Computes the Whittle-Matern correlation function.

This computes the correlation function as derived in Matern model, see Matern (1960) eq. (2.4.7). For nu=1, see Whittle (1954) eq. (68). For the limiting case of nu=0, see Besag (1981) eq. (14-15).

Usage

cWhittleMatern(x, range, nu, kappa = sqrt(8 * nu)/range)

Arguments

x

distance.

range

practical range (our prefered parametrization) given as range = sqrt(8 * nu) / kappa, where kappa is the scale parameter in the specialized references.

nu

process smoothness parameter.

kappa

scale parameter, commonly considered in the specialized literature.

Value

the correlation.

Details

Whittle, P. (1954) On Stationary Processes in the Plane. Biometrika, Vol. 41, No. 3/4, pp. 434-449. http://www.jstor.org/stable/2332724

Matern, B. (1960) Spatial Variation: Stochastic models and their application to some problems in forest surveys and other sampling investigations. PhD Thesis.

Besag, J. (1981) On a System of Two-Dimensional Recurrence Equations. JRSS-B, Vol. 43 No. 3, pp. 302-309. https://www.jstor.org/stable/2984940

Examples

plot(function(x) cWhittleMatern(x, 1, 5),
  bty = "n", las = 1,
  xlab = "Distance", ylab = "Correlation"
)
plot(function(x) cWhittleMatern(x, 1, 1), add = TRUE, lty = 2)
plot(function(x) cWhittleMatern(x, 1, 0.5), add = TRUE, lty = 3)
abline(h = 0.139, lty = 3, col = gray(0.5, 0.5))