WhittleMatern 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 ≥ 0
between two points.
The Whittle model is given by
C(r)=W_{ν}(r)=2^{1 ν} Γ(ν)^{1}r^{ν}K_{ν}(r)
where ν > 0 and K_ν is the modified Bessel function of second kind.
The Matern model is given by
C(r) = 2^{1 ν} Γ(ν)^{1} (√{2ν} r)^ν K_ν(√{2ν} r)
The HandcockWallis parametrisation is given by
C(r) = 2^{1 ν} Γ(ν)^{1} (2√{ν} r)^ν K_ν(2√{ν} r)
Usage
1 2 3 4 5 
Arguments
nu 
a numerical value called “smoothness parameter”; should be greater than 0. 
notinvnu 
logical. If 
var,scale,Aniso,proj 
optional arguments; same meaning for any

Details
The three models are alternative parametrizations of the same covariance function. The Matern model or the HandcockWallis parametrisation should be preferred as they seperate the effects of scaling parameter and the shape parameter.
This WhittleMatern 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 ν > m (see Gelfand et al., 2010, p. 24).
Furthermore, the fractal dimension (see also RFfractaldim
)
D of the Gaussian sample paths
is determined by ν: we have
D = d + 1  ν, 0 < ν < 1
and D = d for ν > 1 where d is the dimension of the random field (see Stein, 1999, p. 32).
If ν=0.5 the Matern model equals RMexp
.
For ν tending to ∞ a rescaled Gaussian
model RMgauss
appears as limit of the Matern model.
For generalisations see section ‘seealso’.
Value
The function return an object of class RMmodel
Note
The WhittleMatern model is a normal scale mixture.
Author(s)
Martin Schlather, schlather@math.unimannheim.de
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 spatiotemporal 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 < ν ≤ 1/2)
Strokorb, K., Ballani, F., and Schlather, M. (2014) Tail correlation functions of maxstable processes: Construction principles, recovery and diversity of some mixing maxstable processes with identical TCF. Extremes, Submitted.
See Also

RMexp
,RMgauss
for special cases of the model (for ν=0.5 and ν=∞, respectively) 
RMhyperbolic
for a univariate generalization 
RMbiwm
for a multivariate generalization 
RMnonstwm
,RMstein
for anisotropic (spacetime) generalizations 
RMmodel
,RFsimulate
,RFfit
for general use.
Examples
1 2 3 4 5 6 7 8 