Statistical Methods for Networks of High-Dimensional Point Processes

Abstract

Fueled in part by recent applications in neuroscience, high-dimensional Hawkes processes are widely used for modeling the network of interactions among multivariate point processes. Despite this popularity, existing methodological and theoretical work has mainly focused on estimation for a single network, assuming all network components are observed. This dissertation aims to develop more flexible estimation and inference tools for high-dimensional Hawkes processes. In Chapter 2, we develop a new statistical inference procedure for high-dimensional Hawkes processes. The key ingredient for this inference procedure is a new concentration inequality on the first- and second-order statistics for integrated stochastic processes, which summarize the entire history of the process. Combining recent results on martingale central limit theory with the new concentration inequality, we then characterize the convergence rate of the test statistics. In Chapter 3, we propose a joint estimation procedure to combine data from multiple experiments for more efficient network estimation. Our procedure incorporates easy-to-compute weights to data-adaptively encourage similarity between the estimated networks. We also develop a powerful hierarchical multiple testing procedure for edges of all estimated networks, taking into account the hierarchical similarity structure of the multi-experiment networks guided by the proposed weights. In Chapter 4, to cope with the challenges of confounding from unobserved components, we develop a deconfounding procedure to estimate high-dimensional point process networks with only a subset of nodes observed. Unlike existing procedures, our method allows flexible connectivity between the observed and unobserved nodes, and unknown number of hidden nodes that can be larger than the observed population. We finish the dissertation with a discussion in Chapter 5, where we outline a possible future research direction that maps brain networks to animal behavior.