Gneiting-Wendland Covariance Models
RMbigneiting is a bivariate stationary isotropic covariance
model family whose elements
are specified by seven parameters.
Let
δ_{ij} = μ + γ_{ij} + 1.
Then,
C_{n}(h) = c_{ij} (C_{n, δ} (h / s_{ij}))_{i,j=1,2}
and C_{n, δ}
is the generalized Gneiting model
with parameters n and δ, see
RMgengneiting, i.e.,
C_{κ=0, δ}(r) = (1 - r)^β 1_{[0,1]}(r), β=δ + 2κ + 1/2;
C_{κ=1, δ}(r) = (1+ β r)(1-r)^β 1_{[0,1]}(r), β = δ + 2κ + 1/2;
C(_{κ=2, δ}(r) = (1 + β r + (β^2-1) r^(2)/3)(1-r)^β 1_{[0,1]}(r), β = δ + 2κ + 1/2;
C_{κ=3, δ}(r) = (1 + β r + (2 β^2-3 )r^(2)/5+(β^2 - 4) β r^(3)/15)(1-r)^β 1_{[0,1]}(r), β=δ + 2κ + 1/2.
RMbigneiting(kappa, mu, s, sred12, gamma, cdiag, rhored, c, var, scale, Aniso, proj)
kappa |
argument that chooses between the four different covariance models and may take values 0,...,3. The model is k times differentiable. |
mu |
|
s |
vector of two elements giving the scale of the models on the diagonal, i.e. the vector (s_{11}, s_{22}). |
sred12 |
value in [-1,1]. The scale on the offdiagonals is
given by s_{12} = s_{21} =
|
gamma |
a vector of length 3 of numerical values; each entry is
positive.
The vector |
cdiag |
a vector of length 2 of numerical values; each entry positive; the vector (c_{11},c_{22}). |
c |
a vector of length 3 of numerical values; the vector (c_{11}, c_{21}, c_{22}). Note that c_{12}= c_{21}. Either
|
rhored |
value in [-1,1]. See also the Details for the corresponding value of c_{12}=c_{21}. |
var,scale,Aniso,proj |
optional arguments; same meaning for any
|
A sufficient condition for the constant c_{ij} is
c_{ij} = ρ_r m (c_{11} c_{22})^{1/2}
where ρ_r in [-1,1].
The constant m in the formula above is obtained as follows:
m = min{1, m_{-1}, m_{+1}}
Let
a = 2 γ_{12} - γ_{11} -γ_{22}
b = -2 γ_{12} (s_{11} + s_{22}) + γ_{11} (s_{12} + s_{22}) + γ_{22} (s_{12} + s_{11})
e = 2 γ_{12} s_{11}s_{22} - γ_{11}s_{12}s_{22} - γ_{22}s_{12}s_{11}
d = b^2 - 4ae
t_j =(-b + j √ d) / (2 a)
If d ≥0 and t_j in (0, s_{12})^c then m_j=∞ else
m_j = \frac{(1 - t_j/s_{11})^{γ_{11}}(1 - t_j/s_{22})^{γ_{22}}}{(1 - t_j/s_{12})^{2 γ_{11}} }{ m_j = (1 - t_j/s_{11})^{γ_{11}} (1 - t_j/s_{22})^{γ_{22}} / (1 - t_j/s_{12})^{2 γ_{11}} }
In the function RMbigneiting, either c is
passed, then the above condition is checked, or rhored is passed;
then c_{12} is calculated by the above formula.
RMbigneiting returns an object of class RMmodel.
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
Bevilacqua, M., Daley, D.J., Porcu, E., Schlather, M. (2012) Classes of compactly supported correlation functions for multivariate random fields. Technical report.
RMbigeneiting is based on this original work.
D.J. Daley, E. Porcu and M. Bevilacqua have published end of
2014 an article intentionally
without clarifying the genuine authorship of RMbigneiting,
in particular,
neither referring to this original work nor to RandomFields,
which has included RMbigneiting since version 3.0.5 (05 Dec
2013).
Gneiting, T. (1999) Correlation functions for atmospherical data analysis. Q. J. Roy. Meteor. Soc Part A 125, 2449-2464.
Wendland, H. (2005) Scattered Data Approximation. Cambridge Monogr. Appl. Comput. Math.
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set ## RFoptions(seed=NA) to make them all random again model <- RMbigneiting(kappa=2, mu=0.5, gamma=c(0, 3, 6), rhored=1) x <- seq(0, 10, 0.02) plot(model) plot(RFsimulate(model, x=x))
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