Smooth and Locally Sparse Estimation for Multiple-Output Functional Linear Regression.

Functional data analysis has attracted substantial research interest and the goal of functional sparsity is to produce a sparse estimate which assigns zero values over regions where the true underlying function is zero, i.e., no relationship between the response variable and the predictor variable. In this paper, we consider a functional linear regression models that explicitly incorporates the interconnections among the responses. We propose a locally sparse (i.e., zero on some subregions) estimator, multiple-smooth and locally sparse (m-SLoS) estimator, for coefficient functions base on the interconnections among the responses. This method is based on a combination of smooth and locally sparse (SLoS) estimator and Laplacian quadratic penalty function, where we used SLoS for encouraging locally sparse and Laplacian quadratic penalty for promoting similar locally sparse among coefficient functions associated with the interconnections among the responses. Simulations show excellent numerical performance of the proposed method in terms of the estimation of coefficient functions especially the coefficient functions are same for all multivariate responses. Practical merit of this modeling is demonstrated by one real application and the prediction shows significant improvements.