\name{RMmodelsAdvanced}
\alias{RMmodelsAdvanced}
\alias{Advanced RMmodels}
\title{Advanced features of the mdoels}
\description{
 Here, further models and advanced comments for \command{\link{RMmodel}}
 are given. See also \command{\link{RFgetModelNames}}.
}

\details{

\bold{Further stationary and isotropic models}

\tabular{ll}{
\command{\link{RMaskey}} \tab Askey model (generalized test or triangle model) \cr
\command{\link{RMbessel}} \tab Bessel family \cr
\command{\link{RMcircular}} \tab circular model \cr
\command{\link{RMcauchy}} \tab modified Cauchy family \cr
\command{\link{RMconstant}} \tab spatially constant model \cr
\command{\link{RMcubic}} \tab cubic model (see Chiles \& Delfiner) \cr
\command{\link{RMdagum}} \tab Dagum model \cr
\command{\link{RMdampedcos}} \tab exponentially damped cosine \cr
\command{\link{RMqexp}} \tab Variant of the exponential model \cr
\command{\link{RMfractdiff}} \tab fractionally differenced process \cr
\command{\link{RMfractgauss}} \tab fractional Gaussian noise \cr
\command{\link{RMgengneiting}} \tab generalized Gneiting model \cr
\command{\link{RMgneitingdiff}} \tab Gneiting model for tapering \cr
\command{\link{RMhyperbolic}} \tab generalised hyperbolic model \cr
\command{\link{RMlgd}} \tab Gneiting's local-global distinguisher\cr
\command{\link{RMma}} \tab one of Ma's model \cr
\command{\link{RMpenta}} \tab penta model (see Chiles \& Delfiner) \cr
\command{\link{RMpower}} \tab Golubov's model \cr
\command{\link{RMwave}} \tab cardinal sine \cr
}

\bold{Variogram models (stationary increments/intrinsically stationary)}

\tabular{ll}{
\command{\link{RMdewijsian}} \tab generalised version of the DeWijsian model \cr
\command{\link{RMgenfbm}} \tab generalized fractal Brownian motion \cr
\command{\link{RMflatpower}} \tab similar to fractal Brownian motion but
always smooth at the origin\cr
}

\bold{General composed models (operators)}

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

\tabular{ll}{
% \command{\link{RMCauchy}} \tab Cauchy like transform -- TO BE PROGRAMMED (includes \code{ma1})) \cr
\command{\link{RMbernoulli}} \tab Correlation function of a binary field
based on a Gaussian field \cr
 \command{\link{RMexponential}} \tab exponential of a covariance model \cr
 \command{\link{RMintexp}} \tab integrated exponential of a covariance model (INCLUDES \code{ma2})\cr
 \command{\link{RMpower}} \tab powered variograms\cr
 \command{\link{RMqam}} \tab Porcu's quasi-arithmetric-mean model\cr
 \command{\link{RMS}} \tab details on the optional transformation
 arguments (\code{var}, \code{scale}, \code{Aniso}, \code{proj}).
}

\bold{Stationary and isotropic composed models (operators)}

\tabular{ll}{
 \command{\link{RMcutoff}} \tab Gneiting's modification towards finite range\cr
 \command{\link{RMintrinsic}} \tab Stein's modification towards finite range\cr
 \command{\link{RMnatsc}} \tab practical range\cr
 \command{\link{RMstein}} \tab Stein's modification towards finite range\cr 
% \command{\link{RMtbm2}} \tab Turning bands operator in two (spatial)
% dimensions\cr % nicht an user exportiert
 \command{\link{RMtbm}}\tab Turning bands operator
}

\bold{Stationary space-time models}

Here, most of the models are composed models (operators).
\tabular{ll}{
 \command{\link{RMave}} \tab space-time moving average model \cr
 \command{\link{RMcoxisham}} \tab Cox-Isham model \cr
 \command{\link{RMcurlfree}} \tab curlfree (spatial) field (stationary and anisotropic)\cr
 \command{\link{RMdivfree}} \tab divergence free (spatial) vector valued field, (stationary and anisotropic)\cr
 \command{\link{RMgennsst}} \tab generalization of Gneiting's non-separable space-time model \cr 
 \command{\link{RMiaco}} \tab non-separabel space-time model\cr
 % obsolete -- included by Cauchy --- should be given by an example
 \command{\link{RMmastein}} \tab Ma-Stein model\cr
 \command{\link{RMnsst}} \tab Gneiting's non-separable space-time model \cr 
 \command{\link{RMstein}} \tab Stein's non-separabel space-time model\cr
 \command{\link{RMstp}} \tab Single temporal process\cr
 \command{\link{RMtbm}} \tab Turning bands operator}

