Grammar and spelling check Chapter 4

chapters/04/04a-various-regression.tex

@@ -67,7 +67,7 @@ \subsection{Multilevel linear modelling}
 That is, every observation for unit $i$ is known to belong to a specific group $j$, and we write $\bx_i^{(j)}$ to indicate this.
 Let $n_j$ denote the sample size for cluster $j$, and the overall sample size be $n = \sum_{j=1}^m n_j$.
 When modelled linearly with the responses $y_i^{(j)}$, the model is known as a multilevel (linear) model, although it is known by many other names: random-effects models, random coefficient models, hierarchical models, and so on.
-As this model is seen as an extension of linear models, applications are plenty, especially in research designs for which the data varies at more than one level.
+As this model is seen as an extension of linear models, application is plentiful, especially in research designs for which the data varies at more than one level.
 
 \index{ANOVA!kernel/RKKS|(}
 Consider a functional ANOVA decomposition of the regression function as follows:
@@ -93,7 +93,7 @@ \subsection{Multilevel linear modelling}
 The standard multilevel random effects assumption is that $(\beta_{0j},\boldsymbol{\beta}_{1j}^\top)^\top$ is normally distributed with mean zero and covariance matrix $\bPhi$.
 In total, there are $p+1$ regression coefficients and $(p+1)(p+2)/2$ covariance parameters in $\bPhi$ to be estimated.
 In contrast, the I-prior model is parameterised by only two RKKS scale parameters---one for $\cF_1$ and one for $\cF_2$---and the error precision $\bPsi$, which is usually proportional to the identity matrix.
-While the estimation procedure for $\bPhi$ in the standard multilevel model can result in non-positive covariance matrices, the I-prior model has the advantage that positive definiteness is taken care of automatically\footnote{By virtue of the estimate of the regression function belonging to $\cF_n$, an RKHS with a positive definite kernel equal to the Fisher information for $f$.}.
+While the estimation procedure for $\bPhi$ in the standard multilevel model can result in non-positive covariance matrices, the I-prior model has the advantage that positive definiteness is taken care of automatically\footnote{By virtue of the estimate of the regression function belonging to $\cF_n$, an RKHS with a positive definite kernel equal to the Fisher information for $f$. The first example in \cref{sec:ipriorexamples} is an instance of such cases.}.
 %This is seen from the calculations for $\Var \beta_{0j}$, $\Var \boldsymbol\beta_{1j}$ and the respective covariances.
 %An example of multilevel modelling using I-priors is given in \hltodo{Section 4.3.1}.
 

chapters/04/04b-iprior-estimation.tex

@@ -132,7 +132,7 @@ \subsection{Direct optimisation}
     \begin{split}
       L(\theta) 
       ={}& -\half[n]\log 2\pi - \half\sum_{i=1}^n\log(\psi u_i^2 + \psi^{-1}) \\
-      &- \half \tilde\by^\top \bV \, \text{diag} \left(\frac{1}{\psi u_1^2 + \psi^{-1}},\dots,\frac{1}{\psi u_n^2 + \psi^{-1}}\right) \, \bV^\top \tilde\by
+      &- \half \tilde\by^\top \bV \, \text{diag} \left(\frac{1}{\psi u_1^2 + \psi^{-1}},\dots,\frac{1}{\psi u_n^2 + \psi^{-1}}\right) \, \bV^\top \tilde\by.
     \end{split}
   \end{align}
 \endgroup
@@ -246,7 +246,7 @@ \subsection{Comparison of estimation methods}
 \label{sec:compareestimation}
 
