# 5. Spatial covariance functions

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 \[ \mathrm{ C}(h; \sigma^2, \phi, \kappa) = \sigma^2 \frac{(h / \phi)^{\kappa} K_{\kappa}( h / \phi )} { 2^{\kappa - 1} \Gamma(\kappa) }. \] 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 \[ \mathrm{ C}(h; \sigma^2, \phi, \kappa) = \sigma^2 \exp \{ - (h / \phi)^{\kappa} \}, \] 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 \[ \mathrm{ C}(h; \sigma^2, \phi) = \sigma^2 \exp \left\{ - \frac{h ^ 2}{2 \phi ^ 2} \right\}, \] 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 \[ \mathrm{ C}(h; \sigma^2, \phi) = \sigma^2 \begin{cases} 1 - \frac{3}{2} (h / \phi) + \frac{1}{2} (h / \phi) ^ 3 & , \, 0 \leq h \leq \phi \\ 0 & , \, h > \phi, \end{cases} \] 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.

# Reference

Diggle, PJ, and Paulo Justiniano Ribeiro. 2007. “Model-Based Geostatistics (Springer Series in Statistics).” In, 51–56. Springer.
Eddelbuettel, Dirk, and Conrad Sanderson. 2014. “RcppArmadillo: Accelerating r with High-Performance c++ Linear Algebra.” Computational Statistics & Data Analysis 71: 1054–63.

1. for this problem, the implementation using the sequential algorithms of the STL library was more efficient than those using the `RcppParallel` package↩︎