RMbigneiting: Gneiting-Wendland Covariance Models
In RandomFields: Simulation and Analysis of Random Fields

Description Usage Arguments Details Value References See Also Examples

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.

1	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`	`mu` has to be greater than or equal to d/2 where d is the (arbitrary) dimension of the random field.
`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} = `sred12 ` min{s_{11}, s_{22}}*.
`gamma`	a vector of length 3 of numerical values; each entry is positive. The vector `gamma` equals (γ_{11},γ_{21},γ_{22}). Note that γ_{12} =γ_{21}.
`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` and `cdiag` or `c` must be given.
`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 `RMmodel`. If not passed, the above covariance function remains unmodified.

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.

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.

RMaskey, RMbiwm, RMgengneiting, RMgneiting, RMmodel, RFsimulate, RFfit.

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))

Loading required package: sp
Loading required package: RandomFieldsUtils

Attaching package: 'RandomFields'

The following object is masked from 'package:RandomFieldsUtils':

    RFoptions

The following objects are masked from 'package:base':

    abs, acosh, asin, asinh, atan, atan2, atanh, cos, cosh, exp, expm1,
    floor, gamma, lgamma, log, log1p, log2, logb, max, min, round, sin,
    sinh, sqrt, tan, tanh, trunc

NULL
NOTE: simulation is performed with fixed random seed 0.
Set 'RFoptions(seed=NA)' to make the seed arbitrary.
New output format of RFsimulate: S4 object of class 'RFsp';
for a bare, but faster array format use 'RFoptions(spConform=FALSE)'.