matern {GPBayes}

The Matérn correlation function proposed by Matérn (1960)

Description

This function computes the Matérn correlation function given a distance matrix. The Matérn correlation function is given by

C(h)=\frac{2^{1-\nu}}{\Gamma(\nu)} \left(\frac{h}{\phi} \right)^{\nu} \mathcal{K}_{\nu}\left( \frac{h}{\phi} \right),

where \phi is the range parameter. \nu is the smoothness parameter. \mathcal{K}_{\nu}(\cdot) is the modified Bessel function of the second kind of order \nu. The form of covariance includes the following special cases by specifying \nu to be 0.5, 1.5, 2.5.

• \nu=0.5 corresponds to the exponential correlation function (exp) of the form

  C(h) = \exp\left\{ - \frac{h}{\phi} \right\}

• \nu=1.5 corresponds to the Matérn correlation function with smoothness parameter 1.5 (matern_3_2) of the form

  C(h) = \left( 1 + \frac{h}{\phi} \right) \exp\left\{ - \frac{h}{\phi} \right\}

• \nu=2.5 corresponds to the Matérn correlation function with smoothness parameter 2.5 (matern_5_2) of the form

  C(h) = \left\{ 1 + \frac{h}{\phi} + \frac{1}{3}\left(\frac{h}{\phi}\right)^2 \right\} \exp\left\{ - \frac{h}{\phi} \right\}

Usage

matern(d, range, nu)

Arguments

d

a matrix of distances

range

a numerical value containing the range parameter

nu

a numerical value containing the smoothness parameter

Value

a numerical matrix

Author(s)

Pulong Ma [e-mail]

See Also

GPBayes-package, GaSP, gp, CH, kernel, ikernel