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.

Spatio-temporal characterization of earthquake sequence parameters and forecasting of strong aftershocks in Xinjiang based on the ETAS model

“…We will use this package to build and run the following models to fit actual observations using the LGCP model, which can accommodate data including points, counts, geographic samples, or distance sample data, and which provides methods for fitting spatial density surfaces and estimating abundance, as well as for mapping and forecasting. Similarly, we will decompose the log-likelihood function of the Hawkes process into multiple parts implemented separately using the INLAbru-based method proposed by Naylor et al (2023) [ 31 ], linearly approximating each single component and applying the Integrated Nested Laplace Approximation (INLA) method to infer the model parameters, a brief description of which is developed below.…”

stelfi: An R package for fitting Hawkes and log‐Gaussian Cox point process models

“…There are a number of R packages that fit temporal Hawkes process models: emhawkes (Lee, 2021) where estimation is based on the maximum likelihood method introduced by Ozaki (1979); hawkesbow (Cheysson, 2021), which fits a Hawkes process to discrete data by minimising the Whittle contrast; hawkes (Zaatour, 2014) that only allows users to evaluate the Hawkes likelihood for their own optimisation technique bayesianETAS (Ross, 2017), and most recently ETAS.inlabru (Naylor & Serafini, 2023; Serafini et al., 2023), allow users to fit epidemic‐type aftershock sequence models (a special case of a Hawkes process typically used to model the evolution of seismicity over time and space; Ogata, 1988) using Bayesian estimation techniques.…”