Bayesian functional linear regression with sparse step functions

TL;DR

This paper introduces a Bayesian approach for functional linear regression that uses sparse step functions to identify influential time regions on a scalar outcome, providing adaptive and interpretable support estimation.

Contribution

It proposes a novel Bayesian model with sparse step functions for support recovery in functional linear regression, including new estimators and a comprehensive performance analysis.

Findings

Effective support recovery on synthetic datasets

Improved interpretability of influential time regions

Application to truffle production data reveals rainfall impact

Abstract

The functional linear regression model is a common tool to determine the relationship between a scalar outcome and a functional predictor seen as a function of time. This paper focuses on the Bayesian estimation of the support of the coefficient function. To this aim we propose a parsimonious and adaptive decomposition of the coefficient function as a step function, and a model including a prior distribution that we name Bayesian functional Linear regression with Sparse Step functions (Bliss). The aim of the method is to recover areas of time which influences the most the outcome. A Bayes estimator of the support is built with a specific loss function, as well as two Bayes estimators of the coefficient function, a first one which is smooth and a second one which is a step function. The performance of the proposed methodology is analysed on various synthetic datasets and is illustrated on a black P\'erigord truffle dataset to study the influence of rainfall on the production.