RandomFields (version 3.0.32)

RMcoxisham: Cox Isham Covariance Model

Description

RMcoxisham is a stationary covariance model which depends on a univariate stationary isotropic covariance model $C_0$, which is a normal scale mixture. The corresponding covariance function only depends on the difference $(h,t) \in {\bf R}^{d+1}={\bf R}^d\times{\bf R}$ between two points in $d+1$-dimensional space and is given by $$C(h,t)=|E + t^\beta D|^{-1/2} C_0([(h - t \mu)^T (E + t^\beta D)^{-1} (h - t \mu)]^{1/2})$$ Here $\mu \in {\bf R}^d$ is a vector in $d$-dimensional space; $E$ is the $d \times d$-identity matrix and $D$ is a $d \times d$-correlation matrix with $|D| > 0$. The parameter $\beta$ is in $(0,2]$. Currently, the implementation is done only for $d=2$.

Usage

RMcoxisham(phi,mu,D,beta,var, scale, Aniso, proj)

Arguments

phi

a univariate stationary isotropic covariance model for random fields on $d$-dimensional space, which is moreover a normal scale mixture, that means an RMmodel whose monotone property e

mu

a vector in $d$-dimensional space

D

a $d \times d$-correlation matrix with $|D| > 0$

beta

numeric in the interval $(0,2]$; default value is 2

var,scale,Aniso,proj

optional arguments; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.

Value

• RMcoxisham returns an object of class RMmodel

Details

This model stems from a rainfall model (cf. Cox, D.R., Isham, V.S. (1988)) and equals the following expectation $$C(h,t)=\bold{E}_V C_0(h-Vt)$$ where the random wind speed vector $V$ follows a $d$-variate normal distribution with expectation $mu$ and covariance matrix $D/2$. (cf. See Schlather, M. (2010), Example 9).

References

• Cox, D.R., Isham, V.S. (1988) A simple spatial-temporal model of rainfall.Proc. R. Soc. Lond. A,415, 317-328.
• Schlather, M. (2010) On some covariance models based on normal scale mixtures.Bernoulli,16, 780-797.

See Also

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 <- RMcoxisham(RMgauss(), mu=1, D=1)
x <- seq(0, 10, if (interactive()) 0.3 else 1) 
plot(model, ylim=c(0, 1), dim=2)
plot(RFsimulate(model, x=x, y=x))
FinalizeExample()