Advanced Nonparametric Bayesian Functional Modeling

Functional analyses have gained more interest as we have easier access to massive data sets. However, such data sets often contain large heterogeneities, noise, and dimensionalities. When generalizing the analyses from vectors to functions, classical methods might not work directly. This dissertation considers noisy information reduction in functional analyses from two perspectives: functional variable selection to reduce the dimensionality and functional clustering to group similar observations and thus reduce the sample size. The complicated data structures and relations can be easily modeled by a Bayesian hierarchical model, or developed from a more generic one by changing the prior distributions. Hence, this dissertation focuses on the development of Bayesian approaches for functional analyses due to their flexibilities.

A nonparametric Bayesian approach, such as the Dirichlet process mixture (DPM) model, has a nonparametric distribution as the prior. This approach provides flexibility and reduces assumptions, especially for functional clustering, because the DPM model has an automatic clustering property, so the number of clusters does not need to be specified in advance. Furthermore, a weighted Dirichlet process mixture (WDPM) model allows for more heterogeneities from the data by assuming more than one unknown prior distribution. It also gathers more information from the data by introducing a weight function that assigns different candidate priors, such that the less similar observations are more separated. Thus, the WDPM model will improve the clustering and model estimation results.

In this dissertation, we used an advanced nonparametric Bayesian approach to study functional variable selection and functional clustering methods. We proposed 1) a stochastic search functional selection method with application to 1-M matched case-crossover studies for aseptic meningitis, to examine the time-varying unknown relationship and find out important covariates affecting disease contractions; 2) a functional clustering method via the WDPM model, with application to three pathways related to genetic diabetes data, to identify essential genes distinguishing between normal and disease groups; and 3) a combined functional clustering, with the WDPM model, and variable selection approach with application to high-frequency spectral data, to select wavelengths associated with breast cancer racial disparities.