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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)
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.

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