Revision c103e17c562b32dcb179c0853306f28a0e456d50 authored by Martin Schlather on 11 April 2005, 00:00:00 UTC, committed by Gabor Csardi on 11 April 2005, 00:00:00 UTC
1 parent e7b694a
CovarianceFct.Rd
\name{CovarianceFct}
\alias{CovarianceFct}
\alias{Variogram}
\alias{PrintModelList}
\alias{GetModelList}
\alias{GetModelNames}
\title{Covariance And Variogram Models}
\description{
\code{CovarianceFct} returns the values of a covariance function

\code{Variogram} returns the values of a variogram model

\code{PrintModelList} prints the list of currently implemented models
including the corresponding simulation methods

\code{GetModelList} returns a matrix of currently implemented models
and their simulation methods

\code{GetModelNames} returns a list of currently implemented models
}
\usage{
CovarianceFct(x, model, param, dim=ifelse(is.matrix(x),ncol(x),1),
fctcall="Covariance")

Variogram(x, model, param, dim=ifelse(is.matrix(x),ncol(x),1))

PrintModelList()

GetModelList(abbr=TRUE)

GetModelNames()
}
\arguments{
\item{x}{vector or \eqn{(n \times \code{dim})}{(n x
\code{dim})}-matrix.  In particular,
if the model is isotropic or \code{dim=1} then \code{x}
is a vector.}

\item{model}{character or list;
if character then name of the covariance function or
variogram model - see below, or type \code{PrintModelList()} for
all options; see Details for the definition of the model by a list.
}
\item{param}{vector or matrix of parameters or missing, see Details
and Examples;
The simplest form is that \code{param} is vector of the form
\code{param=c(NA,variance,nugget,scale,...)}, in this order;\cr
The dots \code{...} stand for additional parameters of the
model.
}
\item{dim}{dimension of the space in which the model is applied}
\item{fctcall}{this parameter should not be changed by the user}
\item{abbr}{logical or numerical. If \code{TRUE} the names for the methods are
abbreviated. If numerical, \code{abbr} gives the number of letters.}
}
\details{
The implemented models are in standard notation for a
covariance function (variance 1, nugget 0, scale 1) and for positive
real arguments \eqn{x} (and \eqn{t}):

\itemize{
\item \code{bessel}
\deqn{C(x)=2^a \Gamma(a+1)x^{-a} J_a(x)}{C(x)=
2^a Gamma(a+1)x^(-a) J_a(x)}
The parameter \eqn{\kappa}{a} is greater than or equal to
\eqn{\frac{d-2}2}{(d-2)/2}, where \eqn{d}{d} is the
dimension of the random field.

\item Brownian motion\cr
see \code{fractalB}

\item cardinal sine\cr
see \code{wave}

\item \code{cauchy}
\deqn{C(x)=\left(1+x^2\right)^{-\kappa}}{C(x)=(1+x^2)^(-a)}
The parameter \eqn{\kappa}{a} is positive.
The model possesses two generalisations, the \code{gencauchy}
model and the \code{hyperbolic} model.

\item \code{cauchytbm}
\deqn{C(x)=\left(1+\left(1-\frac{\kappa_2}{\kappa_3}
\right)x^{\kappa_1}\right)
\left(1+x^{\kappa_1}\right)^{-\frac{\kappa_2}{\kappa_1}-1}}{
C(x)=
(1+(1-b/c)x^a)(1+x^a)^(-b/a-1)}
The parameter \eqn{\kappa_1}{a} is in (0,2],  \eqn{\kappa_2}{b}
is positive, and \eqn{\kappa_3}{c} is an integer.
The model is valid for dimensions \eqn{d\le\kappa_3}{d<=c}.
\cr
It allows for simulating random fields where
fractal dimension and Hurst coefficient can be chosen
independently.
It has negative correlations for \eqn{\kappa_2>1}{b>c} and large
\eqn{x}{x}.

\item \code{circular}
\deqn{C(x)=
\left(1-\frac 2\pi
\left(x \sqrt{1-x^2} +
\arcsin(x)\right)\right)
1_{[0,1]}(x)}{
C(x)=1-2/pi*(x sqrt(1-x^2)+asin(x))   if
0<=x<=1, 0 otherwise}
This isotropic covariance function is valid only for dimensions
less than or equal to 2.

