RandomFields (version 3.0.32)

BrownResnick: Brown-Resnick process

Description

RPbrownresnick defines a Brown-Resnick process.

Usage

RPbrownresnick(phi, tcf, xi, mu, s)

Arguments

phi

specifies the covariance model or variogram, see RMmodel and RMmodelsAdvanced.

tcf

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 extreme value index xi is always a number, i.e. $\xi$ is constant in space. In contrast, $\mu$ 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 $z\xi$, respectively.

The functions RPbrorig, RPbrshifted and RPbrmixed perform the simulation of a Brown-Resnick process, which is defined by $$Z(x) = \max_{i=1}^\infty X_i \exp(W_i(x) - \gamma^2),$$ where the $X_i$ are the points of a Poisson point process on the positive real half-axis with intensity $x^{-2} dx$, $W_i \sim W$ are iid centered Gaussian processes with stationary increments and variogram $\gamma$ given by model. For simulation, internally, one of the methods RPbrorig, RPbrshifted and RPbrmixed is chosen automatically.

References

• Brown, B.M. and Resnick, S.I. (1977). Extreme values of independent stochastic processes.J. Appl. Probab.14, 732-739.
• Buishand, T., de Haan , L. and Zhou, C. (2008). On spatial extremes: With application to a rainfall problem.Ann. Appl. Stat.2, 624-642.
• Kabluchko, Z., Schlather, M. and de Haan, L (2009) Stationary max-stable random fields associated to negative definite functionsAnn. Probab.37, 2042-2065.
• Oesting, M., Kabluchko, Z. and Schlather M. (2012) Simulation of {B}rown-{R}esnick Processes,Extremes,15, 89-107.

See Also

RPbrorig, RPbrshifted, RPbrmixed, RMmodel, 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

\dontrun{
model <- ~ RPbrownresnick(RMfbm(alpha=A), xi=0)
x <- seq(0, 10, if (interactive()) 0.1 else 1)
z <- RFsimulate(model=model, x, x, grid=TRUE, n=4, A=0.9)
plot(z)

z <- RFsimulate(model, x=x, n=4, A=1.9)
plot(z)


## basic model in Buishand, de Haan, Zhou (2008)
model <- RMfbm(proj=1, alpha=1, var=0.5) + RMfbm(proj=2, alpha=1, var=0.5) 
x <- seq(0, 5, if (interactive()) 0.05 else 1)
z <- RFsimulate(RPbrownresnick(model, xi=0), x, x, every=1000)
plot(z)


}

## for some more sophisticated models see 'maxstamableAdvanced'

FinalizeExample()