Smoothing splines estimators for functional linear regression

TL;DR

This paper introduces a smoothing splines estimator for functional linear regression, analyzing its theoretical prediction error rates, their optimality, and applying it to ozone concentration prediction.

Contribution

It proposes a novel smoothing splines estimator with a modified penalty for functional linear regression, including error-in-variables correction and theoretical optimality analysis.

Findings

Prediction error rates depend on slope function smoothness and predictor structure.

The estimator achieves minimax optimal convergence rates.

Application to ozone data demonstrates practical effectiveness.

Abstract

The paper considers functional linear regression, where scalar responses $Y_1,...,Y_n$ are modeled in dependence of random functions $X_1,...,X_n$. We propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty. Theoretical analysis concentrates on the error in an out-of-sample prediction of the response for a new random function $X_{n+1}$. It is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors. We then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves. For the case of models with errors-in-variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves. The methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day.