Whittle-Matern Model

Description

matern calculates the Whittle-Matern covariance function (Soboloev kernel).

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 equals

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

Usage

whittle(x, nu, derivative=0,
       scaling=c("whittle", "matern", "handcockwallis"))
matern(x, nu, derivative=0,
       scaling=c("matern", "whittle", "handcockwallis"))

Arguments

Value

If derivative=0, the function value is returned, otherwise the derivativeth derivative.

A vector of length(x) is returned; nu is recycled; scaling is recycled if numerical.

If scaling has a numerical values s, the covariance model equals

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

The function values are rather precise even for large values of nu.

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de, http://ms.math.uni-mannheim.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 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.

See Also

nonstwm

For more details see also RMmatern.

Examples

x <- 3
confirm(matern(x, 0.5), exp(-x))
confirm(matern(x, Inf), gauss(x/sqrt(2)))
confirm(matern(1:2, c(0.5, Inf)), exp(-(1:2)))