SAPP (version 1.0.9-1)

etarpp: 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)

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).

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 \(\{\Lambda(t_i), i = 1, 2, ..., N\}\).

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)