maxLikelihoodETAS {bayesianETAS}

Estimate the parameters of the ETAS model using maximum likelihood.

Description

The Epidemic Type Aftershock Sequence (ETAS) model is widely used to quantify the degree of seismic activity in a geographical region, and to forecast the occurrence of future mainshocks and aftershocks (Ross 2016). The temporal ETAS model is a point process where the probability of an earthquake occurring at time t depends on the previous seismicity H_t, and is defined by the conditional intensity function:

\lambda(t|H_t) = \mu + \sum_{t[i] < t} \kappa(m[i]|K,\alpha) h(t[i]|c,p)

where

\kappa(m_i|K,\alpha) = Ke^{\alpha \left( m_i-M_0 \right)}

and

h(t_i|c,p) = \frac{(p-1)c^{p-1}}{(t-t_i+c)^p}

where the summation is over all previous earthquakes that occurred in the region, with the i'th such earthquake occurring at time t_i and having magnitude m_i. The quantity M_0 denotes the magnitude of completeness of the catalog, so that m_i \geq M_0 for all i. The temporal ETAS model has 5 parameters: \mu controls the background rate of seismicity, K and \alpha determine the productivity (average number of aftershocks) of an earthquake with magnitude m, and c and p are the parameters of the Modified Omori Law (which has here been normalized to integrate to 1) and represent the speed at which the aftershock rate decays over time. Each earthquake is assumed to have a magnitude which is an independent draw from the Gutenberg-Richter law p(m_i) = \beta e^{\beta(m_i-M_0)}. This function estimates the parameters of the ETAS model using maximum likelihood

Usage

maxLikelihoodETAS(ts, magnitudes, M0, T, initval = NA, displayOutput = TRUE)

Arguments

ts

Vector containing the earthquake times

magnitudes

Vector containing the earthquake magnitudes

M0

Magnitude of completeness.

T

Length of the time window [0,T] the catalog was observed over. If not specified, will be taken as the time of the last earthquake.

initval

Initial value at which to start the estimation. A vector, with elements (mu, K, alpha, c, p)

displayOutput

If TRUE then prints the out the likelihood during model fitting.

Value

A list consisting of

params

A vector containing the estimated parameters, in the order (mu,K,alpha,c,p,beta)

loglik

The corresponding loglikelihood

Author(s)

Gordon J Ross

References

Gordon J. Ross - Bayesian Estimation of the ETAS Model for Earthquake Occurrences (2016), available from http://www.gordonjross.co.uk/bayesianetas.pdf

Examples

## Not run: 
beta <- 2.4; M0 <- 3; T <- 500
catalog <- simulateETAS(0.2, 0.2, 1.5, 0.5, 2, beta, M0, T)
maxLikelihoodETAS(catalog$ts, catalog$magnitudes, M0, 500)

## End(Not run)