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.

针对具有标量响应与函数型预测变量的线性模型拟合任务，一种实用的解决方案是将函数型数据变换至小波域（wavelet domain），并将数据拟合问题转化为变量选择问题。已有研究表明，在此场景下应用套索（LASSO）算法兼具高效性与优异性能。本文针对该方法探索了两类潜在改进方向：预筛选技术与套索型惩罚项的加权构造方法。针对这两个方向，本文提出了若干此前从未在函数型线性回归框架下从数值或理论层面展开研究的策略。本文通过仿真实验与涵盖一维（1D）信号、二维（2D）图像预测变量的真实数据应用，对所提方法的有限样本性能进行了对比分析。此外，本文还探讨了相关渐近性质。研究表明，应用所提流程可有效提升模型的估计与预测性能，并增强其稳定性。