PtProcess (version 3.3-16)

etas_gif: Ground Intensity for ETAS Model

Description

This function calculates the value of the ground intensity of a time-magnitude Epidemic Type Aftershock Sequence (ETAS) model. Spatial coordinates of the events are not taken into account.

Usage

etas_gif(data, evalpts, params, TT=NA, tplus=FALSE)

Value

Two usages are as follows.

etas_gif(data, evalpts, params, tplus=FALSE)
etas_gif(data, evalpts=NULL, params, TT)

The first usage returns a vector containing the values of ${\lambda }_g(t)$ evaluated at the specified points. In the second usage, it returns the value of the integral.

Arguments

data

a data frame containing the event history, where each row represents one event. There must be columns named "time", usually the number of days from some origin; and "magnitude" which is the event magnitude less the magnitude threshold, i.e. $M_i-M_0$.

evalpts

a vector, matrix or data.frame. If a vector, the elements will be assumed to represent the required evaluation times. Other objects must include a column named "time" that can be referred to as evalpts[,"time"], at which the intensity function will be evaluated.

params

vector of parameter values in the following order: $(\mu ,A,\alpha ,c,p)$.

TT

vector of length 2, being the time interval over which the integral of the ground intensity function is to be evaluated.

tplus

logical, ${\lambda }_g(t|{\mathcal{H}}_t)$ is evaluated as ${\lambda }_g(t^{+}|{\mathcal{H}}_t)$ if TRUE, else ${\lambda }_g(t^{-}|{\mathcal{H}}_t)$.

Function Attributes

rate

is "decreasing".

Details

The ETAS model was proposed by Ogata (1988, 1998, 1999) for the modelling of earthquake mainshock-aftershock sequences. The form of the ground intensity function used here is given by $${\lambda }_g(t|{\mathcal{H}}_t)=\mu +A\sum _{i:t_i<t}{e}^{\alpha (M_i-M_0)}{(1+\frac{t-t_i}{c})}^{-p},$$ where $t_i$ denotes the event times and the summation is taken over those $i$ such that $t_i<t$.

References

Cited references are listed on the PtProcess manual page.

See Also

General details about the structure of ground intensity functions are given in the topic gif.

Examples

#  Tangshan: ground intensity and magnitude time plots

data(Tangshan)
p <- c(0.007, 2.3, 0.98, 0.008, 0.94)
bvalue <- 1
TT <- c(0, 4018)

x <- mpp(data=Tangshan,
         gif=etas_gif,
         marks=list(dexp_mark, NULL),
         params=p,
         gmap=expression(params),
         mmap=expression(bvalue*log(10)),
         TT=TT)

par.default <- par(mfrow=c(1,1), mar=c(5.1, 4.1, 4.1, 2.1))
par(mfrow=c(2,1), mar=c(4.1, 4.1, 0.5, 1))

plot(x, log=TRUE, xlab="")

plot(Tangshan$time, Tangshan$magnitude+4, type="h",
     xlim=c(0, 4018),
     xlab="Days Since 1 January 1974", ylab="Magnitude")

par(par.default)