Oblique rotaton in canonical correlation analysis reformulated as maximizing the generalized coefficient of determination. (English) Zbl 1285.62135

Summary: 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 H. Yanai’s [“Explicit expression of projectors on canonical variables and distances between centroids of groups”, 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 J. M. F. ten Berge’s [Psychometrika 48, 519–523 (1983; Zbl 0536.62093)] theorems on suborthonormal matrices are used. A real data example demonstrates that the proposed oblique rotation can provide simple, easily interpreted CCA solutions.


62P15 Applications of statistics to psychology


Zbl 0536.62093
Full Text: DOI


[1] Browne, M.W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111–150.
[2] Cliff, N., & Kruss, D.J. (1976). Interpretation of canonical analysis: rotated vs. unrotated solutions. Psychometrika, 41, 35–42. · Zbl 0332.62043
[3] Darlington, R.B., Weinberg, S.L., & Walberg, H.J. (1973). Canonical variate analysis and related techniques. Review of Educational Research, 43, 433–454.
[4] Gifi, A. (1990). Nonlinear multivariate analysis. Chichester: Wiley. · Zbl 0697.62048
[5] Greenacre, M.J. (1984). Theory and applications of correspondence analysis. London: Academic Press. · Zbl 0555.62005
[6] Hendrickson, A.E., & White, P.O. (1964). PROMAX: a quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17, 65–70.
[7] Izenmann, A.J. (2008). Modern multivariate statistical techniques. New York: Springer.
[8] Jennrich, R.I. (2002). A simple general method for oblique rotation. Psychometrika, 67, 7–20. · Zbl 1297.62232
[9] Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200. · Zbl 0095.33603
[10] Kerlinger, F.N. (1973). Foundations of behavioral research. New York: Holt.
[11] Lattin, J.M., Carroll, J.D., & Green, P.E. (2003). Analyzing multivariate data. Pacific Grove: Duxbury Press.
[12] Levine, M.S. (1977). Canonical analysis and factor comparison. Beverly Hills: Sage.
[13] Lorenzo-Seva, U., Van de Velden, M., & Kiers, H.A.L. (2009). Oblique rotation in correspondence analysis: a step forward in the search for the simplest interpretation. British Journal of Mathematical & Statistical Psychology, 62, 583–600.
[14] MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In L.M. Le Cam & J. Neyman (Eds.), Proceedings of the fifth Berkeley symposium on mathematical statistics and probability (Vol. 1, pp. 281–298). Berkeley: University of California Press. · Zbl 0214.46201
[15] Magnus, J.R., & Neudecker, H. (1991). Matrix differential calculus with applications to statistics and econometrics (revised ed.). New York: Wiley. · Zbl 0651.15001
[16] Mulaik, S.A. (2010). Foundations of factor analysis (2nd ed.). Boca Raton: Chapman & Hall/CRC. · Zbl 1188.62185
[17] Rao, C.R. (2002). Linear statistical inference and its applications (2nd ed.). New York: Wiley.
[18] Reynolds, T.J., & Jackosfsky, E.F. (1981). Interpreting canonical analysis: the use of orthogonal transformations. Educational and Psychological Measurement, 41, 661–671.
[19] Schepers, J.M. (2006). The utility of canonical correlation analysis, coupled with target rotation, in coping with the effects of differential skewness of variables. Journal of Industrial Psychology, 32, 19–22.
[20] Tanaka, Y., & Tarumi, T. (1995). Windows version of handbook of statistical analysis: multivariate analysis. Tokyo: Kyoritsu Shuppan (in Japanese).
[21] ten Berge, J.M.F. (1983). A generalization of Kristof’s theorem on the trace of certain matrix products. Psychometrika, 48, 519–523. · Zbl 0536.62093
[22] ten Berge, J.M.F. (1993). Least squares optimization in multivariate analysis. Leiden: DSWO Press. · Zbl 0937.62542
[23] Thompson, B. (1984). Canonical correlation analysis: uses and interpretation, Newbury Park: Sage.
[24] Thorndike, R.M., & Weiss, D.J. (1973). A study of the stability of canonical correlations and canonical components. Educational and Psychological Measurement, 33, 123–134.
[25] Van de Velden, M., & Kiers, H.A.L. (2005). Rotation in correspondence analysis. Journal of Classification, 22, 251–271. · Zbl 1336.62164
[26] Wang, L., Wang, X., & Feng, J. (2006). Subspace distance analysis with application to adaptive Bayesian algorithm for face recognition. Pattern Recognition, 39, 456–464. · Zbl 1158.68481
[27] Yamashita, N. (2012). Canonical correlation analysis formulated as maximizing sum of squared correlations and rotation of structure matrices. Japanese Journal of Behaviormetrics, 39, 1–9 (in Japanese). · Zbl 1266.62039
[28] Yanai, H. (1974). Unification of various techniques of multivariate analysis by means of generalized coefficient of determination. Japanese Journal of Behaviormetrics, 1, 46–54 (in Japanese).
[29] Yanai, H. (1981). Explicit expression of projectors on canonical variables and distances between centroids of groups. Journal of the Japanese Statistical Society, 11, 43–53. · Zbl 0519.62046
[30] Yanai, H., Takeuchi, K., & Takane, Y. (2011). Projection matrices, generalized inverse matrices, and singular value decomposition. New York: Springer. · Zbl 1279.15003
[31] Yates, A. (1987). Multivariate exploratory data analysis: a perspective on exploratory factor analysis. Albany: State University of New York Press.
[32] Zuccon, G., Azzopardi, L., & Van Rijsbergen, C. (2009). Semantic spaces: measuring the distance between different subspaces. In Lecture notes in computer science (Vol. 5494, pp. 225–236). · Zbl 1229.68073
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