\bold{Multivariate/Multivariable and vector valued models}
See also the vignette \sQuote{\href{../doc/multivariate_jss.pdf}{multivariate}}.
\tabular{ll}{
 \command{\link{RMbiwm}} \tab full bivariate Whittle-Matern model (stationary and isotropic)\cr
 \command{\link{RMbigneiting}} \tab bivariate Gneiting model (stationary and isotropic)\cr
\command{\link{RMcurlfree}} \tab curlfree (spatial) vector-valued field (stationary and anisotropic)\cr
\command{\link{RMdelay}} \tab bivariate delay effect model (stationary)\cr
\command{\link{RMdivfree}} \tab divergence free (spatial) vector valued
field, (stationary and anisotropic)\cr
\command{\link{RMexponential}} \tab functional returning \eqn{e^C}{exp(C)}\cr
 \command{\link{RMkolmogorov}} \tab Kolmogorov's model of turbulence\cr
\command{\link{RMmatrix}} \tab trivial multivariate model\cr
\command{\link{RMmqam}} \tab multivariate quasi-arithmetic mean (stationary)\cr
 \command{\link{RMparswm}} \tab multivariate Whittle-Matern model (stationary and isotropic)\cr
\command{\link{RMschur}} \tab element-wise product with a positive definite
matrix\cr 
\command{\link{RMtbm}} \tab turning bands operator\cr
\command{\link{RMvector}} \tab vector-valued field (combining \command{\link{RMcurlfree}} and \command{\link{RMdivfree}})
}

\bold{Non-stationary models}
\tabular{ll}{
 \command{\link{RMnonstwm}} \tab one of Stein's non-stationary
 Wittle-Matern models \cr
 \command{\link{RMprod}} \tab scalar product\cr
}

\bold{Negative definite models that are not variograms}
\tabular{ll}{
\command{\link{RMsum}} \tab a non-stationary variogram model\cr
}


\bold{Models related to max-stable random fields (tail correlation functions)}
\tabular{ll}{
\command{\link{RMaskey}} \tab Askey model (generalized test or triangle
model) with \eqn{\alpha \ge [dim / 2] +1}\cr
 \command{\link{RMbernoulli}} \tab Correlation function of a binary field
 based on a Gaussian field \cr
 \command{\link{RMbr2bg}} \tab Operator relating a Brown-Resnick process
 to a Bernoulli process\cr
 \command{\link{RMbr2eg}} \tab Operator relating a Brown-Resnick process
 to an extremal Gaussian process\cr
 \command{\link{RMbrownresnick}} \tab tail correlation function
 of Brown-Resnick process\cr
\command{\link{RMgencauchy}} \tab generalized Cauchy family with  \eqn{\alpha\le 1/2}\cr
\command{\link{RMm2r}} \tab shape functions related to max-stable
 processes\cr 
\command{\link{RMm3b}} \tab shape functions related to max-stable
 processes\cr 
 \command{\link{RMmatern}} \tab Whittle-Matern model with \eqn{\nu\le 1}\cr
 \command{\link{RMmps}} \tab shape functions related to max-stable
 processes\cr 
\command{\link{RMschlather}} \tab tail correlation function of the
 extremal Gaussian field \cr
 \command{\link{RMstable}} \tab symmetric stable family or powered
 exponential model with \eqn{\alpha\le 1}\cr
  \command{\link{RMwhittle}} \tab Whittle-Matern model, alternative
 parametrization with \eqn{\nu\le 1/2}\cr

 }


\bold{Other covariance models}
\tabular{ll}{
 \command{\link{RMuser}} \tab User defined model \cr
}

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

\note{
  \itemize{
    \item
    Note that, instead of the named arguments, a single argument \code{k}
    can be passed. This is possible if all the arguments
    are scalar. Then \code{k} must have length equal to the number of
    arguments.
    \item
    If a argument equals \code{NULL} the
    argument is not set (but must be a valid name).
    \item
    \code{Aniso} can be given also by \command{\link{RMangle}}
    instead by a matrix
    \item
    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 \command{PrintModelList()}).
    Here, the argument \code{var}, \code{scale},
    \code{Aniso} and \code{proj} must be passed by the model
    \command{\link{RMS}}. 
    For instance,
    \itemize{
      \item
      \code{model <- RMexp(scale=2, var=5)}
      \cr
      is equivalent to
      \cr
      \code{model <- list("RMS", scale=2, var=5, list("RMexp"))}
      \cr
      The latter definition can be also obtained by
      
