RMhyperbolic {RandomFields}

R Documentation

Generalized Hyperbolic Covariance Model

Description

RMhyperbolic is a stationary isotropic covariance model called “generalized hyperbolic”. The corresponding covariance function only depends on the distance r \ge 0 between two points and is given by

C(r) = \frac{(\delta^2+r^2)^{\nu/2} K_\nu(\xi(\delta^2+r^2)^{1/2})}{\delta^\nu K_\nu(\xi \delta)}

where K_{\nu} denotes the modified Bessel function of second kind.

Usage

RMhyperbolic(nu, lambda, delta, var, scale, Aniso, proj)

Arguments

nu, lambda, delta	numerical values; should either satisfy \delta \ge 0, \lambda > 0 and \nu > 0, or \delta > 0, \lambda > 0 and \nu = 0, or \delta > 0, \lambda \ge 0 and \nu < 0.
var, scale, Aniso, proj	optional arguments; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.

Details

This class is over-parametrized, i.e. it can be reparametrized by replacing the three parameters \lambda, \delta and scale by two other parameters. This means that the representation is not unique.

Each generalized hyperbolic covariance function is a normal scale mixture.

The model contains some other classes as special cases; for \lambda = 0 we get the Cauchy covariance function (see RMcauchy) with \gamma = -\frac{\nu}2 and scale=\delta; the choice \delta = 0 yields a covariance model of type RMwhittle with smoothness parameter \nu and scale parameter \lambda^{-1}.

Value

RMhyperbolic returns an object of class RMmodel.

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/

References

• Shkarofsky, I.P. (1968) Generalized turbulence space-correlation and wave-number spectrum-function pairs. Can. J. Phys. 46, 2133-2153.

• Barndorff-Nielsen, O. (1978) Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5, 151-157.

• Gneiting, T. (1997). Normal scale mixtures and dual probability densities. J. Stat. Comput. Simul. 59, 375-384.

See Also

RMcauchy, RMwhittle, RMmodel, RFsimulate, RFfit.

Examples

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again

model <- RMhyperbolic(nu=1, lambda=2, delta=0.2)
x <- seq(0, 10, 0.02)
plot(model)
plot(RFsimulate(model, x=x))

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