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 $Z (x) = {max}_{i = 1}^{\infty} X_{i} \exp (W_{i} (x) - γ^{2}),$ 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.

Examples

Run this code

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again
StartExample()
\dontrun{
model <- ~ RPbrownresnick(RMfbm(alpha=A), xi=0)
x <- seq(0, 10, 0.2)
z <- RFsimulate(model=model, x, x, n=4, A=0.9) # about 1 min on a fast machine
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, 0.05)
z <- RFsimulate(RPbrownresnick(model, xi=0), x, x, every=1000)
plot(z)


}
## for some more sophisticated models see 'maxstamableAdvanced'

FinalizeExample()

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