R: (Mixed) Moving Maxima

Smith {RandomFields} | R Documentation

## (Mixed) Moving Maxima

### Description

`RPsmith` defines a moving maximum process or a mixed moving maximum process with finite number of shape functions.

### Usage

    RPsmith(shape, tcf, xi, mu, s)

### Arguments

`shape` | an `RMmodel` giving the spectral function
`tcf` | an `RMmodel` specifying the extremal correlation function; either `shape` or `tcf` must be given. If `tcf` is given a shape function is tried to be constructed via the `RMm2r` construction of deterministic, monotone functions.
`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 ξ, respectively.

It simulates max-stable processes Z that are referred to as “Smith model”.

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

where (W_i, X_i) are the points of a Poisson point process on R^d x (0, ∞) with intensity dw * c/x^2 dx and Y_i ~ Y are iid measurable random functions with E[int max(0, Y(x)) dx ] < ∞. The constant c is chosen such that Z has standard Frechet margins.

### Note

IMPORTANT: for consistency reasons with the geostatistical definitions in this package the scale argument differs froms the original definition of the Smith model! See the example below.

`RPsmith` depends on `RRrectangular` and its arguments.

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

### References

- Haan, L. (1984) A spectral representation for max-stable processes. Ann. Probab., 12, 1194-1204.

- Smith, R.L. (1990) Max-stable processes and spatial extremes Unpublished Manuscript.

### See Also

`Advanced RMmodels`, `Auxiliary RMmodels`, `RMmodel`, `RPbernoulli`, `RPgauss`, maxstable `maxstableAdvanced`

### Examples

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

    model <- RMball()
    x <- seq(0, 1000, 0.2)
    z <- RFsimulate(RPsmith(model, xi=0), x)
    plot(z)
    hist(z@data$variable1, 50, freq=FALSE)
    curve(exp(-x) * exp(-exp(-x)), from=-3, to=8, add=TRUE)

    ## 2-dim
    x <- seq(0, 10, 0.1)
    z <- RFsimulate(RPsmith(model, xi=0), x, x)
    plot(z)

    ## original Smith model
    x <- seq(0, 10, 0.05)
    model <- RMgauss(scale = sqrt(2)) # !! cf. definition of RMgauss
    z <- RFsimulate(RPsmith(model, xi=0), x, x)
    plot(z)

    ## for some more sophisticated models see 'maxstableAdvanced'

[Package RandomFields version 3.1.50 Index]