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Component Analysis with Different Sets of Constraints on Different Dimensions

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane ,
Henk A. L. Kiers  and
Jan de Leeuw
Yoshio Takane*
Affiliation:
McGill University
Henk A. L. Kiers
Affiliation:
University of Groningen
Jan de Leeuw
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Yoshio Takane, Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, Quebec H3A 1B1, CANADA.
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Abstract

Many of the “classical” multivariate data analysis and multidimensional scaling techniques call for approximations by lower dimensional configurations. A model is proposed, in which different sets of linear constraints are imposed on different dimensions in component analysis and “classical” multidimensional scaling frameworks. A simple, efficient, and monotonically convergent algorithm is presented for fitting the model to the data by least squares. The basic algorithm is extended to cover across-dimension constraints imposed in addition to the dimensionwise constraints, and to the case of a symmetric data matrix. Examples are given to demonstrate the use of the method.

Keywords

dimensionwise constraintsacross-dimension constraintsrank restrictionsconstrained principal component analysis (CPCA)multidimensional scaling (MDS)

Information

Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 2 , June 1995 , pp. 259 - 280
Copyright
Copyright © 1995 The Psychometric Society

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Footnotes

The work reported in this paper has been supported by the Natural Sciences and Engineering Research Council of Canada, grant number A6394, and by the McGill-IBM Cooperative Grant, both granted to the first author. The research of H. A. L. Kiers has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. We thank Michael Hunter for his helpful comments on earlier drafts of this paper.

