SpatialExtremes (version 1.2-0)

fitextcoeff: Non parametric estimators of the extremal coefficient function

Description

Estimates non parametrically the extremal coefficient function.

Usage

fitextcoeff(data, coord, ..., estim = "ST", marge = "emp", prob = 0,
plot = TRUE, loess = TRUE, method = "BFGS", std.err = TRUE, xlab,
ylab, angles = NULL, identify = FALSE)

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.

...

Additional options to be passed to the plot function.

estim

Character string specifying the estimator to be used. Must be one of "ST" (Schlather and Tawn) or "Smith".

marge

Character string specifying how margins are transformed to unit Frechet. Must be one of "emp", "mle" or "none" - see Details

prob

The probability related to the threshold. Only useful with the ST estimator.

plot

Logical. If TRUE (default), the extremal coefficient function is plotted

loess

If TRUE (default), a local polynomial curve is plotted - see function loess.

method

The optimizer used when fitting the GEV distribution to data. See function gevmle.

std.err

Logical. If TRUE, standard errors are computed. Note that standard errors are not available with the "ST" estimator.

xlab,ylab

The x-axis and y-axis labels. May be missing.

angles

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

identify

Logical. If TRUE, users can use the identify function to identify pairs of stations on the plot.

Value

• Plots the extremal coefficient function and returns the points used for the plot. If loess = TRUE, the output is a list with argument "ext.coeff" and "loess".

Details

During the estimation procedure, data need to be transformed to unit Frechet margins firts. This can be done in two different ways ; by using the empirical CDF or the GEV ML estimates.

If marge = "emp", then the data are transformed using the following relation: $$z_i = - \frac{1}{\log (F(y_i))}$$ where $y_i$ are the observations available at location $i$, $F$ is the empirical CDF and $z_i$ are the observations transformed to unit Frechet scale.

If marge = "mle", then the data are transformed using the MLE of the GEV distribution - see function gev2frech.

Lastly, if data are already supposed to be unit Frechet, then no transformation is performed if one passed the option marge = "frech". If data are already componentwise maxima, prob should be zero. Otherwise, users have to define a threshold $z$ (large enough) where univariate extreme value arguments are relevant. We define prob such that $\Pr[Z \leq z] = prob$.

References

Schlather, M. and Tawn, J. A. (2003) A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90(1):139--156.

Smith, R. L. (1990) Max-stable processes and spatial extremes. Unpublished manuscript.

See Also

madogram

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
sigma <- matrix(c(10, 40, 40, 220),ncol = 2)
sigma.inv <- solve(sigma)
sqrtCinv <- t(chol(sigma.inv))
model <- list(list(model = "gauss", var = 1, aniso = sqrtCinv / 2))
ms0 <- MaxStableRF(locations[,1], locations[,2], grid=FALSE, model=model,
                   maxstable="Bool", n = 40)
ms0 <- t(ms0)

##Plot the extremal coefficient function
op <- par(mfrow=c(1,2))
fitextcoeff(ms0, locations, estim = "Smith")
fitextcoeff(ms0, locations, angles = seq(-pi, pi, length = 4), estim = "Smith")
par(op)