The hardware and bandwidth for this mirror is donated by METANET, the Webhosting and Full Service-Cloud Provider.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]metanet.ch.
In this short vignette, we illustrate the family of spatial
covariance functions implemented in this package. Note that, all of them
are based on the assumptions of stationarity and isotropy of the
underlying Gaussian random field (GRF). Also, the implementations (and
definitions) are based on the families of spatial correlation
functions presented in Diggle and Ribeiro
(2007). The implementations make use of the
RcppArmadillo (Eddelbuettel and
Sanderson 2014) package and the STL library algorithms (C++ 11)1.
The first spatial covariance family implemented is the so-called Matérn Covariance function, which is defined as \[ {\rm C}(h; \sigma^2, \phi, \kappa) = \sigma^2 \frac{(h / \phi)^{\kappa} K_{\kappa}( h / \phi )} { 2^{\kappa - 1} \Gamma(\kappa) }. \] Where \(h\) is the distance between two points, \(\sigma^2\) is a variance parameter, \(\phi\) is the scale parameter that controls the reach of the spatial dependence, and \(\kappa\) is a shape parameter that controls the smoothness of the process. The function \(K_{\kappa}(\cdot)\) is the \(\kappa\)-order Modified Bessel function of Second-Kind. There are special cases of the Matérn family implemented on the package, when setting \(\kappa\) to \(1/2\), \(3/2\), \(5/2\), or \(\to \infty\) the expression simplifies. The first and the last cases yield to the Exponential and Gaussian family, respectively.
The second spatial covariance family implemented is the Powered Exponential Covariance function, it is defined as \[ {\rm C}(h; \sigma^2, \phi, \kappa) = \sigma^2 \exp \{ - (h / \phi)^{\kappa} \}, \] where \(h\) is the distance between two points, \(\sigma^2\), \(\phi\), and \(\kappa\) analogous to the Matérn function.
The penultimate option is the Gaussian family of covariance functions. The expression associated with this family is written as \[ {\rm C}(h; \sigma^2, \phi) = \sigma^2 \exp \left\{ - \frac{h ^ 2}{2 \phi ^ 2} \right\}, \] again, the parameters are analogous to what have defined before in this vignette.
Lastly, we have implemented the Spherical family of covariance functions, defined as \[ {\rm C}(h; \sigma^2, \phi) = \sigma^2 \begin{cases} 1 - \frac{3}{2} (h / \phi) + \frac{1}{2} (h / \phi) ^ 3 & , \, 0 \leq h \leq \phi \\ 0 & , \, h > \phi, \end{cases} \] where, again, \(\sigma^2\) is a variance parameter, while \(\phi > 0\) is a parameter with the same magnitude as the distances on which the function is being evaluated at and controls the speed of decay of the spatial covariances as we increase the distance between two points.
for this problem, the implementation using the
sequential algorithms of the STL library was more efficient than those
using the RcppParallel package↩︎
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.