CREATING AND CLASSIFYING MEASURES OF LINEAR ASSOCIATION BY OPTIMIZATION TECHNIQUES

• Published in: Mathematica scandinavica 2010-01, Vol.106 (1), p.67-87
• Main Author: PESTMAN, WIEBE R.
• Format: Article
• Language: English
• Subjects: Algebra; Coefficients; Correlation coefficients; Covariance matrices; Eigenvalues; Linear algebra; Mathematical theorems; Mathematical vectors; Statistical variance; Symmetry
• ISSN: 0025-5521; 1903-1807
• DOI: 10.7146/math.scand.a-15125

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 Rn 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.