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^{β} D |^{- 1 / 2} C_{0} ([(h - t μ)^{T} (E + t^{β} D)^{- 1} (h - t μ)]^{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

a vector in $d$-dimensional space

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 - V t)

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.

Examples

Run this code

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, dim=2)
plot(RFsimulate(model, x=x, y=x))
FinalizeExample()

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