GpGp (version 0.5.0)

matern_anisotropic3D: Geometrically anisotropic Matern covariance function (three dimensions)

Description

From a matrix of locations and covariance parameters of the form (variance, L11, L21, L22, L31, L32, L33, smoothness, nugget), return the square matrix of all pairwise covariances.

Usage

matern_anisotropic3D(covparms, locs)
d_matern_anisotropic3D(covparms, locs)
d_matern_anisotropic3D_alt(covparms, locs)

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,].

Arguments

covparms

A vector with covariance parameters in the form (variance, L11, L21, L22, L31, L32, L33, smoothness, nugget)

locs

A matrix with n rows and 3 columns. Each row of locs is a point in R^3.

Functions

• d_matern_anisotropic3D(): Derivatives of anisotropic Matern covariance

• d_matern_anisotropic3D_alt(): Derivatives of anisotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, L11, L21, L22, L31, L32, L33, smoothness, nugget) where L11, L21, L22, L31, L32, L33 are the six non-zero entries of a lower-triangular matrix L. The covariances are $$M(x,y)={\sigma }^2{2}^{1-\nu }/\mathrm{\Gamma }(\nu )(||Lx-Ly||{)}^{\nu }{K}_{\nu }(||Lx-Ly||)$$ 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 ${\tau }^2$.