Context-dependent self-exciting point processes: models, methods, and risk bounds in high dimensions

TL;DR

This paper introduces models and methods for estimating high-dimensional, context-dependent self-exciting point processes, capturing how event features influence network interactions, with theoretical guarantees validated by simulations and real data.

Contribution

It develops novel estimators for context-dependent networks in marked point processes, including a convex logistic-normal model, with theoretical analysis and practical validation.

Findings

The logistic-normal model captures dependence across categories effectively.

Both models demonstrate strong theoretical guarantees.

Empirical validation shows practical advantages of the proposed methods.

Abstract

High-dimensional autoregressive point processes model how current events trigger or inhibit future events, such as activity by one member of a social network can affect the future activity of his or her neighbors. While past work has focused on estimating the underlying network structure based solely on the times at which events occur on each node of the network, this paper examines the more nuanced problem of estimating context-dependent networks that reflect how features associated with an event (such as the content of a social media post) modulate the strength of influences among nodes. Specifically, we leverage ideas from compositional time series and regularization methods in machine learning to conduct network estimation for high-dimensional marked point processes. Two models and corresponding estimators are considered in detail: an autoregressive multinomial model suited to categorical marks and a logistic-normal model suited to marks with mixed membership in different categories. Importantly, the logistic-normal model leads to a convex negative log-likelihood objective and captures dependence across categories. We provide theoretical guarantees for both estimators, which we validate by simulations and a synthetic data-generating model. We further validate our methods through two real data examples and demonstrate the advantages and disadvantages of both approaches.