Analysis of Relationships Between Co-Symmetric Dissimilarity Measures of Probability Distributions with Involutive Negations

Abstract

Learning from data in almost any human activity is a very important task, usually using similarity or dissimilarity between data. Recently, it was shown the importance of considering the involution operation defined on the data domain which reflects a symmetry of data structures. This symmetry should be taken into account in data analysis. Co-symmetric similarity and dissimilarity measures defined over a set with involution play an important role in data analysis.  In this paper, four dissimilarity functions over the set of probability distributions are created that meet the property of co-symmetry with respect to the involutive negation of distributions. Scatter graphs are generated from their respective dissimilarity matrices to compare the similarity between them. Additionally, the Pearson, Kendall, and Spearman correlation coefficients are calculated to numerically assess the relationship that exists. Subsequently, four dissimilarity functions are considered due to their higher correlation with those studied in this paper. They are divided into two groups, and an analysis is conducted to determine which are more correlated.