We develop flexible multivariate spatio-temporal Hawkes process models to analyze patterns of terrorism. Previous applications of point process methods to political violence data mainly utilize temporal Hawkes process models, neglecting spatial variation in these attack patterns. This limits what can be learned from these models as any effective counter-terrorism strategy requires knowledge on both when and where attacks are likely to occur. Even the existing work on spatio-temporal Hawkes processes imposes restrictions on the triggering function that are not well-suited for terrorism data. Therefore, we generalize the structure of the spatio-temporal triggering function considerably, allowing for nonseprability, nonstatitionarity, and cross-triggering (i.e., across the groups). To demonstrate the utility of our models, we analyze two samples of real-world terrorism data: Afghanistan (2002-2013) for univariate analysis and Nigeria (2009-2017) for bivariate analysis. Jointly, these two studies demonstrate that our models outperform standard Hawkes process models, besting widely-used alternatives in overall model fit and revealing spatio-temporal patterns that are, by construction, masked in these models (e.g., increasing dispersion in cross-triggering over time).

翻译：我们开发了灵活的多元时空霍克斯过程模型，用于分析恐怖主义的模式。以往将点过程方法应用于政治暴力数据的研究主要采用时间霍克斯过程模型，忽视了攻击模式中的空间变化。这限制了从这些模型中获取的信息，因为任何有效的反恐战略都需要了解攻击可能发生的时间和地点。即便是现有的时空霍克斯过程研究，也对触发函数施加了不适用于恐怖主义数据的限制。因此，我们极大程度上推广了时空触发函数的结构，允许其具有不可分离性、非平稳性以及跨组触发（即跨群体的触发）。为了展示模型的有效性，我们分析了两类真实世界恐怖主义数据样本：阿富汗（2002-2013年，单变量分析）和尼日利亚（2009-2017年，双变量分析）。两项研究共同表明，我们的模型优于标准霍克斯过程模型，在整体模型拟合上超越广泛使用的替代方案，并揭示了这些模型因构造限制而被掩盖的时空模式（例如，跨组触发随时间增加的离散性）。