Hawkes processes are a self-exciting stochastic process used to describe phenomena whereby past events increase the probability of the occurrence of future events. This work presents a flexible approach for modelling a variant of these, namely discrete-time Hawkes processes. Most standard models of Hawkes processes rely on a parametric form for the function describing the influence of past events, referred to as the triggering kernel. This is likely to be insufficient to capture the true excitation pattern, particularly for complex data. By utilising trans-dimensional Markov chain Monte Carlo inference techniques, our proposed model for the triggering kernel can take the form of any step function, affording significantly more flexibility than a parametric form. We first demonstrate the utility of the proposed model through a comprehensive simulation study. This includes univariate scenarios, and multivariate scenarios whereby there are multiple interacting Hawkes processes. We then apply the proposed model to several case studies: the interaction between two countries during the early to middle stages of the COVID-19 pandemic, taking Italy and France as an example, and the interaction of terrorist activity between two countries in close spatial proximity, Indonesia and the Philippines, and then within three regions of the Philippines.

中文翻译：

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用于离散时间霍克斯过程的灵活随机直方图内核

霍克斯过程是一个自激随机过程，用于描述过去事件增加未来事件发生概率的现象。这项工作提出了一种灵活的方法来模拟这些变体，即离散时间霍克斯过程。大多数霍克斯过程的标准模型依赖于描述过去事件影响的函数的参数形式，称为触发内核。这可能不足以捕捉真正的激发模式，特别是对于复杂数据。通过利用跨维马尔可夫链蒙特卡罗推理技术，我们提出的触发内核模型可以采用任何阶跃函数的形式，比参数形式提供更大的灵活性。我们首先通过全面的模拟研究证明了所提出模型的实用性。这包括单变量场景和存在多个交互霍克斯过程的多变量场景。然后，我们将所提出的模型应用于几个案例研究：以意大利和法国为例，两国在 COVID-19 大流行的早期到中期的互动，以及两个空间接近的国家之间的恐怖活动互动，印度尼西亚和菲律宾，然后在菲律宾的三个区域内。