Dissimilarity functions co-symmetry property: a focus on probability distributions with involutive negation

Abstract

In today’s data-rich era, there is a growing need for developing effective similarity and dissimilarity measures to compare vast datasets. It is desirable that these measures reflect the intrinsic structure of the domain of these measures. Recently, it was shown that the space of finite probability distributions has a symmetric structure generated by involutive negation mapping probability distributions into their “opposite” probability distributions and back, such that the correlation between opposite distributions equals –1. An important property of similarity and dissimilarity functions reflecting such symmetry of probability distribution space is the co-symmetry of these functions when the similarity between probability distributions is equal to the similarity between their opposite distributions. This article delves into the analysis of five well-known dissimilarity functions, used for creating new co-symmetric dissimilarity functions. To conduct this study, a random dataset of one thousand probability distributions is employed. From these distributions, dissimilarity matrices are generated that are used to determine correlations similarity between different dissimilarity functions. The hierarchical clustering is applied to better understand the relationships between the studied dissimilarity functions. This methodology aims to identify and assess the dissimilarity functions that best match the characteristics of the studied probability distribution space, enhancing our understanding of data relationships and patterns. The study of these new measures offers a valuable perspective for analyzing and interpreting complex data, with the potential to make a significant impact in various fields and applications.