RMhyperbolic: 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 modifies 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 parameters; same meaning for any
RMmodel. If not passed, the above
covariance function remains unmodified.
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 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 $\lambda^{-1}$.
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.