\item \code{cone}\cr
This model is used only for methods based on marked point processes
(see \code{\link{RFMethods}}); it is defined only in two dimensions.
The corresponding (boolean)
function is a truncated cone with socle. The base has radius
\eqn{\frac12}{1/2}. The model has three parameters, \eqn{\kappa_1}{a},
\eqn{\kappa_2}{b}, and \eqn{\kappa_3}{c}:\cr
\eqn{kappa_1}{a} gives the radius of the top circle of the cone, given
as part of the socle radius; \eqn{kappa_1\in[0,1)}{a in [0,1)}.\cr
\eqn{kappa_2}{b} gives the height of the socle.\cr
\eqn{kappa_3}{c} gives the height of the truncated cone.\cr

\item \code{cubic}
\deqn{C(x)=(1- 7x^2+8.75x^3-3.5x^5+0.75 x^7)1_{[0,1]}(x)}{C(x)=
1- 7 x^2 + 8.75 x^3 - 3.5 x^5 + 0.75 x^7  if 0<=x<=1,
0 otherwise}
This model is valid only for dimensions less than or equal to 3.
It is a 2 times differentiable covariance functions with compact
support. %(See Chiles&Delfiner, 1998)

\item \code{dampedcosine}
\deqn{C(x)=e^{-\kappa x} \cos(x),  \quad x\ge0}{C(x)=
exp(-a x) cos(x) }
This model is valid
for dimension 1 iff \eqn{\kappa\ge1}{a>=0},
for dimension 2 iff \eqn{\kappa\ge1}{a>=1},
and for dimension 3 iff \eqn{\kappa\ge \sqrt{3}}{a >= sqrt(3)}.

\item \code{exponential}
This model is a special case of the \code{whittlematern} model
(for \eqn{\kappa=\frac12}{a=1/2} there)
and the \code{stable} class (for \eqn{\kappa=1}{a=1}).

\item \code{fractalB} (fractal Brownian motion)
\deqn{\gamma(x) = x^\kappa}{gamma(x) = x^a}
The parameter \eqn{\kappa}{a} is in \eqn{[0,2]}.
(Implemented for up to three dimensions)

\item \code{FD}
\deqn{C(k) = \frac{(-1)^k \Gamma(1-\kappa)^2}{\Gamma(1-\kappa+k)
\Gamma(1-\kappa-k),
\qquad k \in {\bf N}}}{C(k) = (-1)^k
G(1-a)^2/(G(1-a+k)G(1-a-k)) for integer k}
and linearly interpolated otherwise.
Here, \eqn{\Gamma}{G} is the Gamma function.
The fractionally differenced process
is a time series model where the grid locations are multiples
of the scale parameter.  The parameter \eqn{\kappa}{a} is in
\eqn{[-1/2, 1/2)}.

\item \code{fractgauss}
\deqn{C(x) = 0.5 (|x+1|^{\kappa_1} - 2|x|^{\kappa_1} +
|x-1|^{\kappa_1})}{C(x) = 0.5 (|x+1|^a - 2|x|^a + |x-1|^a)}
This model is the covariance function for the fractional Gaussian noise
with Hurst parameter \eqn{H=\kappa_1 /2}{H =a/2}, \eqn{\kappa_1 \in
(0,2]}{a in (0,2].
}

\item \code{gauss}
\deqn{C(x)=e^{-x^2}}{C(x)=exp(-x^2)}
This model is a special case of the \code{stable} class
(for \eqn{\kappa=2}{a=2} there).
Note that the corresponding function for the random coins
method (cf. the methods based on marked point processes in
\deqn{e^{- 2 x^2}.}{exp(-2 x^2).}
See \code{gneiting} for an alternative model that does not have
the disadvantages of the Gaussian model.

\item \code{gencauchy} (generalised \code{cauchy})\cr
\deqn{C(x)=(1+x^{\kappa_1})^{-\kappa_2/\kappa_1}}{C(x)=
\left(1+x^a\right)^(-b/a)}
The parameter \eqn{\kappa_1}{a} is in (0,2], and \eqn{\kappa_2}{b}
is positive.
\cr
This model allows for simulating random fields where
fractal dimension and Hurst coefficient can be chosen
independently.

