RandomFields (version 3.0.5)

RMvector: Vector Covariance Model

Description

RMvector is a multivariate covariance model which depends on a univariate covariance model that is stationary in the first $Dspace$ coordinates $h$ and where the covariance function phi(h,t) is twice differentiable in the first component $h$. The corresponding matrix-valued covariance function C of the model only depends on the difference $h$ between two points in the first component. It is given by $$C(h,t)=( -0.5 * (a + 1) \Delta + a \nabla \nabla^T ) C_0(h, t)$$ where the operator is applied to the first component $h$ only.

Usage

RMvector(phi, a, Dspace, var, scale, Aniso, proj)

Arguments

phi

an RMmodel; has two components $h$ (2 or 3 dimensional and stationary) and $t$ (arbitrary dimension)

a

a numerical value; should be in the interval $[-1,1]$.

Dspace

an integer; either 2 or 3; the first $Dspace$ coordinates give the first component $h$

var,scale,Aniso,proj

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

Value

• RMvector returns an object of class RMmodel

Details

$C_0$ is either a spatiotemporal model (then $t$ is the time component) or it is an isotropic model. Then, the first $Dspace$ coordinates are considered as $h$ coordinates and the remaining ones as $t$ coordinates. By default, $Dspace$ equals the dimension of the field (and there is no $t$ component). If $a=-1$ then the field is curl free; if $a=1$ then the field is divergence free.

References

• Scheuerer, M. and Schlather, M. (2012) Covariance Models for Divergence-Free and Curl-Free Random Vector Fields.Stochastic Models28:3.

See Also

RMcurlfree, RMdivfree, RMmodel, RFsimulate, RFfit.

Examples

set.seed(0)model <- RMvector(RMgauss(), scale=0.3)
x <- seq(0, 10, if (interactive()) 0.4 else 3) 
plot(RFsimulate(model, x=x, y=x, z=0), select.variables=list(1:2))