Advanced features of the mdoels

Description

Here, further models and advanced comments for RMmodel are given. See also RFgetModelNames.

Details

Further stationary and isotropic models

Variogram models (stationary increments/intrinsically stationary)

General composed models (operators)

Here, composed models are given that can be of any kind (stationary/non-stationary), depending on the submodel.

Stationary and isotropic composed models (operators)

Stationary space-time models

Here, most of the models are composed models (operators).

Multivariate/Multivariable and vector valued models See also the vignette ‘multivariate’.

Non-stationary models

Negative definite models that are not variograms

Models related to max-stable random fields (tail correlation functions)

Other covariance models

Auxiliary models There are models or better function that are not covariance functions, but can be part of a model definition. See Auxiliary RMmodels.

Note

• Note that, instead of the named arguments, a single argument k can be passed. This is possible if all the arguments are scalar. Then k must have length equal to the number of arguments.

• If a argument equals NULL the argument is not set (but must be a valid name).

• Aniso can be given also by RMangle instead by a matrix

• Note also that a completely different possibility exists to define a model, namely by a list. This format allows for easy flexible models and modifications (and some few more options, as well as some abbreviations to the model names, see PrintModelList()). Here, the argument var, scale, Aniso and proj must be passed by the model RMS. For instance,

  • model <- RMexp(scale=2, var=5)
    is equivalent to
    model <- list("RMS", scale=2, var=5, list("RMexp"))
    The latter definition can be also obtained by

    summary(RMexp(scale=2, var=5))

  • model <- RMnsst(phi=RMgauss(var=7), psi=RMfbm(alpha=1.5), scale=2, var=5)
    is equivalent to
    model <- list("RMS", scale=2, var=5,
list("RMnsst", phi=list("RMS", var=7, list("RMgauss")),
psi=list("RMfbm", alpha=1.5)) ).

• Instead of a deterministic value, a distribution family might be given, see RRmodels. The latter starts with RR or is distribution family, e.g. norm, exp, or unif. Note that the effect of the distribution family varies between the different processes:

  • in Max-stable fields and RPpoisson, a new realisation of the distribution is drawn for each shape function

  • in all the other cases: a realisation is only drawn once. This effects, in particular, Gaussian fields with argument n>1, where all the realisations are based on the same realisation out of the distribution.

  MLE ist not programmed yet.

  Very advanced: In case of a distribution family, its arguments might be again given by a RMmodel. Note that checking the validity of the arguments is rather limited for such complicated models, in general.

  See also RMmodelsAuxiliary and Baysian.

All models have secondary names that stem from RandomFields versions 2 and earlier and that can also be used as strings in the list notation. See RFgetModelNames(internal=FALSE) for the full list.

Author(s)

Alexander Malinowski, malinowski@math.uni-mannheim.de

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

References

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

• Schlather, M. (1999) An introduction to positive definite functions and to unconditional simulation of random fields. Technical report ST 99-10, Dept. of Maths and Statistics, Lancaster University.

• Schlather, M. (2011) Construction of covariance functions and unconditional simulation of random fields. In Porcu, E., Montero, J.M. and Schlather, M., Space-Time Processes and Challenges Related to Environmental Problems. New York: Springer.

• Schlather, M., Malinowski, A., Menck, P.J., Oesting, M. and Strokorb, K. (2015) Analysis, simulation and prediction of multivariate random fields with package RandomFields. Journal of Statistical Software, 63 (8), 1-25, url = ‘http://www.jstatsoft.org/v63/i08/’

• Yaglom, A.M. (1987) Correlation Theory of Stationary and Related Random Functions I, Basic Results. New York: Springer.

• Wackernagel, H. (2003) Multivariate Geostatistics. Berlin: Springer, 3nd edition.

See Also

RFformula, RMmodels, RMmodelsAuxiliary

‘multivariate’, a vignette for multivariate geostatistics

Examples

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again
RFgetModelNames(type="positive", group.by=c("domain", "isotropy"))