matern

This function computes the Matérn correlation function given
a distance matrix. The Matérn correlation function is given by
$$C(h)=\frac{2^{1-\nu}}{\Gamma(\nu)} \left(\frac{h}{\phi} \right)^{\nu} 
\mathcal{K}_{\nu}\left( \frac{h}{\phi} \right),$$
where $\phi$ is the range parameter. $\nu$ is the smoothness parameter. 
$\mathcal{K}_{\nu}(\cdot)$ is the modified Bessel function of the second kind of order $\nu$.
The form of covariance includes the following special cases by specifying $\nu$ to be 0.5, 1.5, 2.5.<ul>
<li>$\nu=0.5$ corresponds to the exponential correlation function (exp) 
of the form
$$C(h) = \exp\left\{ - \frac{h}{\phi} \right\} $$</li>
<li>$\nu=1.5$ corresponds to the Matérn correlation function with smoothness parameter 1.5 (matern_3_2) 
of the form
$$C(h) = \left( 1 + \frac{h}{\phi} \right) \exp\left\{ - \frac{h}{\phi} \right\} $$</li>
<li>$\nu=2.5$ corresponds to the Matérn correlation function with smoothness parameter 2.5 (matern_5_2) 
of the form
$$C(h) = \left\{ 1 + \frac{h}{\phi} + \frac{1}{3}\left(\frac{h}{\phi}\right)^2 \right\} \exp\left\{ - \frac{h}{\phi} \right\} $$</li>
</ul>

Gaussian processes ('GPs') have been widely used to model spatial data, 'spatio'-temporal data, and computer experiments in diverse areas of statistics including spatial statistics, 'spatio'-temporal statistics, uncertainty quantification, and machine learning. This package creates basic tools for fitting and prediction based on 'GPs' with spatial data, 'spatio'-temporal data, and computer experiments. Key characteristics for this GP tool include: (1) the comprehensive implementation of various covariance functions including the 'Matérn' family and the Confluent 'Hypergeometric' family with isotropic form, tensor form, and automatic relevance determination form, where the isotropic form is widely used in spatial statistics, the tensor form is widely used in design and analysis of computer experiments and uncertainty quantification, and the automatic relevance determination form is widely used in machine learning; (2) implementations via Markov chain Monte Carlo ('MCMC') algorithms and optimization algorithms for GP models with all the implemented covariance functions. The methods for fitting and prediction are mainly implemented in a Bayesian framework; (3) model evaluation via Fisher information and predictive metrics such as predictive scores; (4) built-in functionality for simulating 'GPs' with all the implemented covariance functions; (5) unified implementation to allow easy specification of various 'GPs'.

Pulong Ma

GPBayes

Tools for Gaussian Process Modeling in Uncertainty
Quantification

matern function

<dl><dt>d</dt>
<dd>a matrix of distances</dd>
<dt>range</dt>
<dd>a numerical value containing the range parameter</dd>
<dt>nu</dt>
<dd>a numerical value containing the smoothness parameter</dd></dl>

Arguments

Pulong Ma <a href="/link/mpulong%40gmail.com?package=GPBayes&version=0.1.0-5.1" data-mini-rdoc="GPBayes::mpulong@gmail.com">mpulong@gmail.com</a>

Author

The Matérn correlation function proposed by Matérn (1960) — matern

<dl>

<dt>d</dt>
<dd>a matrix of distances</dd>


<dt>range</dt>
<dd>a numerical value containing the range parameter</dd>


<dt>nu</dt>
<dd>a numerical value containing the smoothness parameter</dd>

</dl>

Pulong Ma <a href='mailto:mpulong@gmail.com'>mpulong@gmail.com</a>

matern: The Matérn correlation function proposed by Matérn (1960)

Description

Usage

Value

Arguments

Author

See Also