Abstract
The Hawkes process is a widely used statistical model for point processes which produce clustered event times. A specific version known as the ETAS model is used in seismology to forecast earthquake arrival times under the assumption that mainshocks follow a Poisson process, with aftershocks triggered via a parametric kernel function. However, this Poissonian assumption contradicts several aspects of seismological theory which suggest that the arrival time of mainshocks instead follows alternative renewal distributions such as the Gamma or Brownian Passage Time. We hence show how the standard ETAS/Hawkes process can be extended to allow for non-Poissonian distributions by introducing a dependence based on the underlying process’ behaviour. Direct maximum likelihood estimation of the resulting models is not computationally feasible in the general case, so we also present a novel Bayesian MCMC algorithm for efficient estimation using a latent variable representation.
Highlights
The Epidemic Type Aftershock Sequence (ETAS) model is commonly used for studying and forecasting the occurrence of earthquakes in a geographical region of interest (Ogata 1988)
Since the ETAS model assumes that the immigrant earthquakes follow a Poisson process with constant intensity μ, this implies that they occur completely at random, i.e. that an immigrant event is likely to occur at each point in time, and that the time between each pair of immigrant events follows a time-independent Exponential(μ) distribution
We first compare the performance of the ETAS and SRETAS models on the catalogue of New Madrid earthquakes obtained from The University of Memphis website http:// www.memphis.edu/ceri/seismic/catalog.php
Summary
The Epidemic Type Aftershock Sequence (ETAS) model is commonly used for studying and forecasting the occurrence of earthquakes in a geographical region of interest (Ogata 1988). Since the ETAS model assumes that the immigrant earthquakes follow a Poisson process with constant intensity μ, this implies that they occur completely at random, i.e. that an immigrant event is likely to occur at each point in time, and that the time between each pair of immigrant events (known as the ‘inter-arrival times’) follows a time-independent Exponential(μ) distribution. Stress release models (SRM) were a representation of Reid’s elastic rebound theory (Reid 1910) and were fully described by Isham and Westcott (1979) as a self-correcting point process which is updated after every event occurrence They were introduced to seismology by Vere-Jones (1978) who developed them in order to address Reid’s theory that earthquakes occur due to a release of energy which was previously accumulated strain energy along faults. For simplicity and ease of both simulation and computation, we only consider the original temporal ETAS model in this paper rather than its spatiotemporal extension, our model could be extended to the spatial version without difficulty
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