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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

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