\item \code{gengneiting} (generalised \code{gneiting})\cr
if \eqn{\kappa_1=1}{a=1} then
\deqn{C(x)=\left(1+(\kappa_2+1)x\right) * (1-x)^{\kappa_2+1}
1_{[0,1]}(x)}{C(x)=[1 + (b+1) * x] * (1-x)^(b+1) if 0<=x<=1,
0 otherwise}
if \eqn{\kappa_1=2}{a=2} then
\deqn{C(x)=\left(1+(\kappa_2+2)x+\left((\kappa_2+2)^2-1\right)x^2/3\right)
(1-x)^{\kappa_2+2}  1_{[0,1]}(x)}{C(x)=
[1 + (b+2) * x + ((b+2)^2-1) * x^2 / 3] * (1-x)^(b+2)
if 0<=x<=1, 0 otherwise}
if \eqn{\kappa_1=3}{a=3} then
\deqn{C(x)=\left(1+(\kappa_2+3)x+\left(2(\kappa_2+3)^2-3\right)x^2/5
+\left((\kappa_2+3)^2-4\right)(\kappa_2+3)x^3/15\right)(1-x)^{\kappa_2+3}
1_{[0,1]}(x)}{C(x)=[1 + (b+3) * x +  (2 * (b+3)^2 - 3) * x^2 / 5
+ ((b+3)^2 - 4) * (b+3) * x^3 / 15] * (1-x)^(b+3)
if 0<=x<=1, 0 otherwise}
The parameter \eqn{\kappa_1}{a} is a positive integer; here only the
cases \eqn{\kappa_1=1, 2, 3}{a=1, 2, 3} are implemented.
The parameter \eqn{\kappa_2}{b} is greater than or equal to
\eqn{(d + 2\kappa_1 +1)/2}{(d + 2a +1)/2} where \eqn{d} is the
dimension of the random field.

% the differentiability is ??

\item \code{gneiting}
\deqn{C(x)=\left(1 + 8 sx + 25 (sx)^2 + 32
(sx)^3\right)(1-sx)^8 1_{[0,1]}(sx)}{C(x)=
(1 + 8 s x + 25 s^2 x^2 + 32
s^3 x^3)*(1-s x)^8   if 0<=s x<=1, 0 otherwise}
where
\eqn{s=0.301187465825}.
This isotropic covariance function is valid only for dimensions less
than or equal to 3.
It is a 6 times differentiable covariance functions with compact
support.\cr
It is an alternative to the \code{gaussian} model since
its graph is visually hardly distinguishable from the graph of
the Gaussian model, but possesses neither the mathematical and nor the
numerical disadvantages of the Gaussian model.\cr
This model is a special case of \code{gengneiting} (for
\eqn{\kappa_1=3}{a=3} and \eqn{\kappa_2=5}{b=5} there).
Note that, in the original work by Gneiting (1999),
\eqn{s=\frac{10\sqrt2}{47}\approx 0.3}{s = 10 sqrt(2) / 47 ~=
.3008965}, a numerical value slightly deviating from the
optimal one.

\item gneitingdiff is obsolete, see example below
\deqn{C(x)=\left(1 + 8 \frac x{\kappa_2}
+ 25 \frac {x^2}{\kappa_2^2}
+ 32 \frac {x^3}{\kappa_2^3}\right)
\left(1-\frac{x}{\kappa_2}\right)^8
\;\frac{2^{1-\kappa_1}}{\Gamma(\kappa_1)}
\,x^{\kappa_1} K_{\kappa_1}(x)1_{[0,\kappa_2]}(x)}{
C(x)=(1 + 8 x/b + 25 (x/b)^2 + 32
(x/b)^3)*(1-x/b)^8 * 2^{1-a} Gamma(a)^{-1} x^a K_a(x)   if
0<=x<=b, 0 otherwise}
This isotropic covariance function is valid only for dimensions less
than or equal to 3.
The parameters \eqn{\kappa_1}{a} and \eqn{\kappa_2}{b} are
positive.\cr
This class of models with compact support
allows for smooth parametrisation of the differentiability up to
order 6.

