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Maximization of Sums of Quotients of Quadratic Forms and Some Generalizations

Published online by Cambridge University Press:  01 January 2025

Henk A. L. Kiers
Henk A. L. Kiers*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Henk A. L. Kiers, Department of Psychology (SPA), University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS.
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Abstract

Monotonically convergent algorithms are described for maximizing six (constrained) functions of vectors x, or matrices X with columns x1, ..., xr. These functions are h1(x)=Σk (x′Akx)(x′Ckx)−1, H1(X)=Σk tr (X′AkX)(X′CkX)−1, h1(X)=Σk Σl (x′lAkxl) (x′lCkxl)−1 with X constrained to be columnwise orthonormal, h2(x)=Σk (x′Akx)2(x′Ckx)−1 subject to x′x=1, H2(X)=Σk tr (X′AkX)(X′AkX)′(X′CkX)−1 subject to X′X=I, and h2(X)=Σk Σl (x′lAkxl)2 (x′lCkXl)−1 subject to X′X=I. In these functions the matrices Ck are assumed to be positive definite. The matrices Ak can be arbitrary square matrices. The general formulation of the functions and the algorithms allows for application of the algorithms in various problems that arise in multivariate analysis. Several applications of the general algorithms are given. Specifically, algorithms are given for reciprocal principal components analysis, binormamin rotation, generalized discriminant analysis, variants of generalized principal components analysis, simple structure rotation for one of the latter variants, and set component analysis. For most of these methods the algorithms appear to be new, for the others the existing algorithms turn out to be special cases of the newly derived general algorithms.

Keywords

generalized principal components analysisgeneralized discriminant analysisbinormaminsimple structure rotation
Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 2 , June 1995 , pp. 221 - 245
Copyright
Copyright © 1995 The Psychometric Society

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Footnotes

This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to Jos ten Berge for stimulating this research and for helpful comments on an earlier version of this paper.

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