Extremal Gaussian process
RPschlather defines an extremal Gaussian process.
RPschlather(phi, tcf, xi, mu, s)
The argument xi is always a number, i.e. ξ is constant in space. In contrast, μ 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 |ξ|, respectively, so that it defaults to the standard Frechet case if ξ > 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, 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.
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
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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