The Hawkes process, a self-exciting point process, has a wide range of applications in modeling earthquakes, social networks and stock markets. The established estimation process requires that researchers have access to the exact time stamps and spatial information. However, available data are often rounded or aggregated. We develop a Bayesian estimation procedure for the parameters of a Hawkes process based on aggregated data. Our approach is developed for temporal, spatio-temporal, and mutually exciting Hawkes processes where data are available over discrete time periods and regions. We show theoretically that the parameters of the Hawkes process are identifiable from aggregated data under general specifications. We demonstrate the method on simulated temporal and spatio-temporal data with various model specifications in the presence of one or more interacting processes, and under varying coarseness of data aggregation. Finally, we examine the internal and cross-excitation effects of airstrikes and insurgent violence events from February 2007 to June 2008, with some data aggregated by day.

翻译：霍克斯过程作为一种自激励点过程，在地震建模、社交网络和股票市场等领域具有广泛应用。现有的估计方法要求研究者能够获取精确的时间戳和空间信息。然而，实际可用的数据往往经过取整或聚合处理。本文针对聚合数据，开发了一种用于估计霍克斯过程参数的贝叶斯推断方法。我们的方法适用于时间、时空及互激励霍克斯过程，其中数据以离散时间段和区域的形式提供。我们从理论上证明，在一般设定下，霍克斯过程的参数可以从聚合数据中识别。我们通过模拟的时间与时空数据验证了该方法，涵盖了多种模型设定（包括单个或多个交互过程）以及不同粗细程度的数据聚合。最后，我们以2007年2月至2008年6月期间的空袭事件与叛乱暴力事件（部分数据按日聚合）为例，分析了其内部激励与交叉激励效应。