momori {SAPP}

Maximum Likelihood Estimates of Parameters in the Omori-Utsu (Modified Omori) Formula

Description

Compute the maximum likelihood estimates (MLEs) of parameters in the Omori-Utsu (modified Omori) formula representing for the decay of occurrence rate of aftershocks with time.

Usage

momori(data, mag = NULL, threshold = 0.0, tstart, tend, parami,
       tmpfile = NULL, nlmax = 1000)

Arguments

data

point process data.

mag

magnitude.

threshold

threshold magnitude.

tstart

the start of the target period.

tend

the end of the target period.

parami

the initial estimates of the four parameters B, K, c and p.

tmpfile

a character string naming the file to write the process of minimizing. 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.

Details

The modified Omori formula represent the delay law of aftershock activity in time. In this equation, f(t) represents the rate of aftershock occurrence at time t, where t is the time measured from the origin time of the main shock. B, K, c and p are non-negative constants. B represents constant-rate background seismicity which may be included in the aftershock data.

f(t) = B + K/(t+c)^p

In this function the negative log-likelihood function is minimized by the Davidon-Fletcher-Powell algorithm. Starting from a given set of initial guess of the parameters parai, momori() repeats calculations of function values and its gradients at each step of parameter vector. At each cycle of iteration, the linearly searched step (lambda), negative log-likelihood value (-LL), and two estimates of square sum of gradients are shown (process=1).

The cumulative number of earthquakes at time t since t_0 is given by the integration of f(t) with respect to the time t,

F(t) = B(t-t_0) + K\{c^{1-p}-(t-t_i+c)^{1-p}\} / (p-1)

where the summation of i is taken for all data event.

Value

param

the final estimates of the four parameters B, K, c and p.

ngmle

negative max likelihood.

aic

AIC = -2LL + 2*(number of variables), and the number = 4 in this case.

plist

list of parameters t_i, K, c, p and cls.

References

Ogata, Y. (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
momori(x$time, x$magnitude, threshold = 2.5, tstart = 0.01, tend = 18.68,
       parami = c(0,0.96021e+02, 0.58563e-01, 0.96611e+00))