Constructing Correlation Coefficients from Similarity and Dissimilarity Functions

Correlation coefficients (association measures) were introduced more than one hundred years ago as measures of relationship between variables that usually belong to one of the following basic types: continuous, ordinal, or categorical. Nowadays, it appears the growing demand for the development of new correlation coefficients for measuring associations between variables or objects with more sophisticated structures. The paper presents a non-statistical, functional approach to the study of correlation coefficients. It discusses the methods of construction of correlation coefficients using similarity and dissimilarity measures. Generally, all these measures are considered as functions defined on the underlying (universal) domain and satisfying some sets of properties. The methods of construction of correlation functions on the universal domain can be easily applied for constructing correlation coefficients for specific types of data. The paper introduces a new class of correlation functions, satisfying a weaker set of properties than the previously considered correlation functions (association measures) defined on a set with involution (negation) operation called here strong correlation functions. The methods of constructing both types of correlation functions are discussed. The one-to-one correspondence between the strong correlation functions and the bipolar similarity and dissimilarity functions is established. The theoretical results illustrated by examples of construction of classical Pearson's product-moment correlation coefficient, Spearman's and Kendall's rank correlation coefficients, etc. from similarity and dissimilarity functions.