eptren: Maximum Likelihood Estimates of Intensity Rates

In SAPP: Statistical Analysis of Point Processes

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,\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).

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)   

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