RandomFields (version 3.1.36)

RMwhittlematern: Whittle-Matern Covariance Model

Description

RMmatern is a stationary isotropic covariance model belonging to the Matern family. The corresponding covariance function only depends on the distance $r \ge 0$ between two points.

The Whittle model is given by $$C(r)=W_{\nu}(r)=2^{1- \nu} \Gamma(\nu)^{-1}r^{\nu}K_{\nu}(r)$$ where $\nu > 0$ and $K_\nu$ is the modified Bessel function of second kind.

The Matern model is given by $$C(r) = \frac{2^{1-\nu}}{\Gamma(\nu)} (\sqrt{2\nu}r)^\nu K_\nu(\sqrt{2\nu}r)$$

The Handcock-Wallis parametrisation is given by $$C(r) = \frac{2^{1-\nu}}{\Gamma(\nu)} (2\sqrt{\nu}r)^\nu K_\nu(2 \sqrt{\nu}r)$$

Usage

RMwhittle(nu, notinvnu, var, scale, Aniso, proj)
RMmatern(nu, notinvnu, var, scale, Aniso, proj)
RMhandcock(nu, notinvnu, var, scale, Aniso, proj)

Arguments

nu

a numerical value called “smoothness parameter”; should be greater than 0.

notinvnu

logical. If FALSE then in the definition of the models $\nu$ is replaced by $1/\nu$. This parametrisation seems to be more natural. Default is however FALSE according the definitions in literature.

var,scale,Aniso,proj

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

Value

The function return an object of class RMmodel

Details

The three models are alternative parametrizations of the same covariance function. The Matern model or the Handcock-Wallis parametrisation should be preferred as they seperate the effects of scaling parameter and the shape parameter.

This Whittle-Matern model is the model of choice if the smoothness of a random field is to be parametrized: the sample paths of a Gaussian random field with this covariance structure are $m$ times differentiable if and only if $\nu > m$ (see Gelfand et al., 2010, p. 24).

Furthermore, the fractal dimension (see also RFfractaldim) D of the Gaussian sample paths is determined by $\nu$: we have $$D = d + 1 - \nu, \nu \in (0,1)$$ and $D = d$ for $\nu > 1$ where $d$ is the dimension of the random field (see Stein, 1999, p. 32).

If $\nu=0.5$ the Matern model equals RMexp.

For $\nu$ tending to $\infty$ a rescaled Gaussian model RMgauss appears as limit of the Matern model.

For generalisations see section ‘seealso’.

References

Covariance function

• Chiles, J.-P. and Delfiner, P. (1999) Geostatistics. Modeling Spatial Uncertainty. New York: Wiley.
• Gelfand, A. E., Diggle, P., Fuentes, M. and Guttorp, P. (eds.) (2010) Handbook of Spatial Statistics. Boca Raton: Chapman & Hall/CRL.
• Guttorp, P. and Gneiting, T. (2006) Studies in the history of probability and statistics. XLIX. On the Matern correlation family. Biometrika 93, 989--995.
• Handcock, M. S. and Wallis, J. R. (1994) An approach to statistical spatio-temporal modeling of meteorological fields. JASA 89, 368--378.
• Stein, M. L. (1999) Interpolation of Spatial Data -- Some Theory for Kriging. New York: Springer.

Tail correlation function (for $0 < \nu \le 1/2$)

• Strokorb, K., Ballani, F., and Schlather, M. (2014) Tail correlation functions of max-stable processes: Construction principles, recovery and diversity of some mixing max-stable processes with identical TCF. Extremes, Submitted.

See Also

• RMexp, RMgauss for special cases of the model (for $\nu=0.5$ and $\nu=\infty$, respectively)

• RMhyperbolic for a univariate generalization

• RMbiwm for a multivariate generalization
• RMnonstwm, RMstein for anisotropic (space-time) generalizations

• RMmodel, RFsimulate, RFfit for general use.

Examples

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again

x <- seq(0, 1, len=100)
model <- RMwhittle(nu=1, Aniso=matrix(nc=2, c(1.5, 3, -3, 4)))
plot(model, dim=2, xlim=c(-1,1))
z <- RFsimulate(model=model, x, x)
plot(z)