Structured penalties for functional linear models---partially empirical eigenvectors for regression

TL;DR

This paper introduces a unified method for functional linear models that directly incorporates spatial structure through partially empirical eigenvectors, improving estimation by leveraging the generalized singular value decomposition.

Contribution

It presents a novel approach using GSVD to integrate spatial structure into the estimation process, enhancing the interpretability and performance of functional linear models.

Findings

GSVD clarifies the penalized estimation process

The method improves estimation accuracy in biomedical data

Simulations demonstrate the effectiveness of the approach

Abstract

One of the challenges with functional data is incorporating spatial structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the ill-posed problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates spatial structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are `partially empirical' and the framework is provided by the generalized singular value decomposition (GSVD). The GSVD clarifies the penalized estimation process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance, and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.