Creating and classifying measures of linear association by optimization techniques

Authors

  • Wiebe R. Pestman

DOI:

https://doi.org/10.7146/math.scand.a-15125

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.

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Published

2010-03-01

How to Cite

Pestman, W. R. (2010). Creating and classifying measures of linear association by optimization techniques. MATHEMATICA SCANDINAVICA, 106(1), 67–87. https://doi.org/10.7146/math.scand.a-15125

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Articles