Analysis_Script.Rmd

---
title: "Minimal generators from positive and negative attributes: analysing the knowledge space of a Mathematics course"
description: |
  Script to reproduce the results of the paper.
author:
  - name: Manuel Ojeda-Hernández
    affiliation: Departamento de Matemática Aplicada, Universidad de Málaga
  - name: Francisco Pérez-Gámez
    affiliation: Departamento de Matemática Aplicada, Universidad de Málaga
  - name: Domingo López-Rodríguez
    url: https://dominlopez.netlify.app
    affiliation: Departamento de Matemática Aplicada, Universidad de Málaga
  - name: Nicolás Madrid
    affiliation: Departamento de Matemática Aplicada, Universidad de Málaga
  - name: Ángel Mora
    affiliation: Departamento de Matemática Aplicada, Universidad de Málaga

date: "`r Sys.Date()`"
output:
  distill::distill_article:
    toc: true
    toc_depth: 2
---

```{r setup, include=FALSE}
##%################################################%##
#                                                    #
####      Script for IJIS'2021 on e-learning      ####
#                                                    #
##%################################################%##
knitr::opts_chunk$set(echo = TRUE)
```

# Introduction

This file accompanies the paper of the same title as a means to reproduce the results given in the paper.

We interleave text, code and results to make it simple for the reader to follow the results and check the code.

All the analyses can be reproduced using the commands below in a folder with `code` and `data` subfolders, which store the functions and the dataset and results used in the paper.

# Loading data

First, we must load some `R` libraries:

```{r}
# Load libraries
library(tidyverse)
library(here)
library(Matrix)
library(fcaR)
```

Now, the core of the code is the `fcaR` package, but there are some extensions (for example, to compute the minimal generators) that are, at this moment, outside the package, and are included in the `code` folder. We must load them.

```{r}
# Load functions from "code" folder
code_folder <- here("code")
list.files(path = code_folder,
           pattern = "*.R",
           full.names = TRUE) %>% 
  sapply(source) %>% 
  invisible()  # To keep output clean

```

The data used in this paper is included in the `data` folder, so we must load it (it is a formal context named `context3.rds`):

```{r}
# Import data from data folder
data_folder <- normalizePath(here("data"))
fc <- FormalContext$new(file.path(data_folder, "context3.rds"))
```

# Formal Concept Analysis operations

We use the functions from the `fcaR` package to compute both the concept lattice and the basis of implications of the mixed context.

```{r}
# Find concept lattice and implications
fc$find_implications()
# Number of implications in the basis
fc$implications$cardinality()
# Number of concepts in the lattice
fc$concepts$size()
```

# First analysis: Exploration of the Knowledge Space

First, we build the sublattice formed by the concepts containing the attributes `-P` and `-F` (we use the notation `-X` to denote the negation of attribute `X`).

```{r}
# Concepts and sublattice of those concepts containing -P and -F
selected_attributes <- c("-P", "-F")
id_attr <- which(fc$attributes %in% selected_attributes)
which_attr <- colSums(fc$concepts$intents()[id_attr, ]) == length(selected_attributes)
# Creation of the sublattice
sublattice <- fc$concepts$sublattice(which_attr)
sublattice$plot()
```

# Second analysis: Minimal generators

The computation of the minimal generators is very computationally demanding (and may take hours to days, depending on the hardware), so we include the code but we provide the precomputed minimal generators in form of implication set.

This code would compute the minimal generators.

```{r eval = FALSE}
# Minimal Generators
lsi <- mingen0_minimals(
  attributes = fc$attributes,
  LHS = fc$implications$get_LHS_matrix(),
  RHS = fc$implications$get_RHS_matrix())
```

This code would create the implication set from the minimal generators.

```{r eval = FALSE}
imps <- lsi$to_implications(context = fc$I)
```

Actually, we load the precomputed system of implications.

```{r}
imps <- readRDS(file = file.path(data_folder, "mingen_implications.RDS"))
```

From these implications, we select those that have `-Final` in the right-hand side:

```{r}
fail <- imps$filter(rhs = c("-Final"))
```
 
```{r echo = FALSE}
fail$.__enclos_env__$private$I <- fc$I
fail$.__enclos_env__$private$implication_support <- NULL
```
 
We only keep some of the implications, those whose support is above the 10\%. That is, implications applicable to, at least, 10\% of the students in the course. In addition, we remove some redundancies that appear by using Simplification Logic.

```{r}
fail <- fail[fail$support() > 0.1]

# This gives us 41 implications
fail$cardinality()

# We use simplification logic to remove redundancies:
fail$apply_rules("simp")
```
 
The resulting set of implications is:

```{r}
fail
```