RandomFields (version 3.1.16)

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 via the RMm2r construction of deterministic, monotone functions.

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 argument 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 $\xi$, respectively.

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 $R^d x (0, \infty)$ with intensity $dw * c/x^2 dx$ and $Y_i ~ Y$ 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• Smith, R.L. (1990) Max-stable processes and spatial extremes Unpublished Manuscript.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

Advanced RMmodels, Auxiliary RMmodels, RMmodel, RPbernoulli, 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

model <- RMball()
x <- seq(0, 1000, 0.2)
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)

## 2-dim
x <- seq(0, 10, 0.1) 
z <- RFsimulate(RPsmith(model, xi=0), x, x)
plot(z)

## original Smith model
x <- seq(0, 10, 0.05)
model <- RMgauss(scale = sqrt(2)) # !! cf. definition of RMgauss
z <- RFsimulate(RPsmith(model, xi=0), x, x)
plot(z)


## for some more sophisticated models see 'maxstableAdvanced'