Multivariate Analysis with Optimal Scaling
- Author(s): de Leeuw, Jan
- et al.
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.