Nonparametric estimation of variable productivity Hawkes processes

Abstract

Hawkes models are frequently used to describe point processes that are clustered spatial‐temporally, and have been used in numerous applications including the study of earthquakes, invasive species, and contagious diseases. An extension of the Hawkes model is considered where the productivity is variable. In particular, the case is explored where each point may have its own productivity and a simple analytic formula is derived for the maximum likelihood estimators of these productivities. This estimator is compared with an empirical estimator and ways are explored of stabilizing both estimators by lower truncating, smoothing, and rescaling the estimates. Properties of the estimators are explored in simulations, and the methods are applied to seismological and epidemic datasets to show and quantify substantial variation in productivity.

Flexible Spatio-Temporal Hawkes Process Models for Earthquake Occurrences

“…As an extension, we consider a nonparametric ETAS model whose aftershock productivity depends both on magnitude and location of the mainshock. Schoenberg (2022) suggested a nonparametric method that estimates the aftershock productivity for each event by deriving an analytic form and maximizing the likelihood. But, Schoenberg (2022) smoothed the aftershock productivities only in a magnitude domain without considering their spatial variability.…”

“…Schoenberg (2022) suggested a nonparametric method that estimates the aftershock productivity for each event by deriving an analytic form and maximizing the likelihood. But, Schoenberg (2022) smoothed the aftershock productivities only in a magnitude domain without considering their spatial variability. Furthermore, it requires the invertibility of an (N − 1) × (N − 1) possibly ill-conditioned lower-triangular matrix G whose (i, j)-th element G ij is g(x i+1 − x j , y i+1 − y j , t i+1 − t j ) if i ≥ j, and 0 otherwise.…”

Approximation of Bayesian Hawkes process with inlabru

“…Hawkes processes or self-exciting processes, first introduced by Hawkes (1971aHawkes ( , 1971b, are counting processes often used to model the "arrivals" of some events over time, when each arrival increases the probability of subsequent arrivals in its proximity. Typical applications can be found in seismology (Ogata, 1988(Ogata, , 2011Ogata & Zhuang, 2006;Schoenberg, 2022), capture-recapture (Altieri et al, 2022;Weller et al, 2018), invasive species (Balderama et al, 2012), droughts (Li et al, 2021), crime (Mohler, 2013;Mohler et al, 2011Mohler et al, , 2018, finance (Azizpour et al, 2018;Filimonov & Sornette, 2012;Hawkes, 2018), disease mapping (Chiang et al, 2022;Garetto et al, 2021), wildfires (Peng et al, 2005), and social network analysis (Kobayashi & Lambiotte, 2016;Zhou et al, 2013).…”