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Principal Component Analysis with External Information on both Subjects and Variables

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane  and
Tadashi Shibayama
Yoshio Takane*
Affiliation:
McGill University
Tadashi Shibayama
Affiliation:
McGill University
*
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

A method for structural analysis of multivariate data is proposed that combines features of regression analysis and principal component analysis. In this method, the original data are first decomposed into several components according to external information. The components are then subjected to principal component analysis to explore structures within the components. It is shown that this requires the generalized singular value decomposition of a matrix with certain metric matrices. The numerical method based on the QR decomposition is described, which simplifies the computation considerably. The proposed method includes a number of interesting special cases, whose relations to existing methods are discussed. Examples are given to demonstrate practical uses of the method.

Keywords

orthogonal projection operatortrace-orthogonalitygeneralized singular value decomposition (GSVD)QR decompositionvector preference modelstwo-way CANDELINCdual scalingredundancy analysisGMANOVA (growth curve models)
Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 1 , March 1991 , pp. 97 - 120
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

The work reported in this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. Thanks are due to Jim Ramsay, Haruo Yanai, Henk Kiers, and Shizuhiko Nishisato for their insightful comments on earlier versions of this paper. Jim Ramsay, in particular, suggested the use of the QR decomposition, which simplified the presentation of the paper considerably.

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