Cox-Hawkes: doubly stochastic spatiotemporal Poisson processes

Hawkes processes are a class of point processes that can model self or mutual excitation between events, in which the occurrence of one event triggers additional events, for example: a violent event in one geographical area on a given day encourages another violent event in an area nearby the next day. A unique feature of Hawkes processes is their ability to model exogenous and endogenous "causes" of events. An exogenous cause happens by the external addition of a event, while endogenous events are self-excited from previous events by a triggering kernel. An example of the difference between these two mechanisms is in disease transmission - an exogenous event could be a zoonosis event such as the transmission of Influenza from birds, while endogenous events are subsequent human to human transmission. Due to their flexibility and mathematical tractability, Hawkes processes have been extensively used in the literature in a series of applications. They have modelled among others, neural activity (Linderman et al. 2014), earthquakes (Ogata 1988), violence (Loeffler & Flaxman 2018, Holbrook et al. 2021) and social interactions (Miscouridou et al. 2018). The majority of applied research on Hawkes processes focuses on the purely temporal settings where events occur and are subsequently triggered only in time. However, many practical problems require the inclusion of a spatial dimension. This inclusion is motivated by several factors, first, natural phenomena that self-excite tend to do so both spatial and temporally e.g.