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Multivariate Analysis with Optimal Scaling

Abstract

Suppose K1, … ,Km are convex cones in a Hilbert space H, with unit sphere S and inner product ‹ . | . ›. For a particular choice of quantifications, transformations, or representations of a variable xj in Kj ∩ S we can compute the correlation matrix R(x1, … , xm ) by the rule rij(x1, … , xm) = (xi | xj). Now suppose φ is a real-valued objective function, defined on the space of all correlation matrices. In this paper we study the class of techniques that choose the xj in their feasible regions Kj ∩ S in such a way that φ(R(x1 , … , xm)) is maximized. We discuss typical cases, including linear and nonlinear principal component analysis, canonical correlation analysis, regression analysis. It is shown that correspondence analysis and the Breiman-Friedman ACE-methods are both special cases of this class of techniques. We discuss some choices for the cones Kj, and we indicate that the results simplify greatly if all bivariate regressions can be linearized. A class of iterative projection techniques is suggested, that produces convergent algorithms of simple structure.

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