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Orthogonal Procrustes Rotation for Two or More Matrices

Published online by Cambridge University Press:  01 January 2025

Jos M. F. Ten Berge
Jos M. F. Ten Berge*
Affiliation:
University of Groningen, The Netherlands
*
Requests for reprints should be sent to Jos M. F. Ten Berge, Psychology Department, University of Groningen, Groningen, the Netherlands.
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Abstract

Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower’s method of generalized Procrustes analysis are suggested.

Keywords

factor matchingleast-squares rotation
Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 2 , June 1977 , pp. 267 - 276
Copyright
Copyright © 1977 The Psychometric Society

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References

Reference Notes

Haven, S. Empirical comparison of two methods of simultaneous Procrustes rotation, 1976, Groningen, the Netherlands: University of Groningen, Department of Psychology.Google Scholar
Schonemann, P. H., Bock, R. D., & Tucker, L. R. Some notes on a theorem by Eckart and Young, 1965, Chapel Hill, North Carolina: University of North Carolina Psychometric Laboratory.Google Scholar
Tucker, L. R. A method for synthesis of factor analytic studies, 1951, Washington, D. C.: Department of the Army.CrossRefGoogle Scholar

References

Cliff, N. Orthogonal rotation to congruence. Psychometrika, 1966, 31, 33–42.CrossRefGoogle Scholar
Eckart, C. & Young, G. The approximation of one matrix by another of lower rank. Psychometrika, 1936, 1, 211–218.CrossRefGoogle Scholar
Fischer, G. H. & Roppert, J. Ein Verfahren der Transformationsanalyse faktorenanalytischer Ergebnisse. In Roppert, J. and Fischer, G. H. (Eds.), Lineare Strukturen in Mathematik und Statistik, 1965, Wien/Würzburg: Physika-Verlag.Google Scholar
Gower, J. C. Generalized Procrustes analysis. Psychometrika, 1975, 40, 33–51.CrossRefGoogle Scholar
Green, B. F. The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 1952, 17, 429–440.CrossRefGoogle Scholar
Kettenring, J. R. Canonical analysis of several sets of variables. Biometrika, 1971, 58, 433–451.CrossRefGoogle Scholar
Kristof, W. Die beste orthogonale Transformation zur gegenseitigen Ueberfuehrung zweier Faktormatrizen. Diagnostica, 1964, 10, 87–90.Google Scholar
Kristof, W. & Wingersky, B. Generalization of the orthogonal Procrustes rotation procedure for more than two matrices. Proceedings of the 79th Annual Convention of the American Psychological Association, 1971, 89-90.Google Scholar
Schönemann, P. H. A generalized solution of the orthogonal Procrustes problem. Psychometrika, 1966, 31, 1-10.CrossRefGoogle Scholar