A Multivariate Fractional Hawkes Process for Multiple Earthquake Mainshock Aftershock Sequences

Most point process models for earthquakes in the literature assume that the magnitude is independent and identically distributed. This potentially hinders the ability of the model to describe the main features of datasets containing multiple earthquake mainshock aftershock sequences in succession. This study presents a novel multivariate fractional Hawkes process model designed to capture magnitude dependent triggering behavior by incorporating history dependence into the magnitude distribution. This is done by discretizing the magnitude range into disjoint intervals and modeling events with magnitude in these ranges as the subprocesses of a mutually exciting Hawkes process using the Mittag-Leffler density as the kernel function so that the point process has a history dependent mark distribution. We apply this model to two datasets, Japan and the Middle America Trench, both containing multiple mainshock aftershock sequences and compare it to the existing ETAS model by using information criteria, residual diagnostics and retrospective prediction performance. We find that for both datasets all metrics indicate that the multivariate fractional Hawkes process performs favorably against the ETAS model due to its history dependent magnitude distribution. Furthermore, we are able to infer characteristics of the datasets that cannot be inferred from the ETAS model.