Wavelet-based Weighted LASSO and Screening Approaches in Functional Linear Regression

One useful approach for fitting linear models with scalar outcomes and functional predictors involves transforming the functional data to wavelet domain and converting the data fitting problem to a variable selection problem. Applying the LASSO procedure in this situation has been shown to be efficient and powerful. In this paper we explore two potential directions for improvements to this method: techniques for pre-screening and methods for weighting the LASSO-type penalty. We consider several strategies for each of these directions which have never been investigated, either numerically or theoretically, in a functional linear regression context. The finite-sample performance of the proposed methods are compared through both simulations and real-data applications with both 1D signals and 2D image predictors. We also discuss asymptotic aspects. We show that applying these procedures can lead to improved estimation and prediction as well as better stability.