etasap: Maximum Likelihood Estimates of the ETAS Model

Description

Compute the maximum likelihood estimates of five parameters of ETAS model. This function consists of two (exact and approximated) versions of the calculation algorithm for the maximization of likelihood.

Usage

etasap(time, mag, threshold = 0.0, reference = 0.0, parami, zts = 0.0, 
       tstart, zte, approx = 2, tmpfile = NULL, nlmax = 1000, plot = TRUE)

Value

ngmle: negative max log-likelihood.
param: list of maximum likelihood estimates of five parameters $\mu$, $K$, $c$, $\alpha$ and $p$.
aic2: AIC/2.

Arguments

time: the time measured from the main shock(t=0).
mag: magnitude.
threshold: threshold magnitude.
reference: reference magnitude.
parami: initial estimates of five parameters $\mu$, $K$, $c$, $\alpha$ and $p$.
zts: the start of the precursory period.
tstart: the start of the target period.
zte: the end of the target period.
approx: > 0 : the level for approximation version, which is one of the five levels 1, 2, 4, 8 and 16. The higher level means faster processing but lower accuracy.
= 0 : the exact version.
tmpfile: a character string naming the file to write the process of maximum likelihood procedure. If "" print the process to the standard output and if NULL (default) no report.
nlmax: the maximum number of steps in the process of minimizing.
plot: logical. If TRUE (default) the graph of cumulative number and magnitude of earthquakes against the ordinary time is plotted.

Details

The ETAS model is a point-process model representing the activity of earthquakes of magnitude $M_z$ and larger occurring in a certain region during a certain interval of time. The total number of such earthquakes is denoted by $N$. The seismic activity includes primary activity of constant occurrence rate $\mu$ in time (Poisson process). Each earthquake ( including aftershock of another earthquake) is followed by its aftershock activity, though only aftershocks of magnitude $M_z$ and larger are included in the data. The aftershock activity is represented by the Omori-Utsu formula in the time domain. The rate of aftershock occurrence at time $t$ following the $i$th earthquake (time: $t_i$, magnitude: $M_i$) is given by

$$n_i(t) = K exp[\alpha(M_i-M_z)]/(t-t_i+c)^p,$$

for $ t>t_i $ where $K$, $\alpha$, $c$, and $p$ are constants, which are common to all aftershock sequences in the region. The rate of occurrence of the whole earthquake series at time $t$ becomes

$$\lambda(t) = \mu + \Sigma_i n_i(t).$$

The summation is done for all $i$ satisfying $t_i < t$. Five parameters $\mu$, $K$, $c$, $\alpha$ and $p$ represent characteristics of seismic activity of the region.

References

Ogata, Y. (2006) Computer Science Monographs, No.33, Statistical Analysis of Seismicity - updated version (SASeies2006). The Institute of Statistical Mathematics.

Examples

Run this code

data(main2003JUL26)  # The aftershock data of 26th July 2003 earthquake of M6.2 
x <- main2003JUL26
etasap(x$time, x$magnitude, threshold = 2.5, reference = 6.2, 
       parami = c(0, 0.63348e+02, 0.38209e-01, 0.26423e+01, 0.10169e+01),
       tstart = 0.01, zte = 18.68)

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