Similarity-Based Correlation Functions for Binary Data

The purpose of this study is to survey the correlation and association coefficients introduced previously on the set of binary n-tuples and to determine coefficients satisfying the properties of correlation functions. These functions were recently introduced on the sets with involutive operation as functions generalizing classical correlation coefficients: Pearson’s product-moment correlation, Spearmen’s and Kendall’ rank correlation coefficients, Yule’s Q and Hamann’s association coefficients, etc. It is shown that several, but not all, known correlation and association coefficients defined on the set of binary n-tuples, satisfy the properties of correlation functions. For these association coefficients, there were established similarity measures on the set of binary data that can be used for the generation of these association coefficients. A new parametric family of correlation functions for binary data is proposed. As a particular case, it contains Hamann’s association coefficient.