\item \code{hyperbolic}
\deqn{C(x)=\frac{1}{\kappa_3^{\kappa_2}
K_{\kappa_2}(\kappa_1 \kappa_3)}
\left(\kappa_3^2 +x^2\right)^{{\kappa_2}/2}
K_{{\kappa_2}}\left(
\kappa_1 \left(\kappa_3^2 + x^2\right)^{1/2}\right),  \quad
x>0}{C(x)=
c^(-b) (K_b(a*c))^(-1) * (c^2 +x^2)^(0.5 b) *
K_b(a sqrt(c^2 + x^2))}
The parameters are such that\cr
\eqn{\kappa_3\ge0}{c>=0},  \eqn{\kappa_1>0}{a>0 }  and
or\cr
\eqn{\kappa_3>0}{c>0 },  \eqn{\kappa_1>0}{a>0 }  and
or\cr
\eqn{\kappa_3>0}{c>0 },  \eqn{\kappa_1\ge0}{a>=0}, and
\eqn{\kappa_2<0}{b<0}.\cr
Note that this class is over-parametrised; always one
of the three parameters
\eqn{\kappa_1}{a}, \eqn{\kappa_3}{c}, and scale
can be eliminated in the formula. Therefore, one of these
parameters should be kept fixed in any simulation study.
\cr
The model contains as special cases the \code{whittlematern}
model and the \code{cauchy} model, for
\eqn{\kappa_3=0}{c=0} and \eqn{\kappa_1=0}{a=0}, respectively.

\item J-Bessel\cr
see \code{bessel}

\item K-Bessel\cr
see \code{whittlematern}

\item linear with sill\cr
See \code{power} (\code{a=1} there).

\item lgd1 (local-global distinguisher)
\deqn{C(x)=
\frac\alpha{\alpha+\beta}|x|^{-\beta}, |x|> 1
}{
C(x)=
1- b(a+b)^{-1}|x|^a for |x|\le 1   and
a(a+b)^{-1}|x|^-b for |x|> 1
}
Here \eqn{\beta>0}{b>0} and \eqn{\alpha}{a} is in
\eqn{(0, \frac12 (3 - d)]}{(0, 1.5-d/2]} for dimension \eqn{d=1,2}.
The random field has fractal dimension
\eqn{d + 1 - \frac\alpha2}{d + 1 - a/2}
and Hurst coefficient \eqn{1 - \frac\beta2}{1 - b/2} for
\eqn{\beta\in(0,1]}{b in (0,1]}

\item matern\cr
See \code{whittlematern}.

\item \code{nsst} (Non-Separable Space-Time model)
\deqn{C(x,t)= \psi(t)^{-\kappa_6} \phi(x / \psi(t))}{C(x,t)=
psi(t)^{-f} \phi(x / psi(t))}
This model is used for space-time modelling where the spatial
component is isotropic.  Here\cr
\eqn{\phi} is the \code{stable} model if \eqn{\kappa_2=1}{b=1};\cr
\eqn{\phi} is the \code{whittlematern} model if \eqn{\kappa_2=2}{b=2};\cr
\eqn{\phi} is the \code{cauchy} model if \eqn{\kappa_2=3}{b=3};\cr
here, \eqn{kappa_1}{a} is the respective parameter for the model.
The function \eqn{\psi}{psi} satisfies\cr
\eqn{\psi^2(t) = (t^{\kappa_3}+1)^{\kappa_4}}{psi^2(t) =
(t^c+1)^d} if \eqn{\kappa_5=1}{e=1}\cr
\eqn{\psi^2(t) = \frac{\kappa_4^{-1}t^{\kappa_3}+1}{t^{\kappa_3}+1}
}{psi^2(t) = (d^{-1} t^c+1)/(t^c+1)} if \eqn{\kappa_5=2}{e=2}\cr
\eqn{\psi^2(t)= - \log(t^{\kappa_3}+\kappa_4^{-1})/
\log\kappa_4}{psi^2(t)=-\log(t^c+1/d)/log d}
if \eqn{\kappa_5=3}{e=3}\cr
The parameter \eqn{\kappa_6}{f} must be greater than or equal to
the spatial dimension of the field. Furthermore, \eqn{\kappa_3\in
(0,2]}{c in (0,2]}  and \eqn{\kappa_4\in
(0,1)}{d in (0,1)}.
The spatial dimension must be \code{>=1}.

