Sparse wavelet regression with multiple predictive curves

With the advance of techniques, more and more complicated data are extracted and recorded. In this paper, functional regression models with a scalar response and multiple predictive curves are considered. We transform the functional regression models to multiple linear regression models by using the discrete wavelet transformation. When the number of predictive curves is big, the multiple linear regression model usually has much bigger number of features than the sample size. We apply our correlation-based sparse regression method to the resulted high dimensional regression model. The novel feature of our sparse method is that we impose sparsity penalty on the direction of the estimate of the coefficient vector instead of the estimate itself, and only the direction of the estimate is determined by an optimization problem. The estimation consistency of the coefficient curve for the functional regression model is obtained when both the sample size and the number of curves go to infinity. The effects of the discrete observations are discussed. We compare our method with both functional regression methods and other wavelet based sparse regression methods on both simulated data and four real data sets, including the cases of single and multiple predictive curves. The results indicate that sparse wavelet regression methods are better in extracting local features and our method has good predictive performances in all scenarios.