R: Maximum Likelihood Estimates of Intensity Rates

eptren

R Documentation

Maximum Likelihood Estimates of Intensity Rates

Description

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

Usage

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

Arguments

`data`	point process data.
`mag`	magnitude.
`threshold`	threshold magnitude.
`nparam`	maximum number of parameters.
`nsub`	number of subdivisions in either (0,t) or (0, `cycle`), where t is the length of observed time interval of points.
`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 `NULL` (default) no report.
`nlmax`	the maximum number of steps in the process of minimizing.
`plot`	logical. If `TRUE` (default) intensity rates are plotted.

Details

This function computes the maximum likelihood estimates (MLEs) of the coefficients A_1, A_2,..., 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(2π t/p) + B_2sin(2π t/p) + A_3cos(4π t/p) + B_3sin(4π 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).

Value

`aic`	AIC.
`param`	parameters.
`aicmin`	minimum AIC.
`maice.order`	number of parameters of minimum AIC.
`time`	time ( `cycle` = 0 ) or superposed occurrence time ( `cycle` > 0 ).
`intensity`	intensity rates.

References

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.

Examples

## 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)