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fcaR: Tools for FormalConcept Analysis

The aim of this package is to provide tools to perform fuzzy formalconcept analysis (FCA) from within R. It provides functions to load andsave a Formal Context, extract its concept lattice and implications. Inaddition, one can use the implications to compute semantic closures offuzzy sets and, thus, build recommendation systems.

Details

The fcaR package provides data structures which allow the user towork seamlessly with formal contexts and sets of implications. Moreexplicitly, three main classes are implemented, using theobject-oriented-programming paradigm in R:

• FormalContext encapsulates the definition of a formal context (G, M,I), being G the set of objects, M the set of attributes and I the(fuzzy) relationship matrix, and provides methods to operate on thecontext using FCA tools.
• ImplicationSet represents a set of implications over a specificformal context.
• ConceptLattice represents the set of concepts and theirrelationships, including methods to operate on the lattice.

Two additional helper classes are implemented:

• Set is a class solely used for visualization purposes, since itencapsulates in sparse format a (fuzzy) set.
• Concept encapsulates internally both extent and intent of a formalconcept as Set.

Since fcaR is an extension of the data model in the arules package,most of the methods and classes implemented interoperates with the mainS4 classes in arules (transactions and rules).

Installation

This package is available at CRAN, so its stable version can beeasily installed using:

install.packages("fcaR")

The development version of this package can be installed with

remotes::install_github("neuroimaginador/fcaR", build_vignettes = TRUE)

or

remotes::install_github("Malaga-FCA-group/fcaR", build_vignettes = TRUE)

Example of Use

Here, a value\(x\) in theintersection of a row and a column indicates that the object of thecorresponding row possesses the attribute in the column in a degree ofat least\(x\) (if\(x = 0\), the attribute is absent in theobject, and if\(x = 1\), the attributeis fully present in the object).

We can build a FormalContext object:

fc<- FormalContext$new(I)print(fc)#> FormalContext with 6 objects and 6 attributes.#>      P1   P2   P3   P4   P5  P6#>  O1  0    1   0.5  0.5   1    0#>  O2  1    1    1    0    0    0#>  O3 0.5  0.5   0    0    0    1#>  O4  0    0    0    1   0.5   0#>  O5  0    0    1   0.5   0    0#>  O6 0.5   0    0    0    0    0

With a single function, we can compute the set of concepts:

# Compute all conceptsfc$find_concepts()# The first conceptfc$concepts$sub(1)#> ({O1, O2, O3, O4, O5, O6}, {})# And plot the concept latticefc$concepts$plot()

We can also extract implications from the formal context:

# Extract implicationsfc$find_implications()# Which implications have been extractedfc$implications#> Implication set with 12 implications.#> Rule 1: {P6 [0.5]} -> {P1 [0.5], P2 [0.5], P6}#> Rule 2: {P5 [0.5]} -> {P4 [0.5]}#> Rule 3: {P3 [0.5], P4 [0.5], P5 [0.5]} -> {P2, P5}#> Rule 4: {P3 [0.5], P4} -> {P3}#> Rule 5: {P2 [0.5], P4 [0.5]} -> {P2, P3 [0.5], P5}#> Rule 6: {P2 [0.5], P3 [0.5]} -> {P2}#> Rule 7: {P2, P3, P4 [0.5], P5} -> {P4}#> Rule 8: {P1 [0.5], P4 [0.5]} -> {P1, P2, P3, P4, P5, P6}#> Rule 9: {P1 [0.5], P3 [0.5]} -> {P1, P2, P3}#> Rule 10: {P1 [0.5], P2} -> {P1}#> Rule 11: {P1, P2 [0.5]} -> {P2}#> Rule 12: {P1, P2, P3, P6} -> {P4, P5}

Some fundamental functionalities on the concept lattice associated tothe formal context have been implemented:

• Computing a sublattice.
• Calculating the subconcepts and superconcepts of a givenconcept.
• Finding the join- and meet- irreducible elements, which allows toreduce the context and find thestandard context.

Also, one can compute the support of both implications andconcepts:

fc$implications$support()#>  [1] 0.1666667 0.3333333 0.1666667 0.0000000 0.1666667 0.3333333 0.0000000#>  [8] 0.0000000 0.1666667 0.1666667 0.1666667 0.0000000fc$concepts$support()#>  [1] 1.0000000 0.5000000 0.3333333 0.1666667 0.1666667 0.1666667 0.0000000#>  [8] 0.5000000 0.3333333 0.3333333 0.1666667 0.0000000 0.5000000 0.3333333#> [15] 0.3333333 0.1666667 0.1666667 0.0000000 0.5000000 0.3333333 0.1666667#> [22] 0.1666667 0.1666667 0.0000000 0.1666667 0.0000000

In this package, we have implemented a logic to manage implications.This so-called Simplification Logic allows us to simplify the extractedrules by removing redundancies, as well as computing the closure of agiven fuzzy attribute set.

