RandomFields (version 3.1.16)

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 $\mu \ge d/2$ 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

$0,...,3$

mu

mu has to be greater than or equal to $d / 2$ where $d$ is the dimension of the random field.

var,scale,Aniso,proj

optional arguments; 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. Soc Part A 125, 2449-2464.

• Wendland, H. (2005) Scattered Data Approximation. Cambridge Monogr. Appl. Comput. Math.

See Also

RMaskey, RMbigneiting, RMgneiting, 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 <- RMgengneiting(kappa=1, mu=1.5)
x <- seq(0, 10, 0.02)
plot(model)
plot(RFsimulate(model, x=x))


## same models:
model2 <- RMgengneiting(kappa=3, mu=1.5, scale= 1 / 0.301187465825)
plot(RMgneiting(), model2=model2, type=c("p", "l"), pch=20)