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):=1/2N(r) ∑_{(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
1 2 
Arguments
x 
matrix of coordinates, or vector of x coordinates, or object
of class 
y 
optional vector of y coordinates, ignored if 
z 
optional vector of z coordinates, ignored if 
T 
optional vector of time coordinates,

grid 
logical; determines whether the vectors 
data 
matrix, data.frame or object of class

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 
distances 
object of class 
vdim 
the number of variables of a multivariate data set.
If not given and NOTE: still the argument 
... 
further options and control arguments for the simulation
that are passed to and processed by 
Details
RFempiricalvariogram
computes the empirical
crossvariogram 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
).
Value
RFempiricalvariogram
returns objects of class
RFempVariog
.
Author(s)
Sebastian Engelke, sebastian.engelke@unil.ch
Johannes Martini, jmartin2@unigoettingen.de
Martin Schlather, schlather@math.unimannheim.de http://ms.math.unimannheim.de/de/publications/software
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: SpringerVerlag
See Also
RMstable
,
RMmodel
,
RFsimulate
,
RFfit
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46  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, 0.02)
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, 0.05)
z < RFsimulate(model, x=x, y=x, n=n)
emp.vario < RFempiricalvariogram(data=z, phi=4)
plot(emp.vario, model=model)
## spacetime model
model < RMnsst(phi=RMexp(), psi=RMfbm(alpha=1), delta=2)
x < seq(0, 10, 0.05)
T < c(0, 0.1, 100)
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, 0.1)
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, 0.1)
data < RFsimulate(model, x, x, n=n)
ev < RFempiricalvariogram(data=data, phi=4)
plot(ev, model=model, boundaries=FALSE)

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