Whittle-Matern Model
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)
whittle(x, nu, derivative=0,
scaling=c("whittle", "matern", "handcockwallis"))
matern(x, nu, derivative=0,
scaling=c("matern", "whittle", "handcockwallis"))x |
numerical vector; for negative values the modulus is used |
nu |
numerical vector with positive entries |
derivative |
value in |
scaling |
numerical vector of positive values or character; see Details. |
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.
Martin Schlather, schlather@math.uni-mannheim.de, http://ms.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.
For more details see also RMmatern.
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)))
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