RandomFields (version 3.1.12)

RMbigneiting: Gneiting-Wendland Covariance Models

Description

RMbigneiting is a bivariate stationary isotropic covariance model family whose elements are specified by seven parameters.

Let $$\delta_{ij} = \mu + \gamma_{ij} + 1.$$ Then, $$C_{n}(h) = c_{ij} (C_{n, \delta} (h / s_{ij}))_{i,j=1,2}$$ and $C_{n, \delta}$ is the generalised Gneiting model with parameters $n$ and $\delta$, see RMgengneiting, i.e., $$C_{\kappa=0, \delta}(r) = (1-r)^\beta 1_{[0,1]}(r), \qquad \beta=\delta + 2\kappa + 1/2;$$ $$C_{\kappa=1, \delta}(r) = \left(1+\beta r \right)(1-r)^{\beta} 1_{[0,1]}(r), \qquad \beta = \delta + 2\kappa + 1/2;$$ $$C_{\kappa=2, \delta}(r)=\left( 1 + \beta r + \frac{\beta^{2} - 1}{3}r^{2} \right)(1-r)^{\beta} 1_{[0,1]}(r), \qquad \beta=\delta + 2\kappa + 1/2;$$ $$C_{\kappa=3, \delta}(r)=\left( 1 + \beta r + \frac{(2\beta^{2}-3)}{5} r^{2}+ \frac{(\beta^2 - 4)\beta}{15} r^{3} \right)(1-r)^\beta 1_{[0,1]}(r), \qquad \beta=\delta+2\kappa+1/2.$$

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 $0,\ldots,3$. The model is $k$ times differentiable.

mu

mu has to be greater than or equal to $\frac{d}{2}$ 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 $(\gamma_{11},\gamma_{21},\gamma_{22})$. Note that $\gamma_{12} =\gamma_{21}$.

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.

Value

• RMgengneiting returns an object of class RMmodel

Details

A sufficient condition for the constant $c_{ij}$ is $$c_{12} = \rho_{\rm red} \cdot m \cdot \left(c_{11} c_{22} \prod_{i,j=1,2} \left(\frac{\Gamma(\gamma_{ij} + \mu + 2\kappa + 5/2)}{b_{ij}^{\nu_{ij} + 2\kappa + 1} \Gamma(1 + \gamma_{ij}) \Gamma(\mu + 2\kappa + 3/2)} \right)^{(-1)^{i+j}} \right)^{1/2}$$ where $\rho_{\rm red} \in [-1,1]$.

The constant $m$ in the formula above is obtained as follows: $$m = \min{1, m_{-1}, m_{+1}}$$ Let $$a = 2 \gamma_{12} - \gamma_{11} -\gamma_{22}$$ $$b = -2 \gamma_{12} (s_{11} + s_{22}) + \gamma_{11} (s_{12} + s_{22}) + \gamma_{22} (s_{12} + s_{11})$$ $$e = 2 \gamma_{12} s_{11}s_{22} - \gamma_{11}s_{12}s_{22} - \gamma_{22}s_{12}s_{11}$$ $$d = b^2 - 4ae$$ $$t_j =\frac{- b + j \sqrt d}{2 a}$$ If $d \ge0$ and $t_j \not\in (0, s_{12})$ then $m_j=\infty$ else $$m_j = \frac{(1 - t_j/s_{11})^{\gamma_{11}}(1 - t_j/s_{22})^{\gamma_{22}}}{(1 - t_j/s_{12})^{2 \gamma_{11}} }{ m_j = (1 - t_j/s_{11})^{\gamma_{11}} (1 - t_j/s_{22})^{\gamma_{22}} / (1 - t_j/s_{12})^{2 \gamma_{11}} }$$

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.

References

• Bevilacqua, M., Daley, D.J., Porcu, E., Schlather, M. (2012) Classes of compactly supported correlation functions for multivariate random fields. Technical report.RMbigeneitingis 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 ofRMbigneiting, in particular, neither referring to this original work nor toRandomFields, which has includedRMbigneitingsince version 3.0.5 (05 Dec 2013).
• Gneiting, T. (1999) Correlation functions for atmospherical data analysis.Q. J. Roy. Meteor. SocPart A125, 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
StartExample()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))
FinalizeExample()