madogram {SpatialExtremes} | R Documentation |
Computes the madogram for max-stable processes.
madogram(data, coord, n.lag = 13, gev.param = c(0, 1, 0), which = c("mado", "ext"), xlab, ylab, ...)
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. |
Let Z(x) be a spatial and stationary process. The madogram is defined as follows:
nu(h) = 0.5 * E[|Z(x+h) - Z(x)|]
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:
u_β (μ + nu(h) / Γ(1 - xi)), if xi < 1, exp(nu(h)/σ), otherwise
where Γ(.) is the gamma function and u_β is defined as follows:
(1 + xi (u - μ) / σ )_+^{1/xi}
and β= (μ, σ, xi) i.e the vector of the GEV parameters.
A graphic and (invisibly) a matrix with the lag distances, the estimated madogram values and extremal coefficients.
Mathieu Ribatet
Cooley, D., Naveau, P. and Poncet, P. (2006) Variograms for spatial max-stable random fields. Dependence in Probability and Statistics, 373–390.
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 ## Not run: madogram(ms0, locations)