madogram {SpatialExtremes}

Computes madograms

Description

Computes the madogram for max-stable processes.

Usage

madogram(data, coord, fitted, n.bins, gev.param = c(0, 1, 0), which =
c("mado", "ext"), xlab, ylab, col = c(1, 2), angles = NULL, marge =
"emp", add = FALSE, xlim = c(0, max(dist)), ...)

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.

fitted

An object of class maxstab - usually the output of the fitmaxstab function. May be missing.

n.bins

The number of bins to be used. If missing, pairwise madogram estimates will be 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.

col

The colors used for the points and optionnaly for the fitted curve.

angles

A numeric vector. A partition of the interval (0, \pi) to help detecting anisotropy.

marge

Character string. If 'emp', the observation are first transformed to the unit Frechet scale by using the empirical CDF. If 'mle' (default), maximum likelihood estimates are used.

add

Logical. If TRUE, the plot is added to the current figure; otherwhise (default) a new plot is computed.

xlim

A numeric vector of length 2 specifying the x coordinate range.

...

Additional options to be passed to the plot function.

Details

Let Z(x) be a 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 random field with GEV marginals. Provided the GEV shape parameter \xi is such that \xi < 1. The extremal coefficient \theta(h) satisfies:

\theta(h) = \left\{ \begin{array}{ll} u_\beta \left(\mu + \frac{\nu(h)}{\Gamma(1 - \xi)} \right), &\xi \neq 0\\ \exp\left(\frac{\nu(h)}{\sigma}\right), &\xi = 0 \end{array} \right.

where \Gamma 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.

Value

A graphic and (invisibly) a matrix with the lag distances, the madogram and extremal coefficient estimates.

Author(s)

Mathieu Ribatet

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

fmadogram, lmadogram

Examples

n.site <- 15
locations <- matrix(runif(2*n.site, 0, 10), ncol = 2)
colnames(locations) <- c("lon", "lat")

##Simulate a max-stable process - with unit Frechet margins
data <- rmaxstab(40, locations, cov.mod = "whitmat", nugget = 0, range = 1,
smooth = 2)

##Compute the madogram
madogram(data, locations)

##Compare the madogram with a fitted max-stable model
fitted <- fitmaxstab(data, locations, "whitmat", nugget = 0)
madogram(fitted = fitted, which = "ext")