RFempiricalvariogram: Empirical (Cross-)Variogram

Description

Calculates the empirical (cross-)variogram. The empirical (cross-)variogram of two random fields $X$ and $Y$ is given by

γ (r) := \frac{1}{2 N (r)} \sum_{(t_{i}, t_{j}) | t_{i, j} = r} (X (t_{i}) - X (t_{j})) (Y (t_{i}) - Y (t_{j}))

where $t_{i,j}:=t_{i}-t_{j}$, and where $N(r)$ denotes the number of pairs of data points with distancevector $t_{i,j}=r$.

Usage

RFempiricalvariogram(x, y = NULL, z = NULL, T = NULL, data, grid,
 bin, phi, theta, deltaT, distances, vdim, ...)

Arguments

matrix of coordinates, or vector of x coordinates, or object of class GridTopology or raster. If matrix, ncol(x)<

optional vector of y coordinates, ignored if x is a matrix

optional vector of z coordinates, ignored if x is a matrix

optional vector of time coordinates, T must always be an equidistant vector or given in a gridtriple format (see grid); for each component of T, the random field is simulated at all location points; the argument

grid

logical; determines whether the vectors x, y, and z or the columns of x should be interpreted as a grid definition (see Details). If grid=TRUE, either x, y, and

data

matrix, data.frame or object of class RFsp;

bin

a vector giving the borders of the bins; If not specified an array describing the empirical (pseudo-)(cross-) variogram in every direction is returned.

phi

an integer defining the number of sectors one half of the X/Y plane shall be devided into. If not specified, either an array is returned (if bin missing) or isotropy is assumed (if bin specified)

theta

an integer defining the number of sectors one half of the X/Z plane shall be devided into. Use only for dimension $d=3$ if phi is already specified

deltaT

vector of length 2, specifying the temporal bins. The internal bin vector becomes seq(from=0, to=deltaT[1], by=deltaT[2])

distances

object of class dist representing the upper trianguar part of the matrix of Euclidean distances between the points at which the field is to be simulated; only applicable for stationary and isotropic mo

vdim

the number of variables of a multivariate data set. If not given and data is an RFsp object created by RandomFields, the information there is taken from there. Otherwise vdim is assumed to be o

...

further options and control arguments for the simulation that are passed to and processed by RFoptions.

Value

RFempiricalvariogram returns objects of class RFempVariog.

Details

RFempiricalvariogram computes the empirical cross-variogram for given (multivariate) spatial data.

The spatial coordinates x, y, z should be vectors. For random fields of spatial dimension $d > 3$ write all vectors as colums of matrix x. In this case do neither use y, nor z and write the colums in gridtriple notation.

If the data is spatially located on a grid a fast algorithm based on the fast Fourier transformed (fft) will be used. As advanced option the calculation method can also be changed for grid data (see RFoptions.) It is also possible to use RFempiricalvariogram to calulate the pseudovariogram (see RFoptions).

References

Gelfand, A. E., Diggle, P., Fuentes, M. and Guttorp, P. (eds.) (2010) Handbook of Spatial Statistics. Boca Raton: Chapman & Hall/CRL.

Stein, M. L. (1999) Interpolation of Spatial Data. New York: Springer-Verlag

Examples

Run this code

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

n <- 1 ## use n <- 2 for better results

## isotropic model
model <- RMexp()
x <- seq(0, 10, if (interactive()) 0.02 else 1) 
z <- RFsimulate(model, x=x, n=n)
emp.vario <- RFempiricalvariogram(data=z)
plot(emp.vario, model=model)


## anisotropic model
model <- RMexp(Aniso=cbind(c(2,1), c(1,1)))
x <- seq(0, 10, if (interactive()) 0.05 else 1)
z <- RFsimulate(model, x=x, y=x, n=n)
emp.vario <- RFempiricalvariogram(data=z, phi=4)
plot(emp.vario, model=model)


## space-time model
model <- RMexp()
x <- seq(0, 10, if (interactive()) 0.05 else 1)
T <- c(0, if (interactive()) 0.1 else 1, if (interactive()) 100 else 10)
z <- RFsimulate(x=x, T=T, model=model, n=n)
emp.vario <- RFempiricalvariogram(data=z, deltaT=c(10, 1))
plot(emp.vario, model=model, nmax.T=3)


## multivariate model
model <- RMbiwm(nudiag=c(1, 2), nured=1, rhored=1, cdiag=c(1, 5), 
                s=c(1, 1, 2))
x <- seq(0, 20, if (interactive()) 0.1 else 2)
z <- RFsimulate(model, x=x, y=x, n=n)
emp.vario <- RFempiricalvariogram(data=z)
plot(emp.vario, model=model)


## multivariate and anisotropic model
model <- RMbiwm(A=matrix(c(1,1,1,2), nc=2),
                nudiag=c(0.5,2), s=c(3, 1, 2), c=c(1, 0, 1))
x <- seq(0, 20, if (interactive()) 0.1 else 2)
data <- RFsimulate(model, x, x, n=n)
ev <- RFempiricalvariogram(data=data, phi=4)
plot(ev, model=model, boundaries=FALSE)


FinalizeExample()

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