simulate: Simulate a Point Process

In PtProcess: Time Dependent Point Process Modelling

Description

Provides methods for the generic function simulate.

Usage

## S3 method for class 'mpp'
simulate(object, nsim=1, seed=NULL, max.rate=NA,
         stop.condition=NULL, ...)
## S3 method for class 'linksrm'
simulate(object, nsim=1, seed=NULL, max.rate=NA,
         stop.condition=NULL, ...)

Arguments

object	an object with class "mpp" or "linksrm".
nsim	has no effect, and is only included for compatibility with the generic function simulate. See section “Length of Simulated Series” below for control information.
seed	seed for the random number generator.
max.rate	maximum rate, only used if the attribute of object$gif is "bounded". It is the maximum value of object$gif on the simulation interval object$TT.
stop.condition	a function returning a logical value. It is called after the addition of each simulated event. The simulation continues until either object$TT[2] is exceeded or stopping.condition returns TRUE. See section “Length of Simulated Series” below for further information.
...	other arguments.

Details

The thinning method (Ogata, 1981; Lewis & Shedler, 1979) is used to simulate a point process with specified ground intensity function. The method involves calculating an upper bound for the intensity function, simulating a value for the time to the next possible event using a rate equal to this upper bound, and then calculating the intensity at this simulated point; hence these “events” are simulated too frequently. The ratio of this rate with the upper bound is compared with a uniform random number to randomly determine whether the simulated time is retained or not (i.e. thinned).

The functions need to calculate an upper bound for the intensity function. The ground intensity functions will usually be discontinuous at event times, but may be monotonically increasing or decreasing at other times. The ground intensity functions have an attribute called rate with values of "bounded", "increasing" or "decreasing". This information is used to determine the required upper bounded.

The function simulate.linksrm is currently only used in conjunction with linksrm_gif, or a variation of that function. It expects the gif function to have an attribute called regions, which may be an expression, being the number of regions. The method used by the function simulate.linksrm also assumes that the function is “increasing” (i.e. rate, summed over all regions, apart from discontinuous jumps), hence a positive tectonic input over the whole system.

Value

The returned value is an object of the same class as object. It will contain all events prior to object$TT[1] in object$data and all subsequently simulated events. Variables (columns) in object$data will be restricted to "time" and those for which a mark is simulated.

Length of Simulated Series

The interval of time over which events are simulated is determined by object$TT. Simulation starts at object$TT[1] and stops at object$TT[2]. The “current” dataset will consist of all events prior to object$TT[1] in object, plus subsequently simulated events. A more complicated stopping condition can be formulated by using the argument stop.condition.

The argument stop.condition can be assigned a function that returns a logical value. The assigned function is a function of the “current” dataset. It is executed near the bottom of simulate.mpp (check by printing the function). Simulation will then continue until either the stopping condition has been met or the current time exceeds object$TT[2].

For example, we may want to simulate until the first earthquake with a magnitude of 8. Assume that the current dataset contains a variable with name "magnitude" (untransformed). We would then assign Inf to object$TT[2], and write this condition as a function:

    stop.cond <- function(data){
        n <- nrow(data)
        #   most recent event is the nth
        return(data$magnitude[n] >= 8)
    }

References

Cited references are listed on the PtProcess manual page.

Examples

TT <- c(0, 1000)
bvalue <- 1
params <- c(-2.5, 0.01, 0.8, bvalue*log(10))

x <- mpp(data=NULL,
         gif=srm_gif,
         marks=list(NULL, rexp_mark),
         params=params,
         gmap=expression(params[1:3]),
         mmap=expression(params[4]),
         TT=TT)
x <- simulate(x, seed=5)

y <- hist(x$data$magnitude, xlab="Magnitude", main="")

#   overlay with an exponential density
magn <- seq(0, 3, length.out=100)
points(magn, nrow(x$data)*(y$breaks[2]-y$breaks[1])*
             dexp(magn, rate=1/mean(x$data$magnitude)),
       col="red", type="l")

Example output

PtProcess documentation built on May 4, 2021, 1:06 a.m.