Cox-Hawkes: doubly stochastic spatiotemporal Poisson processes

• URL: http://arxiv.org/abs/2210.11844v1
• Date: Fri, 21 Oct 2022 09:47:34 GMT
• Title: Cox-Hawkes: doubly stochastic spatiotemporal Poisson processes
• Authors: Xenia Miscouridou, Samir Bhatt, George Mohler, Seth Flaxman, Swapnil Mishra
• Abstract summary: We develop a new class of inference Hawkes processes that can both trigger and clustering and behavior we provide an efficient method for performing inference. We show the efficacy and flexibility of our approach in experiments on simulated data and use our methods to uncover the trends in a dataset of reported crimes in the US.
• Score: 2.6599014990168834
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Hawkes processes are point process models that have been used to capture self-excitatory behavior in social interactions, neural activity, earthquakes and viral epidemics. They can model the occurrence of the times and locations of events. Here we develop a new class of spatiotemporal Hawkes processes that can capture both triggering and clustering behavior and we provide an efficient method for performing inference. We use a log-Gaussian Cox process (LGCP) as prior for the background rate of the Hawkes process which gives arbitrary flexibility to capture a wide range of underlying background effects (for infectious diseases these are called endemic effects). The Hawkes process and LGCP are computationally expensive due to the former having a likelihood with quadratic complexity in the number of observations and the latter involving inversion of the precision matrix which is cubic in observations. Here we propose a novel approach to perform MCMC sampling for our Hawkes process with LGCP background, using pre-trained Gaussian Process generators which provide direct and cheap access to samples during inference. We show the efficacy and flexibility of our approach in experiments on simulated data and use our methods to uncover the trends in a dataset of reported crimes in the US.

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