Models the conditional intensity of clustered space-time events – near-miss
airspace violations, crime, seismic aftershocks – where each event
transiently raises the risk of further events nearby and soon after. The
existing morie_hawkes_fit is purely temporal; this adds the
spatial dimension:
$$\lambda(t,x,y) = \mu + \sum_{t_i < t} \alpha\,\beta e^{-\beta(t-t_i)}
\, \frac{1}{2\pi\sigma^2} e^{-((x-x_i)^2+(y-y_i)^2)/(2\sigma^2)}$$
with background rate \(\mu\) (per unit area per unit time), branching ratio
\(\alpha\) (expected offspring per event; \(\alpha<1\) for stability),
temporal decay \(\beta\), and Gaussian spatial spread \(\sigma\). The
triggering kernel integrates to \(\alpha\) in time and 1 in space, so the
spatial factor is a proper density (Reinhart 2018).
References
Reinhart A (2018). A review of self-exciting spatio-temporal point processes. Statistical Science 33(3), 299–318. doi:10.1214/17-STS629
Ogata Y (1988). Statistical models for earthquake occurrences. JASA 83(401), 9–27. doi:10.1080/01621459.1988.10478560