Adaptive Smoothing Spline Estimator for the Function-on-Function Linear Regression Model

TL;DR

This paper introduces an adaptive smoothing spline estimator for function-on-function linear regression that automatically adjusts to the true coefficient function's curvature, improving estimation and prediction accuracy.

Contribution

The paper proposes a novel adaptive smoothing spline estimator with spatially adaptive penalties and a new evolutionary algorithm for tuning, enhancing flexibility and performance.

Findings

Outperforms existing methods in estimation accuracy

Achieves better prediction results in simulations

Demonstrates effectiveness on real data examples

Abstract

In this paper, we propose an adaptive smoothing spline (AdaSS) estimator for the function-on-function linear regression model where each value of the response, at any domain point, depends on the full trajectory of the predictor. The AdaSS estimator is obtained by the optimization of an objective function with two spatially adaptive penalties, based on initial estimates of the partial derivatives of the regression coefficient function. This allows the proposed estimator to adapt more easily to the true coefficient function over regions of large curvature and not to be undersmoothed over the remaining part of the domain. A novel evolutionary algorithm is developed ad hoc to obtain the optimization tuning parameters. Extensive Monte Carlo simulations have been carried out to compare the AdaSS estimator with competitors that have already appeared in the literature before. The results show that our proposal mostly outperforms the competitor in terms of estimation and prediction accuracy. Lastly, those advantages are illustrated also on two real-data benchmark examples.