Point processes are widely used statistical models for uncovering the temporal patterns in dependent event data. In many applications, the event time cannot be observed exactly, calling for the incorporation of time uncertainty into the modeling of point process data. In this work, we introduce a framework to model time-uncertain point processes possibly on a network. We start by deriving the formulation in the continuous-time setting under a few assumptions motivated by application scenarios. After imposing a time grid, we obtain a discrete-time model that facilitates inference and can be computed by first-order optimization methods such as Gradient Descent or Variation inequality (VI) using batch-based Stochastic Gradient Descent (SGD). The parameter recovery guarantee is proved for VI inference at an $O(1/k)$ convergence rate using $k$ SGD steps. Our framework handles non-stationary processes by modeling the inference kernel as a matrix (or tensor on a network) and it covers the stationary process, such as the classical Hawkes process, as a special case. We experimentally show that the proposed approach outperforms previous General Linear model (GLM) baselines on simulated and real data and reveals meaningful causal relations on a Sepsis-associated Derangements dataset.

翻译：点过程是广泛用于揭示依赖事件数据中时间模式的统计模型。在许多应用中，事件时间无法被精确观测，这要求在点过程数据的建模中纳入时间不确定性。在本工作中，我们引入了一个框架，用于对可能位于网络上的时间不确定点过程进行建模。我们首先在由应用场景驱动的若干假设下，推导出连续时间设置中的公式。在施加时间网格后，我们获得了一个便于推断的离散时间模型，该模型可通过一阶优化方法（如梯度下降或使用基于批次的随机梯度下降（SGD）的变分不等式（VI））进行计算。对于使用 $k$ 个 SGD 步的 VI 推断，我们证明了其参数恢复保证具有 $O(1/k)$ 的收敛率。我们的框架通过将推断核建模为一个矩阵（或在网络上的张量）来处理非平稳过程，并将平稳过程（如经典的霍克斯过程）作为特例涵盖在内。我们通过实验表明，所提出的方法在模拟数据和真实数据上优于先前的广义线性模型（GLM）基线，并在脓毒症相关紊乱数据集上揭示了有意义的因果关系。