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.
momori(data, mag = NULL, threshold = 0.0, tstart, tend, parami,
tmpfile = NULL, nlmax = 1000)
the final estimates of the four parameters \(B\), \(K\), \(c\) and \(p\).
negative max likelihood.
AIC = -2\(LL\) + 2*(number of variables), and the number = 4 in this case.
list of parameters \(t_i\), \(K\), \(c\), \(p\) and \(cls\).
point process data.
magnitude.
threshold magnitude.
the start of the target period.
the end of the target period.
the initial estimates of the four parameters \(B\), \(K\), \(c\) and \(p\).
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.
the maximum number of steps in the process of minimizing.
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.
Ogata, Y. (2006) Computer Science Monographs, No.33, Statistical Analysis of Seismicity - updated version (SASeies2006). The Institute of Statistical Mathematics.
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))
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