Description Usage Arguments Details Value References Examples
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.
1 2 |
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 |
nlmax |
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.
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. |
Ogata, Y. (2006) Computer Science Monographs, No.33, Statistical Analysis of Seismicity - updated version (SASeies2006). The Institute of Statistical Mathematics.
1 2 3 4 | 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))
|
$param
[1] 0.00000000 95.37593200 0.05960031 0.97406207
$ngmle
[1] -1802.324
$aic
[1] -1799.324
$plist
$plist$t_i
[1] 0.01
$plist$K
[1] 95.37593
$plist$c
[1] 0.05960031
$plist$p
[1] 0.9740621
$plist$cls
[1] 549.7778
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