matern: Computer Values of the Matern Correlation Function

In rundel/geoR: Analysis of Geostatistical Data

Description

This function computes values of the Matern correlation function for given distances and parameters.

Usage

matern(u, phi, kappa)

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.

Details

The Matern model is defined as:

rho(u;phi,kappa) =(2^(kappa-1) Gamma(kappa))^(-1) (u/phi)^κ K_κ(u/φ)

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

Value

A vector matrix or array, according to the argument u, with the values of the Matern correlation function for the given distances.

Author(s)

Paulo J. Ribeiro Jr. [e-mail],
Peter J. Diggle [e-mail].

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))

rundel/geoR documentation built on May 18, 2019, 11:28 p.m.