R: Full Bivariate Whittle Matern Model

RMbiwm {RandomFields} | R Documentation

## Full Bivariate Whittle Matern Model

### Description

`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.

### 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 (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.

### Details

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))

### Value

`RMbiwm` returns an object of class `RMmodel`.

### Author(s)

Martin Schlather, schlather@math.uni-mannheim.de

### References

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

### See Also

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

### Examples

    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))

[Package RandomFields version 3.1.50 Index]