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Tools for Formal Concept Analysis

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# fcaR: Tools for Formal Concept Analysis

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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)

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:
<table>

<thead>

<tr>

</th>

P1
</th>

P2
</th>

P3
</th>

P4
</th>

P5
</th>

P6
</th>

</tr>

</thead>

<tbody>

<tr>

O1
</td>

0.0
</td>

1.0
</td>

0.5
</td>

0.5
</td>

1.0
</td>

0
</td>

</tr>

<tr>

O2
</td>

1.0
</td>

1.0
</td>

1.0
</td>

0.0
</td>

0.0
</td>

0
</td>

</tr>

<tr>

O3
</td>

0.5
</td>

0.5
</td>

0.0
</td>

0.0
</td>

0.0
</td>

1
</td>

</tr>

<tr>

O4
</td>

0.0
</td>

0.0
</td>

0.0
</td>

1.0
</td>

0.5
</td>

0
</td>

</tr>

<tr>

O5
</td>

0.0
</td>

0.0
</td>

1.0
</td>

0.5
</td>

0.0
</td>

0
</td>

</tr>

<tr>

O6
</td>

0.5
</td>

0.0
</td>

0.0
</td>

0.0
</td>

0.0
</td>

0
</td>

</tr>

</tbody>

</table>

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:

``` r
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:

``` r
# 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:

``` r
# 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:

``` r
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.

``` r
# 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.

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