Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-10T00:54:03.570Z Has data issue: false hasContentIssue false
  • English
  • Français

Matrix Correlation

Published online by Cambridge University Press:  01 January 2025

J. O. Ramsay ,
Jos ten Berge  and
G. P. H. Styan
J. O. Ramsay*
Affiliation:
McGill University
Jos ten Berge
Affiliation:
University of Groningen
G. P. H. Styan
Affiliation:
McGill University
*
Reprint requests should be sent to the first author at Department of Psychology, 1205 Dr. Penfield Ave., Montreal, Quebec, Canada H3A 1B1.
Get access Rights & Permissions [Opens in a new window]

Abstract

A correlational measure for an n by p matrix X and an n by q matrix Y assesses their relation without specifying either as a fixed target. This paper discusses a number of useful measures of correlation, with emphasis on measures which are invariant with respect to rotations or changes in singular values of either matrix. The maximization of matrix correlation with respect to transformations XL and YM is discussed where one or both transformations are constrained to be orthogonal. Special attention is focussed on transformations which cause XL and YM to be n by s, where s may be any number between 1 and min (p, q). An efficient algorithm is described for maximizing the correlation between XL and YM where analytic solutions do not exist. A factor analytic example is presented illustrating the advantages of various coefficients and of varying the number of columns of the transformed matrices.

Information

Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 3 , September 1984 , pp. 403 - 423
Copyright
Copyright © 1984 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.)

Article purchase

Temporarily unavailable

Footnotes

This research was supported by grant APA 0320 from the Natural Sciences and Engineering Research Council of Canada. The authors wish to acknowledge valuable discussion of this problem with Jan de Leeuw, University of Leiden.

References

Berge, J. M. F. (1977). Optimizing factorial invariance, Netherlands: University of Groningen.Google Scholar
Berge, J. M. F. (1983). A generalization of Kristof's theorem on the trace of certain matrix products. Psychometrika, 48, 519–523.CrossRefGoogle Scholar
Besse, P. (1979). Etude descriptive d'un processus, France: l'Université Paul-Sabatier de Toulouse.Google Scholar
Cailliez, F. & Pages, JP. (1976). Introduction à l'Analyse des Données, Paris: Société de Mathématiques Appliquées et de Sciences Humaines, 9 rue Duban, 75016 Paris.Google Scholar
Cliff, N. (1966). Orthogonal rotation to congruence. Psychometrika, 31, 33–42.CrossRefGoogle Scholar
Cramer, E., Nicewander, W. A. (1979). Some symmetric, invariant measures of multivariate association. Psychometrika, 44, 43–54.CrossRefGoogle Scholar
Dauxois, J. & Pousse, A. (1976). Les analyses factorielles en calcul des probabilités et en statistique: Essai d'étude synthètique, France: l'Université Paul-Sabatier de Toulouse.Google Scholar
Escoufier, Y. (1973). Le traitment des variables vectorielles. Biometrics, 29, 751–760.CrossRefGoogle Scholar
Escoufier, Y. (1977). Operators related to a data matrix. In Barra, J. R. et al. (Eds.), Recent Developments in Statistics, Amsterdam: North-Holland.Google Scholar
Evans, G. T. (1971). Transformation of factor matrices to achieve congruence. British Journal of Mathematical and Statistical Psychology, 24, 22–48.CrossRefGoogle Scholar
Fan, Ky (1951). Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Nat. Acad. Sci., 37, 760–766.CrossRefGoogle ScholarPubMed
Green, B. F. (1952). The orthogonal approximation of an oblique simple structure in factor analysis. Psychometrika, 17, 429–440.CrossRefGoogle Scholar
Hotelling, H. (1936). Relations between two sets of variates. Biometrika, 28, 321–377.CrossRefGoogle Scholar
Jaffrenou, P. A. (1978). Sur l'analyse des familles finies de variables vectorielles: Bases algébrique et application à la description statistique, France: l'Université de Saint-Etienne.Google Scholar
Kettenring, J. R. (1971). Canonical analysis of several sets of variables. Biometrika, 58, 433–451.CrossRefGoogle Scholar
Kristof, W. (1970). A theorem on the trace of certain matrix products and some applications. Journal of Mathematical Psychology, 7, 515–530.CrossRefGoogle Scholar
Lancaster, H. O. (1969). The chi square distribution, New York: Wiley.Google Scholar
Lingoes, J. C. & Schönemann, P. H. (1974). Alternative measures of fit for the Schönemann-Carroll matrix fitting algorithm. Psychometrika, 39, 423–427.CrossRefGoogle Scholar
Pages, J. P., Ramsay, J. O., & Tenenhaus, M. (1984). A geometrical approach to multivariate analysis, Unpublished manuscript submitted for publication.Google Scholar
Ramsay, J. O. (1980). Exploring multivariate data with inside-out plots. Presented at the Psychometric Society Meetings, Iowa City.Google Scholar
Ramsay, J. O. & Novick, M. R. (1980). PLU robust Bayesian decision theory: Point estimation. Journal of American Statistical Association, 75, 901–907.CrossRefGoogle Scholar
Rao, C. R. & Yanai, H. (1979). General definition and decomposition of projectors and some applications to statistical problems. Journal of Statistical Planning and Inference, 3, 1–17.CrossRefGoogle Scholar
Rao, C. R. (1979). Separation theorems for singular values of matrices and their applications in multivariate analysis. Journal of Multivariate Analysis, 9, 362–377.CrossRefGoogle Scholar
Renyi, A. (1959). On measures of dependence. Acta Mathematica of the Academy of Science of Hungary, 10, 441–451.CrossRefGoogle Scholar
Richter, H. (1958). Zur Abschatzung von Matrizennormen. Dieze Nachr., 18, 178–187.CrossRefGoogle Scholar
Robert, P., Escoufier, Y. (1976). A unifying tool for linear multivariate statistical methods: The RV-coefficient. Applied Statistics, 25, 257–265.CrossRefGoogle Scholar
SAS Institute Inc. (1982). SAS User's Guide: Statistics 1982 Edition,, Cary, NC: SAS Institute Inc..Google Scholar
Schönemann, P. H. (1966). A generalized solution of the orthogonal Procrustes problem. Psychometrika, 31, 1–10.CrossRefGoogle Scholar
Styan, G. P. H. (1976). Separation theorems. Unpublished manuscript, McGill University.Google Scholar
Theobald, C. M. (1975). An inequality with application to multivariate analysis. Biometrika, 62, 461–466.CrossRefGoogle Scholar
von Neumann, J. (1937). Some matrix-inequalities and metrization of matric-space. Tomask Univ. Rev., 1, 286–300.Google Scholar
Wainer, H. & Thissen, D. (1981). Graphical data analysis. Annual Review of Psychology, 32, 191–241.CrossRefGoogle Scholar
Yanai, H. (1974). Unification of various techniques of multivariate analysis by means of generalized coefficient of determination (G.C.D.). J. of Behaviormetrics, 1, 45–54.Google Scholar
Yanai, H. (1980). A proposition of generalized method for forward selection of variables. Behaviormetrika, 7, 95–107.CrossRefGoogle Scholar