Interpretable and Scalable Graphical Models for Complex Spatio-temporal Processes

TL;DR

This thesis develops scalable, interpretable probabilistic graphical models for complex spatio-temporal data, introducing new tensor-variate Gaussian models, optimization methods, and applications to diverse real-world datasets.

Contribution

It introduces a novel class of tensor-variate Gaussian graphical models, scalable optimization techniques, and a framework for interpretable topic modeling of time-varying data.

Findings

Enhanced interpretability and accuracy in brain-connectivity analysis.

Scalable modeling of space weather forecasting data.

Improved analysis of public opinion dynamics over time.

Abstract

This thesis focuses on data that has complex spatio-temporal structure and on probabilistic graphical models that learn the structure in an interpretable and scalable manner. We target two research areas of interest: Gaussian graphical models for tensor-variate data and summarization of complex time-varying texts using topic models. This work advances the state-of-the-art in several directions. First, it introduces a new class of tensor-variate Gaussian graphical models via the Sylvester tensor equation. Second, it develops an optimization technique based on a fast-converging proximal alternating linearized minimization method, which scales tensor-variate Gaussian graphical model estimations to modern big-data settings. Third, it connects Kronecker-structured (inverse) covariance models with spatio-temporal partial differential equations (PDEs) and introduces a new framework for ensemble Kalman filtering that is capable of tracking chaotic physical systems. Fourth, it proposes a modular and interpretable framework for unsupervised and weakly-supervised probabilistic topic modeling of time-varying data that combines generative statistical models with computational geometric methods. Throughout, practical applications of the methodology are considered using real datasets. This includes brain-connectivity analysis using EEG data, space weather forecasting using solar imaging data, longitudinal analysis of public opinions using Twitter data, and mining of mental health related issues using TalkLife data. We show in each case that the graphical modeling framework introduced here leads to improved interpretability, accuracy, and scalability.