  Most point process models for earthquakes currently in the literature assume
the magnitude distribution is i.i.d. potentially hindering the ability of the
model to describe the main features of data sets containing multiple earthquake
mainshock aftershock sequences in succession. This study presents a novel
multidimensional fractional Hawkes process model designed to capture magnitude
dependent triggering behaviour by incorporating history dependence into the
magnitude distribution. This is done by discretising the magnitude range into
disjoint intervals and modelling events with magnitude in these ranges as the
subprocesses of a mutually exciting Hawkes process using the Mittag-Leffler
density as the kernel function. We demonstrate this model's use by applying it
to two data sets, 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 data sets all metrics indicate that the
multidimensional fractional Hawkes process performs favourably against the ETAS
model. Furthermore, using the multidimensional fractional Hawkes process we are
able to infer characteristics of the data sets that are consistent with results
currently in the literature and that cannot be found by using the ETAS model.
