Topological analysis of intuitionistic fuzzy distance measures with applications in classification and clustering. (November 2022) - Khan, Mohd Shoaib; Lohani, Q.M. Danish
Abstract: The distance measure is a vital classifier used to solve classification and clustering problems in a metric space. In this paper, we discussed the types of distance measures and elaborated that they generate uniquely shaped balls. Due to their balls, they cannot guarantee a satisfactory classification outcome for a given dataset. To address this limitation, researchers used intuitionistic fuzzy sets (IFSs) to lose the boundary of belongingness of data points in a given metric space with respect to the chosen membership and non-membership functions and accomplished some novel intuitionistic fuzzy distance measures (IFDMs). IFDMs are successfully used in various applications; however, they also predict counterintuitive cases. To address this issue, in this paper, we conducted a topological analysis of intuitionistic fuzzy distance measures. To do that, we constructed three categories of distance measures; Type 1 distance measures, Type 2 distance measures, and Intuitionistic fuzzy distance measures (in this paper, we called them Intuitive Distance Measures (IDMs)). We further sub-categorized Intuitive Distance Measures into Type 1 Intuitive Distance Measures (T1IDMs) and Type 2 Intuitive Distance Measures (T2IDMs). We established a homeomorphic relation between the topological spaces generated by the T1IDMs and T2IDMs. Using the proposed homeomorphic relation and proving the necessary lemma and theorems, we presented the type 2 distance measures of well-known normable T1IDMs. We applied the proposed type 2 variants to solve some classification and clustering problems of machine learning. The result analysis shows that the proposed type 2 variants overcome the drawbacks of their type 1 counterparts. Highlights: We did topological analysis of intuitionistic fuzzy distance measures. Some categorization of distance measures are suggested. We established homeomorphic relations (TMT) between constructed categories. Using TMT, we obtained the variants of some distance measures. We compared the proposed variants over some classification and clustering problems.