fmadogram: Computes the F-madogram

Description

Computes the F-madogram for max-stable processes.

Usage

fmadogram(data, coord, fitted, n.bins, 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 F-madogram estimates will be computed.

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 the fitted curve.

angles

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

marge

Character string. If 'emp', the probabilities of non exceedances are estimated 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.

Value

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

Details

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

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

The extremal coefficient $\theta(h)$ satisfies:

$$\theta(h) = \frac{1 + 2 \nu(h)}{1 - 2 \nu(h)}$$

References

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

Examples

Run this code

# NOT RUN {
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 F-madogram
fmadogram(data, locations)

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

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