SpatialExtremes (version 1.0-0)

madogram: Computes madograms

Description

Computes the madogram for max-stable processes.

Usage

madogram(data, coord, n.lag = 13, gev.param = c(0, 1, 0), which =
c("mado", "ext"), xlab, ylab, ...)

Arguments

data

A matrix representing the data. Each column corresponds to one location.

coord

A matrix that gives the coordinates of each location. Each row corresponds to one location.

n.lag

Integer. The number of distance lags for which the madogram is computed.

gev.param

Numeric vector of length 3 specifying the location, scale and shape parameters for the GEV.

which

A character vector of maximum size 2. It specifies if the madogram and/or the extremal coefficient functions have to be plotted.

xlab,ylab

The x-axis and y-axis labels. May be missing. Note that ylab must have the same length has which.

...

Additional options to be passed to the plot function.

Value

• A graphic and (invisibly) a matrix with the lag distances, the estimated madogram values and extremal coefficients.

Details

Let $Z(x)$ be a spatial and stationary process. The madogram is defined as follows:

$$\nu(h) = \frac{1}{2}\mbox{E}\left[|Z(x+h) - Z(x)| \right]$$

If now $Z(x)$ is a stationary max-stable spatial random field with GEV marginals. Provided the GEV shape parameter $\xi$ is such that $\xi < 1$. The extremal coefficient $\theta(h)$ satisfies:

$$\theta(h) = \begin{cases} u_\beta \left(\mu + \frac{\nu(h)}{\Gamma(1 - \xi)} \right), & \xi \neq 0\ \exp\left(\frac{\nu(h)}{\sigma}\right), & \xi = 0 \end{cases}$$ where $\Gamma(\cdot)$ is the gamma function and $u_\beta$ is defined as follows:

$$u_\beta(u) = \left(1 + \xi \frac{u - \mu}{\sigma} \right)_+^{1/\xi}$$ and $\beta = (\mu, \sigma, \xi)$ i.e the vector of the GEV parameters.

References

Cooley, D., Naveau, P. and Poncet, P. (2006) Variograms for spatial max-stable random fields. Dependence in Probability and Statistics, 373--390.

See Also

fitextcoeff

Examples

require(RandomFields)
n.site <- 30
locations <- matrix(runif(2*n.site, 0, 10), ncol = 2)
colnames(locations) <- c("lon", "lat")

##Simulate a max-stable process - with unit Frechet margins
ms0 <- MaxStableRF(locations[,1], locations[,2], grid=FALSE, model="wh",
                   param=c(0,1,0,30, .5), maxstable="extr",
                   n = 40)
ms0 <- t(ms0)

##Compute the madogram
madogram(ms0, locations)