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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: adachi@hus.osaka-u.ac.jp
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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

Information

Type
Theory and Methods
Information
Psychometrika , Volume 74 , Issue 4 , December 2009, pp. 667 - 683
Copyright
Copyright © 2009 The Psychometric Society

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