On Generalized Canonical Correlation Analysis

In generalized canonical correlation analysis several sets of variables are analyzed simultaneously. This makes the method suited for the analysis of various types of data. For example, in marketing research, subjects may be asked to rate a set of objects on a set of attributes. For each individual, a data matrix can then be constructed where the objects are represented row-wise and the attributes columnwise. Then, using generalized canonical correlation analysis a graphical representation, sometimes referred to as a perceptual map, can be made on the basis of the individuals’observation matrices. Note that, the observation matrices do not necessarily contain the same attributes. Several generalizations of canonical correlation analysis have been proposed. Some of these are discussed and compared in Kettenring (1971) and Gower (1989). In this paper, we shall concern ourselves with the generalization proposed by Carroll (1968). Carroll’s approach has some attractive properties that makes the method well fit for the analysis of multiple-set data. First of all, computationally, the method is straightforward as its solution is based on an eigenequation. Secondly, the method is closely related to several well-known multivariate techniques. In particular, principal component analysis and multiple correspondence analysis. Thirdly, Carroll’s generalization takes ordinary canonical correlation analysis as a special case. Although this last property is well known and already mentioned by Carroll (1968), a formal proof in the context of generalized canonical correlation analysis is not easy to find in the literature. Ten Berge (1979) does provide a proof in the context of factor rotation. In this paper, we will present a new proof of the equivalence. In addition, we propose a new generalized canonical correlation analysis approach that takes classical canonical correlationa analysis as a special case and always yields orthogonal canonical variates.