R: Computes madograms

madogram {SpatialExtremes} | R Documentation

## 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, π) 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:

ν(h) = 0.5 * E[|Z(x+h) - Z(x)|]

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

u_β (μ + ν(h) / Γ(1 - ξ)), if ξ < 1, exp(ν(h)/σ), otherwise

where Γ is the gamma function and u_β is defined as follows:

(1 + ξ (u - μ) / σ )_+^{1/ξ}

and β= (μ, σ, ξ), 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")

[Package SpatialExtremes version 2.0-5 Index]