\item \code{nsst2}
\deqn{C(x,t)= \psi(t)^{-\kappa_7} \phi(x /\psi(t))}{C(x,t)=
psi(t)^{-g} \phi(x / psi(t))}
This model is used for space-time modelling where the spatial
component is isotropic.  Here\cr
\eqn{\phi} is the \code{gencauchy} model if \eqn{\kappa_3=1}{c=1}.\cr
The parameters \eqn{kappa_1}{a} and \eqn{kappa_2}{b}
are the respective parameters for the model.
The function \eqn{\psi}{psi} satisfies\cr
\eqn{\psi^2(t) = (t^{\kappa_4}+1)^{\kappa_5}}{psi^2(t) =
(t^d+1)^e} if \eqn{\kappa_6=1}{f=1}\cr
\eqn{\psi^2(t) = \frac{\kappa_5^{-1}t^{\kappa_4}+1}{t^{\kappa_4}+1}
}{psi^2(t) = (e^{-1} t^d+1)/(t^d+1)} if \eqn{\kappa_6=2}{f=2}\cr
\eqn{\psi^2(t) =-\log(t^{\kappa_4}+\kappa_5^{-1})/
\log\kappa_5}{psi^2(t)=-\log(t^d+1/e)/log e}
if \eqn{\kappa_6=3}{f=3}\cr
The parameter \eqn{\kappa_7}{g} must be greater than or equal to
the spatial dimension of the field. Furthermore, \eqn{\kappa_4\in
(0,2]}{d in (0,2]}  and \eqn{\kappa_5\in
(0,1]}{e in (0,1]}.
Necessarily, \code{dim>=2}.
The spatial dimension must be \code{>=1}.

\item \code{nugget}
\deqn{C(x)=1_{\{0\}}(x)}{1(x==0)}
Here, either \code{param[2]}, the \code{variance},
or \code{param[3]}, the \code{nugget}, must be zero.

\item \code{penta}
\deqn{C(x)= \left(1 - \frac{22}3 x^2 +33 x^4 -
\frac{77}2 x^5 + \frac{33}2
x^7 -\frac{11}2 x^9 + \frac 56 x^{11}
\right)1_{[0,1]}(x)}{C(x)=
1 - 22/3 x^2 +33 x^4
- 77/2 x^5 + 33/2 x^7 - 11/2 x^9 + 5/6 x^11  if 0<=x<=1,
0 otherwise}
valid only for dimensions less than or equal to 3.
This is a 4 times differentiable covariance functions with compact
support.
%(See Chiles&Delfiner, 1998)

\item \code{power}
\deqn{C(x)= (1-x)^\kappa 1_{[0,1]}(x)}{C(x)=
(1-x)^a   if 0<=x<=1, 0 otherwise}
This covariance function is valid for dimension \eqn{d}{d} if
\eqn{\kappa\ge\frac{d+1}2}{a >= (d+1)/2}.
For \eqn{\kappa=1}{a=1} we get the well-known triangle (or tent)
model, which is valid on the real line, only.
% proposition 3.8 in phd thesis tilmann gneiting
% Golubov, Zastavnyi

\item powered exponential\cr
See \code{stable}.

\item \code{qexponential}
\deqn{C(x)=\frac{2 e^{-x}-\kappa e^{-2x}}{2-\kappa}}{
C(x) = (2 exp(-x)-a exp(-2x))/(2-a)}
The parameter \eqn{\kappa}{a} takes values in \eqn{[0,1]}{[0,1]}.

% \item rational quadratic model\cr
%  See \code{cauchy} for \eqn{\kappa=1}{a=1}.
% (Cressie)

\item \code{spherical}
\deqn{C(x)=\left(1- \frac32 x+\frac 12 x^3\right)
1_{[0,1]}(x)}{C(x)=
1 - 1.5 x + 0.5 x^3  if 0<=x<=1, 0 otherwise}
This isotropic covariance function is valid only for dimensions
less than or equal to 3.