      \code{summary(RMexp(scale=2, var=5))}
      
      \item
      \code{model <- RMnsst(phi=RMgauss(var=7), psi=RMfbm(alpha=1.5),
	scale=2, var=5)}
       \cr
       is equivalent to
        \cr
	\code{model <- list("RMS", scale=2, var=5,} \cr
	\code{list("RMnsst", phi=list("RMS", var=7, list("RMgauss")),} \cr
	\code{psi=list("RMfbm", alpha=1.5))
	)}.
  }
  
  \item
  Instead of a deterministic value, a distribution family might be
  given, see \command{\link{RRmodels}}.
  The latter starts with \code{RR} or is distribution family,
  e.g. \code{norm}, \code{exp}, or \code{unif}. Note that the
  effect of the distribution family varies between the different processes:

  \itemize{
    \item in Max-stable fields and
    \command{\link{RPpoisson}}, a new realisation of the
    distribution is drawn for each shape function
    \item in all the other cases: a realisation is only drawn once.
    This effects, in particular, Gaussian fields with argument
    \code{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 \link{RMmodel}. Note that checking the validity of the
  arguments is rather limited for such complicated models, in general.
  
  See also
  \link{RMmodelsAuxiliary} and \link{Baysian}.
  }
  
  All models have secondary names that stem from 
  \pkg{RandomFields} versions 2 and earlier and
  that can also be used as strings in the list notation.
  See \code{\link{RFgetModelNames}(internal=FALSE)} for
  the full list. 
}


%\section{Methods}{
% \describe{
% \item{[}{\code{signature(x = "RFgridDataFrame")}: selects
% slot by name}
% \item{[<-}{\code{signature(x = "RFgridDataFrame")}: replaces
% slot by name}
% \item{as}{\code{signature(x = "RFgridDataFrame")}:
% converts into other formats, only implemented for target class
% \command{\link[=RFpointsDataFrame-class]{RFpointsDataFrame}} } 
% \item{cbind}{\code{signature(...)}: if arguments have identical
% topology, combine their attribute values}
% }
%}


\references{
 \itemize{
 \item Chiles, J.-P. and Delfiner, P. (1999)
 \emph{Geostatistics. Modeling Spatial Uncertainty.}
 New York: Wiley.
 % \item Gneiting, T. and Schlather, M. (2004)
 % Statistical modeling with covariance functions.
 % \emph{In preparation.}
 \item Schlather, M. (1999) \emph{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.
 \item Schlather, M. (2011) Construction of covariance functions and
 unconditional simulation of random fields. In Porcu, E., Montero, J.M.
 and Schlather, M., \emph{Space-Time Processes and Challenges Related
 to Environmental Problems.} New York: Springer.
 % \item Schlather, M. (2002) Models for stationary max-stable
 % random fields. \emph{Extremes} \bold{5}, 33-44.
 \item
   Schlather, M., Malinowski, A., Menck, P.J., Oesting, M. and
    Strokorb, K. (2015) 
    Analysis, simulation and prediction of multivariate
    random fields with package \pkg{RandomFields}. \emph{
      Journal of Statistical Software}, \bold{63} (8), 1-25,
    url          = \sQuote{http://www.jstatsoft.org/v63/i08/}
 \item Yaglom, A.M. (1987) \emph{Correlation Theory of Stationary and
 Related Random Functions I, Basic Results.}
 New York: Springer.
 \item Wackernagel, H. (2003) \emph{Multivariate Geostatistics.} Berlin:
 Springer, 3nd edition.
 }
}

\seealso{\command{\link{RFformula}}, \command{\link{RMmodels}},
  \command{\link{RMmodelsAuxiliary}}

 \sQuote{\href{../doc/multivariate_jss.pdf}{multivariate}}, a vignette
 for multivariate geostatistics
}

\author{
 Alexander Malinowski, \email{malinowski@math.uni-mannheim.de}
 
 Martin Schlather, \email{schlather@math.uni-mannheim.de}
}
\keyword{spatial}

\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"))
\dontshow{FinalizeExample()}
}