Generalized Canonical Variables

Often the random vector variable, X, being encountered, for example, in atmospheric, biological, economic and environmental and other research, is of a large dimension but admits of meaningful grouping(s) into two or more mutually exclusive and exhaustive subvectors. Dimension reduction techniques are then sought to obtain “representative” new variables for each group to be formed by taking suitable compounds of the components within that group. This results in the substantially reduced dimensional vector variable Y to represent X. This greatly facilitates the associated computational and inferential statistical analyses. Hotelling’s construction of canonical variables for the case of two groups of quantitative variables has been generalized in various directions to yield Generalized Canonical Variables that encompass more than two groups, both quantitative and qualitative components in X, order constraints imposed by the user on the compounding coefficients, and so on. The availability of •packages makes the imple•Q1 mentation of such techniques for real-life problems quite feasible and attractive. On the inferential side, a variety of new types of hypotheses is posed: determination of optimal number of groups, best grouping for the same number of groups, and so on. While the corresponding distribution theory is quite involved, some interesting results are, nevertheless, emerging also. This topic is a fertile area and is in need of further theoretical and applied research, which should be very useful to a wide variety of practi-