geoR (version 1.8-1)

cov.spatial: Computes Value of the Covariance Function

Description

Computes the covariances for pairs variables, given the separation distance of their locations. Options for different correlation functions are available. The results can be seen as a change of metric, from the Euclidean distances to covariances.

Usage

cov.spatial(obj, cov.model= "matern",
            cov.pars=stop("no cov.pars argument provided"),
            kappa = 0.5)

Arguments

obj

a numeric object (vector or matrix), typically with values of distances between pairs of spatial locations.

cov.model

string indicating the type of the correlation function. Available choices are: "matern", "exponential", "gaussian", "spherical", "circular", "cubic", "wave", "power", "powered.exponential", "cauchy", "gencauchy", "gneiting", "gneiting.matern", "pure.nugget". See section DETAILS for available options and expressions of the correlation functions.

cov.pars

a vector with 2 elements or an \(ns \times 2\) matrix with the covariance parameters. The first element (if a vector) or first column (if a matrix) corresponds to the variance parameter \(\sigma^2\). The second element or column corresponds to the range parameter \(\phi\) of the correlation function. If a matrix is provided, each row corresponds to the parameters of one spatial structure (see DETAILS below).

kappa

numerical value for the additional smoothness parameter of the correlation function. Only required by the following correlation functions: "matern", "powered.exponential", "cauchy", "gencauchy" and "gneiting.matern".

Value

The function returns values of the covariances corresponding to the given distances. The type of output is the same as the type of the object provided in the argument obj, typically a vector, matrix or array.

Details

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 Chil<e8>s \& 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.

References

For a review on correlation functions: Schlather, M. (1999) An introduction to positive definite functions and to unconditional simulation of random fields. Technical report ST 99-10, Dept. of Maths and Statistics, Lancaster University.

Chil<e8>s, J.P. and Delfiner, P. (1999) Geostatistics: Modelling Spatial Uncertainty, Wiley.

Further information on the package geoR can be found at: http://www.leg.ufpr.br/geoR.

See Also

matern for computation of the \(\mbox{Mat\'{e}rn}\) model, besselK for computation of the Bessel function and varcov.spatial for computations related to the covariance matrix.

Examples

# NOT RUN {
#
# Variogram models with the same "practical" range:
#
v.f <- function(x, ...){1-cov.spatial(x, ...)}
#
curve(v.f(x, cov.pars=c(1, .2)), from = 0, to = 1,
      xlab = "distance", ylab = expression(gamma(h)),
      main = "variograms with equivalent \"practical range\"")
curve(v.f(x, cov.pars = c(1, .6), cov.model = "sph"), 0, 1,
      add = TRUE, lty = 2)
curve(v.f(x, cov.pars = c(1, .6/sqrt(3)), cov.model = "gau"),
      0, 1, add = TRUE, lwd = 2)
legend("topleft", c("exponential", "spherical", "gaussian"),
       lty=c(1,2,1), lwd=c(1,1,2))
#
# Matern models with equivalent "practical range"
# and varying smoothness parameter
#
curve(v.f(x, cov.pars = c(1, 0.25), kappa = 0.5),from = 0, to = 1,
      xlab = "distance", ylab = expression(gamma(h)), lty = 2,
      main = "models with equivalent \"practical\" range")
curve(v.f(x, cov.pars = c(1, 0.188), kappa = 1),from = 0, to = 1,
      add = TRUE)      
curve(v.f(x, cov.pars = c(1, 0.14), kappa = 2),from = 0, to = 1,
      add = TRUE, lwd=2, lty=2)      
curve(v.f(x, cov.pars = c(1, 0.117), kappa = 2),from = 0, to = 1,
      add = TRUE, lwd=2)      
legend("bottomright",
       expression(list(kappa == 0.5, phi == 0.250), 
         list(kappa == 1, phi == 0.188), list(kappa == 2, phi == 0.140),
         list(kappa == 3, phi == 0.117)), lty=c(2,1,2,1), lwd=c(1,1,2,2))
# plotting a nested variogram model
curve(v.f(x, cov.pars = rbind(c(.4, .2), c(.6,.3)),
          cov.model = c("sph","exp")), 0, 1, ylab='nested model')
# }