R: Gneiting-Wendland Covariance Models

RMbigneiting {RandomFields} | R Documentation

## Gneiting-Wendland Covariance Models

### Description

`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 generalised 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.

### Usage

    RMbigneiting(kappa, mu, s, sred12, gamma, cdiag, rhored, c, var, scale, Aniso, proj)

### Arguments

`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 randomfield.
`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.

### Details

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.

### Value

`RMgengneiting` returns an object of class `RMmodel`

### Author(s)

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

### References

- 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.

### See Also

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

### Examples

    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))

[Package RandomFields version 3.1.50 Index]