cov.spatial | R Documentation |
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.
cov.spatial(obj, cov.model= "matern", cov.pars=stop("no cov.pars argument provided"), kappa = 0.5)
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 |
cov.pars |
a vector with 2 elements or an ns x 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: |
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_κ(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+(h/phi)^2]^(-kappa)
gencauchy (generalised Cauchy)
rho(h) = [1+(h/phi)^kappa2]^(-kappa1/kappa2), kappa1 > 0, 0 < kappa2 <=0
circular
Let theta = min(h/phi,1) and
gamma(h)= 2 * ((θ * √{1-θ^2} + \sin^{-1} √{theta}))/pi.
Then, the circular model is given by:
rho(h) = 1 - gamma(h) if h < phi , 0 otherwise
cubic
rho(h) = 1 - (7 * ((h/phi)^2) - 8.75 * ((h/phi)^3) + 3.5 * ((h/phi)^5) - 0.75 * ((h/phi)^7)) if h < phi , 0 otherwise.
gaussian
rho(h) = exp(-(h/phi)^2)
exponential
rho(h) = exp(-h/phi)
matern
ρ(h) = (1/(2^(κ-1) * Γ(κ))) * ((h/φ)^κ) * K_{κ}(h/φ)
spherical
rho(h) = 1 - 1.5 * (h/phi) + 0.5(h/phi)^3 if h < phi , 0 otherwise
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:
sigma^2 * h^phi , 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[-(h/phi)^kappa] if 0 < kappa <= 2
gneiting
C(x)= (1 + 8 s h + 25 s^2 h^2 + 32 s^3 h^3)*(1-s h)^8 if 0<=s h<=1, 0 otherwise
where
.
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(.) denotes the Matern model
and rho_g(.) the Gneiting model. Then the
Gneiting-Matern is given by
rho(h) = rho_g(h | phi=alpha) * rho_m(h | phi=phi, kappa = kappa_1)
wave
(phi/h) * sin(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.
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.
Paulo J. Ribeiro Jr. paulojus@leg.ufpr.br,
Peter J. Diggle p.diggle@lancaster.ac.uk.
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.
Chilès, 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/.
matern
for computation of the
Matern model, besselK
for
computation of the Bessel function and
varcov.spatial
for computations related to the covariance matrix.
# # 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')
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