RMbiwm: Full Bivariate Whittle Matern Model
In RandomFields: Simulation and Analysis of Random Fields

Description Usage Arguments Details Value References See Also Examples

RMbiwm is a bivariate stationary isotropic covariance model whose corresponding covariance function only depends on the distance r ≥ 0 between two points and is given for i,j = 1,2 by

C_{ij}(r)=c_{ij} W_{ν_{ij}}(r/s_{ij}).

Here W_ν is the covariance of the RMwhittle model. For constraints on the constants see Details.

1 2	RMbiwm(nudiag, nured12, nu, s, cdiag, rhored, c, notinvnu, var, scale, Aniso, proj)

`nudiag`	a vector of length 2 of numerical values; each entry positive; the vector (ν_{11},ν_{22})
`nured12`	a numerical value in the interval [1,∞); ν_{21} is calculated as 0.5 (ν_{11} + ν_{22})ν_{red}*.
`nu`	alternative to `nudiag` and `nured12`: a vector of length 3 of numerical values; each entry positive; the vector (ν_{11},ν_{21},ν_{22}). Either `nured` and `nudiag`, or `nu` must be given.
`s`	a vector of length 3 of numerical values; each entry positive; the vector (s_{11},s_{21},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` model applies. See there for details.
`var,scale,Aniso,proj`	optional arguments; same meaning for any `RMmodel`. If not passed, the above covariance function remains unmodified.

Constraints on the constants: For the diagonal elements we have

ν_{ii}, s_{ii}, c_{ii} > 0.

For the offdiagonal elements we have

s_{12}=s_{21} > 0,

ν_{12} =ν_{21} = 0.5 (ν_{11} + ν_{22}) * ν_{red}

for some constant ν_{red} \in [1,∞) and

c_{12} =c_{21} = ρ_{red} √{f m c_{11} c_{22}}

for some constant ρ_{red} in [-1,1].

The constants f and m in the last equation are given as follows:

f = (Γ(ν_{11} + d/2) Γ(ν_{22} + d/2)) / (Γ(ν_{11}) Γ(ν_{22})) * (Γ(ν_{12}) / Γ(ν_{12}+d/2))^2 * ( s_{12}^{2*ν_{12}} / (s_{11}^{ν_{11}} s_{22}^{ν_{22}}) )^2

where Γ is the Gamma function and d is the dimension of the space. The constant m is the infimum of the function g on [0,∞) where

g(t) = (1/s_{12}^2 +t^2)^{2ν_{12} + d} (1/s_{11}^2 + t^2)^{-ν_{11}-d/2} (1/s_{22}^2 + t^2)^{-ν_{22}-d/2}

(cf. Gneiting, T., Kleiber, W., Schlather, M. (2010), Full Bivariate Matern Model (Section 2.2)).

RMbiwm returns an object of class RMmodel.

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

RMparswm, RMwhittle, RMmodel, RFsimulate, RFfit, Multivariate RMmodels.

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again

x <- y <- seq(-10, 10, 0.2)
model <- RMbiwm(nudiag=c(0.3, 2), nured=1, rhored=1, cdiag=c(1, 1.5), 
                s=c(1, 1, 2))
plot(model)
plot(RFsimulate(model, x, y))