etarpp: Residual Point Process of the ETAS Model

In SAPP: Statistical Analysis of Point Processes

Residual Point Process of the ETAS Model

Description

Compute the residual data using the ETAS model with MLEs.

Usage

etarpp(time, mag, threshold = 0.0, reference = 0.0, parami, zts = 0.0, tstart,
       zte, ztend = NULL, plot = TRUE)

etarpp2(etas, threshold = 0.0, reference = 0.0, parami, zts = 0.0, tstart, zte,
        ztend = NULL, plot = TRUE)

Arguments

time	the time measured from the main shock(t = 0).
mag	magnitude.
etas	a etas-format dataset on 9 variables (no., longitude, latitude, magnitude, time, depth, year, month and days).
threshold	threshold magnitude.
reference	reference magnitude.
parami	initial estimates of five parameters \mu, K, c, \alpha and p.
zts	the start of the precursory period.
tstart	the start of the target period.
zte	the end of the target period.
ztend	the end of the prediction period. If NULL (default) the last time of available data is set.
plot	logical. If TRUE (default) the graphs of cumulative number and magnitude against the ordinary time and transformed time are plotted.

Details

The cumulative number of earthquakes at time t since t_0 is given by the integration of \lambda(t) ( see etasap ) with respect to the time t,

\Lambda(t) = \mu(t-t_0) + K \Sigma_i \exp[\alpha(M_i-M_z)]\{c^{(1-p)}-(t-t_i+c)^{(1-p)}\}/(p-1),

where the summation of i is taken for all data event. The output of etarpp2 is given in a res-format dataset which includes the column of \{\Lambda(t_i), i = 1, 2, ..., N\}.

Value

trans.time	transformed time \Lambda(t_i), i = 1, 2, ..., N.
no.tstart	data number of the start of the target period.
resData	a res-format dataset on 7 variables (no., longitude, latitude, magnitude, time, depth and transformed time).

References

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

Examples

data(main2003JUL26)  # The aftershock data of 26th July 2003 earthquake of M6.2

## output transformed times and cumulative numbers
x <- main2003JUL26
etarpp(time = x$time, mag = x$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 = 7, ztend = 18.68)

## output a res-format dataset
etarpp2(main2003JUL26, 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 = 7, ztend = 18.68)

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