Approximation of Bayesian Hawkes process with inlabru

Abstract

Hawkes process are very popular mathematical tools for modeling phenomena exhibiting a self-exciting or self-correcting behavior. Typical examples are earthquakes occurrence, wild-fires, drought, capture-recapture, crime violence, trade exchange, and social network activity. The widespread use of Hawkes process in different fields calls for fast, reproducible, reliable, easy-to-code techniques to implement such models. We offer a technique to perform approximate Bayesian inference of Hawkes process parameters based on the use of the R-package inlabru . The inlabru R-package, in turn, relies on the INLA methodology to approximate the posterior of the parameters. Our Hawkes process approximation is based on a decomposition of the log-likelihood in three parts, which are linearly approximated separately. The linear approximation is performed with respect to the mode of the parameters' posterior distribution, which is determined with an iterative gradient-based method. The approximation of the posterior parameters is therefore deterministic, ensuring full reproducibility of the results. The proposed technique only requires the user to provide the functions to calculate the different parts of the decomposed likelihood, which are internally linearly approximated by the R-package inlabru . We provide a comparison with the bayesianETAS  R-package which is based on an MCMC method. The two techniques provide similar results but our approach requires two to ten times less computational time to converge, depending on the amount of data.