| RMwhittlematern {RandomFields} | R Documentation |
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)
RMwhittle(nu, notinvnu, var, scale, Aniso, proj) RMmatern(nu, notinvnu, var, scale, Aniso, proj)
nu |
a numerical value called “smoothness parameter”; should be greater than 0. |
notinvnu |
logical, if not given the model is defined as above. (default). This argument should not be set by users. See the Notes. |
var,scale,Aniso,proj |
optional arguments; same meaning for any
|
RMwhittle and RMmatern
are two alternative parametrizations of the same covariance function.
The Matern model should be preferred as this model seperates the
effects of scaling parameter and the shape parameter.
This is the normal scale mixture 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’.
The function return an object of class RMmodel
The Matern model called by C(r √(2)) equals the Handcock-Wallis (1994) parametrisation.
The model allows further to be reparameterized by substituting
ν for ν^{-1} setting the argument
invnu=TRUE. Note that the inversion of ν does not really
make sense for the Whittle model. Due to this fact, if the argument
invnu is given, the Whittle model changes its definition and
becomes identical to the Matern model.
Martin Schlather, schlather@math.uni-mannheim.de
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 < ν ≤ 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.
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 (space-time) generalizations
RMmodel,
RFsimulate,
RFfit for general use.
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=if (interactive()) 100 else 3) 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)