Abstract
Monotonically convergent algorithms are described for maximizing six (constrained) functions of vectors x, or matricesX with columns x1, ..., x r . These functions are h1(x)=Σ k (x′A kx)(x′C kx)−1, H1(X)=Σ k tr (X′A k X)(X′C k X)−1, h1(X)=Σ k Σ l (x′ l A kx l ) (x′ l C kx l )−1 withX constrained to be columnwise orthonormal, h2(x)=Σ k (x′A kx)2(x′C kx)−1 subject to x′x=1, H2(X)=Σ k tr(X′A kX)(X′AkX)′(X′CkX)−1 subject toX′X=I, and h2(X)=Σ k Σ l (x′ l A kx l )2 (x′ l C kX l )−1 subject toX′X=I. In these functions the matricesC k are assumed to be positive definite. The matricesA k 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
de Leeuw, J. (1982). Generalized eigenvalue problems with positive semi-definite matrices.Psychometrika, 47, 87–93.
de Leeuw, J. & Heiser, W. (1980). Multidimensional scaling with restrictions on the configuration. In P. R. Krishnaiah (Ed.),Multivariate analysis V (pp. 501–522). Amsterdam: North Holland.
Dinkelbach, W. (1967). On nonlinear fractional programming.Management Science, 13, 492–498.
Harman, H. H. (1976).Modern factor analysis (3rd ed.). Chicago: University of Chicago Press.
Henderson, H. V., & Searle, S. R. (1981). The Vec-permutation matrix, the Vec operator and Kronecker products: A review.Linear and multilinear algebra, 9, 271–288.
Kaiser, H. F., & Dickman, K. W. (1959).Analytic determination of common factors. Unpublished manuscript, University of Illinois.
Kiers, H. A. L. (1990). Majorization as a tool for optimizing a class of matrix functions.Psychometrika, 55, 417–428.
Kiers, H. A. L. (1991). Simple structure in components analysis techniques for mixtures of qualitative and quantitative variables.Psychometrika, 56, 197–212.
Kiers, H. A. L., & ten Berge, J. M. F. (1989). Alternating least squares algorithms for simultaneous components analysis with equal component weight matrices for all populations.Psychometrika, 54, 467–473.
Kiers, H. A. L., & ten Berge, J. M. F. (1992). Minimization of a class of matrix trace functions by means of refined majorization.Psychometrika, 57, 371–382.
Kiers, H. A. L., & ten Berge, J. M. F. (1994). Hierarchical relations between methods for simultaneous component analysis and a technique for rotation to a simultaneous structure.British Journal of Mathematical and Statistical Psychology,47, 109–126.
Kroonenberg, P. M., & de Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms.Psychometrika, 45, 69–97.
McDonald, R. P. (1968). A unified treatment of the weighting problem.Psychometrika, 33, 351–384.
McDonald, R. P., Torii, Y., & Nishisato, S. (1979). Some results on proper eigenvalues and eigenvectors with applications to scaling.Psychometrika, 44, 211–227.
Meulman, J. J. (1986).A distance approach to nonlinear multivariate analysis. Leiden: DSWO Press.
Millsap, R. E., & Meredith, W. (1988). Component analysis in cross-sectional and longitudinal data.Psychometrika, 53, 123–134.
Nierop, A. F. M. (1993).Multidimensional analysis of grouped variables: An integrated approach. Leiden: DSWO Press.
Ostrowski, A. M. (1969).Solutions of equations and systems of equations. New York: Academic Press.
Tatsuoka, M. M. (1971).Multivariate analysis: Techniques for educational and psychological research. New York: Wiley.
ten Berge, J. M. F. (1983). A generalization of Kristof's theorem on the trace of certain matrix products.Psychometrika, 48, 519–523.
ten Berge, J. M. F., Knol, D. L., & Kiers, H. A. L. (1988). A treatment of the orthomax rotation family in terms of diagonalization, and a re-examination of a singular value approach to varimax rotation.Computational Statistics Quarterly, 3, 207–217.
Author information
Authors and Affiliations
Additional information
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.
Rights and permissions
About this article
Cite this article
Kiers, H.A.L. Maximization of sums of quotients of quadratic forms and some generalizations. Psychometrika 60, 221–245 (1995). https://doi.org/10.1007/BF02301414
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02301414