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

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  • Hironori Satomura
  • Kohei Adachi

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. Copyright The Psychometric Society 2013

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  • Hironori Satomura & Kohei Adachi, 2013. "Oblique Rotaton in Canonical Correlation Analysis Reformulated as Maximizing the Generalized Coefficient of Determination," Psychometrika, Springer;The Psychometric Society, vol. 78(3), pages 526-537, July.
    • Handle: RePEc:spr:psycho:v:78:y:2013:i:3:p:526-537
    DOI: 10.1007/s11336-012-9310-4
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    References listed on IDEAS

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    1. Robert Jennrich, 2002. "A simple general method for oblique rotation," Psychometrika, Springer;The Psychometric Society, vol. 67(1), pages 7-19, March.
    2. Jos Berge, 1983. "A generalization of Kristof's theorem on the trace of certain matrix products," Psychometrika, Springer;The Psychometric Society, vol. 48(4), pages 519-523, December.
    3. Norman Cliff & David Krus, 1976. "Interpretation of canonical analysis: Rotated vs. unrotated solutions," Psychometrika, Springer;The Psychometric Society, vol. 41(1), pages 35-42, March.
    4. Michel Velden & Henk A.L. Kiers, 2005. "Rotation in Correspondence Analysis," Journal of Classification, Springer;The Classification Society, vol. 22(2), pages 251-271, September.
    5. Henry Kaiser, 1958. "The varimax criterion for analytic rotation in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 23(3), pages 187-200, September.
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    Cited by:

    1. Naomichi Makino, 2022. "Rotation in Correspondence Analysis from the Canonical Correlation Perspective," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 1045-1063, September.
    2. Naoto Yamashita & Kohei Adachi, 2020. "A Modified k-Means Clustering Procedure for Obtaining a Cardinality-Constrained Centroid Matrix," Journal of Classification, Springer;The Classification Society, vol. 37(2), pages 509-525, July.

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