RMgencauchy: Generalized Cauchy Family Covariance Model
Description
RMgencauchy is a stationary isotropic covariance model
belonging to the generalized Cauchy family.
The corresponding covariance function only depends on the distance $r \ge 0$ between
two points and is given by
$$C(r) = (1 + r^\alpha)^(-\beta/\alpha)$$
where $\alpha \in (0,2]$ and $\beta > 0$.
See also RMcauchy.
Usage
RMgencauchy(alpha, beta, var, scale, Aniso, proj)
Arguments
alpha
a numerical value; should be in the interval (0,2]
to provide a valid covariance function for a random field of any dimension.
beta
a numerical value; should be positive to provide a valid
covariance function for a random field of any dimension.
var,scale,Aniso,proj
optional parameters; same meaning for any
RMmodel. If not passed, the above
covariance function remains unmodified.
This model has a smoothness parameter $\alpha$ and a
paramater $\beta$ which determines the asymptotic power law.
More precisely, this model admits simulating random fields where RFfractaldimension
D of the Gaussian sample and Hurst coefficient H
can be chosen independently (compare also RMlgd.): Here, we have
$$D = d + 1 - \alpha/2, \alpha \in (0,2]$$
and
$$H = 1 - \beta/2, \beta > 0.$$
I. e. the smaller $\beta$, the longer the long-range
dependence.
The covariance function is very regular near the origin, because its
Taylor expansion only contains even terms and reaches its sill slowly.
Each covariance function of the Cauchy family is a normal scale mixture.
Note that the Cauchy Family (see RMcauchy) is included
in this family for the choice $\alpha = 2$ and
$\beta = 2 \gamma$.