Introduction

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

We provide the code, data in the folders and the script Analysis_Script.Rmd which, once executed, provides the HTML document Analysis_Script.html.

Below, we copy the code and results to facilitate checking the results. 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:

# 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.

# 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):

# 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.

# Find concept lattice and implications
fc$find_implications()
# Number of implications in the basis
fc$implications$cardinality()
#> [1] 403
# Number of concepts in the lattice
fc$concepts$size()
#> [1] 1769

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).

# 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.

# 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.

imps <- lsi$to_implications(context = fc$I)

Actually, we load the precomputed system of implications.

imps <- readRDS(file = file.path(data_folder, "mingen_implications.RDS"))

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

fail <- imps$filter(rhs = c("-Final"))

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 the Simplification Logic.

fail <- fail[fail$support() > 0.1]

# This gives us 41 implications
fail$cardinality()
#> [1] 41

# We use simplification logic to remove redundancies:
fail$apply_rules("simp")
#> Processing batch
#> --> Simplification: from 41 to 20 in 0.042 secs.
#> Batch took 0.044 secs.

The resulting set of implications is:

fail
#> Implication set with 20 implications.
#> Rule 1: {-2nd T, -3rd T} -> {-Final, -M}
#> Rule 2: {-F, -2nd T, -M} -> {-3rd T, -Final}
#> Rule 3: {-F, -2nd T, -G} -> {-3rd T, -Final, -M}
#> Rule 4: {-1st T, -3rd T, -M} -> {-2nd T, -Final}
#> Rule 5: {-1st T, -F} -> {-I, -D, -P, -2nd T, -3rd T, -Final, -M}
#> Rule 6: {D, -2nd T, -M} -> {-Final}
#> Rule 7: {-P, -2nd T, -3rd T} -> {-1st T}
#> Rule 8: {-D, -2nd T, -3rd T} -> {-1st T}
#> Rule 9: {-I, -D, -P, -2nd T, -3rd T} -> {-F}
#> Rule 10: {-P, -F, -3rd T} -> {-I, -D, -1st T, -2nd T, -Final, -M}
#> Rule 11: {-D, -F, -3rd T} -> {-I, -P, -1st T, -2nd T, -Final, -M}
#> Rule 12: {-D, -F, -2nd T, -M} -> {-I, -P, -1st T}
#> Rule 13: {-I, -D, -P, -3rd T, -M} -> {-F}
#> Rule 14: {-P, -3rd T, -M} -> {-1st T, -2nd T, -Final}
#> Rule 15: {-P, -F, -M} -> {-I, -D, -1st T, -2nd T, -3rd T, -Final}
#> Rule 16: {-I, -P, -F} -> {-D, -1st T, -2nd T, -3rd T, -Final, -M}
#> Rule 17: {-D, -P, -F} -> {-I, -1st T, -2nd T, -3rd T, -Final, -M}
#> Rule 18: {-I, -F, -2nd T} -> {-D, -P, -1st T, -3rd T, -Final, -M}
#> Rule 19: {-I, -2nd T, -G} -> {-Final, -M}
#> Rule 20: {-I, -D, -F, -M} -> {-P, -1st T, -2nd T, -3rd T, -Final}