SAPP (version 1.0.9-1)

eptren: 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)

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.

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).

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)