Functional Data Analysis Tools for the Analysis of High-Dimensional Brain Imaging Data

This dissertation develops methodology and presents applications of functional data analysis tools used in high-dimensional functional data settings. In particular, the tools detailed were intended for use when analyzing electroencephalography (EEG) measurements, which records spontaneous electrical activity in the brain at electrodes placed across the scalp, resulting in rich multidimensional functional data. EEG data is typically analyzed in either the time and/or frequency domains depending on the application: resting-state experiments are typically analyzed in the frequency domain, and task-related experiments are typically analyzed in the time domain. In the first chapter, we develop an algorithm for analyzing EEG data jointly in both the time and frequency and results in a method for analyzing high-dimensional EEG data that adds an additional level of specificity to our data application than is available in single-domain analysis alone. The second chapter of this dissertation showcases a Bayesian functional principal component analysis (BFPCA) model applied to a resting-state EEG experiment analyzed in the frequency domain. We develop a fully data-driven tool that relies on functional depth, a method to order a set of functional observations from the center outwards, to flexibly visualize uncertainty in the estimated posterior samples. The final chapter extends this visualization tool from use in BFPCA to Bayesian longitudinal FPCA (B-LFPCA) for analysis of longitudinal functional data, which is conceptualized as functional datum measured repeatedly over a set of longitudinal time points. We apply our flexible depth-based visualization tool in the higher-dimensional setting to an event-related EEG experiment analyzed in the time domain