Ma-Stein operator
RMmastein is a univariate stationary covariance model
depending on a variogram or covariance model on the real axis.
The corresponding covariance function only depends on the difference
h between two points and is given by
C(h, t)= [ Gamma(nu + phi(t))Gamma(nu + delta) ] / [Gamma(nu + phi(t) + delta) Gamma(nu) ] W_{nu + phi(t)}(|h - Vt|)
if φ is a variogram model. It is given by
C(h, t)= [ Gamma(nu + phi(0)-phi(t))Gamma(nu + delta) ] / [Gamma(nu + phi(0)-phi(t) + delta) Gamma(nu) ] W_{nu + phi(0)-phi(t)}(|h - Vt|)
if φ is a covariance model.
Here Γ is the Gamma function; W is the Whittle-Matern model (RMwhittle).
RMmastein(phi, nu, delta, var, scale, Aniso, proj)
phi |
an |
nu |
numerical value; positive; smoothness parameter of the Whittle-Matern model (for t=0) |
delta |
a numerical value; δ must be greater than or equal to half the dimension of h |
var,scale,Aniso,proj |
optional arguments; same meaning for any
|
See Stein (2005), formula (12).
Instead of the velocity parameter V in the original model description, a preceding anisotropy matrix is chosen appropriately:
matrix(c(A, -V, 0, 1), nr=2, by=TRUE)
A is a spatial transformation matrix. (I.e. (x,t) is multiplied from the left on the above matrix and the first elements of the obtained vector are interpreted as new spatial components and only these components are used to form the argument in the Whittle-Matern function.) The last component in the new coordinates is the time which is passed to phi. (Velocity is assumed to be zero in the new coordinates.)
Note, that for numerical reasons, ν+φ+d may not exceed the value 80.0. If exceeded the algorithm fails.
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
Ma, C. (2003) Spatio-temporal covariance functions generated by mixtures. Math. Geol., 34, 965-975.
Stein, M.L. (2005) Space-time covariance functions. JASA, 100, 310-321.
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set ## RFoptions(seed=NA) to make plotthem all random again model <- RMmastein(RMgauss(), nu=1, delta=10) plot(RMexp(), model.mastein=model, dim=2) x <- seq(0, 10, 0.1) plot(RFsimulate(model, x=x, y=x))
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