Robust Graphical Modeling of High-Dimensional Gaussian and Non-Gaussian Processes: Theory and Applications

Abstract

This dissertation develops robust graphical models for high-dimensional, non-Gaussian data. Sparse precision matrix estimation, typically via the GLASSO, relies heavily on Gaussian assumptions, which limit its use when data exhibit heavy tails, complex dependencies, or irregularities. This research extends sparse graphical modeling beyond the normality assumption while maintaining sparsity, interpretability, and computational speed.

First, clarifying the theoretical link between GLASSO estimation and conditional independence structures. Then, extending the framework to selected elliptical distributions. This improves robustness under non-Gaussian conditions. Comprehensive simulations assess estimation accuracy and structural recovery across diverse scenarios.

The proposed methods are validated through interdisciplinary applications. In statistical process control, we integrate precision matrix estimation into multivariate monitoring to improve the sensitivity of the corresponding control charts. In behavioral science, these models uncover the conditional dependence structures among psychological variables. Finally, we analyze nutrient interactions in baobab leaves to map chemical relationships critical to food system research.

This dissertation strengthens the theoretical foundation of sparse graphical models. It bridges Gaussian and non-Gaussian frameworks and proves their practical utility across high-dimensional, diverse domains.