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Hawkes processes are a self-exciting stochastic process used to describephenomena whereby past events increase the probability of the occurrence offuture events. This work presents a flexible approach for modelling a variantof these, namely discrete-time Hawkes processes. Most standard models of Hawkesprocesses rely on a parametric form for the function describing the influenceof past events, referred to as the triggering kernel. This is likely to beinsufficient to capture the true excitation pattern, particularly for complexdata. By utilising trans-dimensional Markov chain Monte Carlo inferencetechniques, our proposed model for the triggering kernel can take the form ofany step function, affording significantly more flexibility than a parametricform. We first demonstrate the utility of the proposed model through acomprehensive simulation study. This includes univariate scenarios, andmultivariate scenarios whereby there are multiple interacting Hawkes processes.We then apply the proposed model to several case studies: the interactionbetween two countries during the early to middle stages of the COVID-19pandemic, taking Italy and France as an example, and the interaction ofterrorist activity between two countries in close spatial proximity, Indonesiaand the Philippines, and then within three regions of the Philippines.

A flexible, random histogram kernel for discrete-time Hawkes processes