RandomFields (version 3.0.10)

RMmastein: Ma-Stein operator

Description

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

$$C(h,t)=\frac{\mathrm{\Gamma }(\nu +\varphi (t))\mathrm{\Gamma }(\nu +\delta )}{\mathrm{\Gamma }(\nu +\varphi (t)+\delta )\mathrm{\Gamma }(\nu )}{W}_{\nu +\varphi (t)}(\Vert h-Vt\Vert )$$ if $\phi$ is a variogram model. It is given by $$C(h,t)=\frac{\mathrm{\Gamma }(\nu +\varphi (0)-\varphi (t))\mathrm{\Gamma }(\nu +\delta )}{\mathrm{\Gamma }(\nu +\varphi (0)-\varphi (t)+\delta )\mathrm{\Gamma }(\nu )}{W}_{\nu +\varphi (t)}(\Vert h-Vt\Vert )$$

if $\phi$ is a covariance model.

Here $\Gamma$ is the Gamma function; $W$ is the Whittle-Matern model (RMwhittle).

Usage

RMmastein(phi, nu, delta, var, scale, Aniso, proj)

Arguments

phi

an RMmodel on the real axis

nu

numerical value; positive; smoothness parameter of the Whittle-Matern model (for $t=0$)

delta

a numerical value; $\delta$ must be greater than or equal to half the dimension of $h$

var,scale,Aniso,proj

optional parameters; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.

Value

• RMmastein returns an object of class RMmodel

Details

See Stein (2005) formula (12). Instead of the velocity parameter $V$ in the original model description, a preceeding anisotropy matrix is chosen appropriately: $$\left(\begin{array}{ccc}A& -V0& 1\end{array}\right)$$ A is a spatial transformation matrix. (I.e. (x,t) is multiplied from left on the above matrix and the first elements of the obtained vector are intepreted 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, $\nu+\phi+d$ may not exceed the value 80.0. If exceeded the algorithm fails.

References

• 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.

See Also

RMwhittle, RMmodel, RFsimulate, RFfit.

Examples

RFoptions(seed=0)
model <- RMmastein(RMgauss(), nu=1, delta=10)
x <- seq(0, 10, if (interactive()) 0.1 else 3) 
plot(RFsimulate(model, x=x, y=x))
RFoptions(seed=NA)