R: Whittle-Matern Covariance Model

RMwhittlematern {RandomFields} | R Documentation

## Whittle-Matern Covariance Model

### Description

`RMmatern` is a stationary isotropic covariance model belonging to the Matern family. The corresponding covariance function only depends on the distance r ≥ 0 between two points.

The Whittle model is given by

C(r)=W_{ν}(r)=2^{1- ν} Γ(ν)^{-1}r^{ν}K_{ν}(r)

where ν > 0 and K_ν is the modified Bessel function of second kind.

The Matern model is given by

C(r) = 2^{1- ν} Γ(ν)^{-1} (√{2ν} r)^ν K_ν(√{2ν} r)

The Handcock-Wallis parametrisation is given by

C(r) = 2^{1- ν} Γ(ν)^{-1} (2√{ν} r)^ν K_ν(2√{ν} r)

### Usage

    RMwhittle(nu, notinvnu, var, scale, Aniso, proj)

    RMmatern(nu, notinvnu, var, scale, Aniso, proj)

    RMhandcock(nu, notinvnu, var, scale, Aniso, proj)

### Arguments

`nu` | a numerical value called “smoothness parameter”; should be greater than 0.
`notinvnu` | logical. If `FALSE` then in the definition of the models ν is replaced by 1/ν. This parametrisation seems to be more natural. Default is however `FALSE` according the definitions in literature.
`var,scale,Aniso,proj` | optional arguments; same meaning for any `RMmodel`. If not passed, the above covariance function remains unmodified.

### Details

The three models are alternative parametrizations of the same covariance function. The Matern model or the Handcock-Wallis parametrisation should be preferred as they seperate the effects of scaling parameter and the shape parameter.

This Whittle-Matern model is the model of choice if the smoothness of a random field is to be parametrized: the sample paths of a Gaussian random field with this covariance structure are m times differentiable if and only if ν > m (see Gelfand et al., 2010, p. 24).

Furthermore, the fractal dimension (see also `RFfractaldim`) D of the Gaussian sample paths is determined by ν: we have

D = d + 1 - ν, 0 < ν < 1

and D = d for ν > 1 where d is the dimension of the random field (see Stein, 1999, p. 32).

If ν=0.5 the Matern model equals `RMexp`.

For ν tending to ∞ a rescaled Gaussian model `RMgauss` appears as limit of the Matern model.

For generalisations see section ‘seealso’.

### Value

The function return an object of class `RMmodel`

### Note

The Whittle-Matern model is a normal scale mixture.

### Author(s)

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

### References

Covariance function

- Chiles, J.-P. and Delfiner, P. (1999) Geostatistics. Modeling Spatial Uncertainty. New York: Wiley.

- Gelfand, A. E., Diggle, P., Fuentes, M. and Guttorp, P. (eds.) (2010) Handbook of Spatial Statistics. Boca Raton: Chapman & Hall/CRL.

- Guttorp, P. and Gneiting, T. (2006) Studies in the history of probability and statistics. XLIX. On the Matern correlation family. Biometrika 93, 989–995.

- Handcock, M. S. and Wallis, J. R. (1994) An approach to statistical spatio-temporal modeling of meteorological fields. JASA 89, 368–378.

- Stein, M. L. (1999) Interpolation of Spatial Data – Some Theory for Kriging. New York: Springer.

Tail correlation function (for 0 < ν ≤ 1/2)

- Strokorb, K., Ballani, F., and Schlather, M. (2014) Tail correlation functions of max-stable processes: Construction principles, recovery and diversity of some mixing max-stable processes with identical TCF. Extremes, Submitted.

### See Also

- `RMexp`, `RMgauss` for special cases of the model (for ν=0.5 and ν=∞, respectively)

- `RMhyperbolic` for a univariate generalization

- `RMbiwm` for a multivariate generalization

- `RMnonstwm`, `RMstein` for anisotropic (space-time) generalizations

- `RMmodel`, `RFsimulate`, `RFfit` for general use.

### Examples

    RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
    ##                   RFoptions(seed=NA) to make them all random again

    x <- seq(0, 1, len=100)
    model <- RMwhittle(nu=1, Aniso=matrix(nc=2, c(1.5, 3, -3, 4)))
    plot(model, dim=2, xlim=c(-1,1))
    z <- RFsimulate(model=model, x, x)
    plot(z)

[Package RandomFields version 3.1.50 Index]