# momori: Maximum Likelihood Estimates of Parameters in the Omori-Utsu... In SAPP: Statistical Analysis of Point Processes

## 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

 ```1 2``` ```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

 ```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)) ```

### Example output

```\$param
  0.00000000 95.37593200  0.05960031  0.97406207

\$ngmle
 -1802.324

\$aic
 -1799.324

\$plist
\$plist\$t_i
 0.01

\$plist\$K
 95.37593

\$plist\$c
 0.05960031

\$plist\$p
 0.9740621

\$plist\$cls
 549.7778
```

SAPP documentation built on July 2, 2020, 2:59 a.m.