# Reduce the number of implications using two simple# rules. The algorithm applies the specified rules# in batches, if the number of rules is high.fc$implications$apply_rules(rules =c("composition","generalization"))#> Processing batch#> --> Composition: from 12 to 12 in 0.001 secs.#> --> Generalization: from 12 to 12 in 0.001 secs.#> Batch took 0.004 secs.# Reduced set of implicationsfc$implications#> Implication set with 12 implications.#> Rule 1: {P6 [0.5]} -> {P1 [0.5], P2 [0.5], P6}#> Rule 2: {P5 [0.5]} -> {P4 [0.5]}#> Rule 3: {P3 [0.5], P4 [0.5], P5 [0.5]} -> {P2, P5}#> Rule 4: {P3 [0.5], P4} -> {P3}#> Rule 5: {P2 [0.5], P4 [0.5]} -> {P2, P3 [0.5], P5}#> Rule 6: {P2 [0.5], P3 [0.5]} -> {P2}#> Rule 7: {P2, P3, P4 [0.5], P5} -> {P4}#> Rule 8: {P1 [0.5], P4 [0.5]} -> {P1, P2, P3, P4, P5, P6}#> Rule 9: {P1 [0.5], P3 [0.5]} -> {P1, P2, P3}#> Rule 10: {P1 [0.5], P2} -> {P1}#> Rule 11: {P1, P2 [0.5]} -> {P2}#> Rule 12: {P1, P2, P3, P6} -> {P4, P5}

All these functions work natively with fuzzy and with binarydatasets.

For more details on the methods implemented and further examples, seethe vignettes in this package.

Changelog

With respect to the CRAN version, the development version has thefollowing changes.

fcaR 1.2.2

Enhancements:

• Added more unit tests.
• Minor changes to the plotting of formal contexts.
• Now thefc$scale() function admits a new argumentbg (default: FALSE) which, if set to TRUE, avoids computingthe background knowledge of the scales.

fcaR 1.2.1

Enhancements:

• Other logics have been implemented. Now, we can usefc$use_logic() to select one of theavailable_logics().
• Improved export to LaTeX.

Bugfixes:

• Some rounding errors might induce errors in the computations. Thesehas been fixed.

fcaR 1.2.0

Bugfixes:

• Fixes required by the new version of Matrix and the new use of HTMLTidy in R 4.2.

fcaR 1.1.1

Enhancements:

• The user can control the number of decimal digits when exporting toLaTeX or when printing formal contexts, concept lattices andimplications. Just use fcaR_options(decimal_places = n), where n is thenumber of desired decimal digits.

New functionality:

• Now the package uses thesettings package to manage severaloptions. Currently, the only option is the number of decimal digits touse when printing or exporting to LaTeX.

Bugfixes:

• Fixed exporting to latex with special characters such as $, _,etc.

References

Guigues J, Duquenne V (1986). “Familles minimales d’implicationsinformatives résultant d’un tableau de données binaires.”Mathématiques et Sciences humaines,95, 5-18.

Ganter B, Wille R (1999).Formal concept analysis : mathematicalfoundations. Springer. ISBN 3540627715.

Cordero P, Enciso M, Mora Á, Pérez de Guzman I (2002). “SLFD Logic:Elimination of Data Redundancy in Knowledge Representation.”Advances in Artificial Intelligence - IBERAMIA 2002,2527, 141-150. doi: 10.1007/3-540-36131-6_15 (URL:https://doi.org/10.1007/3-540-36131-6_15).

Belohlavek R (2002). “Algorithms for fuzzy concept lattices.” InProc. Fourth Int. Conf. on Recent Advances in Soft Computing.Nottingham, United Kingdom, 200-205.

Hahsler M, Grun B, Hornik K (2005). “arules - a computationalenvironment for mining association rules and frequent item sets.”JStat Softw,14, 1-25.

Mora A, Cordero P, Enciso M, Fortes I, Aguilera G (2012). “Closurevia functional dependence simplification.”International Journal ofComputer Mathematics,89(4), 510-526.

Belohlavek R, Cordero P, Enciso M, Mora Á, Vychodil V (2016).“Automated prover for attribute dependencies in data with grades.”International Journal of Approximate Reasoning,70,51-67.