RandomFields (version 1.3.27)

MaxStableRF: 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

x

matrix of coordinates, or vector of x coordinates

y

vector of y coordinates

z

vector of z coordinates

grid

logical; determines whether the vectors x, y, and z should be interpreted as a grid definition, see Details.

model

string; see CovarianceFct, or type PrintModelList() to get all options; interpretation depends on the value of

param

parameter vector: param=c(mean, variance, nugget, scale,...); the parameters must be given in this order; further parameters are to be added in case of a parametrised class of covariance functions, see

maxstable

string. Either 'extremalGauss' or 'BooleanFunction'; see Details.

method

NULL or string; method used for simulating, see RFMethods, or type PrintMethodList() to get all optio

n

number of realisations to generate

register

0:9; place where intermediate calculations are stored; the numbers are aliases for 10 internal registers

gridtriple

logical; if gridtriple==FALSE ascending sequences for the parameters x, y, and z are expected; if gridtriple==TRUE triples of form c(start,end,step) expec

...

RFparameters that are locally used only.

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 repetitions. * grid==TRUE. An array of dimension $d+1$, where $d$ is the dimension of the random field, is returned. The last dimension contains the repetitions.

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 vectorparam, themodel, and themethodare interpreted in the same way as for Gaussian random fields, seeGaussRF.
• maxstable="BooleanFunction" Deterministic or random, upper semi-continuous$L_1$-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, seeRFMethods. Hence, only models that are suitable for the random coin method are suitable for this technique, seePrintModelList()for a complete list of suitable covariance models. The only value allowed formethodis 'max.MPP' (andNULL), seePrintMethodList(). In the parameter listparamthe first two entries, namelymeanandvariance, are ignored. If the nugget is positive, for each point an additional independent unit Frechet variable with scale parameternuggetis involved when building the maximum over all functions.

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

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))