Point processes with event time uncertainty
Xiuyuan Cheng, Tingnan Gong, Yao Xie
TL;DR
The paper develops a principled framework for point processes with event-time uncertainty, combining a continuous-time Hawkes model with uncertain observation windows and a discrete-time grid that yields a learnable kernel matrix or tensor. It provides two efficient inference mechanisms, gradient descent for MLE and a monotone variational-inequality approach with an $O(1/k)$ recovery-rate guarantee, applicable to both time-only and networked settings. The framework accommodates non-stationary kernels and offers low-rank or time-invariant structures to reduce complexity, with VI shown to be more numerically stable in practice. Empirical results on synthetic data and real datasets (SADs and Atlanta crime) demonstrate accurate kernel recovery, interpretable dynamic influence networks, and superior predictive performance over standard GLMs and Hawkes approaches. This work enables flexible, time-uncertainty-aware modeling of complex event dynamics in domains ranging from healthcare to criminology.
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.