Many modern spatio-temporal data sets, in sociology, epidemiology or seismology, for example, exhibit self-exciting characteristics, triggering and clustering behaviors both at the same time, that a suitable Hawkes space-time process can accurately capture. This paper aims to develop a fast and flexible parametric inference technique to recover the parameters of the kernel functions involved in the intensity function of a space-time Hawkes process based on such data. Our statistical approach combines three key ingredients: 1) kernels with finite support are considered, 2) the space-time domain is appropriately discretized, and 3) (approximate) precomputations are used. The inference technique we propose then consists of a $\ell_2$ gradient-based solver that is fast and statistically accurate. In addition to describing the algorithmic aspects, numerical experiments have been carried out on synthetic and real spatio-temporal data, providing solid empirical evidence of the relevance of the proposed methodology.

翻译：许多现代时空数据集，例如在社会学、流行病学或地震学中，同时表现出自激特性、触发和聚集行为，而合适的霍克斯时空过程能够准确捕捉这些特征。本文旨在开发一种快速灵活的参数推断技术，以基于此类数据恢复时空霍克斯过程强度函数中涉及的核函数参数。我们的统计方法结合了三个关键要素：1) 考虑具有有限支撑的核函数，2) 对时空域进行适当离散化，3) 使用（近似）预计算。我们提出的推断技术随后包含一个基于 $\ell_2$ 梯度的求解器，该求解器兼具快速性与统计准确性。除描述算法细节外，我们在合成与真实时空数据上进行了数值实验，为所提方法的相关性提供了坚实的经验证据。