Kowal, Daniel R2024-05-222024-05-222024-052024-04-18May 2024Sun, Thomas. Efficient Bayesian Regression Methods for Dependent, Sparse Functional Data. (2024). PhD diss., Rice University. https://hdl.handle.net/1911/116177https://hdl.handle.net/1911/116177Functional data analysis is widely useful for analysis of high resolution measurements over a continuous domain. However, Bayesian inference for functional regression models can be computationally challenging, especially in the presence of dependent or sparsely observed data. We provide novel Bayesian methods for functional regression across three specific areas. Existing algorithms for Bayesian inference with functional mixed models only provide either scalable computing or accurate approximations to the posterior distribution, but not both. We first introduce a new MCMC strategy for highly efficient and fully Bayesian regression with longitudinal functional data. Using a novel blocking structure paired with an orthogonalized basis reparametrization, our joint sampler optimizes efficiency for key parameters while preserving computational scalability. We surpass state-of-the-art algorithms for frequentist estimation and variational Bayes approximations while also providing accurate posterior uncertainty quantification. Next, we propose a fully Bayesian scalar-on-function regression model for sparse functional predictors measured with error. Estimation of sparsely-observed and noisy longitudinal data require careful consideration as to provide adequate uncertainty quantification and avoid overfitting, especially when used as covariates in regression models. We utilize functional factor models to parsimoniously represent the sparse curves while maintaining an efficient MCMC sampler that is stable under high missingness. We demonstrate the benefits of our modeling and algorithm design through simulations and applications on a bone mineral density study and actigraphy data. We also develop software to implement this approach and the longitudinal functional regression. Finally, we consider a challenging application of functional regression to study the dynamics between wastewater concentrations of SARS-CoV-2 and COVID-19 infection rates in the community. Wastewater-based surveillance yields a low-cost, noninvasive method for tracking disease transmissions and provides early warning signs of upcoming outbreaks. There is tremendous interest in understanding the exact dynamics between wastewater viral loads and infection rates in the population, but numerous complexities and dependency structures are present in the datasets. We propose a novel Bayesian functional concurrent regression model that accounts for both spatial and temporal correlations while estimating the dynamic effects and time lag between wastewater concentrations and positivity rates over time.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.functional data analysisfactor modelswastewater-based epidemiologyThesis2024-05-22