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Generalized Constrained Canonical Correlation Analysis

A method for generalized constrained canonical correlation analysis (GCCANO) is proposed that incorporates external information on both rows and columns of data matrices. In this method each set of variables is first decomposed into the sum of several submatrices according to the external information, and then canonical correlation analysis is applied to pairs of derived submatrices, one from each set, to explore linear relationships between them. Technically, the former amounts to projections of the data matrix onto the spaces spanned by matrices of external information, while the latter involves the generalized singular value decomposition of a matrix with certain metric matrices. GCCANO subsumes a number of existing methods as special cases. It generalizes various kinds of linearly constrained correspondence analysis as well as multivariate analysis of variance/canonical discriminant analysis. Permutation tests are applied to test the significance of canonical correlations obtained from GCCANO. Examples are given to illustrate the proposed method.