An approach to extract topological information from Intuitionistic Fuzzy Sets and their application in obtaining a Natural Hierarchical Clustering Algorithm

Abstract

Topological data analysis (TDA) is a powerful mathematical framework that extracts valuable insights about the shape and structure of complex datasets by identifying and analyzing underlying topological features, including connected components, holes, voids, and higher-dimensional counterparts. TDA achieves this by constructing a simplicial complex from the data, comprising simple shapes like points, lines, triangles, and higher-dimensional counterparts. The Vietoris–Rips complex (VRC), a widely used method, constructs simplicial complexes by measuring the distances between data points in a metric space. While the link between Intuitionistic Fuzzy Sets (IFSs) and TDA is clear through intuitionistic fuzzy distance measures, the literature has not explored the application of TDA for extracting topological features from uncertain and vague datasets using IFSs. In this paper, we propose a novel approach that leverages Intuitionistic Fuzzy Distance Measures (IFDMs) to generate the Intuitionistic Fuzzy Vietoris–Rips Complex (IFVRC) of IFSs. Specifically, we investigate the persistence of invariant relations of topological features between IFVRCs constructed using different distances, including Atanassov’s Euclidean distance (l2IFS), Boran and Akay’s distance (D), and Liu’s distance measure (dL), over artificially generated IFSs. Furthermore, we demonstrate that the generation of IFVRC can be viewed as a Natural Hierarchical Clustering Algorithm (NHCA) for IFSs. Finally, we apply the proposed IFVRC as a Natural Hierarchical Clustering Algorithm to cluster intuitionistic fuzzy datasets related to Car and Building materials. Moreover, we incorporate a relatively large dataset, the GTZAN dataset from the Kaggle datasets repository, to classify music genres based on their topological information.