Dimension Reduction Based on Canonical Correlation

Dimension reduction is helpful and often necessary in exploring nonlinear or nonparametric regression structures with a large number of predictors. We consider using the canonical variables from the design space whose correlations with a spline basis in the response space are significant. The method can be viewed as a variant of sliced inverse regression (SIR) with simple slicing replaced by Bspline basis functions. The asymptotic distribution theory we develop extends to weakly dependent stationary sequences and enables us to consider asymptotic tests that are useful in determining the number of significant dimensions for modeling. We compare several tests for dimensionality and make specific recommendations for dimension selection based on our theoretical and empirical studies. These tests apply to any form of SIR. The methodology and some of the practical issues are illustrated through a tuition study of American colleges.