Advertisement

Psychometrika

, 76:257 | Cite as

Regularized Generalized Canonical Correlation Analysis

  • Arthur Tenenhaus
  • Michel Tenenhaus
Article

Abstract

Regularized generalized canonical correlation analysis (RGCCA) is a generalization of regularized canonical correlation analysis to three or more sets of variables. It constitutes a general framework for many multi-block data analysis methods. It combines the power of multi-block data analysis methods (maximization of well identified criteria) and the flexibility of PLS path modeling (the researcher decides which blocks are connected and which are not). Searching for a fixed point of the stationary equations related to RGCCA, a new monotonically convergent algorithm, very similar to the PLS algorithm proposed by Herman Wold, is obtained. Finally, a practical example is discussed.

Keywords

generalized canonical correlation analysis multi-block data analysis PLS path modeling regularized canonical correlation analysis 

References

  1. Barker, M., & Rayens, W. (2003). Partial least squares for discrimination. Journal of Chemometrics, 17, 166–173. CrossRefGoogle Scholar
  2. Bougeard, S., Hanafi, M., & Qannari, E.M. (2007). ACPVI multibloc. Application en épidémiologie animale. Journal de la Société Française de Statistique, 148, 77–94. Google Scholar
  3. Bougeard, S., Hanafi, M., & Qannari, E.M. (2008). Continuum redundancy-PLS regression: a simple continuum approach. Computational Statistics & Data Analysis, 52, 3686–3696. CrossRefGoogle Scholar
  4. Burnham, A.J., Viveros, R., & MacGregor, J.F. (1996). Frameworks for latent variable multivariate regression. Journal of Chemometrics, 10, 31–45. CrossRefGoogle Scholar
  5. Carroll, J.D. (1968a). A generalization of canonical correlation analysis to three or more sets of variables. In Proc. 76th conv. Am. Psych. Assoc. (pp. 227–228). Google Scholar
  6. Carroll, J.D. (1968b). Equations and tables for a generalization of canonical correlation analysis to three or more sets of variables. Unpublished companion paper to Carroll, J.D. (1968a). Google Scholar
  7. Chessel, D., & Hanafi, M. (1996). Analyse de la co-inertie de K nuages de points. Revue de Statistique Appliquée, 44, 35–60. Google Scholar
  8. Chu, M.T., & Watterson, J.L. (1993). On a multivariate eigenvalue problem: I. Algebraic theory and power method. SIAM Journal on Scientific and Statistical Computing, 14, 1089–1106. CrossRefGoogle Scholar
  9. Dahl, T., & Næs, T. (2006). A bridge between Tucker-1 and Carroll’s generalized canonical analysis. Computational Statistics & Data Analysis, 50, 3086–3098. CrossRefGoogle Scholar
  10. Fornell, C., & Bookstein, F.L. (1982). Two structural equation models: LISREL and PLS applied to consumer exit-voice theory. Journal of Marketing Research, 19, 440–452. CrossRefGoogle Scholar
  11. Gifi, A. (1990). Nonlinear multivariate analysis. Chichester: Wiley. Google Scholar
  12. Hanafi, M. (2007). PLS Path modelling: computation of latent variables with the estimation mode B. Computational Statistics, 22, 275–292. CrossRefGoogle Scholar
  13. Hanafi, M., & Kiers, H.A.L. (2006). Analysis of K sets of data, with differential emphasis on agreement between and within sets. Computational Statistics & Data Analysis, 51, 1491–1508. CrossRefGoogle Scholar
  14. Hanafi, M., & Lafosse, R. (2001). Généralisations de la régression simple pour analyser la dépendance de K ensembles de variables avec un K+1ème. Revue de Statistique Appliquée, 49, 5–30. Google Scholar
  15. Horst, P. (1961). Relations among m sets of variables. Psychometrika, 26, 126–149. CrossRefGoogle Scholar
  16. Jöreskog, K.G. (1970). A general method for the analysis of covariance structure. Biometrika, 57, 239–251. Google Scholar
  17. Kettenring, J.R. (1971). Canonical analysis of several sets of variables. Biometrika, 58, 433–451. CrossRefGoogle Scholar
  18. Krämer, N. (2007). Analysis of high-dimensional data with partial least squares and boosting. Doctoral dissertation, Technischen Universität Berlin. Google Scholar
  19. Ledoit, O., & Wolf, M. (2004). A well conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88, 365–411. CrossRefGoogle Scholar
