Functional wavelet regression for linear function-on-function models

Abstract

We consider linear function-on-function regression models with multiple predictive curves. We first apply the wavelet transformation to the predictive curves and transform the original model to a linear model with functional response and high dimensional multivariate predictors. Based on the best finite dimensional approximation to the signal part in the response curve, we find an expansion of the vector of coefficient functions, which enjoys a good predictive property. To estimate this expansion, we propose a penalized generalized eigenvalue problem followed by a penalized least squares problem. We establish the sparse oracle inequalities for our estimates in the high-dimensional settings. The choices of tuning parameters and the number of components are provided. Simulations studies and application to real datasets demonstrate that our method has good predictive performance and is efficient in dimension reduction.