fcaR: Tools for Formal Concept Analysis

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

Details

The fcaR package provides data structures which allow the user to work seamlessly with formal contexts and sets of implications. More explicitly, three main classes are implemented, using the object-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 the context using FCA tools.
• ImplicationSet represents a set of implications over a specific formal context.
• ConceptLattice represents the set of concepts and their relationships, including methods to operate on the lattice.

Two additional helper classes are implemented:

• Set is a class solely used for visualization purposes, since it encapsulates in sparse format a (fuzzy) set.
• Concept encapsulates internally both extent and intent of a formal concept 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 main S4 classes in arules (transactions and rules).

Installation

This package is available at CRAN, so its stable version can be easily 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

Let us start with a fuzzy dataset (stored in a matrix I) as follows:

	P1	P2	P3	P4	P5	P6
O1	0.0	1.0	0.5	0.5	1.0	0
O2	1.0	1.0	1.0	0.0	0.0	0
O3	0.5	0.5	0.0	0.0	0.0	1
O4	0.0	0.0	0.0	1.0	0.5	0
O5	0.0	0.0	1.0	0.5	0.0	0
O6	0.5	0.0	0.0	0.0	0.0	0

Here, a value $x$ in the intersection of a row and a column indicates that the object of the corresponding row possesses the attribute in the column in a degree of at least $x$ (if $x = 0$, the attribute is absent in the object, and if $x = 1$, the attribute is 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 concepts
fc$find_concepts()

# The first concept
fc$concepts$sub(1)
#> ({O1, O2, O3, O4, O5, O6}, {})

# And plot the concept lattice
fc$concepts$plot()

We can also extract implications from the formal context:

# Extract implications
fc$find_implications()

# Which implications have been extracted
fc$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 to the formal context have been implemented:

• Computing a sublattice.
• Calculating the subconcepts and superconcepts of a given concept.
• Finding the join- and meet- irreducible elements, which allows to reduce the context and find the standard context.

Also, one can compute the support of both implications and concepts:

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 0.0000000
fc$concepts$support()
#>  [1] 1.00000000 0.50000000 0.33333333 0.25000000 0.33333333 0.25000000
#>  [7] 0.16666667 0.50000000 0.33333333 0.41666667 0.25000000 0.16666667
#> [13] 0.50000000 0.41666667 0.33333333 0.16666667 0.25000000 0.08333333
#> [19] 0.50000000 0.33333333 0.16666667 0.33333333 0.25000000 0.08333333
#> [25] 0.16666667 0.00000000

In this package, we have implemented a logic to manage implications. This so-called Simplification Logic allows us to simplify the extracted rules by removing redundancies, as well as computing the closure of a given 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"))

# Reduced set of implications
fc$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 binary datasets.

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

Changelog

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

fcaR 1.4.0

Major Enhancements:

• Tidyverse integration: Implemented S3 methods to support dplyr verbs, allowing for a fluent, grammar-based manipulation of FCA objects:
  • FormalContext: Support for select() (attributes), filter() (objects), mutate() (feature engineering), arrange() (sorting), and rename(). Includes support for tidyselect helpers (e.g., starts_with()).
  • ImplicationSet: Support for filter() (based on metrics or attributes), arrange() (sorting rules), and slice() (subsetting by index).
• Semantic Rule Filtering: Introduced helper functions for ImplicationSet filtering to query rules based on attribute presence/absence: lhs(), rhs(), not_lhs(), lhs_any(), etc. This allows querying rules like filter(rhs("Attribute_A"), support > 0.2).
• Mining Causal Association Rules: Implemented the find_causal_rules() method in FormalContext, enabling the discovery of causal rules by controlling for confounding variables using the “Fair Odds Ratio” on matched pairs.

Improvements:

• Robust Subsetting: Completely rewritten subcontext() method in FormalContext. It now robustly handles negative indices, logical vectors, and character vectors, and prevents dimension collapsing issues (using drop = FALSE) that previously caused errors with the Matrix package.
• Metadata Preservation: Rewritten [ and related methods of ImplicationSet. These ensure that critical context metadata (such as the number of objects $N$ for support calculation) is preserved when filtering or sorting rules, fixing previous issues where metadata was lost.
• Data Safety: Enhanced type safety in internal functions to strictly handle integer indices, preventing errors with dplyr attributes.

Documentation:

• New Vignette: Added fcaR_dplyr vignette illustrating the new data manipulation workflow.
• New Vignette: Added causal vignette explaining the new causal mining functionality and its application to Simpson’s Paradox.

Fixes:

• Fixed Matrix coercion errors (dgCMatrix to data.frame) in R 4.x when using internal incidence matrices.
• Fixed fixupDN.if.valid errors from the Matrix package when filtering operations resulted in empty contexts (0 objects or 0 attributes).
• Resolved floating-point precision issues in unit tests when comparing support values.
• Reduced dependencies: moved ggplot2, ggraph, igraph, rstudioapi, and yaml to Suggests.
• Removed forcats and magrittr dependencies by using base R equivalents and the native pipe |>.

fcaR 1.3.1

• Added tests

References

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

Ganter B, Wille R (1999). Formal concept analysis : mathematical foundations. 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.” In Proc. Fourth Int. Conf. on Recent Advances in Soft Computing. Nottingham, United Kingdom, 200-205.

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

Mora A, Cordero P, Enciso M, Fortes I, Aguilera G (2012). “Closure via functional dependence simplification.” International Journal of Computer 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.