Network Estimation from Point Process Data

TL;DR

This paper develops a novel framework for estimating network structures from self-exciting point process data, incorporating non-linear effects, structural assumptions, and long-range dependencies, with theoretical guarantees and practical demonstrations.

Contribution

It introduces a comprehensive approach that models saturation, structural assumptions, and long-term memory in point processes, providing new theoretical bounds and practical algorithms.

Findings

The method ensures stability through saturation modeling.

It handles high-dimensional data with sparsity, group sparsity, and low-rank assumptions.

The approach is validated with simulations and real-world data.

Abstract

Consider observing a collection of discrete events within a network that reflect how network nodes influence one another. Such data are common in spike trains recorded from biological neural networks, interactions within a social network, and a variety of other settings. Data of this form may be modeled as self-exciting point processes, in which the likelihood of future events depends on the past events. This paper addresses the problem of estimating self-excitation parameters and inferring the underlying functional network structure from self-exciting point process data. Past work in this area was limited by strong assumptions which are addressed by the novel approach here. Specifically, in this paper we (1) incorporate saturation in a point process model which both ensures stability and models non-linear thresholding effects; (2) impose general low-dimensional structural assumptions that include sparsity, group sparsity and low-rankness that allows bounds to be developed in the high-dimensional setting; and (3) incorporate long-range memory effects through moving average and higher-order auto-regressive components. Using our general framework, we provide a number of novel theoretical guarantees for high-dimensional self-exciting point processes that reflect the role played by the underlying network structure and long-term memory. We also provide simulations and real data examples to support our methodology and main results.