Constructing Correlation Coefficients from Similarity and Dissimilarity Functions

Abstract

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.

Contrast Similarity Measures of Fuzzy Sets

“…The definition of a similarity function coincides with the definition of a fuzzy (valued) proximity relation [12,1,9] and the properties of valued proximity relations can be extended also on similarity functions [4]. Generally, similarity functions have deep relationship with correlation functions (association measures) taking values in interval [-1,1], and similarity functions satisfying some additional properties can be used for constructing correlation functions [2,4,5].…”

Bipolar Dissimilarity and Similarity Correlations of Numbers

“…The concept of correlation function (association measure) was introduced and studied in [12][13][14][15][16][17][18][19]. Let Ω be a nonempty set.…”

“…Example 1. It is shown [12][13][14][15][16][17][18][19] that most of the known in statistics correlation and association coefficients [20][21][22][23][24][25][26] satisfy these properties on specific domain Ω with suitably defined involution operation. Let us show that Pearson's linear correlation coefficient satisfies A1-A3.…”

“…It was proposed several general methods of construction of such correlation functions using suitable similarity and dissimilarity functions. One of the important results obtained in [19] establishes the one-to-one correspondence between bipolar similarity or dissimilarity functions (fuzzy similarity or dissimilarity relations) and correlation functions.…”

“…Another reason to consider bipolar similarity functions is that these functions define measures of correlation (association) between elements of the domain. The paper gives a short introduction to the theory of correlation functions (association measures) defined on sets with involutive operation [12][13][14][15][16][17][18][19]. The correlation function (association measure) is defined on such sets as a function satisfying several simple properties.…”