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Oblique Rotaton in Canonical Correlation Analysis Reformulated as Maximizing the Generalized Coefficient of Determination

Published online by Cambridge University Press:  01 January 2025

Hironori Satomura  and
Kohei Adachi
Hironori Satomura
Affiliation:
Osaka University
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

To facilitate the interpretation of canonical correlation analysis (CCA) solutions, procedures have been proposed in which CCA solutions are orthogonally rotated to a simple structure. In this paper, we consider oblique rotation for CCA to provide solutions that are much easier to interpret, though only orthogonal rotation is allowed in the existing formulations of CCA. Our task is thus to reformulate CCA so that its solutions have the freedom of oblique rotation. Such a task can be achieved using Yanai’s (Jpn. J. Behaviormetrics 1:46–54, 1974; J. Jpn. Stat. Soc. 11:43–53, 1981) generalized coefficient of determination for the objective function to be maximized in CCA. The resulting solutions are proved to include the existing orthogonal ones as special cases and to be rotated obliquely without affecting the objective function value, where ten Berge’s (Psychometrika 48:519–523, 1983) theorems on suborthonormal matrices are used. A real data example demonstrates that the proposed oblique rotation can provide simple, easily interpreted CCA solutions.

Keywords

canonical correlation analysisoblique rotationgeneralized coefficient of determinationsuborthonormal matrices
Type
Original Paper
Information
Psychometrika , Volume 78 , Issue 3 , July 2013 , pp. 526 - 537
Copyright
Copyright © 2012 The Psychometric Society

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