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.

翻译：点过程是广泛用于连续时间离散事件数据的统计模型，例如医疗记录、犯罪报告和社交网络互动，用于捕捉历史事件对未来发生事件的影响。然而，在许多应用中，事件时间并非被精确观测到，这促使了将时间不确定性纳入点过程建模的需求。在本工作中，我们提出了一个用于建模具有时间不确定性的自激励点过程（即Hawkes过程）的框架，该过程可能定义在网络之上。我们首先基于现实场景的假设，在连续时间下构建模型。通过施加时间网格，我们获得了一个离散时间模型，该模型便于推断，并能通过一阶优化方法（如梯度下降和变分不等式（VI））进行计算。我们为VI推断建立了参数恢复保证，其收敛速度为$O(1/k)$，其中$k$为迭代步数。我们的框架通过将影响核表示为矩阵（或网络上的张量）来容纳非平稳过程，同时也将平稳过程（如经典的Hawkes过程）作为特例包含在内。在实证研究中，我们证明了所提出的方法在模拟和真实世界数据集（包括脓毒症相关紊乱预测挑战和亚特兰大警察犯罪数据集）上均优于现有基线。