R: Cox Isham Covariance Model

RMcoxisham {RandomFields} | R Documentation

## 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) 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 μ is a vector in d-dimensional space; E is the d x d-identity matrix and D is a d x d-correlation matrix with |D| > 0. The parameter β 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 equals `'normal mixture'`, see `RFgetModelNames(monotone="normal mixture")` and whose `maxdim` is at least 2.
`mu` | a vector in d-dimensional space
`D` | a d x 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.

### 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).

### Value

`RMcoxisham` returns an object of class `RMmodel`

### Author(s)

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

### 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, 0.3)
    plot(model, dim=2)
    plot(RFsimulate(model, x=x, y=x))

[Package RandomFields version 3.1.50 Index]