RMgengneiting {RandomFields}

R Documentation

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 being 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.

Usage

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

Arguments

kappa

0,\ldots,3

; it chooses between the three different covariance models above.

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

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.

Value

RMgengneiting returns an object of class RMmodel.

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/

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)

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