eptren | R Documentation |

Compute the maximum likelihood estimates of intensity rates of either exponential polynomial or exponential Fourier series of non-stationary Poisson process models.

```
eptren(data, mag = NULL, threshold = 0.0, nparam, nsub, cycle = 0,
tmpfile = NULL, nlmax = 1000, plot = TRUE)
```

`data` |
point process data. |

`mag` |
magnitude. |

`threshold` |
threshold magnitude. |

`nparam` |
maximum number of parameters. |

`nsub` |
number of subdivisions in either ( |

`cycle` |
periodicity to be investigated days in a Poisson process model. If zero (default) fit an exponential polynomial model. |

`tmpfile` |
a character string naming the file to write the process of minimizing by
Davidon-Fletcher-Powell 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 |

This function computes the maximum likelihood estimates (MLEs) of the
coefficients `A_1, A_2,\ldots A_n`

is an exponential
polynomial

` f(t) = exp(A_1 + A_2t + A_3t^2 + ... ) `

or `A_1, A_2, B_2, ..., A_n, B_n`

in a Poisson process model with an
intensity taking the form of an exponential Fourier series

`f(t) = exp\{ A_1 + A_2cos(2\pi t/p) + B_2sin(2\pi t/p) + A_3cos(4\pi t/p) + B_3sin(4\pi t/p) +... \}`

which represents the time varying rate of occurrence (intensity function) of earthquakes in a region.

These two models belong to the family of non-stationary Poisson process. The
optimal order `n`

can be determined by minimize the value of the Akaike
Information Criterion (AIC).

`aic` |
AIC. |

`param` |
parameters. |

`aicmin` |
minimum AIC. |

`maice.order` |
number of parameters of minimum AIC. |

`time` |
time ( |

`intensity` |
intensity rates. |

Ogata, Y., Katsura, K. and Zhuang, J. (2006)
*Computer Science Monographs, No.32, TIMSAC84: STATISTICAL ANALYSIS OF
SERIES OF EVENTS (TIMSAC84-SASE) VERSION 2*.
The Institute of Statistical Mathematics.

Ogata, Y. (2006)
*Computer Science Monographs, No.33, Statistical Analysis of Seismicity -
updated version (SASeies2006).*
The Institute of Statistical Mathematics.

```
## The Occurrence Times Data of 627 Blastings
data(Brastings)
# exponential polynomial trend fitting
eptren(Brastings, nparam = 10, nsub = 1000)
# exponential Fourier series fitting
eptren(Brastings, nparam = 10, nsub = 1000, cycle = 1)
## Poisson Process data
data(PoissonData)
# exponential polynomial trend fitting
eptren(PoissonData, nparam = 10, nsub = 1000)
# exponential Fourier series fitting
eptren(PoissonData, nparam = 10, nsub = 1000, cycle = 1)
## The aftershock data of 26th July 2003 earthquake of M6.2
data(main2003JUL26)
x <- main2003JUL26
# exponential polynomial trend fitting
eptren(x$time, mag = x$magnitude, nparam = 10, nsub = 1000)
# exponential Fourier series fitting
eptren(x$time, mag = x$magnitude, nparam = 10, nsub = 1000, cycle = 1)
```

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