exponential_anisotropic3D_alt: Geometrically anisotropic exponential covariance function...

In GpGp: Fast Gaussian Process Computation Using Vecchia's Approximation

Geometrically anisotropic exponential 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

exponential_anisotropic3D_alt(covparms, locs)

d_exponential_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,].

Functions

• d_exponential_anisotropic3D_alt(): Derivatives of anisotropic Matern covariance

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 ]

(B13,B23) can be interpreted as a drift vector in space over time if first two dimensions are space and third is time. Assuming x is transformed to u and y transformed to v, the covariances are

M(x,y) = \sigma^2 exp( - || 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 .

GpGp documentation built on Oct. 16, 2024, 9:06 a.m.