# matern: Computer Values of the Matern Correlation Function In geoR: Analysis of Geostatistical Data

 matern 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)^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. paulojus@leg.ufpr.br,
Peter J. Diggle p.diggle@lancaster.ac.uk.

`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,