发表机构
The Graduate University for Advanced Studies, SOKENDAI; The Institute of Statistical Mathematics（advanced studies大学，SOKENDAI; 统计数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对时空聚合计数数据，提出时空粗粒化霍克斯过程，通过构建有效激发核实现直接建模，提供高阶近似，并开发矩估计方法，在地震数据上验证了高效性。
AI 中文摘要
时空霍克斯过程被广泛用于对自激事件数据进行建模，但传统的推断方法需要访问每个事件的发生时间和位置。然而，在许多应用中，观测数据仅以在时间区间和空间区域上聚合的计数形式提供。我们提出了一种时空粗粒化霍克斯过程，这是一种离散时间计数模型，用于近似由底层时空霍克斯过程生成的聚合观测数据。所提出的模型通过在时空箱上对时间和空间触发效应进行平均来构建有效的激发核，同时明确纳入同一时间箱内发生的激发。这种表述使得能够直接对聚合计数数据进行建模，而无需引入潜在的事件时间或位置。我们刻画了所提出过程的一阶和二阶性质，并在时间和空间离散化的联合极限下，推导了其平稳均值和自协方差相对于相应聚合霍克斯过程的渐近近似误差。分析表明，所提出的模型通过考虑箱内激发，提供了比传统分箱泊松近似更高阶的近似。数值实验证实了理论结果，并展示了在广泛的聚合尺度上的精确近似。我们进一步开发了一种基于矩的估计程序，并通过聚合的流行型余震序列（ETAS）模型将所提出的框架应用于地震发生数据。结果表明，所提出的方法为从时空聚合事件数据进行推断和预测提供了一种计算高效的替代方案。
英文摘要
Spatio-temporal Hawkes processes are widely used to model self-exciting event data, but conventional inference methods require access to the occurrence time and location of every event. In many applications, however, observations are available only as counts aggregated over temporal intervals and spatial regions. We propose a spatio-temporal coarse-grained Hawkes process, a discrete-time count model that approximates aggregated observations generated by an underlying spatio-temporal Hawkes process. The proposed model constructs effective excitation kernels by averaging the temporal and spatial triggering effects over spatio-temporal bins while explicitly incorporating excitation occurring within the same temporal bin. This formulation enables direct modeling of aggregated count data without introducing latent event times or locations. We characterize the first- and second-order properties of the proposed process and derive asymptotic approximation errors for the stationary mean and autocovariance relative to those of the corresponding aggregated Hawkes process in the joint limit of fine temporal and spatial discretization. The analysis shows that the proposed model provides a higher-order approximation than the conventional binned Poisson approximation by accounting for intra-bin excitation. Numerical experiments confirm the theoretical results and demonstrate accurate approximation over a broad range of aggregation scales. We further develop a moment-based estimation procedure and apply the proposed framework to earthquake occurrence data through an aggregated Epidemic-Type Aftershock Sequence (ETAS) model. The results indicate that the proposed approach provides a computationally efficient alternative for inference and prediction from spatio-temporally aggregated event data.
Comments72 pages, 4 figures
