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

\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 generalized 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 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 (\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.

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.

Value

RMbigneiting returns an object of class RMmodel.

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/

References

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 3.3.14 Index]