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Orthogonal Rotations to Maximal Agreement for Two or More Matrices of Different Column Orders

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge  and
Dirk L. Knol
Jos M. F. ten Berge*
Affiliation:
University of Groningen
Dirk L. Knol*
Affiliation:
University of Groningen
*
Reprint requests should be addressed to Jos ten Berge and Dirk Knol, Subfakulteit Psychologie R.U. Groningen, Grote Markt 31/32, 9712 HV Groningen, The Netherlands.
Reprint requests should be addressed to Jos ten Berge and Dirk Knol, Subfakulteit Psychologie R.U. Groningen, Grote Markt 31/32, 9712 HV Groningen, The Netherlands.
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Abstract

Methods for orthogonal Procrustes rotation and orthogonal rotation to a maximal sum of inner products are examined for the case when the matrices involved have different numbers of columns. An inner product solution offered by Cliff is generalized to the case of more than two matrices. A nonrandom start for a Procrustes solution suggested by Green and Gower is shown to give better results than a random start. The Green-Gower Procrustes solution (with nonrandom start) is generalized to the case of more than two matrices. Simulation studies indicate that both the generalized inner product solution and the generalized Procrustes solution tend to attain their global optima within acceptable computation times. A simple procedure is offered for approximating simple structure for the rotated matrices without affecting either the Procrustes or the inner product criterion.

Keywords

Procrustes rotationtarget rotationcongruence
Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 1 , March 1984 , pp. 49 - 55
Copyright
Copyright © 1984 The Psychometric Society

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Footnotes

The authors are obliged to Charles Lewis for helpful comments on a previous draft of this paper and to Frank Brokken for preparing a computer program that was used in this study.

References

Reference Notes

Green, B. F. & Gower, J. C. (1979). A problem with congruence. Paper presented at the Annual Meeting of the Psychometric Society, Monterey, California.Google Scholar
Ten Berge, J. M. F. (1977). Optimizing factorial invariance. Unpublished doctoral dissertation, University of Groningen.Google Scholar
Tucker, L. R. (1951). A method of synthesis of factor analysis studies, Washington, D.C.: Department of the Army.CrossRefGoogle Scholar

References

Cliff, N. (1966). Orthogonal rotation to congruence. Psychometrika, 31, 33–42.CrossRefGoogle Scholar
Green, B. F. (1952). The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 17, 429–440.CrossRefGoogle Scholar
Kettenring, J. R. (1971). Canonical analysis of several sets of variables. Biometrika, 58, 433–451.CrossRefGoogle Scholar
Ten Berge, J. M. F. (1977). Orthogonal Procrustes rotation for two or more matrices. Psychometrika, 42, 267–276.CrossRefGoogle Scholar
Ten Berge, J. M. F. & Nevels, K. (1977). A general solution for Mosier's oblique Procrustes problem. Psychometrika, 42, 593–600.CrossRefGoogle Scholar