# Bayesian Wavelet-packet Historical Functional Linear Models

## TL;DR

This paper introduces a novel Bayesian approach using wavelet-packet transformations for historical functional linear models, enabling improved inference on temporal exposure-outcome relationships.

## Contribution

It proposes the first use of wavelet-packet transformations in Bayesian HFLMs and evaluates two Bayesian inference methods in this context.

## Key findings

- Wavelet-packet HFLM performs well in simulations.
- Bayesian inference procedures provide reliable uncertainty quantification.
- Application reveals effects of particulate matter exposure on heart rate variability.

## Abstract

Historical Functional Linear Models (HFLM) quantify associations between a functional predictor and functional outcome where the predictor is an exposure variable that occurs before, or at least concurrently with, the outcome. Current work on the HFLM is largely limited to frequentist estimation techniques that employ spline-based basis representations. In this work, we propose a novel use of the discrete wavelet-packet transformation, which has not previously been used in functional models, to estimate historical relationships in a fully Bayesian model. Since inference has not been an emphasis of the existing work on HFLMs, we also employ two established Bayesian inference procedures in this historical functional setting. We investigate the operating characteristics of our wavelet-packet HFLM, as well as the two inference procedures, in simulation and use the model to analyze data on the impact of lagged exposure to particulate matter finer than 2.5$\mu$g on heart rate variability in a cohort of journeyman boilermakers over the course of a day's shift.