SAPP (version 1.0.4)

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)

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

write the process of maximum likelihood procedure to $tmpfile$. 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.

Value

ngmle

negative max log-likelihood.

param

list of maximum likelihood estimates of five parameters $mu$, $K$, $c$, $alpha$ and $p$.

aic2

AIC/2.

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-Usu 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

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

Examples

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