 \index{smoothing model}
-Consider a one-dimensional smoothing example, for which $n=150$ data pairs $(y_i,x_i)$ have been generated according to the relationship 
+Consider a one-dimensional smoothing example, for which $n=150$ data pairs $\{(y_i,x_i)\}_{i=1}^n$ have been generated according to the relationship 
 \begin{align}\label{eq:examplesmoothingdata}
   \begin{gathered}
     y_i = \const + \myoverbrace{
@@ -256,7 +256,7 @@ \subsection{Comparison of estimation methods}
   \end{gathered}
 \end{align}
 where $\phi(\cdot|\mu,\sigma^2)$ is the probability density function of a normal distribution with mean $\mu$ and variance $\sigma^2$.
-The observed $y_i$'s are thought to be noisy versions of the true points, in which $\epsilon_i$ follows an indescript, not necessarily normal, distribution.
+The observed $y_i$'s are thought to be noisy versions of the true points, in which $\epsilon_i$ follows an indescript, non-normal, distribution.
 The predictors $x_1,\dots,x_n$ have been sampled roughly from the interval $(-1,6)$, and the sampling was intentionally not uniform so that there is slight sparsity in the middle.
 \cref{fig:examplesmoothingdata} plots the sampled points and the true regression function.
 

chapters/04/04x-examples.tex

@@ -70,7 +70,7 @@
 \IfFileExists{upquote.sty}{\usepackage{upquote}}{}
 \begin{document}
 
-We demonstrate I-prior modelling on a toy data set to illustrate the Nyström method, as well as three other real-data examples.
+We demonstrate I-prior modelling using three real-data examples, as well as on a toy data set to illustrate the Nyström method.
 All of the analyses were conducted in \proglang{R}, and I-prior model estimation was done using the \pkg{iprior} package \citep{jamil2017iprior}.
 The \pkg{iprior} package comes documented with usage examples in the vignette.
 The complete source code for replication is found at \url{http://myphdcode.haziqj.ml}.

chapters/04/05a-naive.tex

@@ -11,8 +11,8 @@
     (\epsilon_{i1},\dots,\epsilon_{im})^\top \iid \N_m(\bzero,\bPsi^{-1}).
   \end{gathered}
 \end{equation}
-The idea here being that we attempt to model the class responses $y_{ij}$ using class-specific regression functions, and the class responses are assumed to be independent among individuals, but may or may not be correlated among classes for each individual.
-The class correlations are manifest themselves in the variance of the errors $\bPsi^{-1}$, which is an $m\times m$ matrix.
+The idea here is to model the class responses $y_{ij}$ using class-specific regression functions, in which class responses are assumed to be independent among individuals, but may or may not be correlated among classes for each individual.
+The class correlations manifest themselves in the variance of the errors $\bPsi^{-1}$, which is an $m\times m$ matrix.
 
 \index{ANOVA!kernel/RKKS|)}
 Denote the regression function $f$ in \cref{eq:naiveclassmod} on the set $\cX\times\cM$ as $f(x_i,j) = \alpha_j + f_j(x_i)$.

chapters/04/chapter4.tex

@@ -29,7 +29,7 @@
 Careful considerations of the computational aspects are required to ensure efficient estimation of I-prior models, and these are discussed in \cref{sec:ipriorcompcons}.
 The culmination of the computational work on I-prior estimation is the \pkg{iprior} package \citep{jamil2017iprior}, which is a publicly available \proglang{R} package that has been published to the Comprehensive \proglang{R} Archive Network (CRAN).
 
-Finally, several examples of I-prior modelling are presented in  \cref{sec:ipriorexamples}: in particular, a multilevel data set, a longitudinal data set, and a data set involving a functional covariate, are analysed using the I-prior methodology.
+Finally, several examples of I-prior modelling are presented in  \cref{sec:ipriorexamples}, in particular, a multilevel data set, a longitudinal data set, and a data set involving a functional covariate, are analysed using the I-prior methodology.
 Code for replication is available at \url{http://myphdcode.haziqj.ml}.
 
 \section{Various regression models}

chapters/R/04x-examples.Rnw

@@ -29,7 +29,7 @@ knitr::opts_chunk$set(myspacing = TRUE)  # single spacing for code chunks
 
 \begin{document}
 
-We demonstrate I-prior modelling on a toy data set to illustrate the Nyström method, as well as three other real-data examples.
+We demonstrate I-prior modelling using three real-data examples, as well as on a toy data set to illustrate the Nyström method.
 All of the analyses were conducted in \proglang{R}, and I-prior model estimation was done using the \pkg{iprior} package \citep{jamil2017iprior}.
 The \pkg{iprior} package comes documented with usage examples in the vignette.
 The complete source code for replication is found at \url{http://myphdcode.haziqj.ml}.