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.
Summary Operational earthquake forecasting for risk management and communication during seismic sequences depends on our ability to select an optimal forecasting model. To do this, we need to compare the performance of competing models in prospective experiments, and to rank their performance according to the outcome using a fair, reproducible, and reliable method, usually in a low-probability environment. The Collaboratory for the Study of Earthquake Predictability (CSEP) conducts prospective earthquake forecasting experiments around the globe. In this framework, it is crucial that the metrics employed to rank the competing forecasts are ‘proper’, meaning that, on average, they prefer the data generating model. We prove that the Parimutuel Gambling score, proposed, and in some cases applied, as a metric for comparing probabilistic seismicity forecasts, is in general ‘improper’. In the special case where it is proper, we show it can still be used improperly. We demonstrate the conclusions both analytically and graphically providing a set of simulation based techniques that can be used to assess if a score is proper or not. They only require a data generating model and, at least two forecasts to be compared. We compare the Parimutuel Gambling score’s performance with two commonly-used proper scores (the Brier and logarithmic scores) using confidence intervals to account for the uncertainty around the observed score difference. We suggest that using confidence intervals enables a rigorous approach to distinguish between the predictive skills of candidate forecasts, in addition to their rankings. Our analysis shows that the Parimutuel Gambling score is biased, and the direction of the bias depends on the forecasts taking part in the experiment. Our findings suggest the Parimutuel Gambling score should not be used to distinguishing between multiple competing forecasts, and for care to be taken in the case where only two are being compared.
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