High-Dimensional Functional Graphs and Inference for Unknown Heterogeneous Populations
| dc.contributor.author | Chen, Han | en |
| dc.contributor.committeechair | Kim, Inyoung | en |
| dc.contributor.committeemember | Franck, Christopher Thomas | en |
| dc.contributor.committeemember | Deng, Xinwei | en |
| dc.contributor.committeemember | Higdon, David | en |
| dc.contributor.department | Statistics | en |
| dc.date.accessioned | 2024-11-22T09:00:18Z | en |
| dc.date.available | 2024-11-22T09:00:18Z | en |
| dc.date.issued | 2024-11-21 | en |
| dc.description.abstract | 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. | en |
| dc.description.abstractgeneral | This study introduces innovative techniques for analyzing complex, high-dimensional functional data, such as functional magnetic resonance imaging (fMRI) data from the brain. Our goal is to reveal underlying patterns and network connections, particularly in the context of autism spectrum disorder (ASD). In functional data, we treat each signal curve from various locations as a single data point. These datasets are characterized by high dimensionality, with the number of model parameters exceeding the sample size. We employ functional graphical models (FGMs) to investigate the conditional dependencies among data elements. Our approach combines FGMs with finite mixture of regression models (FMR), allowing us to uncover hidden patterns that traditional methods assuming homogeneity might miss. Additionally, we introduce a new method for selecting relevant variables in high-dimensional regression contexts. This method enhances prediction accuracy by utilizing shared information among regression components. Furthermore, we develop a robust testing framework to facilitate the comparison of network structures between groups without relying on distribution assumptions. This enables precise evaluations of functional graphs. Hence, our research works contribute to a deeper understanding of complex, diverse functional data, paving the way for novel insights across various fields. | en |
| dc.description.degree | Doctor of Philosophy | en |
| dc.format.medium | ETD | en |
| dc.identifier.other | vt_gsexam:41942 | en |
| dc.identifier.uri | https://hdl.handle.net/10919/123646 | en |
| dc.language.iso | en | en |
| dc.publisher | Virginia Tech | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject | Functional Data | en |
| dc.subject | Graphical Model | en |
| dc.subject | Joint Variable Selection | en |
| dc.subject | Mixture Regression | en |
| dc.subject | Permutation Test | en |
| dc.title | High-Dimensional Functional Graphs and Inference for Unknown Heterogeneous Populations | en |
| dc.type | Dissertation | en |
| thesis.degree.discipline | Statistics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
| thesis.degree.level | doctoral | en |
| thesis.degree.name | Doctor of Philosophy | en |
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