On the granular representation of fuzzy quantifier-based fuzzy rough sets

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. Alternatively, this property of granular representability can be defined as being free from inconsistencies, i.e. there are no elements within the set that are indiscernible to elements outside the set. 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. Additionally, we provide counterexamples for the granular representability of other FQFRS models, and show that despite this, these approaches still demonstrate effectiveness in mitigating inconsistencies in real-life datasets. This observation highlights the potential of these models for resolving data inconsistencies and managing noise.