ExtremalGaussian {RandomFields}

R Documentation

Extremal Gaussian process

Description

RPschlather defines an extremal Gaussian process.

Usage

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

Arguments

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.

Details

The argument xi is always a number, i.e. \xi is constant in space. In contrast, \mu and s might be constant numerical values or (in future!) be given by an RMmodel, in particular by an RMtrend model.
For xi=0, the default values of mu and s are 0 and 1, respectively. For xi\not=0, the default values of mu and s are 1 and |\xi|, respectively, so that it defaults to the standard Frechet case if \xi > 0.

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}^\infty 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 \sim 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.

Note

Advanced options are maxpoints and max_gauss, see RFoptions.

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/

See Also

RMmodel, RPgauss, maxstable, maxstableAdvanced.

Examples

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) 

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