RandomFields (version 3.0.5)

RFcov: Evaluate Covariance and Variogram Functions

Description

RFcov returns the values of a covariance function; RFvariogram returns the values of a variogram; RFpseudovariogram returns the values of a pseudovariogram; RFcovmatrix returns the covariance matrix for a set of points RFfctn returns the values of a shape function;

Usage

RFcov(model, x, y = NULL, z = NULL, T = NULL, grid, distances, dim,       ...)
RFvariogram(model, x, y = NULL, z = NULL, T = NULL, grid, distances, dim, ...)
RFpseudovariogram(model, x, y = NULL, z = NULL, T = NULL, grid, distances, dim, ...) 
RFcovmatrix(model, x, y = NULL, z = NULL, T = NULL, grid, distances, dim,...)
RFfctn(model, x, y = NULL, z = NULL, T = NULL, grid, distances, dim,...)

Arguments

model

object of class RMmodel; the covariance or variogram model, which is to be evaluated

x

vector or $(n \times \code{dim})$-matrix, where $n$ is the number of points at which the covariance function is to be evaluated; in particular, if the model is isotropic or dim=1 then x is a vector. x

y

second vector or matrix for non-stationary covariance functions

z

z-component of point if xyzT-specification of points is used

T

T-component of point if xyzT-specification of points is used

grid

boolean; whether xyzT specify a grid

distances

vector; only if the function RFcovmatrix is used; the lower triangular part of the distance matrix column-wise; equivalently the upper triangular part of the distance matrix row-wise; either

dim

dimension of the coordinate space in which the model is applied

...

arguments passed to RFcov (RFcovmatrix) and arguments passed to RFoptions

Value

• RFcov returns a vector of values of the covariance function. RFvariogram returns a vector of values of the variogram model.

  RFpseudovariogram returns a vector of values of the variogram model.

  RFcovmatrix returns a covariance matrix.

Details

RFcovmatrix returns a covariance matrix. Here a matrix of of coordinates (x) or a vector or a matrix of distances is expected. RFcovmatrix allows also for variogram models. Then the negative of the variogram matrix is returned.

See Also

RMmodel, RFsimulate, RFfit.

Examples

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again

# locations:
x <- matrix(runif(15), ncol=3)
 # coordinate matrix of 5 arbitrary points
 # p1, p2, p3, p4, p5 in 3-dimensional space
y <- matrix(runif(15),ncol=3)
 # coordinate matrix of 5 arbitrary points
 # q1, q2, q3, q4, q5 in 3-dimensional space


#####################################################
# get available models
RFgetModelNames(type=c("tail correlation", "positive definite",
               "negative definite", "shape function", "trend"))

#####################################################
# Example 1: a stationary covariance model

model <- RMexp()

# covariance only depends on differences
# of locations hi=pi-qi in 3-dimensional space
# therefore, the following 2 commands yield the same

RFcov(model=model, x=x, y=y)
RFcov(model=model, x=x-y)
 # yields 5 values C(hi)=C(pi,qi) for i=1,2,3,4,5



##################################################
# Example 2: get covariance matrix C(x_i,x_j)    
# at given locations x_i, i=1,...,n 
#
# here for an isotropic stationary covariance model
# yields a 4 times 4 covariance matrix of the form
# C(0)   C(5)   C(3)   C(2.5)
# C(5)   C(0)   C(4)   C(2.5)
# C(3)   C(4)   C(0)   C(2.5)
# C(2.5) C(2.5) C(2.5) C(0)

model <- RMexp() # the covariance function C(x,y)=C(r) of this model
#                  depends only on the distance r between x and y
RFcovmatrix(model=model, distances=c(5,3,2.5,4,2.5,2.5), dim=4)



######################################################
# Example 3: distinguish the different uses of x and y

x <- c(1,2,1)
y <- c(4,5,6)

# coordinate space 1-dimensional, evaluated at 3 points:
RFcov(model=model, x=as.matrix(x), y=as.matrix(y))
# coordinate space is 3-dimensional, evaluated at a pair of points
RFcov(model=model, x=t(x), y=t(y))



FinalizeExample()