(Cross-)Covariance function
Calculates both the empirical and the theoretical (cross-)covariance function.
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 |
object of class |
x |
vector of x coordinates, or object of class |
y,z |
optional vectors of y (z) coordinates, which should not be given if |
T |
optional vector of time coordinates, |
grid |
logical; the function finds itself the correct value in nearly all cases, so that usually |
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-) covariance function in every direction is returned. |
phi |
an integer defining the number of sectors one half of the X/Y plane shall be divided 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 divided 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,dim |
another alternative for the argument |
vdim |
the number of variables of a multivariate data set. If not given and |
... |
for advanced use: further options and control arguments for the simulation that are passed to and processed by |
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.)
RFcov returns objects of class
RFempVariog.
Jonas Auel; Sebastian Engelke; Johannes Martini; Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
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
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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.