On Measuring Internal Dependence in a Set of Random Variables

Abstract

To measure dependence in a set of random variables, a multivariate analog of maximal correlation is considered. This consists of transforming each of the variables so that the largest partial sums of the eigenvalues of the resulting correlation matrix is maximized. A "maximalized" measure of association obtained in this manner permits statements to be made about the strength of internal dependence exhibited by the random variables. It is shown, under a weak regularity condition, that optimizing transformations exist and that they satisfy a geometrically interpretable fixed point property. If the variables are jointly Gaussian, then the identity transformation is shown to be optimal, which extends Kolmogorov's result for canonical correlation to the principal components setting.