Covariance functions return the value of the covariance
\(C(h)\) between a pair variables located at points separated by the
distance \(h\).
The covariance function can be written as a product of a variance
parameter \(\sigma^2\) times a positive definite
correlation function \(\rho(h)\):
$$C(h) = \sigma^2 \rho(h).$$
The expressions of the covariance functions available in geoR
are given below. We recommend the LaTeX (and/or the corresponding
.dvi, .pdf or .ps) version of this document for
better visualization of the formulas.
Denote \(\phi\) the basic parameter of the correlation
function and name it the range parameter.
Some of the correlation functions will have an extra parameter
\(\kappa\), the smoothness parameter.
\(K_\kappa(x)\) denotes the modified Bessel
function of the third kind of order \(\kappa\). See
documentation of the function besselK for further details.
In the equations below the functions are valid for \(\phi>0\) and \(\kappa>0\), unless stated otherwise.
cauchy
$$\rho(h) = [1+(\frac{h}{\phi})^2]^{-\kappa}$$
gencauchy (generalised Cauchy)
$$\rho(h) = [1+(\frac{h}{\phi})^{\kappa_{2}}]^{-{\kappa_1}/{\kappa_2}},
\kappa_1 > 0, 0 < \kappa_2 \leq 2 $$
circular
Let \(\theta = \min(\frac{h}{\phi},1)\) and
$$g(h)= 2\frac{(\theta\sqrt{1-\theta^2}+
\sin^{-1}\sqrt{\theta})}{\pi}.$$
Then, the circular model is given by:
$$\rho(h) = \left\{ \begin{array}{ll}
1 - g(h) \mbox{ , if $h < \phi$}\cr
0 \mbox{ , otherwise}
\end{array} \right.$$
cubic
$$\rho(h) = \left\{ \begin{array}{ll}
1 - [7(\frac{h}{\phi})^2 - 8.75(\frac{h}{\phi})^3 +
3.5(\frac{h}{\phi})^5-0.75(\frac{h}{\phi})^7] \mbox{ , if $h<\phi$} \cr
0 \mbox{ , otherwise.}
\end{array} \right.$$
gaussian
$$\rho(h) = \exp[-(\frac{h}{\phi})^2]$$
exponential
$$\rho(h) = \exp(-\frac{h}{\phi})$$
matern
$$\rho(h) =
\frac{1}{2^{\kappa-1}\Gamma(\kappa)}(\frac{h}{\phi})^\kappa
K_{\kappa}(\frac{h}{\phi})$$
spherical
$$\rho(h) = \left\{ \begin{array}{ll}
1 - 1.5\frac{h}{\phi} + 0.5(\frac{h}{\phi})^3
\mbox{ , if $h$ < $\phi$} \cr
0 \mbox{ , otherwise}
\end{array} \right.$$
power (and linear)
The parameters of the this model
\(\sigma^2\) and \(\phi\) can not be
interpreted as partial sill and range
as for the other models.
This model implies an unlimited dispersion and,
therefore, has no sill and corresponds to a process which is only
intrinsically stationary.
The variogram function is given by:
$$\gamma(h) = \sigma^2 {h}^{\phi} \mbox{ , } 0 < \phi < 2,
\sigma^2 > 0$$
Since the corresponding process is not second order stationary the
covariance and correlation functions are not defined.
For internal calculations the geoR
functions uses the fact the this model possesses locally
stationary representations with covariance functions of the form:
$$C_(h) = \sigma^2 (A - h^\phi)$$ ,
where \(A\) is a suitable constant as given in
Chiles & Delfiner (pag. 511, eq. 7.35).
The linear model corresponds a particular case with
\(\phi = 1\).
powered.exponential (or stable)
$$\rho(h) = \exp[-(\frac{h}{\phi})^\kappa] \mbox{ , } 0 < \kappa
\leq 2$$
gneiting
$$C(h)=\left(1 + 8 sh + 25 (sh)^2 + 32
(sh)^3\right)(1-sh)^8 1_{[0,1]}(sh)$$
where
\(s=0.301187465825\).
For further details see documentation of the function
CovarianceFct
in the package
RandomFields from where we extract the following :
It is an alternative to the gaussian model since
its graph is visually hardly distinguishable from the graph of
the Gaussian model, but possesses neither the mathematical and nor the
numerical disadvantages of the Gaussian model.
gneiting.matern
Let \(\alpha=\phi\kappa_2\), \(\rho_m(\cdot)\) denotes the \(\mbox{Mat\'{e}rn}\) model
and \(\rho_g(\cdot)\) the Gneiting model. Then the
\(\mbox{Gneiting-Mat\'{e}rn}\) is given by
$$\rho(h) = \rho_g(h|\phi=\alpha) \,
\rho_m(h|\phi=\phi,\kappa=\kappa_1)$$
wave
$$\rho(h) = \frac{\phi}{h}\sin(\frac{h}{\phi})$$
pure.nugget
$$\rho(h) = k$$
where k is a constant value. This model corresponds to
no spatial correlation.
Nested models
Models with several structures
usually called nested models
in the geostatistical literature are also allowed.
In this case the argument cov.pars takes a matrix and
cov.model and lambda can either have length equal to
the number of rows of this matrix or length 1.
For the latter cov.model and/or lambda are recycled, i.e. the same
value is used for all structures.