RandomFields (version 3.0.5)

RMgengneiting: Gneiting-Wendland Covariance Models

Description

RMgengneiting is a stationary isotropic covariance model family whose elements are specified by the two parameters $\kappa$ and $\mu$ with $n$ a non-negative integer and $\mu \ge \frac{d}{2}$ with $d$ denoting the dimension of the random field (the models can be used for any dimension). A corresponding covariance function only depends on the distance $r \ge 0$ between two points. For the case $\kappa = 0$ the Gneiting-Wendland model equals the Askey model RMaskey, $$C(r) = (1-r)^\beta 1_{[0,1]}(r),\qquad\beta = \mu +1/2 = \mu + 2\kappa + 1/2.$$ For $\kappa = 1$ the Gneiting model is given by $$C(r) = \left(1+\beta r \right)(1-r)^{\beta} 1_{[0,1]}(r), \qquad \beta = \mu +2\kappa+1/2.$$ If $\kappa = 2$ $$C(r) = \left(1 + \beta r + \frac{\beta^{2} - 1}{3}r^{2} \right)(1-r)^{\beta} 1_{[0,1]}(r), \qquad \beta = \mu+2\kappa+1/2.$$ In the case $\kappa = 3$ $$C(r) = \left( 1 + \beta r + \frac{(2\beta^{2}-3)}{5} r^{2}+ \frac{(\beta^2 - 4)\beta}{15} r^{3} \right)(1-r)^\beta 1_{[0,1]}(r), \qquad \beta = \mu+2\kappa + 1/2.$$ A special case of this model is RMgneiting. $\ell$

Usage

RMgengneiting(kappa, mu, var, scale, Aniso, proj)

Arguments

kappa

parameter that chooses between the three different covariance models and may take values $0,\ldots,3$

mu

mu has to be greater than or equal to $\frac{d}{2}$ where $d$ is the dimension of the random field.

var,scale,Aniso,proj

optional parameters; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.

Value

• RMgengneiting returns an object of class RMmodel

Details

This isotropic family of covariance functions is valid for any dimension of the random field.

A special case of this family is RMgneiting (with $s = 1$ there) for the choice $\kappa = 3, \mu = 3/2$.

References

• Gneiting, T. (1999) Correlation functions for atmospherical data analysis.Q. J. Roy. Meteor. SocPart A125, 2449-2464.
• Wendland, H. (2005)Scattered Data Approximation.Cambridge Monogr. Appl. Comput. Math.

See Also

RMaskey, RMbigneiting, RMgneiting, RMmodel, RFsimulate, RFfit.

Examples

set.seed(0)
model <- RMgengneiting(kappa=1, mu=1.5)
x <- seq(0, 10, if (interactive()) 0.02 else 1) 
plot(model, ylim=c(0,1))
plot(RFsimulate(model, x=x))


scale <- 0.4
model <- RMgengneiting(kappa=3, mu=1.5, scale= 1 / 0.301187465825 * scale)
x <- seq(0, 10, if (interactive()) 0.02 else 1) 
plot(model, ylim=c(0,1))
points(x,  RFcov(RMgneiting(scale = scale), x))