etas_gif {PtProcess} | R Documentation |
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.
etas_gif(data, evalpts, params, tplus=FALSE) etas_gif(data, evalpts=NULL, params, TT)
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. Mi - M0.
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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.
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params |
vector of parameters values in the following order: c (mu, A, alpha, c, p).
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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|Ht) is evaluated as lambda_g(t^+|Ht) if TRUE , else lambda_g(t^-|Ht).
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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|Ht) = mu + A sum{exp[alpha(Mi - M0)](1 + (t-ti)/c)^(-p)},
where ti denotes the event times and the summation is taken over those i such that ti < t.
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.
rate
"decreasing"
.type
"default"
.
General details about the structure of ground intensity functions are given in the topic gif
.
# 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, mark=list(rexp_mark, rexp_mark), 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)