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Impact of local congruences in variable selection from datasets

Roberto G. Aragón, Jesús Medina, Eloísa Ramírez-Poussa

TL;DR

This paper addresses how local congruences interact with attribute reduction in Formal Concept Analysis. It proves that the quotient set from attribute reduction, $ ext{C}(A,B,R)/\rho_D$, is isomorphic to the reduced concept lattice $\text{C}(D,B,R_{|D\times B})$ via explicit maps, and introduces a partial order $\leq_\delta$ on quotient sets induced by a local congruence $\delta$, showing that the quotient need not form a lattice. It then derives conditions under which the lattice structure is preserved when removing elements: removing join-irreducible concepts (with a constructive context transform) and its dual for meet-irreducible concepts; for elements that are neither join-irreducible nor meet-irreducible, Dedekind–MacNeille completion is required to recover a complete lattice. Collectively, the results provide a traceable, structure-preserving framework for variable selection in FCA and elucidate when and how to modify contexts to retain lattice isomorphism after local congruence-based clustering, with implications for robust data analysis and interpretability.

Abstract

Formal concept analysis (FCA) is a useful mathematical tool for obtaining information from relational datasets. One of the most interesting research goals in FCA is the selection of the most representative variables of the dataset, which is called attribute reduction. Recently, the attribute reduction mechanism has been complemented with the use of local congruences in order to obtain robust clusters of concepts, which form convex sublattices of the original concept lattice. Since the application of such local congruences modifies the quotient set associated with the attribute reduction, it is fundamental to know how the original context (attributes, objects and relationship) has been modified in order to understand the impact of the application of the local congruence in the attribute reduction.

Impact of local congruences in variable selection from datasets

TL;DR

This paper addresses how local congruences interact with attribute reduction in Formal Concept Analysis. It proves that the quotient set from attribute reduction, , is isomorphic to the reduced concept lattice via explicit maps, and introduces a partial order on quotient sets induced by a local congruence , showing that the quotient need not form a lattice. It then derives conditions under which the lattice structure is preserved when removing elements: removing join-irreducible concepts (with a constructive context transform) and its dual for meet-irreducible concepts; for elements that are neither join-irreducible nor meet-irreducible, Dedekind–MacNeille completion is required to recover a complete lattice. Collectively, the results provide a traceable, structure-preserving framework for variable selection in FCA and elucidate when and how to modify contexts to retain lattice isomorphism after local congruence-based clustering, with implications for robust data analysis and interpretability.

Abstract

Formal concept analysis (FCA) is a useful mathematical tool for obtaining information from relational datasets. One of the most interesting research goals in FCA is the selection of the most representative variables of the dataset, which is called attribute reduction. Recently, the attribute reduction mechanism has been complemented with the use of local congruences in order to obtain robust clusters of concepts, which form convex sublattices of the original concept lattice. Since the application of such local congruences modifies the quotient set associated with the attribute reduction, it is fundamental to know how the original context (attributes, objects and relationship) has been modified in order to understand the impact of the application of the local congruence in the attribute reduction.
Paper Structure (12 sections, 18 theorems, 21 equations, 6 figures, 3 tables, 2 algorithms)

This paper contains 12 sections, 18 theorems, 21 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Formal concept analysis
  4. Lattice theory
  5. Local congruences
  6. Attribute reduction quotient sets and local congruences
  7. Attribute reduction: quotient set versus reduced concept lattice
  8. The poset associated with a local congruence
  9. Impact of removing elements in a concept lattices
  10. Removing join-irreducible elements
  11. Removing non-join-irreducible elements
  12. Conclusions and future work

Key Result

Proposition 1

If $C$ is a join-irreducible concept of $\mathcal{C}(A,B, R)$, then $\text{Obg}(C)$ is a nonempty set. Equivalently, if $C$ is a meet-irreducible concept of $\mathcal{C}(A,B, R)$, then $\text{Atg}(C)$ is a nonempty set.

Figures (6)

  • Figure 1: The local congruence $\delta$ on $\mathcal{C}(A, B, R)$ of Example \ref{['Example2']} (left) and its quotient set $(\mathcal{C}(A, B, R)/\delta, \leq_\delta)$ (right).
  • Figure 2: The associated concept lattice $\mathcal{C}(A, B, R)$ (left) and the induced partition $\rho_D$ (right) of Example \ref{['ex:contrae0']}.
  • Figure 3: The local congruence $\delta$ on $\mathcal{C}(A, B, R)$ of Example \ref{['ex:contrae0']} (left) and its effect on $\mathcal{C}(A, B, R)/\rho_D$ (right).
  • Figure 4: The concept lattice $(\mathcal{C}(D,B\backslash{\{b_4\}},R_{|D\times B\backslash{\{b_4\}}}),\leq_D)$ (left) and the quotient set $(\mathcal{C}(D, B, R)/\delta, \leq_{\delta})$ (right) of Example \ref{['ex:contrae0']}.
  • Figure 5: The concept lattice $\mathcal{C}(A, B, R)$ (left) and the induced partition $\rho_D$ (right) of Example \ref{['ex:contrae1']}.
  • ...and 1 more figures

Theorems & Definitions (35)

  • definition 1
  • definition 2
  • Proposition 1
  • proof
  • Theorem 2
  • Proposition 3: bmrd:RSTFCA:c
  • Proposition 4: bmrd:RSTFCA:c
  • definition 3: DaveyPriestley
  • Theorem 5: DaveyPriestley
  • definition 4: DaveyPriestley
  • ...and 25 more