Topological analysis of intuitionistic fuzzy distance measures with applications in classification and clustering

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.