Statistical Inference for Networks of High-Dimensional Point Processes

TL;DR

This paper introduces a novel statistical inference method for high-dimensional Hawkes processes, enabling uncertainty quantification in network estimates, with theoretical guarantees and practical validation on neuroscience data.

Contribution

It develops a new concentration inequality and inference procedure for high-dimensional Hawkes processes, bridging the gap between estimation and uncertainty quantification.

Findings

Validates inference method through extensive simulations

Demonstrates utility on neuron spike train data

Provides convergence rate analysis for test statistics

Abstract

Fueled in part by recent applications in neuroscience, the multivariate Hawkes process has become a popular tool for modeling the network of interactions among high-dimensional point process data. While evaluating the uncertainty of the network estimates is critical in scientific applications, existing methodological and theoretical work has primarily addressed estimation. To bridge this gap, this paper develops 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. We illustrate finite sample validity of our inferential tools via extensive simulations and demonstrate their utility by applying them to a neuron spike train data set.