Description Usage Arguments Details Value References Examples

Compute the maximum likelihood estimates of five parameters of ETAS model. This function consists of two (exact and approximated) versions of the calculation algorithm for the maximization of likelihood.

1 2 |

`time` |
the time measured from the main shock(t=0). |

`mag` |
magnitude. |

`threshold` |
threshold magnitude. |

`reference` |
reference magnitude. |

`parami` |
initial estimates of five parameters |

`zts` |
the start of the precursory period. |

`tstart` |
the start of the target period. |

`zte` |
the end of the target period. |

`approx` |
>0 : the level for approximation version, which is one of the five levels 1, 2, 4, 8 and 16.
The higher level means faster processing but lower accuracy. |

`tmpfile` |
write the process of maximum likelihood procedure to |

`nlmax` |
the maximum number of steps in the process of minimizing. |

`plot` |
logical. If TRUE (default) the graph of cumulative number and magnitude of earthquakes against the ordinary time is plotted. |

The ETAS model is a point-process model representing the activity of earthquakes of magnitude *M_z* and larger occurring in a certain region during a certain interval of time.
The total number of such earthquakes is denoted by *N*. The seismic activity includes primary activity of constant
occurrence rate *mu* in time (Poisson process). Each earthquake ( including aftershock of another earthquake)
is followed by its aftershock activity, though only aftershocks of magnitude *M_z* and larger are included in the data.
The aftershock activity is represented by the Omori-Usu formula in the time domain. The rate of aftershock occurrence
at time *t* following the *i*th earthquake (time: *t_i*, magnitude: *M_i*) is given by

*
n_i(t) = K exp[alpha(M_i-M_z)]/(t-t_i+c)^p,*

for * t>t_i * where *K*, *alpha*, *c*, and *p* are constants, which are common to all aftershock sequences
in the region. The rate of occurrence of the whole earthquake series at time *t* becomes

*
lambda(t) = mu + Sigma_i n_i(t).*

The summation is done for all *i* satisfying *t_i < t*. Five parameters *mu*, *K*, *c*, *alpha* and *p*
represent characteristics of seismic activity of the region.

`ngmle` |
negative max log-likelihood. |

`param` |
list of maximum likelihood estimates of five parameters |

`aic2` |
AIC/2. |

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
etasap(x$time, x$magnitude, 2.5, 6.2,
c(0, 0.63348E+02, 0.38209E-01, 0.26423E+01, 0.10169E+01),, 0.01, 18.68)
``` |

```
$ngmle
[1] -1806.161
$aic2
[1] -1801.161
$param
$param$mu
[1] 0
$param$K
[1] 69.84539
$param$c
[1] 0.04076131
$param$alpha
[1] 2.826344
$param$p
[1] 1.002435
```

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