Spatiotemporal ETAS model with a renewal main-shock arrival process

We propose a spatiotemporal point process model that enhances the classical Epidemic-Type Aftershock Sequence (ETAS) model. This is achieved with the introduction of a renewal main-shock arrival process and we call this extension the renewal ETAS (RETAS) model. This modification is similar in spirit to the renewal Hawkes (RHawkes) process but the conditional intensity process supports a spatial component. It empowers the main-shock intensity to reset upon the arrival of main-shocks. This allows for heavier clustering of main-shocks than the classical spatiotemporal ETAS model. We introduce a likelihood evaluation algorithm for parameter estimation and provide a novel procedure to evaluate the fitted model's goodness-of-fit (GOF) based on a sequential application of the Rosenblatt transformation. A simulation algorithm for the RETAS model is outlined and used to validate the numerical performance of the likelihood evaluation algorithm and GOF test procedure. We illustrate the proposed model and methods on various earthquake catalogues around the world each with distinctly different seismic activity. These catalogues demonstrate the RETAS model's additional flexibility in comparison to the classical spatiotemporal ETAS model and emphasizes the potential for superior modelling and forecasting of seismicity.