Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2025-01-05T03:36:17.966Z Has data issue: false hasContentIssue false
  • English
  • Français

A Monotonically Convergent Algorithm for Orthogonal Congruence Rotation

Published online by Cambridge University Press:  01 January 2025

Henk A. L. Kiers  and
Patrick Groenen
Henk A. L. Kiers*
Affiliation:
University of Groningen
Patrick Groenen
Affiliation:
University of Leiden
*
Requests for reprints should be sent to Henk A. L. Kiers, Department of Psychology (SPA), Grute Kruisstraat 2/1, 9712TS Groningen, THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

Abstract

Brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix. This method employs an algorithm for which convergence from every starting point is not guaranteed. In the present paper, an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point. Specifically, it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero. In addition to the better convergence properties, another advantage of the present algorithm over Brokken's one is that it is easier to program. The algorithms are compared on 80 simulated data sets, and it turned out that the new algorithm performed well in all cases, whereas Brokken's algorithm failed in almost half the cases. The derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts.

Keywords

optimizationmajorizationmatchingProcrustes rotation
Type
Original Paper
Information
Psychometrika , Volume 61 , Issue 2 , June 1996 , pp. 375 - 389
Copyright
Copyright © 1996 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to Willem J. Heiser and Jos M. F. ten Berge for useful comments on an earlier version of this paper.

References

Brokken, F.B. (1983). Orthogonal Procrustes rotation maximizing congruence. Psychometrika, 48, 343–352.CrossRefGoogle Scholar
Brokken, F.B. (1985). The simultaneous maximization of congruence for two or more matrices under orthogonal rotation. Psychometrika, 50, 51–56.CrossRefGoogle Scholar
Cliff, N. (1966). Orthogonal rotation to congruence. Psychometrika, 31, 33–42.CrossRefGoogle Scholar
de Leeuw, J., Heiser, W. (1980). Multidimensional scaling with restrictions on the configuration. In Krishnaiah, P. R. (Eds.), Multivariate Analysis V (pp. 501–522). Amsterdam: North Holland.Google Scholar
Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society, Series B, 53, 285–339.CrossRefGoogle Scholar
Green, B.F. (1952). The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 17, 429–440.CrossRefGoogle Scholar
Groenen, P.J.F. (1993). The majorization approach to multidimensional scaling: Some problems and extensions, Leiden: DSWO Press.Google Scholar
Groenen, P.J.F., Heiser, W.J. (1991). An improved tunnelling function for finding a decreasing series of local minima in MDS, Leiden: Department of Data Theory.Google Scholar
Heiser, W.J. (1987). Correspondence Analysis with least absolute residuals. Computational Statistics and Data Analysis, 5, 337–356.CrossRefGoogle Scholar
Kiers, H.A.L. (1990). Majorization as a tool for optimizing a class of matrix functions. Psychometrika, 55, 417–428.CrossRefGoogle Scholar
Kiers, H.A.L. (1995). Maximization of sums of quotients of quadratic forms and some genralizations. Psychometrika, 60, 221–245.CrossRefGoogle Scholar
Korth, B., Tucker, L.R. (1976). Procrustes matching by congruence coefficients. Psychometrika, 41, 531–535.CrossRefGoogle Scholar
Koschat, M.A., Swayne, D.F. (1991). A weighted Procrustes criterion. Psychometrika, 56, 229–239.CrossRefGoogle Scholar
Meulman, J.J. (1986). A distance approach to nonlinear multivariate analysis, Leiden: DSWO Press.Google Scholar
Ostrowski, A.M. (1969). Solutions of equations and systems of equations, New York: Academic Press.Google Scholar
ten Berge, J.M.F. (1986). Some relationships between descriptive comparisons of components from different studies. Multivariate Behavioral Research, 21, 29–40.CrossRefGoogle ScholarPubMed
Tucker, L.R. (1951). A method for synthesis of factor analysis studies, Washington DC: Department of the Army.CrossRefGoogle Scholar