RandomFields (version 3.0.5)

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.

Value

• RMgencauchy returns an object of class RMmodel

Details

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 $\beta = 2 \gamma$.

References

% \item Gneiting, T. (2002) Compactly supported correlation % functions. \emph{J. Multivariate Anal.} \bold{83} 493--508.
• Gneiting, T. and Schlather, M. (2004) Stochastic models which separate RFfractaldimension and Hurst effect.SIAM review46, 269--282.
• Strokorb, K., Ballani, F., and Schlather, M. (2014) In Preparation.

See Also

RMcauchy, RMcauchytbm, RMmodel, RFsimulate, RFfit.

Examples

set.seed(0)
model <- RMgencauchy(alpha=1.5, beta=1.5, scale=0.3)
x <- seq(0, 10, if (interactive()) 0.02 else 1) 
plot(model, ylim=c(0,1))
plot(RFsimulate(model, x=x))