etasap | R Documentation |

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.

```
etasap(time, mag, threshold = 0.0, reference = 0.0, parami, zts = 0.0,
tstart, zte, approx = 2, tmpfile = NULL, nlmax = 1000, plot = TRUE)
```

`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` |
a character string naming the file to write the process of maximum likelihood procedure.
If "" print the process to the standard output and if |

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

`plot` |
logical. If |

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-Utsu 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.

```
data(main2003JUL26) # The aftershock data of 26th July 2003 earthquake of M6.2
x <- main2003JUL26
etasap(x$time, x$magnitude, threshold = 2.5, reference = 6.2,
parami = c(0, 0.63348e+02, 0.38209e-01, 0.26423e+01, 0.10169e+01),
tstart = 0.01, zte = 18.68)
```

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

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