Functional additive regression

TL;DR

The paper introduces Functional Additive Regression (FAR), a novel method for high-dimensional nonlinear functional regression that employs penalized least squares and efficient algorithms, demonstrating superior performance over existing methods.

Contribution

FAR extends functional regression to high-dimensional, nonlinear additive models using penalized optimization and efficient algorithms, with theoretical support and empirical validation.

Findings

FAR outperforms competing methods in simulations.

FAR effectively handles high-dimensional functional predictors.

Theoretical results support the FAR optimization approach.

Abstract

We suggest a new method, called Functional Additive Regression, or FAR, for efficiently performing high-dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, $X(t)$, and a scalar response, $Y$, in two key respects. First, FAR uses a penalized least squares optimization approach to efficiently deal with high-dimensional problems involving a large number of functional predictors. Second, FAR extends beyond the standard linear regression setting to fit general nonlinear additive models. We demonstrate that FAR can be implemented with a wide range of penalty functions using a highly efficient coordinate descent algorithm. Theoretical results are developed which provide motivation for the FAR optimization criterion. Finally, we show through simulations and two real data sets that FAR can significantly outperform competing methods.