Partition-Based Functional Ridge Regression for High-Dimensional Data

TL;DR

This paper introduces a partition-based functional ridge regression method that improves stability and interpretability in high-dimensional functional linear models by adaptively penalizing different functional components.

Contribution

It develops a novel partition-based ridge regression framework with three estimators, providing theoretical guarantees and demonstrating superior performance in simulations and real data.

Findings

FRSM performs best in small samples due to variance reduction

FRFM achieves higher accuracy in larger samples by preserving functional structure

Empirical application shows improved prediction and interpretability

Abstract

This paper proposes a partition-based functional ridge regression framework to address multicollinearity, overfitting, and interpretability in high-dimensional functional linear models. The coefficient function vector \( \boldsymbol{\beta}(s) \) is decomposed into two components, \( \boldsymbol{\beta}_1(s) \) and \( \boldsymbol{\beta}_2(s) \), representing dominant and weaker functional effects. This partition enables differential ridge penalization across functional blocks, so that important signals are preserved while less informative components are more strongly shrunk. The resulting approach improves numerical stability and enhances interpretability without relying on explicit variable selection. We develop three estimators: the Functional Ridge Estimator (FRE), the Functional Ridge Full Model (FRFM), and the Functional Ridge Sub-Model (FRSM). Under standard regularity conditions, we establish consistency and asymptotic normality for all estimators. Simulation results reveal a clear bias--variance trade-off where FRSM performs best in small samples through strong variance reduction, whereas FRFM achieves superior accuracy in moderate to large samples by retaining informative functional structure through adaptive penalization. An empirical application to Canadian weather data further demonstrates improved predictive performance, reduced variance inflation, and clearer identification of influential functional effects. Overall, partition-based ridge regularization provides a practical and theoretically grounded method for high-dimensional functional regression.