Representation Learning for High Dimensional Stochastic Processes and Rare Events

Abstract

We are interested in using probability theory to characterize the evolution of different phenomena and the distribution of rare events within these processes with particular emphasis on high dimensional data that appear in biomedical applications.Inferring and simulating stochastic processes are important for many aspects of health care, such as describing the trajectory of a patient's prognosis. However, a lack of mature computational frameworks that are both flexible and mathematically interpretable limits the broad application of such techniques within a health care setting. In this thesis, we make strides towards developing computational methods for achieving this goal. In Chapters 2 and 3, we begin by describing interpretable machine learning frameworks for inferring differential equations from data. In Chapters 4 and 5, we then discuss how to extend this to the case of nonlinear stochastic differential equations and their related partial differential equations. In Chapter 6, we shift our focus to developing methods that infer high dimensional tail events. We develop a particular neural network architecture that preserves the properties of extreme value distributions. In Chapter 7, we connect the work in Chapters 3 and 6 by describing arrivals of tail events as the excursions of a latent stochastic differential equation. In Chapter 8, we then describe how we can infer representations of high dimensional tail data. Throughout the chapters, we apply the proposed methods to several biomedical applications such as disease forecasting, neural spike inference, and modeling electroencephalography data. The methods in this thesis provide a direction for describing stochastic processes and tail distributions through flexible frameworks that scale to high dimensional covariates.