High-Dimensional Functional Graphs and Inference for Unknown Heterogeneous Populations

In this dissertation, we develop innovative methods for analyzing high-dimensional, heterogeneous functional data, focusing specifically on uncovering hidden patterns and network structures within such complex data. We utilize functional graphical models (FGMs) to explore the conditional dependence structure among random elements. We mainly focus on the following three research projects. The first project combines the strengths of FGMs with finite mixture of regression models (FMR) to overcome the challenges of estimating conditional dependence structures from heterogeneous functional data. This novel approach facilitates the discovery of latent patterns, proving particularly advantageous for analyzing complex datasets, such as brain imaging studies of autism spectrum disorder (ASD). Through numerical analysis of both simulated data and real-world ASD brain imaging, we demonstrate the effectiveness of our methodology in uncovering complex dependencies that traditional methods may miss due to their homogeneous data assumptions. Secondly, we address the challenge of variable selection within FMR in high-dimensional settings by proposing a joint variable selection technique. This technique employs a penalized expectation-maximization (EM) algorithm that leverages shared structures across regression components, thereby enhancing the efficiency of identifying relevant predictors and improving the predictive ability. We further expand this concept to mixtures of functional regressions, employing a group lasso penalty for variable selection in heterogeneous functional data. Lastly, we recognize the limitations of existing methods in testing the equality of multiple functional graphs and develop a novel, permutation-based testing procedure. This method provides a robust, distribution-free approach to comparing network structures across different functional variables, as illustrated through simulation studies and functional magnetic resonance imaging (fMRI) analysis for ASD. Hence, these research works provide a comprehensive framework for functional data analysis, significantly advancing the estimation of network structures, functional variable selection, and testing of functional graph equality. This methodology holds great promise for enhancing our understanding of heterogeneous functional data and its practical applications.