Creating and classifying measures of linear association by optimization techniques

Abstract

The idea of measures of linear association, such as Pearson's correlation coefficient, can be put in a general framework by axiomization. Groups of linear transformations on $\mathsf{R}^n$ can be exploited to create new and classify existing measures according to their invariance properties. Thus invariance under the Euclidean transformation group leads to the class of so-called geometric measures. Similarly, a measure is called algebraic if it is invariant under scalings. Pearson's coefficient is an example of an algebraic measure; it is not geometric. It is proved that, generally, a measure of linear association cannot possibly be both geometric and algebraic. A procedure is developed to convert a geometric measure into an algebraic and vice versa. Thus a kind of a duality between algebraic and geometric measures arises. In this duality measures can be reflexive or not.