# RMmatern: Whittle-Matern Covariance Model In RandomFields: Simulation and Analysis of Random Fields

## 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

 ```1 2 3 4 5``` ```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 parametrization seems to be more natural. Default is, however, `TRUE` according with 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 the scaling parameter and the shape parameter.

The 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` C(r) = -r^2 appears as limit of the above Handcock-Wallis parametrisation.

## Value

The functions return an object of class `RMmodel`.

## Note

The Whittle-Matern model is a normal scale mixture.

## 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.

• `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

 ```1 2 3 4 5 6 7 8``` ```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) ```

### Example output

```Loading required package: sp

Attaching package: 'RandomFields'

The following object is masked from 'package:RandomFieldsUtils':

RFoptions

The following objects are masked from 'package:base':

abs, acosh, asin, asinh, atan, atan2, atanh, cos, cosh, exp, expm1,
floor, gamma, lgamma, log, log1p, log2, logb, max, min, round, sin,
sinh, sqrt, tan, tanh, trunc

NULL
NOTE: simulation is performed with fixed random seed 0.
Set 'RFoptions(seed=NA)' to make the seed arbitrary.
New output format of RFsimulate: S4 object of class 'RFsp';
for a bare, but faster array format use 'RFoptions(spConform=FALSE)'.
```

RandomFields documentation built on Feb. 6, 2020, 5:13 p.m.