Full Bivariate Whittle Matern Model
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.
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 |
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
|
notinvnu |
logical or |
var,scale,Aniso,proj |
optional arguments; same meaning for any
|
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)).
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
Gneiting, T., Kleiber, W., Schlather, M. (2010) Matern covariance functions for multivariate random fields JASA
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))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.