Granular representation of OWA-based fuzzy rough sets

Abstract

Granular representations of crisp and fuzzy sets play an important role in rule induction algorithms based on rough set theory. In particular, arbitrary fuzzy sets can be approximated using unions of simple fuzzy sets called granules. These granules, in turn, have a straightforward interpretation in terms of human-readable fuzzy “if..., then...” rules. In this paper, we are considering a fuzzy rough set model based on ordered weighted average (OWA) aggregation over considered values. We show that this robust extension of the classical fuzzy rough set model, which has been applied successfully in various machine learning tasks, also allows for a granular representation. In particular, we prove that when approximations are defined using a directionally convex t-norm and its residual implicator, the OWA-based lower and upper approximations are definable as unions of fuzzy granules. This result has practical implications for rule induction from such fuzzy rough approximations.