Model Form Identification in High-Dimensional Functional Linear Regressions

TL;DR

This paper introduces MoFI-FLR, a novel RKHS-based two-step estimation method for high-dimensional functional linear regression, enabling covariate screening and interpretability of functional effects.

Contribution

The paper proposes MoFI-FLR, a new framework combining elastic-net screening and functional decomposition for interpretability in high-dimensional functional regression.

Findings

MoFI-FLR accurately recovers active covariates in simulations.

The method distinguishes simple and complex functional effects.

Application to EEG data demonstrates practical utility.

Abstract

High-dimensional functional data are becoming increasingly common in fields such as environmental monitoring and neuroimaging. This paper studies high-dimensional functional linear regression models that relate a scalar response to ultra-high-dimensional functional predictors, where each predictor is treated as a random element in an infinite-dimensional functional space. To address the dual challenges of high-dimensionality and model interpretability, we propose MoFI-FLR, a novel two-step estimation framework rooted in reproducing kernel Hilbert space (RKHS) theory. The first step employs a functional elastic-net penalty to screen out irrelevant covariates, while the second step decomposes each selected predictor's functional coefficient into an interpretable finite-dimensional simple component and an infinite-dimensional complementary complement. By penalizing only the complementary component, our method automatically distinguishes simple effects, which consist only of the simple component, from complex effects, which also include complementary deviations. Under mild regularity conditions, we establish non-asymptotic theoretical guarantees, demonstrating that MoFI-FLR consistently recovers the active covariates and accurately identifies their true functional forms. We develop a computationally efficient algorithm to implement the proposed method and evaluate its performance through comprehensive simulation studies and an application to Psychomotor Vigilance Task EEG data.