R: Extremal Gaussian process

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. ξ is constant in space. In contrast, μ and s might be constant numerical value or given a `RMmodel`, in particular by a `RMtrend` model. The default values of mu and s are 1 and zξ, respectively.

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

It simulates 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 `model`, and c is chosen such that Z has standard Frechet margins. `model` 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 http://ms.math.uni-mannheim.de/de/publications/software

### 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-staable 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 the following defintion 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)

[Package RandomFields version 3.1.50 Index]