RandomFields (version 3.0.5)

Smith: (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

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 parameter 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.

It simulates max-stable processes $Z$ that are referred to as Smith model. $$Z(x) = \max_{i=1}^\infty X_i Y_i(x-W_i),$$ where $(W_i, X_i)$ are the points of a Poisson point process on $\R^d \times (0, \infty)$ with intensity $dw * c/x^2 dx$ and $Y_i \sim Y$ are iid measurable random functions with $E[\int \max(0, Y(x)) dx] < \infty$. The constant $c$ is chosen such that $Z$ has standard Frechet margins.

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• Smith, R.L. (1990) Max-stable processes and spatial extremes Unpublished Manuscript.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

RMmodel, RPbernoulli, RPgauss, maxstable maxstableAdvanced

Examples

set.seed(0)

model <- RMball()
x <- if (interactive()) seq(0, 1000, 0.02) else seq(0, 100, 10)
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)

if (!.C("isAuthor", a=integer(1))$a) {
}


## for some more sophisticated models see 'maxstableAdvanced'