maxLikelihoodETAS: Estimate the parameters of the ETAS model using maximum...

In bayesianETAS: Bayesian Estimation of the Temporal and Spatio-Temporal ETAS Models for Earthquake Occurrences

Estimate the parameters of the ETAS model using maximum likelihood.

Description

The 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 2019). 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-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}

and s() is a spatial triggering kernel. The public spatio-temporal fitting routines currently use a Gaussian spatial triggering kernel with a homogeneous spatial background over the supplied area and no spatial edge correction. s() can also be omitted to fit the temporal ETAS model without a spatial component. The summation is over all previous earthquakes that occurred in the region, with the i'th such earthquake occurring at time t_i at spatial location ( x_i,, y_i) aand 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)}.

Usage

maxLikelihoodETAS(
  ts,
  ms,
  M0,
  maxTime = NULL,
  xs = NULL,
  ys = NULL,
  area = NULL,
  initval = NA,
  displayOutput = FALSE,
  zeromu = FALSE,
  handle_ties = c("error", "jitter"),
  c_lower = NULL
)

Arguments

ts	Vector containing the earthquake times
ms	Vector containing the earthquake magnitudes
M0	Magnitude of completeness.
maxTime	Length of the time window [0,maxTime] the catalog was observed over. If not specified, will be taken as the time of the last earthquake.
xs	(For the spatio-temporal ETAS model only), a vector containing the x coordinates of the earthquakes
ys	(For the spatio-temporal ETAS model only), a vector containing the y coordinates of the earthquakes
area	(For the spatio-temporal ETAS model only), a single number containing the area of the rectangular spatial region containing the earthquakes. For example if the earthquakes are defined on [28,35]x[-119,-128] then the area would be 7x9 = 63. If this is not specified then it will be calculated based on the distance between the furthest apart earthquakes in the catalog.
initval	Initial value at which to start the estimation. A vector, with elements (mu, K, alpha, c, p) for temporal fitting or (mu, K, alpha, c, p, sigma2x, sigma2y) for spatio-temporal fitting.
displayOutput	If TRUE, display likelihood values during model fitting.
zeromu	If the ETAS model is fit to a single aftershock sequence, we may want to force mu=0 in the model to have no background rate. Set zeromu=TRUE and the ETAS model will be fit under the assumption that mu=0
handle_ties	How to handle duplicate event times. The default "error" stops because continuous-time ETAS requires strictly increasing event times. "jitter" applies a deterministic, order-preserving jitter before fitting. This is a preprocessing convention, not part of the ETAS model.
c_lower	Lower bound for the Omori c parameter. If NULL, a data-scale default equal to one tenth of the smallest positive event-time gap after tie handling is used.

Details

In the spatio-temporal version of the model, s() is the multivariate Gaussian with diagonal covariance parameters \sigma^2_x and \sigma^2_y. The spatial background is homogeneous over the supplied area and no spatial edge correction is implemented. This function estimates the parameters of the ETAS model (both temporal and spatio-temporal) using maximum likelihood. Parameters are optimized on constrained transformed scales from multiple starting values, but the result is still a numerical maximum of a non-convex likelihood.

The likelihood assumes strictly increasing event times. By default duplicate times are rejected. With handle_ties = "jitter", tied event times are separated deterministically by a small amount before fitting. The Omori c parameter is constrained to be greater than c_lower; this prevents finite-resolution catalogues with exact or near ties from driving c to zero.

Value

A list consisting of

params	A vector containing the estimated parameters, in the order (mu,K,alpha,c,p) for the temporal model or (mu,K,alpha,c,p,sigma2x,sigma2y) for the spatio-temporal model
loglik	The corresponding loglikelihood

Author(s)

Gordon J Ross

References

Gordon J. Ross - Bayesian Estimation of the ETAS Model for Earthquake Occurrences (2019)

Examples

#temporal model
ts <- c(1, 2, 3, 5, 8)
ms <- c(3.0, 3.1, 3.3, 3.2, 3.0)
maxLikelihoodETAS(ts, ms, M0=3, maxTime=10)

#spatio-temporal model
xs <- c(1, 2, 3, 4, 5)
ys <- c(1, 1.5, 2, 1.5, 1)
maxLikelihoodETAS(ts, ms, M0=3, maxTime=10, xs=xs, ys=ys, area=25)

bayesianETAS documentation built on May 26, 2026, 1:06 a.m.