RFcov: (Cross-)Covariance function
In RandomFields: Simulation and Analysis of Random Fields

Description Usage Arguments Details Value Author(s) References See Also Examples

Calculates both the empirical and the theoretical (cross-)covariance function.

1 2	RFcov(model, x, y = NULL, z = NULL, T=NULL, grid, params, distances, dim, ..., data, bin=NULL, phi=NULL, theta = NULL, deltaT = NULL, vdim=NULL)

`model,params`	\argModel
`x`	\argX
`y,z`	\argYz
`T`	\argT
`grid`	\argGrid
`data`	\argData
`bin`	\argBin
`phi`	\argPhi
`theta`	\argTheta
`deltaT`	\argDeltaT
`distances,dim`	\argDistances
`vdim`	\argVdim
`...`	\argDots

RFcov computes the empirical cross-covariance function for given (multivariate) spatial data.

The empirical (cross-)covariance function of two random fields X and Y is given by

γ(r):=1/N(r) ∑_{(t_{i},t_{j})|t_{i,j}=r} (X(t_{i})Y(t_{j})) - m_{X} m_{Y}

where t_{i,j}:=t_{i}-t_{j}, N(r) denotes the number of pairs of data points with distancevector t_{i,j}=r and where m_{X} := \frac{1}{N(r)} ∑_{(t_{i},t_{j})|t_{i,j}=r} X_{t_{i}} and m_{Y} := 1/N(r) ∑_{(t_{i},t_{j})|t_{i,j}=r} Y_{t_{i}} denotes the mean of data points with distancevector t_{i,j}=r.

The spatial coordinates x, y, z should be vectors. For random fields of spatial dimension d > 3 write all vectors as columns of matrix x. In this case do neither use y, nor z and write the columns 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 RFcov to calculate the pseudocovariance function (see RFoptions).

RFcov returns objects of class RFempVariog.

Jonas Auel; Sebastian Engelke; Johannes Martini; \martin

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

RFvariogram, RFmadogram, RMstable, RMmodel, RFsimulate, RFfit.

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 <- RFcov(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 <- RFcov(data=z, phi=4)
plot(emp.vario, model=model)


## space-time 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 <- RFcov(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 <- RFcov(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)
dta <- RFsimulate(model, x, x, n=n)
ev <- RFcov(data=dta, phi=4)
plot(ev, model=model, boundaries=FALSE)