Towards a general theory of similarity and association measures: Similarity, dissimilarity and correlation functions

Constructing Correlation Coefficients from Similarity and Dissimilarity Functions

“…The paper considers similarity, dissimilarity, and correlation measures or coefficients as functions defined on some nonempty underlying set Ω [8]. As such set one can use, for example, any specific domain: the set of all real-valued ntuples, the set of binary vectors, the set of membership values, the set of images, etc.…”

“…Due to the duality of similarity and dissimilarity functions, one can consider only one of these functions, but they have different interpretations and methods of construction; hence, we consider them together. These functions studied in [8]. For short, we call them also (dis)similarity functions.…”

“…Here these correlation functions called strong correlation functions. The methods of construction of correlation functions using similarity functions have been proposed, and it was shown how the Pearson's correlation and Yule's Q association coefficient could be constructed using suitable similarity functions [2,5,8]. In the following sections, we will consider correlation functions satisfying a weaker set of properties.…”

“…Symmetric function R satisfying these properties will be called a co-symmetric function [8]. Further, we will consider co-symmetric similarity, dissimilarity, and correlation functions.…”

“…The similarity, dissimilarity, and correlation measures are considered as functions satisfying some sets of properties [3,4,8]. This paper introduces a new class of correlation functions, satisfying a weaker set of properties than the previously considered correlation functions (association measures) [4,8] called here strong correlation functions and defined on a set with involution (negation) operation. We discuss the methods of constructing both types of correlation functions using (dis)similarity functions.…”