matern_anisotropic3D_alt

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.

Functions for fitting and doing predictions with
Gaussian process models using Vecchia's (1988) approximation.
Package also includes functions for reordering input locations,
finding ordered nearest neighbors (with help from 'FNN' package),
grouping operations, and conditional simulations.
Covariance functions for spatial and spatial-temporal data
on Euclidean domains and spheres are provided. The original
approximation is due to Vecchia (1988)
<http://www.jstor.org/stable/2345768>, and the reordering and
grouping methods are from Guinness (2018)
<doi:10.1080/00401706.2018.1437476>.
Model fitting employs a Fisher scoring algorithm described
in Guinness (2019) <doi:10.48550/arXiv.1905.08374>.

Joseph Guinness

GpGp

Fast Gaussian Process Computation Using Vecchia's Approximation

Matthias Katzfuss

Youssef Fahmy

matern_anisotropic3D_alt function

<dl><dt>covparms</dt>
<dd>A vector with covariance parameters
in the form (variance, B11, B12, B13, B22, B23, B33, smoothness, nugget)</dd>
<dt>locs</dt>
<dd>A matrix with <code>n</code> rows and <code>3</code> columns.
Each row of locs is a point in R^3.</dd></dl>

Arguments

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 $.

Parameterization

Geometrically anisotropic Matern covariance function (three dimensions, alternate parameterization) — matern_anisotropic3D_alt

<dl>

<dt>covparms</dt>
<dd>A vector with covariance parameters
in the form (variance, B11, B12, B13, B22, B23, B33, smoothness, nugget)</dd>


<dt>locs</dt>
<dd>A matrix with <code>n</code> rows and <code>3</code> columns.
Each row of locs is a point in R^3.</dd>

</dl>

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 $.

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matern_anisotropic3D_alt: Geometrically anisotropic Matern covariance function (three dimensions, alternate parameterization)

Description

Usage

Value

Arguments

Parameterization