Historical functional linear models (HFLMs) 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. Prior work on the HFLM has largely focused on estimation of a surface that represents a time-varying association between the functional outcome and the functional exposure. This existing work has employed frequentist and spline-based estimation methods, with little attention paid to formal inference or adjustment for multiple testing and no approaches that implement wavelet bases. In this work, we propose a new functional regression model that estimates the time-varying, lagged association between a functional outcome and a functional exposure. Building off of recently developed function-on-function regression methods, the model employs a novel use the wavelet-packet decomposition of the exposure and outcome functions that allows us to strictly enforce the temporal ordering of exposure and outcome, which is not possible with existing wavelet-based functional models. Using a fully Bayesian approach, we conduct formal inference on the time-varying lagged association, while adjusting for multiple testing. We investigate the operating characteristics of our wavelet-packet HFLM and compare them to those of two existing estimation procedures in simulation. We also assess several inference techniques and use the model to analyze data on the impact of lagged exposure to particulate matter finer than 2.5\(\upmu \)g, or PM\(_{2.5}\), on heart rate variability in a cohort of journeyman boilermakers during the morning of a typical day’s shift.

历史功能线性模型 (HFLM) 量化功能预测变量和功能结果之间的关联，其中预测变量是在结果之前或至少与结果同时发生的暴露变量。之前关于 HFLM 的工作主要集中在表面的估计上，该表面代表了功能结果和功能暴露之间的时变关联。这项现有的工作采用了频率论和基于样条的估计方法，很少关注形式推断或多次测试的调整，也没有实现小波基的方法。在这项工作中，我们提出了一种新的功能回归模型，该模型估计功能结果和功能暴露之间的时变、滞后关联。基于最近开发的函数对函数回归方法，该模型采用了一种新颖的暴露和结果函数的小波包分解，使我们能够严格执行暴露和结果的时间顺序，这在现有的基于小波的函数模型中是不可能的。使用完全贝叶斯方法，我们对随时间变化的滞后关联进行正式推断，同时针对多次测试进行调整。我们研究了我们的小波包 HFLM 的操作特性，并将它们与模拟中两个现有估计程序的操作特性进行了比较。我们还评估了几种推理技术，并使用该模型分析有关滞后暴露于小于 2.5 的颗粒物的影响的数据 我们对随时间变化的滞后关联进行正式推断，同时针对多次测试进行调整。我们研究了我们的小波包 HFLM 的操作特性，并将它们与模拟中两个现有估计程序的操作特性进行了比较。我们还评估了几种推理技术，并使用该模型分析有关滞后暴露于小于 2.5 的颗粒物的影响的数据 我们对随时间变化的滞后关联进行正式推断，同时针对多次测试进行调整。我们研究了我们的小波包 HFLM 的操作特性，并将它们与模拟中两个现有估计程序的操作特性进行了比较。我们还评估了几种推理技术，并使用该模型分析有关滞后暴露于小于 2.5 的颗粒物的影响的数据\(\upmu \) g 或 PM \(_{2.5}\)，在典型的一天轮班的早晨，一组熟练的锅炉制造商的心率变异性。