matern_anisotropic2D {GpGp}

Geometrically anisotropic Matern covariance function (two dimensions)

Description

From a matrix of locations and covariance parameters of the form (variance, L11, L21, L22, smoothness, nugget), return the square matrix of all pairwise covariances.

Usage

matern_anisotropic2D(covparms, locs)

d_matern_anisotropic2D(covparms, locs)

Arguments

covparms

A vector with covariance parameters in the form (variance, L11, L21, L22, smoothness, nugget)

locs

A matrix with n rows and 2 columns. Each row of locs is a point in R^2.

Value

A matrix with n rows and n columns, with the i,j entry containing the covariance between observations at locs[i,] and locs[j,].

Functions

• d_matern_anisotropic2D(): Derivatives of anisotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, L11, L21, L22, smoothness, nugget) where L11, L21, L22, are the three non-zero entries of a lower-triangular matrix L. The covariances are

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (|| L x - L y || )^\nu K_\nu(|| L x - L y ||)

This means that L11 is interpreted as an inverse range parameter in the first dimension. The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix. NOTE: the nugget is \sigma^2 \tau^2 , not \tau^2 .