References

Andersen, E. B. (1980). Discrete statistical models with social science applications, Amsterdam: North-Holland.Google Scholar
Anderson, T. W. (1951). Estimating linear restrictions on regression coefficients for multivariate normal distribution. Annals of Mathematical Statistics, 22, 327–351.CrossRefGoogle Scholar
Böckenholt, U., Böckenholt, I. (1990). Canonical analysis of contingency tables with linear constraints. Psychometrika, 55, 633–639.CrossRefGoogle Scholar
Carroll, J. D. (1972). Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K., Nerlove, S. B. (Eds.), Multidimensional scalling, Vol. 1 (pp. 105–155). New York: Seminar Press.Google Scholar
Carroll, J. D., Pruzansky, S., Kruskal, J. B. (1980). CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters. Psychometrika, 45, 3–24.CrossRefGoogle Scholar
Coombs, C. H. (1964). A theory of data, New York: Wiley.Google Scholar
Delbeke, L. (1978). Enkele analyses op voorkeuroordelen voor gezinssamenstellingen [Some analyses of preference judgments for family compositions]. Centrum voor Mathematische Psychologie en Psychologische Methodologie, Katholieke Universiteit Leuven.Google Scholar
de Leeuw, J. (1984). The Gifi-system of nonlinear multivariate analysis. In Diday, E., Jambu, M., Lebart, L., Pagès, J., Thomassone, R. (Eds.), Data analysis and informatics III (pp. 415–424). Amsterdam: North Holland.Google Scholar
de Leeuw, J., Young, F. W., Takane, Y. (1976). Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometrika, 41, 471–503.CrossRefGoogle Scholar
DeSarbo, W. S., Rao, V. R. (1984). GENFOLDS: A set of models and algorithms for the GENeral UnFOLDing analysis of preference/dominance data. Journal of Classification, 1, 147–186.CrossRefGoogle Scholar
Efron, B. (1979). Bootstrap methods: Another look at the Jackknife. Annals of Statistics, 7, 1–26.CrossRefGoogle Scholar
Gabriel, K. R. (1971). The biplot graphic display of matrices with application to principal component analysis. Biometqrika, 58, 453–467.CrossRefGoogle Scholar
Golub, G. H. (1973). Some modified eigenvalue problems. SIAM Review, 15, 318–335.CrossRefGoogle Scholar
Greenacre, M. J. (1984). Theory and applications of correspondence analysis, London: Academic Press.Google Scholar
Greenacre, M. J. (1993). Correspondence analysis in practice, London: Academic Press.Google Scholar
Heiser, W. J. (1981). Unfolding analysis of proximity data. Unpublished doctoral dissertation, University of Leiden.Google Scholar
Heiser, W. J. (1987). Joint ordination of species and sites: The unfolding technique. In Legendre, P., Legendre, L. (Eds.), Developments in numerical ecology (pp. 189–221). Berlin: Springer Verlag.CrossRefGoogle Scholar
Heiser, W. J., Meulamn, J. (1983). Constrained multidimensional scaling, including confirmation. Applied Psychological Measurement, 7, 381–404.CrossRefGoogle Scholar
Jackson, J. E. (1991). A user's guide to principal components, New York: Wiley.CrossRefGoogle Scholar
Jolliffe, I. T. (1986). Principal component analysis, Berlin: Springer Verlag.CrossRefGoogle Scholar
Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202.CrossRefGoogle Scholar
Jöreskog, K. G. (1970). A general method for analysis of covariance structures. Biometrika, 57, 239–251.CrossRefGoogle Scholar
Kiefer, J., Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Mathematical Statistics, 27, 887–906.CrossRefGoogle Scholar
Kiers, H. A. L., Krijnen, W. P. (1991). An efficient algorithm for PARAFAC of three-way data with large numbers of observation units. Psychometrika, 56, 147–152.CrossRefGoogle Scholar
Kiers, H. A. L., Kroonenberg, P. M., ten Berge, J. M. F. (1992). An efficient algorithm for TUCKALS 3 on data with large numbers of observation units. Psychometrika, 57, 415–422.CrossRefGoogle Scholar
Kiers, H. A. L., Takane, Y. (1993). Constrained DEDICOM. Psychometrika, 58, 339–355.CrossRefGoogle Scholar
Lebart, L., Morineau, A., Warwick, K. (1984). Multivariate descriptive statistical analysis, New York: Wiley.Google Scholar
Lohmöller, J. (1989). Latent variable path modelling with partial least squares, Heidelberg: Physica Verlag.CrossRefGoogle Scholar
Meredith, W., Millsap, R. E. (1985). On component analysis. Psychometrika, 50, 495–507.CrossRefGoogle Scholar
Meulman, J. (1986). A distance approach to nonlinear multivariate analysis, Leiden: DSWO Press.Google Scholar
Meulman, J., Heiser, W. et al. (1984). Constrained multidimensional scaling: More directions than dimensions. In Havranek, T. et al. (Eds.), COMPSTAT 84 (pp. 137–142). Vienna: Physica Verlag.Google Scholar
Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications, Toronto: University of Toronto Press.CrossRefGoogle Scholar
Ramsay, J. O. (1980). Joint analysis of direct ratings, pairwise preferences and dissimilarities. Psychometrika, 45, 149–165.CrossRefGoogle Scholar
Rao, C. R. (1964). The use and interpretation of principal component analysis in applied research. Sankhya A, 26, 329–358.Google Scholar
Rao, C. R. (1980). Matrix approximations and reduction of dimensionality in multivariate statistical analysis. In Krishnaiah, P. R. (Eds.), Multivariate analysis (pp. 3–22). Amsterdam: North Holland.Google Scholar
Takane, Y. (1978). A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings are independent theory and evaluations. Japanese Psychological Research, 20, 7–17.CrossRefGoogle Scholar
Takane, Y. (1981). Multidimensional successive categories scaling: A maximum likelihood method. Psychometrika, 46, 9–28.CrossRefGoogle Scholar
Takane, Y., Shibayama, T. (1991). Principal component analysis with external information on both subjects and variables. Psychometrika, 56, 97–120.CrossRefGoogle Scholar
Takane, Y., Yanai, H., Mayekawa, S. (1991). Relationships among several methods of linearly constrained correspondence analysis. Psychometrika, 56, 667–684.CrossRefGoogle Scholar
Takane, Y., Young, F. W., de Leeuw, J. (1980). An individual differences additive model: An alternating least squares method with optimal scaling features. Psychometrika, 45, 183–209.CrossRefGoogle Scholar
Berge, J. M. F. (1986). Rotation to perfect congruence and the cross-validation of component weights across populations. Multivariate Behavioral Research, 21, 41–64.CrossRefGoogle ScholarPubMed
Berge, J. M. F., Kiers, H. A. L. (1991). Some clarifications of the CANDECOMP algorithm applied to INDSCAL. Psychometrika, 56, 317–326.CrossRefGoogle Scholar
ter Braak, C. J. F. (1986). Canonical correspondence analysis: A new eigenvector technique for multivariate direct gradient analysis. Ecology, 67, 1167–1179.CrossRefGoogle Scholar
ter Braak, C. J. F. (1990). Interpreting canonical correlation analysis through biplots of structural correlations and weights. Psychometrika, 55, 519–531.CrossRefGoogle Scholar
Torgerson, W. S. (1958). Theory and methods of scalling, New York: Wiley.Google Scholar
van den Wollenberg, A. L. (1977). Redundancy analysis: An alternative for canonical correlation analysis. Psychometrika, 42, 207–219.CrossRefGoogle Scholar
van der Burg, E., de Leeuw, J., Verdegaal, R. (1988). Homogeneity analysis: An alternating least squares method with optimal scaling features. Psychometrika, 53, 177–197.CrossRefGoogle Scholar
van der Lans, I. A. (1992). Nonlinear multivariate analysis for multivariate preference data, Leiden: DSWO Press.Google Scholar
van der Leeden, R. (1990). Reduced rank regression with structured residuals, Leiden, The Netherlands: DSWO Press.Google Scholar
Velicer, W. F., Jackson, D. N. (1990). Component analysis versus common factor analysis: Some issues in selecting an appropriate procedure. Multivariate Behavioral Research, 25, 1–28.CrossRefGoogle ScholarPubMed
Velu, R. P. (1991). Reduced rank models with two sets of regressors. Applied Statistics, 40, 159–170.CrossRefGoogle Scholar
Wold, H. (1982). Soft modelling: The basic design and some extensions. Systems under indirect observations, II (pp. 1–54). Amsterdam: North-Holland.Google Scholar
Young, G., Householder, A. S. (1938). Discussion of a set of points in terms of their mutual distances. Psychometrika, 3, 19–22.CrossRefGoogle Scholar