Advertisement

Psychometrika

, Volume 42, Issue 2, pp 267–276 | Cite as

Orthogonal procrustes rotation for two or more matrices

  • Jos M. F. Ten Berge
Article

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.

Key words

factor matching least-squares rotation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference Notes

  1. Haven, S.Empirical comparison of two methods of simultaneous Procrustes rotation (Heymans Bulletin 76-245 EX). Groningen, the Netherlands: University of Groningen, Department of Psychology, 1976.Google Scholar
  2. Schonemann, P. H., Bock, R. D., & Tucker, L. R.Some notes on a theorem by Eckart and Young (Res. Memorandum No. 25). Chapel Hill, North Carolina: University of North Carolina Psychometric Laboratory, 1965.Google Scholar
  3. Tucker, L. R.A method for synthesis of factor analytic studies (Personnel Research Section Report No. 984). Washington, D. C.: Department of the Army, 1951.Google Scholar

References

  1. Cliff, N. Orthogonal rotation to congruence.Psychometrika, 1966,31, 33–42.Google Scholar
  2. Eckart, C. & Young, G. The approximation of one matrix by another of lower rank.Psychometrika, 1936,1, 211–218.Google Scholar
  3. Fischer, G. H. & Roppert, J. Ein Verfahren der Transformationsanalyse faktorenanalytischer Ergebnisse. In J. Roppert and G. H. Fischer,Lineare Strukturen in Mathematik und Statistik. Wien/Würzburg: Physika-Verlag, 1965.Google Scholar
  4. Gower, J. C. Generalized Procrustes analysis.Psychometrika, 1975,40, 33–51.Google Scholar
  5. Green, B. F. The orthogonal approximation of an oblique structure in factor analysis.Psychometrika, 1952,17, 429–440.Google Scholar
  6. Kettenring, J. R. Canonical analysis of several sets of variables.Biometrika, 1971,58, 433–451.Google Scholar
  7. Kristof, W. Die beste orthogonale Transformation zur gegenseitigen Ueberfuehrung zweier Faktormatrizen.Diagnostica, 1964,10, 87–90.Google Scholar
  8. 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
  9. Schönemann, P. H. A generalized solution of the orthogonal Procrustes problem.Psychometrika, 1966,31, 1–10.Google Scholar

Copyright information

© Psychometric Society 1977

Authors and Affiliations

  • Jos M. F. Ten Berge
    • 1
  1. 1.Psychology DepartmentUniversity of GroningenGroningenthe Netherlands

Personalised recommendations