Nonlinear Hawkes Process with Gaussian Process Self Effects

Sequences of self exciting, or inhibiting, temporal events are frequent footmarks of natural phenomena: Earthquakes are known to be temporally clustered as aftershocks are commonly triggered following the occurrence of a main event [Ogata, 1988]; in social networks, the propagation of news can be modeled in terms of information cascades over the edges of a graph [Zhao et al., 2015]; and in neuronal activity, the occurrence of one spike may increase or decrease the probability of the occurrence of the next spike over some time period [Dayan and Abbott, 2001]. Traditionally, sequences of events in continuous time are modeled by Point processes, of which Cox processes [Cox, 1955], or doubly stochastic processes, use a stochastic process for the intensity function, which depends only on time and is not effected by the occurrences of the events. The Hawkes process [Hawkes and Oakes, 1974] extends the Cox process to capture phenomena in which the past events affects future arrivals, by introducing a memory dependence via a memory kernel. When incorporating dependence of the process on its own history, due to the superposition theorem of point process, new events will depend on either an exogenous rate, which is independent of the history, or an endogenous rate from past arrivals. This results in a branching structure, where new events that originate from previous events can be seen as "children" of the past events.