Scalar measure of the interdependence between random vectors

Abstract

The problem of assessing tightness of the interdependence between random vectors of different dimensionality is considered. These random vectors can obey arbitrary multidimensional continuous distribution laws. An analytical expression is derived for the coefficient of tightness of the interdependence between random vectors. It is expressed in terms of the coefficients of determination of conditional regressions between the components of random vectors. For the case of Gaussian random vectors, a simpler formula is obtained, expressed through the determinants of each of the random vectors and determinant of their association. It is shown that the introduced coefficient meets all the basic requirements imposed on the degree of tightness of the interdependence between random vectors. This approach is more preferable compared to the method of canonical correlations providing determination of the actual tightness of the interdependence between random vectors. Moreover, it can also be used in case of non-linear correlation dependence between the components of random vectors. The measure thus introduced is rather simply interpretable and can be applied in practice to real data samplings. Examples of calculating the tightness of the interdependence between Gaussian random vectors of different dimensionality are given.