Granular structures of covering-based rough set approximations through the Mapper algorithm

Abstract

Covering-based rough sets (CBRS) offer a flexible extension of classical rough set theory, making them well-suited for analyzing diverse types of complex and possibly inconsistent data. A requirement for the effective use of CBRS is the availability of high-quality coverings of the dataset. A promising approach for generating such coverings is the Mapper algorithm, which builds on the underlying topological structure of the data. However, Mapper's performance depends heavily on the choice of its parameters. In this paper, we study how the structure of the parameter space is transferred to the structure of the coverings generated by these parameters, focusing on the weak and strong approximation operators. We prove important monotonicity results that give insights into the inner workings of the Mapper algorithm and how it can generate coverings. We give an overview of how each parameter influences the output covering and illustrate this using real-world datasets.