Functional contour regression

In this paper, we propose functional contour regression (FCR) for dimension reduction in the functional regression context. FCR achieves dimension reduction using the empirical directions on the functional predictor in contours defined on the response variable. It is more efficient than the functional variants of the sliced inverse regression (SIR) method by exploiting inter-slice information. A modified BIC is used to determine the dimensionality of the effective dimension reduction space. We prove that FCR is consistent in estimating the functional regression parameters, and simulations show that the estimates given by our FCR method provide better prediction accuracy than other existing methods such as functional sliced inverse regression, functional inverse regression and wavelet SIR. The merit of FCR is further demonstrated by two real data examples.