Towards a theory of granular sets

TL;DR

This paper introduces the concept of granular sets, a generalization of rough sets, motivated by sensor fusion and hierarchical classification problems, aiming to develop a comprehensive theory with broad applications.

Contribution

It proposes the novel concept of granular sets, extending rough set theory, and explores their natural emergence in hierarchical and graded classification contexts.

Findings

Granular sets naturally arise in hierarchical classification problems.

Target sets forming graded sets lead to graded approximations.

The concept generalizes traditional rough set theory.

Abstract

Motivated by the application problem of sensor fusion the author introduced the concept of graded set. It is reasoned that in classification problem arising in an information system (represented by information table), a novel set called Granular set naturally arises. It is realized that in any hierarchical classification problem, Granular set naturally arises. Also when the target set of objects forms a graded set the lower and upper approximations of target sets form a graded set. This generalizes the concept of rough set. It is hoped that a detailed theory of granular/ graded sets finds several applications.