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

 momori R Documentation

## 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 = -2`LL` + 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))
``````

SAPP documentation built on June 7, 2023, 5:45 p.m.