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On the Granular Representation of Fuzzy Quantifier-Based Fuzzy Rough Sets

Adnan Theerens, Chris Cornelis

TL;DR

The paper investigates granular representations in fuzzy quantifier-based fuzzy rough sets (FQFRS). It proves that Choquet-based FRS preserve granular representability under the same conditions as OWA-based FRS, and that Sugeno-based FRS are granularity-friendly without additional constraints, supporting robust rule induction in noisy or inconsistent data. It also shows that certain extended weighting models (WOWA, YWI) are not granularly representable in general, though they can still mitigate inconsistencies in practical datasets. An empirical study on 20 UCI datasets quantifies inconsistencies, finding near-zero maximal errors (often dominated by floating-point effects) and that Sugeno-based FQFRS exhibit the cleanest granular behavior among the tested models. Overall, the results guide the selection of FQFRS models for granular rule-based reasoning and highlight directions for achieving granularity under broader conditions.

Abstract

Rough set theory is a well-known mathematical framework that can deal with inconsistent data by providing lower and upper approximations of concepts. A prominent property of these approximations is their granular representation: that is, they can be written as unions of simple sets, called granules. The latter can be identified with "if. . . , then. . . " rules, which form the backbone of rough set rule induction. It has been shown previously that this property can be maintained for various fuzzy rough set models, including those based on ordered weighted average (OWA) operators. In this paper, we will focus on some instances of the general class of fuzzy quantifier-based fuzzy rough sets (FQFRS). In these models, the lower and upper approximations are evaluated using binary and unary fuzzy quantifiers, respectively. One of the main targets of this study is to examine the granular representation of different models of FQFRS. The main findings reveal that Choquet-based fuzzy rough sets can be represented granularly under the same conditions as OWA-based fuzzy rough sets, whereas Sugeno-based FRS can always be represented granularly. This observation highlights the potential of these models for resolving data inconsistencies and managing noise.

On the Granular Representation of Fuzzy Quantifier-Based Fuzzy Rough Sets

TL;DR

The paper investigates granular representations in fuzzy quantifier-based fuzzy rough sets (FQFRS). It proves that Choquet-based FRS preserve granular representability under the same conditions as OWA-based FRS, and that Sugeno-based FRS are granularity-friendly without additional constraints, supporting robust rule induction in noisy or inconsistent data. It also shows that certain extended weighting models (WOWA, YWI) are not granularly representable in general, though they can still mitigate inconsistencies in practical datasets. An empirical study on 20 UCI datasets quantifies inconsistencies, finding near-zero maximal errors (often dominated by floating-point effects) and that Sugeno-based FQFRS exhibit the cleanest granular behavior among the tested models. Overall, the results guide the selection of FQFRS models for granular rule-based reasoning and highlight directions for achieving granularity under broader conditions.

Abstract

Rough set theory is a well-known mathematical framework that can deal with inconsistent data by providing lower and upper approximations of concepts. A prominent property of these approximations is their granular representation: that is, they can be written as unions of simple sets, called granules. The latter can be identified with "if. . . , then. . . " rules, which form the backbone of rough set rule induction. It has been shown previously that this property can be maintained for various fuzzy rough set models, including those based on ordered weighted average (OWA) operators. In this paper, we will focus on some instances of the general class of fuzzy quantifier-based fuzzy rough sets (FQFRS). In these models, the lower and upper approximations are evaluated using binary and unary fuzzy quantifiers, respectively. One of the main targets of this study is to examine the granular representation of different models of FQFRS. The main findings reveal that Choquet-based fuzzy rough sets can be represented granularly under the same conditions as OWA-based fuzzy rough sets, whereas Sugeno-based FRS can always be represented granularly. This observation highlights the potential of these models for resolving data inconsistencies and managing noise.
Paper Structure (13 sections, 17 theorems, 80 equations, 2 figures, 3 tables)

This paper contains 13 sections, 17 theorems, 80 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Implicator-conjunctor based fuzzy rough sets
  4. Choquet and Sugeno integral
  5. Fuzzy quantifier-based fuzzy rough sets
  6. Granularity of Choquet-based fuzzy rough sets
  7. Granularity of Sugeno-based fuzzy rough sets
  8. Granularity of other fuzzy quantifier based fuzzy rough sets
  9. Counterexamples of the granularity of YWI-FQFRS and WOWA-FQFRS
  10. Granularity on realistic datasets
  11. Setup
  12. Results and discussion
  13. Conclusion and future work

Key Result

Proposition 2.1

If $\mathcal{T}$ is a left-continuous t-norm and $\mathcal{I}$ its R-implicatorThe residual implicator (R-implicator) of a t-norm $\mathcal{T}$ is defined as I_\mathcal{T} (x,y) = \sup\{\lambda \in [0,1] | \mathcal{T}(x,\lambda)\leq y\},for all $x,y \in [0,1]$., we have for all $x,y,z \in [0,1]$.

Figures (2)

  • Figure 1: Plot of the error with respect to the $\alpha$-parameter for the 4 datasets with the highest maximal error.
  • Figure 2: Plot of the percentage of inconsistencies with respect to the $\alpha$-parameter for the 4 datasets with the highest maximal percentage of inconsistencies.

Theorems & Definitions (39)

  • Proposition 2.1: klement2013triangular
  • Proposition 2.2: d2015comprehensive
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Definition 2.9: wang2010generalized
  • ...and 29 more