Dimension reduction in functional regression using mixed data canonical correlation analysis

We propose a new dimension reduction method, mixed data canonical correlation (MDCANCOR), for functional regression with a scalar response and a functional predictor. MDCANCOR achieves dimension reduction using the canonical correlation analysis between the functional predictor and a set of B-spline basis functions that represent the transformed response space. And we propose a modified version of BIC to determine the dimensionality of the effective dimension reduction (EDR) space. This criterion is generally applicable to dimension reduction problems in functional regression. Asymptotically, we prove that MDCANCOR consistently estimates the directions when the dimensionality of the EDR space is given, and the modified BIC consistently estimates the dimensionality of the EDR space. Both simulation and real data examples show that the MDCANCOR method performs similarly as the regularized functional sliced inverse regression and better than other existing dimension reduction methods.

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