\name{EmpiricalVariogram}
\alias{EmpiricalVariogram}
\title{Empirical (Semi-)Variogram}
\description{
\code{EmpiricalVariogram} calculates the empirical (semi-)variogram
of a random field realisation
}
\usage{
EmpiricalVariogram(x, y=NULL, z=NULL, T=NULL, data, grid, bin,
gridtriple=FALSE, phi, theta, deltaT)}
\arguments{
\item{x}{vector of x-coordinates, or matrix}
\item{y}{vector of y-coordinates}
\item{z}{vector of z-coordinates}
\item{T}{vector of time components; here T is given in grid
format, see \code{\link{GaussRF}}.}
\item{data}{vector or matrix of data; if \code{data} has a multiple
number of components as expected by the definition of the
coordinates then it is assumed that the data stem from repeated,
independent measurements at the given locations; the empirical
variogram is calculated for the repeated data.}
\item{grid}{logical; if \code{TRUE} then
\code{x}, \code{y}, and \code{z} define a grid; otherwise
\code{x}, \code{y}, and \code{z} are interpreted as points}
\item{bin}{vector of ascending values giving the bin boundaries}
\item{gridtriple}{logical. Only relevant if \code{grid==TRUE}.
If \code{gridtriple==TRUE}
then \code{x}, \code{y}, and \code{z} are of the
form \code{c(start,end,step)}; if
\code{gridtriple==FALSE} then \code{x}, \code{y}, and \code{z}
must be vectors of ascending values}
\item{phi}{vector of two components. First component gives the angle
for the first line of midpoints of an angular variogram.
The second component gives the number of directions (on the half circle).
The spatial dimension must be at least 2.
}
\item{theta}{vector of two components. First component gives the angle
for the first line of midpoints of an angular variogram
(angle is zero for the xy-plane).
The second component gives the number of directions (on the half circle).
The spatial dimension must be at least 3.
}
\item{deltaT}{vector of two components.
First component gives the largest temporal distance;
the second component the grid length, that must be a multiple of
\code{T[3]}.
}
}
\value{
The function returns a list:
\item{centers}{central points of the bins}
\item{emp.vario}{empirical variogram; vector or matrix or array,
depending on the anisotropy definitions. The sequence is distances,
phi, theta, Tbins. If phi, theta, or Tbins below are not given,
the respective dimensions are missing.}
\item{sd}{sd of the variogram cloud within each bin}
\item{n.bin}{number of points within a bin}
\item{phi}{vector of angles in xy plane}
\item{theta}{vector of angles in the third dimensions}
\item{Tbins}{vector of temporal distances}
The first four elements are vectors of length \code{(length(bin)-1)}.
}
\details{
% A version of the algorithm is custom-tailored to large grids with
% constant grid length in each direction, and invoked in such cases.
Comments on specific parameters:
\itemize{
\item \code{data}: the number of values must match the number of
points (given by \code{x}, \code{y}, \code{z}, \code{grid}, and
\code{gridtriple}). That is, it must equal the number of points or be
a multiple of it. In case the number of data equals \eqn{n}{n} times the
number of points, the data are interpreted as \eqn{n}{n} independent
realisations for the given set of points.
\item \code{(grid==FALSE)}: the vectors \code{x}, \code{y}, and
\code{z}, are interpreted as
vectors of coordinates
\item \code{(grid==TRUE) && (gridtriple==FALSE)}: the vectors \
code{x}, \code{y}, and
\code{z}
are increasing sequences with identical lags for each sequence.
A corresponding
grid is created (as given by \code{expand.grid}).
\item \code{(grid==TRUE) && (gridtriple==TRUE)}: the vectors
\code{x}, \code{y}, and \code{z}
are triples of the form (start,end,step) defining a grid (as given by
\code{expand.grid(seq(x$start,x$end,x$step),
seq(y$start,y$end,y$step),
seq(z$start,z$end,z$step))})
\item
The bins are left open, right closed intervals, i.e.,
\eqn{(b_i,b_{i+1}]}{(\code{b[i]},\code{bin[i+1]}]} for
\eqn{i=1,\ldots,}{i=1,...,}\code{length(bin)}\eqn{-1}{-1}.
Hence, to include zero, \code{bin[1]} must be negative.
}
}
\author{Martin Schlather, \email{schlath@hsu-hh.de}
\url{http://www.unibw-hamburg.de/WWEB/math/schlath/schlather.html}
}
\seealso{\code{\link{GaussRF}}, \code{\link{fitvario},
and \code{\link{RandomFields}}}}
\examples{
% library(RandomFields, lib="~/TMP"); RFparameters(Print=6)
#############################################################
## this example checks whether a certain simulation method ##
## works well for a specified covariance model and ##
## a configuration of points ##
#############################################################
x <- seq(0, 10, 0.5)
y <- seq(0, 10, 0.5)
gridtriple <- FALSE ## see help("GaussRF")
model <- "whittle" ## whittlematern
bins <- seq(0, 5, 0.001)
repetition <- 5 ## by far too small to get reliable results!!
## It should be of order 500,
## but then it will take some time
## to do the simulations
param <- c(mean=1, variance=10, nugget=5, scale=2, alpha=2)
f <- GaussRF(x=x, y=y, grid=TRUE, gridtriple=gridtriple,
model=model, param=param, method="TBM3",
n=repetition)
binned <- EmpiricalVariogram(x=x, y=y, data=f, grid=TRUE,
gridtriple=gridtriple, bin=bins)
truevariogram <- Variogram(binned$c, model, param)
matplot(binned$c, cbind(truevariogram,binned$e), pch=c("*","e"))
##black curve gives the theoretical values
% #############################################################
% ## this example shows under which conditions the algorithm ##
% ## still works reasonably fast (about 15 sec. at 500 MHz), ##
% ## using all information available (and not only a random ##
% ## subset of the points) ##
% #############################################################
% grid <- TRUE;
% y <- x <- seq(0,50, 0.1) ## large grid (of size 500 x 500)
% bins <- seq(0, 2,0.001) ## 2 << 50 -- this is the key point
%
% model <- "sph" ## spherical
% mean <- 1
% variance <- 10
% nugget <- 5
% scale <- 1.5
% param <- c(mean, variance, nugget, scale)
% f <- GaussRF(x=x, y=y, grid=grid, model=model, param=param)
% binned <-
% EmpiricalVariogram(x=x, y=y, data=f, grid=grid, bin=bins)
% truevariogram <- Variogram(binned$c, model, param)
% matplot(midbin, cbind(truevariogram,binned$e), pch=c("*","e"))
%
}
\keyword{spatial}
