Granular Generalized Variable Precision Rough Sets and Rational Approximations

TL;DR

This paper develops a formal framework for rational approximations in granular variable precision rough sets, extending previous concepts, and explores their applications in clustering, image segmentation, and dynamic sorting.

Contribution

It introduces uniform generalizations of granular VPRS, establishes connections with granular graded rough sets, and extends the framework to probabilistic rough sets.

Findings

Formal characterization of conditions for rationality in granular VPRS

Connections established between granular VPRS and graded rough sets

Framework extended to probabilistic rough sets

Abstract

Rational approximations are introduced and studied in granular graded rough sets and generalizations thereof by the first author in recent research papers. The concept of rationality is determined by related ontologies and coherence between granularity, mereology and approximations in the context. In addition, a framework for rational approximations is introduced by her in the mentioned paper(s). Granular approximations constructed as per the procedures of variable precision rough sets (VPRS) are likely to be more rational than those constructed from a classical perspective under certain conditions. This may continue to hold for some generalizations of the former. However, a formal characterization of such conditions is not available in the previously published literature. In this research, theoretical aspects of the problem are critically examined, uniform generalizations of granular VPRS are introduced, new connections with granular graded rough sets are proved, appropriate concepts of substantial parthood are introduced, their extent of compatibility with the framework is accessed, and the framework is extended. Basic assumptions are explained in detail, and additional examples are constructed for readability. Furthermore, meta applications to cluster validation, image segmentation and dynamic sorting are invented. Extensions to direct generalizations of VPRS such as probabilistic rough sets are a natural consequence of the work.