Symmetries in Gaussian Graphical Models

Abstract

This thesis explores the application of Gaussian Graphical Models (GGMs) with a specific focus on identifying symmetries within EEG data. A significant challenge in using GGMs is achieving accurate estimations with limited observations, which is common in medical data. This research proposes a methodology for detecting and integrating symmetric structures in GGMs, thereby reducing the number of parameters and improving model interpretability. We follow developments presented by Højsgaard en Lauritzen (2008).
The study includes a detailed explanation of the multivariate Gaussian distribution, Maximum Likelihood Estimation (MLE), and the implementation of Iterative Partial Maximization for parameter estimation. Additionally, hierarchical clustering is employed to systematically identify symmetry classes. Results indicate that incorporating symmetry constraints enhances the accuracy and interpretability of GGMs.
The research shows that symmetry constraints simplify models, making them more robust and easier to understand.
A simulation study was conducted to test the efficiency and accuracy of the developed algorithm for symmetry detection. The findings from these simulations validate the proposed approach, demonstrating significant improvements in model performance.
Finally, the methodology was applied to an EEG dataset, highlighting practical applications in neuroscience. The results from the EEG data analysis further confirm that symmetry constraints can reveal underlying patterns in brain connectivity, offering valuable insights into the neural dynamics.
This thesis contributes to the existing literature by providing a systematic approach to detect symmetries in high-dimensional data models, particularly its practical utility with real-world EEG data.