Max-Stable Random Fields

Description

These functions simulate stationary and isotropic max-stable random fields with unit Frechet margins.

Usage

MaxStableRF(x, y=NULL, z=NULL, grid, model, param, maxstable,
            method=NULL, n=1, register=0, gridtriple=FALSE,...)

InitMaxStableRF(x, y=NULL, z=NULL, grid, model, param, maxstable,
               method=NULL, register=0, gridtriple=FALSE)

Arguments

Details

There are two different kinds of models for max-stable processes implemented:

• maxstable="extremalGauss"
  Gaussian random fields are multiplied by independent random factors, and the maximum is taken. The random factors are such that the resulting random field has unit Frechet margins; the specification of the random factor is uniquely given by the specification of the random field. The parameter vector param, the model, and the method are interpreted in the same way as for Gaussian random fields, see GaussRF.
• maxstable="BooleanFunction"
  Deterministic or random, upper semi-continuous L1-functions are randomly centred and multiplied by suitable, independent random factors; the pointwise maximum over all these functions yields a max-stable random field. The simulation technique is related to the random coin method for Gaussian random field simulation, see RFMethods. Hence, only models that are suitable for the random coin method are suitable for this technique, see PrintModelList() for a complete list of suitable covariance models.
  The only value allowed for method is 'max.MPP' (and NULL), see PrintMethodList(). In the parameter list param the first two entries, namely mean and variance, are ignored. If the nugget is positive, for each point an additional independent unit Frechet variable with scale parameter nugget is involved when building the maximum over all functions.

  The model may be defined alternatively in one of the two extended ways as introduced in CovarianceFct and GaussRF. However only a single model may be given! The model may be anisotropic.

Value

InitMaxStableRF returns 0 if no error has occurred, and a positive value if failed.

MaxStableRF and DoSimulateRF return NULL if an error has occurred; otherwise the returned object depends on the parameters:
n=1:
* grid=FALSE. A vector of simulated values is returned (independent of the dimension of the random field)
* grid=TRUE. An array of the dimension of the random field is returned.

n>1:
* grid=FALSE. A matrix is returned. The columns contain the realisations.
* grid=TRUE. An array of dimension d+1, where d is the dimension of the random field, is returned. The last dimension contains the realisations.

Author(s)

Martin Schlather, martin.schlather@math.uni-goettingen.de http://www.stochastik.math.uni-goettingen.de/institute

References

Schlather, M. (2002) Models for stationary max-stable random fields. Extremes 5, 33-44.

See Also

CovarianceFct, GaussRF, RandomFields, RFMethods, RFparameters, DoSimulateRF, .

Examples

n <- 30 ## nicer, but time consuming if n <- 100
x <- y <- 1:n
ms0 <- MaxStableRF(x, y, grid=TRUE, model="exponen",
                 param=c(0,1,0,40), maxstable="extr"
                 ,CE.force = TRUE                 )
image(x,y,ms0)

############################################################
##                                                        ##
##  Plots used in M. Schlather, Extremes 5:1, 33-44, 2002 ##
##                                                        ##
############################################################

pts <- if (interactive()) 512 else 32
x <- (1:pts) / pts * 10
scalegauss <- 1.5
RFparameters(MPP.radius=2*scalegauss, Print=3)
runif(1)
save.seed <- .Random.seed
ms1 <- MaxStableRF(x, x, model="gauss", param=c(0, 1, 0, scalegauss), 
                  maxstable="Bool", grid=TRUE)
image(x, x, sqrt(ms1))
 
scalecone <- 3.7
.Random.seed <- save.seed
ms2 <- MaxStableRF(x, x, model="cone", param=c(0, 1, 0, scalecone,
           0, 0, 1), maxstable="Bool", grid=TRUE)  
image(x, x, sqrt(ms2))
   
.Random.seed <- save.seed  
ms3 <- MaxStableRF(x, x, model="exponen", param=c(0, 1, 0, 1),
           maxstable="extr", grid=TRUE, method="ci")
image(x, x, sqrt(ms3))

.Random.seed <- save.seed 
ms4 <- MaxStableRF(x, x, model="gauss", param=c(0, 1, 0, 1),
           maxstable="extr", grid=TRUE, method="ci")
image(x, x, sqrt(ms4))