geoR (version 1.9-4)

matern: Computer Values of the Matern Correlation Function

Description

This function computes values of the \(\mbox{Mat\'{e}rn}\) correlation function for given distances and parameters.

Usage

matern(u, phi, kappa)

Value

A vector matrix or array, according to the argument u, with the values of the \(\mbox{Mat\'{e}rn}\)

correlation function for the given distances.

Arguments

u

a vector, matrix or array with values of the distances between pairs of data locations.

phi

value of the range parameter \(\phi\).

kappa

value of the smoothness parameter \(\kappa\).

Author

Paulo J. Ribeiro Jr. ,
Peter J. Diggle .

Details

The \(\mbox{Mat\'{e}rn}\) model is defined as:

$$\rho(u;\phi,\kappa) =\{2^{\kappa-1} \Gamma(\kappa)\}^{-1} (u/\phi)^\kappa K_\kappa(u/\phi)$$

where \(\phi\) and \(\kappa\) are parameters and \(K_\kappa(\cdot)\) denotes the modified Bessel function of the third kind of order \(\kappa\). The family is valid for \(\phi>0\) and \(\kappa>0\).

See Also

cov.spatial for the correlation functions implemented in geoR, and besselK for computation of the Bessel functions.

Examples

#
# Models with fixed range and varying smoothness parameter
#
curve(matern(x, phi= 0.25, kappa = 0.5),from = 0, to = 1.5,
      xlab = "distance", ylab = expression(rho(h)), lty = 2,
      main=expression(paste("varying  ", kappa, "  and fixed  ", phi)))
curve(matern(x, phi= 0.25, kappa = 1),from = 0, to = 1.5, add = TRUE)
curve(matern(x, phi= 0.25, kappa = 2),from = 0, to = 1.5, add = TRUE,
      lwd = 2, lty=2)
curve(matern(x, phi= 0.25, kappa = 3),from = 0, to = 1.5, add = TRUE,
      lwd = 2)
legend("topright", expression(kappa==0.5, kappa==1.5, kappa==2, kappa==3),
    lty=c(2,1,2,1), lwd=c(1,1,2,2))

#
# Correlations with equivalent "practical range"
# and varying smoothness parameter
#
curve(matern(x, phi = 0.25, kappa = 0.5),from = 0, to = 1,
      xlab = "distance", ylab = expression(gamma(h)), lty = 2,
      main = "models with equivalent \"practical\" range")
curve(matern(x, phi = 0.188, kappa = 1),from = 0, to = 1, add = TRUE)      
curve(matern(x, phi = 0.14, kappa = 2),from = 0, to = 1,
      add = TRUE, lwd=2, lty=2)      
curve(matern(x, phi = 0.117, kappa = 2), from = 0, to = 1,
      add = TRUE, lwd=2)      
legend("topright", expression(list(kappa == 0.5, phi == 0.250),
       list(kappa == 1, phi == 0.188), list(kappa == 2, phi == 0.140),
       list(kappa == 3, phi == 0.117)), lty=c(2,1,2,1), lwd=c(1,1,2,2))