\item \code{stable}
\deqn{C(x)=\exp\left(-x^\kappa\right)}{C(x)=exp(-x^a)}
The parameter \eqn{\kappa}{a} is in \eqn{[0,2]}{[0,2]}.
See \code{exponential} and \code{gaussian} for special cases.

\item symmetric stable\cr
See \code{stable}.

\item tent model\cr
See \code{power}.

\item triangle\cr
See \code{power}.

\item \code{wave}
\deqn{C(x)=\frac{\sin x}x, \quad x>0}{C(x)=sin(x)/x if x>0}
This isotropic covariance function is valid only for dimensions less
than or equal to 3.
It is a special case of the \code{bessel} model
(for \eqn{\kappa}{a}\eqn{=0.5}).

\item \code{whittlematern}
\deqn{C(x)=2^{1-\kappa} \Gamma(\kappa)^{-1} x^\kappa
K_\kappa(x)}{C(x)=2^{1-a} Gamma(a)^{-1} x^a K_a(x),
}
The parameter \eqn{\kappa}{a} is positive.
\cr
This is the model of choice if the smoothness of a random field is to
be parametrised: if \eqn{\kappa\ge}{a>=}\eqn{(2m+1)/2} then the
graph is \eqn{m} times differentiable.

The model is a special case of the
\code{hyperbolic} model (for \eqn{\kappa_3=0}{c=0} there).

}

Let \eqn{\rm cov}{cov} be a model given in standard notation.
Then the covariance model
applied with arbitrary variance and scale equals
\deqn{\rm \qquad variance * \rm cov( (\cdot)/ scale).
}{variance * cov( (.)/scale).}

For a given covariance function \eqn{\rm cov}{cov} the variogram
\eqn{\gamma}{gamma} equals
\deqn{\gamma(x) = {\rm cov}(0) - {\rm cov}(x).}{
gamma(x) = cov(0) - cov(x).}

Note that the value of the covariance function or variogram
depends also on
\code{\link{RFparameters}()\$PracticalRange}.  If the latter is
\code{TRUE} and the covariance model is isotropic
then the covariance function is internally
rescaled such that cov\eqn{(1)\approx 0.05}{(1)~=0.05} for standard
parameters (\code{scale==1}).

The model and the parameters can be specified by three different
forms; the first \sQuote{standard} form allows for the specification of the
covariance model as given above for an isotropic random
field.  The second form defines isotropic nested models using
matrices.  The third
form allows for defining anisotropic and/or space-time models
using lists;
here any basic models can arbitrarily be combined by multiplication and
summation.
\itemize{
\item \code{model} is a string; \code{param} is a vector of the form
\code{param=c(mean,variance,nugget,scale,...)}. (These components
might be given separately or bound to a simple list passed to
\code{model}.)

The first component of \emph{param} is reserved for the \code{mean}
of a random field and thus ignored in the evaluation of the covariance
function or variogram.  The parameters mean, variance, nugget, and scale
must be given in this order; additional
parameters have to be supplied in case of a parametrised class of
models (e.g. \code{hyperbolic}, see below),
in the order \eqn{\kappa_1}{a}, \eqn{\kappa_2}{b}, \eqn{\kappa_3}{c}.

Let \eqn{\rm cov}{cov} be a model given in standard notation.
Then the covariance model
applied with arbitrary variance,
nugget, and scale equals
\deqn{\rm \qquad nugget + variance * \rm cov( (\cdot)/ scale).
}{nugget + variance * cov( (.)/scale).}

Some models allow certain parameter combinations only for certain
dimensions.  As any model valid in \eqn{d}{d} dimensions is also valid in 1
dimension, the default in \code{CovarianceFct} and \code{Variogram}
is \code{dim=1}.

\item \code{model} is a string; \code{param} is a matrix with columns
of the form \code{c(variance, scale, ...)}.

Except that the entries for the \code{mean} and the \code{nugget}
are missing all explanations given above also apply here.
Each column defines a summand of the nested model. A nugget effect
is indicated by \code{scale=0}; possibly additional parameters
are ignored.

