RPschlather: Extremal Gaussian process
In RandomFields: Simulation and Analysis of Random Fields

Description Usage Arguments Details Note See Also Examples

RPschlather defines an extremal Gaussian process.

1	RPschlather(phi, tcf, xi, mu, s)

`phi`	an `RMmodel`, see Details.
`tcf`	an `RMmodel` specifying the extremal correlation function; either `phi` or `tcf` must be given.
`xi,mu,s`	the extreme value index, the location parameter and the scale parameter, respectively, of the generalized extreme value distribution. See Details.

\GEV

The argument phi can be any random field for which the expectation of the positive part is known at the origin.

It simulates an Extremal Gaussian process Z (also called “Schlather model”), which is defined by

Z(x) = max_{i=1, 2, ...} X_i * max(0, Y_i(x)),

where the X_i are the points of a Poisson point process on the positive real half-axis with intensity c/x^2 dx, Y_i ~ Y are iid stationary Gaussian processes with a covariance function given by phi, and c is chosen such that Z has standard Frechet margins. phi must represent a stationary covariance model.

Advanced options are maxpoints and max_gauss, see RFoptions.

RMmodel, RPgauss, maxstable, maxstableAdvanced.

RFoptions(seed=0, xi=0)
## seed=0: *ANY* simulation will have the random seed 0; set
##         RFoptions(seed=NA) to make them all random again

## xi=0: any simulated max-stable random field has extreme value index 0
x <- seq(0, 2,0.01)

## standard use of RPschlather (i.e. a standardized Gaussian field)
model <- RMgauss()
z1 <- RFsimulate(RPschlather(model), x)
plot(z1, type="l")

## the following refers to the generalized use of RPschlather, where
## any random field can be used. Note that 'z1' and 'z2' have the same
## margins and the same .Random.seed (and the same simulation method),
## hence the same values
model <- RPgauss(RMgauss(var=2))
z2 <- RFsimulate(RPschlather(model), x)
plot(z2, type="l")
all.equal(z1, z2) # true

## Note that the following definition is incorrect
try(RFsimulate(model=RPschlather(RMgauss(var=2)), x=x))


## check whether the marginal distribution (Gumbel) is indeed correct:
model <- RMgauss()
z <- RFsimulate(RPschlather(model, xi=0), x, n=100)
plot(z)
hist(unlist(z@data), 50, freq=FALSE)
curve(exp(-x) * exp(-exp(-x)), from=-3, to=8, add=TRUE)