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 Handcock-Wallis parametrisation is given by

C(r) = 2^{1- ν} Γ(ν)^{-1} (2√{ν} r)^ν K_ν(2√{ν} r)

Usage

Arguments

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 the scaling parameter and the shape parameter.

The 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 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 C(r) = -r^2 appears as limit of the above Handcock-Wallis parametrisation.

For generalizations see section ‘See Also’.

Value

The functions return an object of class RMmodel.

Note

The Whittle-Matern model is a normal scale mixture.

Author(s)

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

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 < 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.

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 (space-time) generalizations

• RMmodel, RFsimulate, RFfit for general use.

Examples