matern_anisotropic3D_alt {GpGp}

Geometrically anisotropic Matern covariance function (three dimensions, alternate parameterization)

Description

From a matrix of locations and covariance parameters of the form (variance, B11, B12, B13, B22, B23, B33, smoothness, nugget), return the square matrix of all pairwise covariances.

Usage

matern_anisotropic3D_alt(covparms, locs)

Arguments

covparms

A vector with covariance parameters in the form (variance, B11, B12, B13, B22, B23, B33, smoothness, nugget)

locs

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

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,].

Parameterization

The covariance parameter vector is (variance, B11, B12, B13, B22, B23, B33, smoothness, nugget) where B11, B12, B13, B22, B23, B33, transform the three coordinates as

u_1 = B11[ x_1 + B12 x_2 + (B13 + B12 B23) x_3]

u_2 = B22[ x_2 + B23 x_3]

u_3 = B33[ x_3 ]

NOTE: the u_1 transformation is different from previous versions of this function. NOTE: now (B13,B23) can be interpreted as a drift vector in space over time. Assuming x is transformed to u and y transformed to v, the covariances are

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

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 .