Directional regression for functional data

We propose a method of sufficient dimension reduction for functional data based on directional regression. Functional principal component analysis with truncation acting as the regularization mechanism is used in the development. We establish optimal convergence rate for the estimation of the dimension reduction directions, in both L² estimation error and prediction risk. Somewhat surprisingly, the optimal rates for both error measures are achieved at the same truncation level. Effect of discrete sampling of functional predictor on a dense grid is also considered. Simulations and a real data application illustrate the favorable finite sample performance of the method.