A family of correlation function describing realizations of Gaussian spatial processes with different smoothnesses
(i.e. either smooth or rugged surfaces). Also includes a scaling and a 'nugget' parameter.
A scaling factor for distance. The 'range' considered in some
formulations is the reciprocal of this scaling factor
smoothness
The smoothness parameter, >0. $\nu=0.5$ corresponds to the exponential correlation function,
and the limit function when $\mu$ goes to $\infty$ is the squared exponential function (as in a Gaussian).
nu
Same as smoothness
Nugget
(Following the jargon of Kriging) a parameter describing a discontinuous decrease in
correlation at zero distance. Correlation will always be 1 at $d=0$, and from which it immediately drops to
(1-Nugget)
Value
Scalar/vector/matrix depending on input.
Details
The correlation at distance $d>0$ is
$$(1-\textrm{Nugget}) \frac{(\rho d)^\nu K_\nu(\rho d)}{2^{(\nu - 1)} \Gamma(\nu)}$$
where
$K_\nu$ is the besselK function of order $\nu$.
References
Stein, M.L. (1999) Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
See Also
By default the Nugget is set to 0. See one of the examples on data set Loaloa
for a fit including the estimation of the Nugget.
## The Matérn function can be used in Euclidian spaces of any dimension:set.seed(123)
randpts <- matrix(rnorm(20),nrow=5)
distMatrix <- as.matrix(proxy::dist(randpts))
Matern.corr(distMatrix,nu=2)