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Regularized Partial and/or Constrained Redundancy Analysis

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane  and
Sunho Jung
Yoshio Takane*
Affiliation:
McGill University
Sunho Jung
Affiliation:
McGill University
*
Requests for reprints should be sent to Yoshio Takane, Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC, H3A 1B1, Canada. E-mail: [email protected]
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Abstract

Methods of incorporating a ridge type of regularization into partial redundancy analysis (PRA), constrained redundancy analysis (CRA), and partial and constrained redundancy analysis (PCRA) were discussed. The usefulness of ridge estimation in reducing mean square error (MSE) has been recognized in multiple regression analysis for some time, especially when predictor variables are nearly collinear, and the ordinary least squares estimator is poorly determined. The ridge estimation method was extended to PRA, CRA, and PCRA, where the reduced rank ridge estimates of regression coefficients were obtained by minimizing the ridge least squares criterion. It was shown that in all cases they could be obtained in closed form for a fixed value of ridge parameter. An optimal value of the ridge parameter is found by G-fold cross validation. Illustrative examples were given to demonstrate the usefulness of the method in practical data analysis situations.

Keywords

reduced rank approximationscovariateslinear constraintsleast squares estimationridge least squares estimationgeneralized singular value decomposition (GSVD)G-fold cross validationbootstrap method
Type
Original Paper
Information
Psychometrika , Volume 73 , Issue 4 , December 2008 , pp. 671 - 690
Copyright
Copyright © 2008 The Psychometric Society

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Footnotes

We thank Jim Ramsay for his insightful comments on an earlier draft of this paper. The work reported in this paper is supported by Grants 10630 from the Natural Sciences and Engineering Research Council of Canada to the first author.

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