Ridge regression with adaptive additive rectangles and other piecewise functional templates

TL;DR

This paper introduces an adaptive penalization method for functional linear regression that shrinks coefficients toward a data-driven, piecewise rectangular shape template, enhancing interpretability and predictive accuracy.

Contribution

It presents a novel $L_{2}$-based penalization algorithm that adaptively positions rectangles in the coefficient function, simplifying the nonconvex knot placement problem.

Findings

Method improves predictive performance in simulations

Enhances interpretability through piecewise rectangular templates

Demonstrates effectiveness on real-world case studies

Abstract

We propose an $L_{2}$-based penalization algorithm for functional linear regression models, where the coefficient function is shrunk towards a data-driven shape template $\gamma$, which is constrained to belong to a class of piecewise functions by restricting its basis expansion. In particular, we focus on the case where $\gamma$ can be expressed as a sum of $q$ rectangles that are adaptively positioned with respect to the regression error. As the problem of finding the optimal knot placement of a piecewise function is nonconvex, the proposed parametrization allows to reduce the number of variables in the global optimization scheme, resulting in a fitting algorithm that alternates between approximating a suitable template and solving a convex ridge-like problem. The predictive power and interpretability of our method is shown on multiple simulations and two real world case studies.