Modelling and inferring the triggering function in a self-exciting point process

Self-exciting spatio-temporal point processes offer a flexible class of models that have found success in a range of applications. They involve a triggering effect that accounts for the clustering patterns observed in many natural and sociological applications. In this work, we focus on the key step of inferring or designing the form of the triggering function. In the inference setting, we use a nonparametric approach to fit a process to a range of datasets arising in criminology. By analysing this public domain data we find that the inferred trigger shape varies across different categories of crime. Motivated by these observations, and also by hypotheses from the criminology literature, we then propose a variation on the classical Epidemic-Type Aftershock Sequences trigger, which we call the Delayed Response trigger. After calibrating both parametric models, we show that Delayed Response is comparable with Epidemic-Type Aftershock Sequences in terms of predicting future events, and additionally provides an estimate of the time lag before the risk of triggering is maximised.