Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-08T10:08:46.810Z Has data issue: false hasContentIssue false
  • English
  • Français

Joint Procrustes Analysis for Simultaneous Nonsingular Transformation of Component Score and Loading Matrices

Published online by Cambridge University Press:  01 January 2025

Kohei Adachi
Kohei Adachi*
Affiliation:
Osaka University
*
Requests for reprints should be sent to Kohei Adachi, Graduate School of Human Sciences, Osaka University, 1-2, Yamadaoka, Suita, Osaka 565-0871, Japan. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In component analysis solutions, post-multiplying a component score matrix by a nonsingular matrix can be compensated by applying its inverse to the corresponding loading matrix. To eliminate this indeterminacy on nonsingular transformation, we propose Joint Procrustes Analysis (JPA) in which component score and loading matrices are simultaneously transformed so that the former matrix matches a target score matrix and the latter matches a target loading matrix. The loss function of JPA is a function of the nonsingular transformation matrix and its inverse, and is difficult to minimize straightforwardly. To deal with this difficulty, we reparameterize those matrices by their singular value decomposition, which reduces the minimization to alternately solving quartic equations and performing the existing multivariate procedures. This algorithm is assessed in a simulation study. We further extend JPA for cases where targets are linear functions of unknown parameters. We also discuss how the application of JPA can be extended in different fields.

Keywords

joint Procrustes analysiscomponent analysisnonsingular transformationsingular value decomposition
Type
Theory and Methods
Information
Psychometrika , Volume 74 , Issue 4 , December 2009 , pp. 667 - 683
Copyright
Copyright © 2009 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.)

References

Benzécri, J.-P. (1992). Correspondence analysis handbook, New York: Marcel Dekker.CrossRefGoogle Scholar
Browne, M.W. (1967). On oblique Procrustes rotation. Psychometrika, 32, 125–132.CrossRefGoogle ScholarPubMed
Gower, J.C., Dijksterhuis, G.B. (2004). Procrustes problems, Oxford: Oxford University Press.CrossRefGoogle Scholar
Green, B.F. (1952). The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 17, 429–440.CrossRefGoogle Scholar
Greenacre, M.J. (1984). Theory and applications of correspondence analysis, London: Academic Press.Google Scholar
Hendrickson, A.E., White, P.O. (1964). PROMAX: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17, 65–70.CrossRefGoogle Scholar
Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200.CrossRefGoogle Scholar
Kiers, H.A.L. (1990). Majorization as a tool for optimizing a class of matrix functions. Psychometrika, 55, 417–428.CrossRefGoogle Scholar
Kiers, H.A.L. (1994). SIMPLIMAX: Oblique rotation to an optimal target with simple structure. Psychometrika, 59, 567–579.CrossRefGoogle Scholar
Kiers, H.A.L. (1998). Joint orthomax rotation of the core and component matrices resulting from three-mode principal component analysis. Journal of Classification, 15, 245–263.CrossRefGoogle Scholar
Kiers, H.A.L. (1998). Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure. Computational Statistics and Data Analysis, 28, 307–324.CrossRefGoogle Scholar
Kiers, H.A.L., ten Berge, J.M.F. (1992). Minimization of a class of matrix trace functions by means of refined majorization. Psychometrika, 57, 371–382.CrossRefGoogle Scholar
Kroonenberg, P.M. (2008). Applied multiway data analysis, Hoboken: Wiley.CrossRefGoogle Scholar
Kroonenberg, P.M., de Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 45, 69–97.CrossRefGoogle Scholar
Lorenzo-Seva, U., ten Berge, J.M.F. (2006). Tucker’s congruence coefficient as a meaningful index of factor similarity. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 2, 57–64.CrossRefGoogle Scholar
Mori, K. (1990). Nonlinear equations. In Ohno, Y., Isoda, K. (Eds.), Handbook of Numerical Computations: A New Version (pp. 589–688). Tokyo: Ohm-Sha (in Japanese)Google Scholar
Mosier, C.I. (1939). Determining a simple structure when loadings for certain tests are known. Psychometrika, 4, 149–162.CrossRefGoogle Scholar
Mulaik, S.A. (1972). The foundations of factor analysis, New York: McGraw-Hill.Google Scholar
Osgood, C.E., Suci, G.J., Tannenbaum, P.H. (1957). The measurement of meaning, Urbana: University of Illinois Press.Google Scholar
Takane, Y., Shibayama, T. (1991). Principal component analysis with external information of both subjects and variables. Psychometrika, 56, 97–120.CrossRefGoogle Scholar
ten Berge, J.M.F. (1993). Least squares optimization in multivariate analysis, Leiden: DSWO Press.Google Scholar
ten Berge, J.M.F., Kiers, H.A.L. (1996). Optimality criteria for principal component analysis and generalizations. British Journal of Mathematical and Statistical Psychology, 49, 335–345.CrossRefGoogle Scholar
ten Berge, J.M.F., Nevels, K. (1977). A general solution of Mosier’s oblique procrustes problem. Psychometrika, 42, 593–600.CrossRefGoogle Scholar
Tucker, L.R. (1951). A method for synthesis of factor analysis studies (Personnel Research Section Report No. 984). Department of the Army, Washington, DC.Google Scholar
Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.CrossRefGoogle ScholarPubMed
van de Velden, M., Kiers, H.A.L. (2003). An application of rotation in correspondence analysis. In Yanai, H., Okada, A., Shigemasu, K., Kano, Y., Meulman, J.J. (Eds.), New developments in psychometrics (pp. 471–478). Tokyo: Springer.CrossRefGoogle Scholar
van de Velden, M., Kiers, H.A.L. (2005). Rotation in correspondence analysis. Journal of Classification, 22, 251–271.CrossRefGoogle Scholar
Weisstein, E.W. (1998). CRC concise encyclopedia of mathematics, New York: CRC Press.Google Scholar