RandomFields (version 2.0.71)

EmpiricalVariogram: Empirical (Semi-)Variogram

Description

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

x

vector of x-coordinates, or matrix

y

vector of y-coordinates

z

vector of z-coordinates

T

vector of time components; here T is given in grid format, see GaussRF.

data

vector or matrix of data; if 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 empir

grid

logical; if TRUE then x, y, and z define a grid; otherwise x, y, and z are interpreted as points

bin

vector of ascending values giving the bin boundaries

gridtriple

logical. Only relevant if grid=TRUE. If gridtriple=TRUE then x, y, and z are of the form c(start,end,step); if gridtriple=FALSE then x,

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.

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 dime

deltaT

vector of two components. First component gives the largest temporal distance; the second component the grid length, that must be a multiple of T[3].

Value

• The function returns a list:
• centers
  central points of the bins
• 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.
• sd
  sd of the variogram cloud within each bin
• n.bin
  number of points within a bin
• phi
  vector of angles in xy plane
• theta
  vector of angles in the third dimensions
• Tbins
  vector of temporal distances
• The first four elements are vectors of length (length(bin)-1).

Details

Comments on specific parameters:

• data: the number of values must match the number of points (given byx,y,z,grid, andgridtriple). That is, it must equal the number of points or be a multiple of it. In case the number of data equals$n$times the number of points, the data are interpreted as$n$independent realisations for the given set of points.
• (grid=FALSE): the vectorsx,y, andz, are interpreted as vectors of coordinates
• (grid=TRUE) && (gridtriple=FALSE): the vectors \ code{x},y, andzare increasing sequences with identical lags for each sequence. A corresponding grid is created (as given byexpand.grid).
• (grid=TRUE) && (gridtriple=TRUE): the vectorsx,y, andzare triples of the form (start,end,step) defining a grid (as given byexpand.grid(seq(x$start,x$end,x$step), seq(y$start,y$end,y$step), seq(z$start,z$end,z$step)))
• The bins are left open, right closed intervals, i.e.,$(b_i,b_{i+1}]$for$i=1,\ldots,$length(bin)$-1$. Hence, to include zero,bin[1]must be negative.

See Also

GaussRF, fitvario, and RandomFields

Examples

#############################################################
  ## 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)
  realisations <- 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=realisations)
  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