  20. Leurgans, S.E., Moyeed, R.A., & Silverman, B.W. (1993). Canonical correlation analysis when the data are curves. Journal of the Royal Statistical Society, Series B, 55, 725–740. Google Scholar
  21. Lohmöller, J.-B. (1989). Latent variables path modeling with partial least squares. Heildelberg: Physica-Verlag. Google Scholar
  22. Noonan, R., & Wold, H. (1982). PLS path modeling with indirectly observed variables: a comparison of alternative estimates for the latent variable. In K.G. Jöreskog & H. Wold (Eds.), Systems under indirect observation, Part 2 (pp. 75–94). Amsterdam: North-Holland. Google Scholar
  23. Qannari, E.M., & Hanafi, M. (2005). A simple continuum regression approach. Journal of Chemometrics, 19, 387–392. CrossRefGoogle Scholar
  24. R Development Core Team (2009). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. http://www.R-project.org. Google Scholar
  25. Russett, B.M. (1964). Inequality and instability: the relation of land tenure to politics. World Politics, 16, 442–454. CrossRefGoogle Scholar
  26. Schäfer, J., & Strimmer, K. (2005). A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. Statistical Applications in Genetics and Molecular Biology, 4(1), Article 32. Google Scholar
  27. Shawe-Taylor, J., & Cristianini, N. (2004). Kernel methods for pattern analysis. New York: Cambridge University Press. Google Scholar
  28. Takane, Y., & Hwang, H. (2007). Regularized linear and kernel redundancy analysis. Computational Statistics & Data Analysis, 52, 394–405. CrossRefGoogle Scholar
  29. Takane, Y., Hwang, H., & Abdi, H. (2008). Regularized multiple-set canonical correlation analysis. Psychometrika, 73, 753–775. CrossRefGoogle Scholar
  30. Ten Berge, J.M.F. (1988). Generalized approaches to the MAXBET problem and the MAXDIFF problem, with applications to canonical correlations. Psychometrika, 53, 487–494. CrossRefGoogle Scholar
  31. Tenenhaus, A. (2010). Kernel generalized canonical correlation analysis. In 42ièmes journées de statistique (JdS’10), Marseille, France, May 24–28. Google Scholar
  32. Tenenhaus, M. (2008). Component-based structural equation modelling. Total Quality Management & Business Excellence, 19, 871–886. CrossRefGoogle Scholar
  33. Tenenhaus, M., Esposito Vinzi, V., Chatelin, Y.-M., & Lauro, C. (2005). PLS path modeling. Computational Statistics & Data Analysis, 48, 159–205. CrossRefGoogle Scholar
  34. Tenenhaus, M., & Hanafi, M. (2010). A bridge between PLS path modeling and multi-block data analysis. In V. Esposito Vinzi, J. Henseler, W. Chin, & H. Wang (Eds.), Handbook of partial least squares (PLS): concepts, methods and applications (pp. 99–123). Berlin: Springer. CrossRefGoogle Scholar
  35. Tucker, L.R. (1958). An inter-battery method of factor analysis. Psychometrika, 23, 111–136. CrossRefGoogle Scholar
  36. Van de Geer, J.P. (1984). Linear relations among k sets of variables. Psychometrika, 4(9), 70–94. Google Scholar
  37. Vinod, H.D. (1976). Canonical ridge and econometrics of joint production. Journal of Econometrics, 4, 147–166. CrossRefGoogle Scholar
  38. Vivien, M., & Sabatier, R. (2003). Generalized orthogonal multiple co-inertia analysis (-PLS): new multiblock component and regression methods. Journal of Chemometrics, 17, 287–301. CrossRefGoogle Scholar
  39. Westerhuis, J.A., Kourti, T., & MacGregor, J.F. (1998). Analysis of multiblock and hierarchical PCA and PLS models. Journal of Chemometrics, 12, 301–321. CrossRefGoogle Scholar
  40. Wold, H. (1982). Soft modeling: the basic design and some extensions. In K.G. Jöreskog & H. Wold (Eds.), Systems under indirect observation, Part 2 (pp. 1–54). Amsterdam: North-Holland. Google Scholar
  41. Wold, H. (1985). In S. Kotz & N.L. Johnson (Eds.), Encyclopedia of statistical sciences. Partial least squares (Vol. 6, pp. 581–591). New York: Wiley. Google Scholar
  42. Wold, S., Martens, H., & Wold, H. (1983). The multivariate calibration problem in chemistry solved by the PLS method. In A. Ruhe & B. Kåstrøm (Eds.), Lecture notes in mathematics. Proc. conf. matrix pencils, March 1982, (pp. 286–293). Heidelberg: Springer. Google Scholar

Copyright information

© The Psychometric Society 2011

Authors and Affiliations

  1. 1.Department of Signal Processing and Electronics SystemsSupelec, Gif-sur-YvetteGif-sur-Yvette cedexFrance
  2. 2.HEC ParisJouy-en-JosasFrance

Personalised recommendations