Scalar measure of the relationship between several random vectors

Abstract

The problem of estimating the closeness of co-relation (interdependence) between several random vectors of arbitrary dimension is considered. These random vectors can have arbitrary multidimensional continuous distribution laws. Earlier, within the framework of the entropy approach, indicators were obtained for estimating the closeness of the correlation relationship between the components of one random vector and between two random vectors. The goal of the study is to generalize the previously obtained results to the case of several random vectors. The analytical expression for the coefficient of the closeness of co-relation between several random vectors is obtained. This coefficient is expressed through the indices of determination of conditional regressions between the components of random vectors. For the introduced scalar measure of the relationship, a number of particular results were obtained, which turned out to be known correlation coefficients. A rather simple formula expressed through the determinants of each random vector and the determinant of their combination is derived for the case of Gaussian random vectors. The proposed coefficient can be used to study the network structures consisting of many subsystems. In particular, the interpretation of the correlation coefficient between the elements of the network structure and the other elements can be introduced as the correlation coefficient of the system at the vertex. The introduced measure is quite simply interpretable and allows an unambiguous assessing of the closeness of co-relation between several random vectors of arbitrary dimensions and can be used on real data samples. An example of calculating the closeness of the co-relation between three Gaussian random vectors is presented.