RandomFields (version 3.0.5)

RMbiwm: Full Bivariate Whittle Matern Model

Description

RMbiwm is a bivariate stationary isotropic covariance model whose corresponding covariance function only depends on the distance $r \ge 0$ between two points and is given for $i,j \in {1,2}$ by $$C_{ij}(r)=c_{ij} W_{\nu_{ij}}(r/s_{ij}).$$ Here $W_\nu$ is the covariance of the RMwhittle model. For constraints on the constants see details.

Usage

RMbiwm(nudiag, nured12, nu, s, cdiag, rhored, c, notinvnu, var,
 scale, Aniso, proj)

Arguments

nudiag

a vector of length 2 of numerical values; each entry positive; the vector $(\nu_{11},\nu_{22})$

nured12

a numerical value in the interval $[1,\infty)$; $\nu_{21}$ is calculated as $0.5 (\nu_{11} + \nu_{22})*\nu_{red}$.

nu

alternative to nudiag and nured12: a vector of length 3 of numerical values; each entry positive; the vector $(\nu_{11},\nu_{21},\nu_{22})$. Either nured and nudiag or nu must be given.

s

a vector of length 2 of numerical values; each entry positive; the vector $(s_{11},s_{22})$

cdiag

a vector of length 2 of numerical values; each entry positive; the vector $(c_{11},c_{22})$

rhored

a numerical value; in the interval $[-1,1]$. See also the Details for the corresponding value of $c_{12}=c_{21}$.

c

a vector of length 3 of numerical values; the vector $(c_{11},c_{21}, c_{22})$. Either rhored and cdiag or c must be given.

notinvnu

logical or NULL. If not given (default) then the formula of the (RMwhittle) model applies. If logical then the formula for the RMmatern

var,scale,Aniso,proj

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

Value

• RMbiwm returns an object of class RMmodel.

Details

Constraints on the constants: For the diagonal elements we have $$\nu_{ii}, s_{ii}, c_{ii} > 0.$$ For the offdiagonal elements we have $$s_{12}=s_{21} > 0,$$ $$\nu_{12} =\nu_{21} = 0.5 (\nu_{11} + \nu{22}) / \nu_{red}$$ for some constant $\nu_{red} \in (0,1]$ and $$c_{12} =c_{21} = \rho_{red} \sqrt{f m c_{11} c_{22}}$$ for some constant $\rho_{red}$ in $[-1,1]$. The constants $f$ and $m$ in the last equation are given as follows: $$f = (\Gamma(\nu_{11} + d/2) \Gamma(\nu_{22} + d/2)) / (\Gamma(\nu_{11}) \Gamma(\nu_{22})) * (\Gamma(\nu_{12}) / \Gamma(\nu_{12}+d/2))^2 * ( s_{12}^{2*\nu_{12}} / (s_{11}^{\nu_{11}} s_{22}^{\nu_{22}}) )^2$$ where $\Gamma$ is the Gamma function and $d$ is the dimension of the space. The constant $m$ is the infimum of the function $g$ on $[0,\infty)$ where $$g(t) = (1/s_{12}^2 +t^2)^{2\nu_{12} + d} (1/s_{11}^2 + t^2)^{-\nu_{11}-d/2} (1/s_{22}^2 + t^2)^{-\nu_{22}-d/2}$$ (cf. Gneiting, T., Kleiber, W., Schlather, M. (2010), Full Bivariate Matern Model (Section 2.2))

References

• Gneiting, T., Kleiber, W., Schlather, M. (2010) Matern covariance functions for multivariate random fieldsJASA

See Also

RMparswm, RMwhittle, RMmodel, RFsimulate, RFfit.

Examples

set.seed(0)
x <- y <- seq(-10, 10, if (interactive()) 0.2 else 5)
model <- RMbiwm(nudiag=c(1, 2), nured=1, rhored=1, cdiag=c(1, 5), 
                s=c(1, 1, 2))
simu <- RFsimulate(model, x, y, grid=TRUE)
plot(simu)