Point processes with event time uncertainty

TL;DR

This paper introduces a novel framework for modeling time-uncertain Hawkes processes, enabling better inference and prediction in real-world applications with uncertain event timings.

Contribution

The work develops a continuous-to-discrete modeling approach for time-uncertain Hawkes processes, with provable parameter recovery guarantees and applicability to non-stationary networks.

Findings

Outperforms existing baselines on simulated data

Effective on real-world datasets like sepsis prediction and crime analysis

Provides convergence guarantees for inference methods

Abstract

Point processes are widely used statistical models for continuous-time discrete event data, such as medical records, crime reports, and social network interactions, to capture the influence of historical events on future occurrences. In many applications, however, event times are not observed exactly, motivating the need to incorporate time uncertainty into point process modeling. In this work, we introduce a framework for modeling time-uncertain self-exciting point processes, known as Hawkes processes, possibly defined over a network. We begin by formulating the model in continuous time under assumptions motivated by real-world scenarios. By imposing a time grid, we obtain a discrete-time model that facilitates inference and enables computation via first-order optimization methods such as gradient descent and variational inequality (VI). We establish a parameter recovery guarantee for VI inference with an $O(1/k)$ convergence rate using $k$ steps. Our framework accommodates non-stationary processes by representing the influence kernel as a matrix (or tensor on a network), while also encompassing stationary processes, such as the classical Hawkes process, as a special case. Empirically, we demonstrate that the proposed approach outperforms existing baselines on both simulated and real-world datasets, including the sepsis-associated derangement prediction challenge and the Atlanta Police Crime Dataset.