0d641973e7657 Mon Sep 17 00:00:00 2001 From: Christopher Kenny Date: Fri, 5 Jul 2024 01:09:17 -0400 Subject: [PATCH 28/34] rerender book, fix date --- _book/prefresher.pdf | Bin 4353780 -> 4398620 bytes 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