Interpretable and Scalable Graphical Models for Complex Spatio-temporal Processes

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.