matern {geoR} | R Documentation

## Computer Values of the Matern Correlation Function

### 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)^kappa K_kappa(u/phi)

where phi and kappa are parameters and K_kappa(...) 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. Paulo.Ribeiro@est.ufpr.br,
Peter J. Diggle p.diggle@lancaster.ac.uk.

### 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,
          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, add = TRUE)
    curve(matern(x, phi= 0.25, kappa = 2),from = 0, to = 1, add = TRUE,
          lwd = 2, lty=2)
    curve(matern(x, phi= 0.25, kappa = 3),from = 0, to = 1, add = TRUE,
          lwd = 2)
    legend(0.6,1, c(expression(kappa == 0.5), expression(kappa == 1),
        expression(kappa == 2), expression(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(0.4,1, c(expression(paste(kappa == 0.5, "  and  ",
           phi == 0.250)),
           expression(paste(kappa == 1, "  and  ", phi == 0.188)),
           expression(paste(kappa == 2, "  and  ", phi == 0.140)),
           expression(paste(kappa == 3, "  and  ", phi == 0.117))),
           lty=c(2,1,2,1), lwd=c(1,1,2,2))

[Package geoR version 1.5-7 Index]