\item \code{model} is a list as specified below; \code{param} is
missing.

\code{model = list(l.1, OP.1, l.2, OP.2, ..., l.n)}
where \eqn{n} is at most 10. The lists \code{l.i}
are all either of the  form \code{l.i =
list(model=,var=,kappas=,scale=,method=)}
or of the form  \code{l.i = list(model=,var=,kappas=,aniso=,method=)}.
\code{model} is a string; \code{var} gives the variance;
\code{scale} is a scalar whereas \code{aniso} is a \eqn{d \times
d}{d x d} matrix, which is multiplied from the right to the
\eqn{(n\times d)}{n x d} matrix of points;
at the transformed points the values of the (isotropic)
random field (with scale 1) are
calculated. The dimension \eqn{d} of matrix must match the
number of columns of \code{x}. The models given by \code{l.i}
can be combined by \code{OP.i="+"} or \code{OP.i="*"}.
\code{method} is ignored here; it can be set in \code{\link{GaussRF}}.
}
}
\value{
\code{CovarianceFct} returns a vector of values of the covariance function.

\code{Variogram} returns a vector of values of the variogram model.

\code{PrintModelList} prints a table of the currently implemented covariance
functions and the matching methods.
\code{PrintModelList} returns \code{NULL}.

\code{GetModelNames} returns a list of implemented models
}
\references{
Overviews:
\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. (2002) Models for stationary max-stable
random fields. \emph{Extremes} \bold{5}, 33-44.

\item Yaglom, A.M. (1987) \emph{Correlation Theory of Stationary and
Related Random Functions I, Basic Results.}
New York: Springer.

\item Wackernagel, H. (1998) \emph{Multivariate Geostatistics.} Berlin:
Springer, 2nd edition.
}

Cauchy models, generalisations and extensions
\itemize{
\item Gneiting, T. and Schlather, M. (2004)
Stochastic models which separate fractal dimension and Hurst effect.
\emph{SIAM review} \bold{46}, 269-282.% see also lgd
}

Gneiting's models
\itemize{
\item Gneiting, T. (1999)
Correlation functions for atmospheric data analysis.
\emph{Q. J. Roy. Meteor. Soc., Part A} \bold{125}, 2449-2464.
}

Holeeffect model
\itemize{
\item Zastavnyi, V.P. (1993)
Positive definite functions depending on a norm.
\emph{Russian Acad. Sci. Dokl. Math.} \bold{46}, 112-114.
}

Hyperbolic model
\itemize{
\item Shkarofsky, I.P. (1968) Generalized turbulence space-correlation and
wave-number spectrum-function pairs. \emph{Can. J. Phys.} \bold{46},
2133-2153.
}

lgd
\itemize{
\item Gneiting, T. and Schlather, M. (2004)
Stochastic models which separate fractal dimension and Hurst effect.
}

nsst
\itemize{
\item Gneiting, T. (2001) Nonseparable, stationary covariance
functions for space-time data, \emph{JASA} \bold{97}, 590-600.

\item Gneiting, T. and  Schlather, M. (2001)
Space-time covariance models.
In El-Shaarawi, A.H. and Piegorsch, W.W.:
\emph{The Encyclopedia of Environmetrics.} Chichester: Wiley.
}

Power model
\itemize{
\item Golubov, B.I. (1981) On Abel-Poisson type and Riesz means,
\emph{Analysis Mathematica} \bold{7}, 161-184.

\item  Zastavnyi, V.P. (2000) On positive definiteness of some
functions, \emph{J. Multiv. Analys.} \bold{73}, 55-81.
}

fractalB
\itemize{
\item  Stein, M.L. (2002)
Fast and exact simulation of fractional Brownian surfaces.
{\emph J. Comput. Graph. Statist.} \bold{11}, 587-599.
}
}
\author{Martin Schlather, \email{schlath@hsu-hh.de}
\url{http://www.unibw-hamburg.de/WWEB/math/schlath/schlather.html};

Yindeng Jiang \email{jiangyindeng@gmail.com} (circulant embedding
methods \sQuote{cutoff} and \sQuote{intrinsic})}

\examples{
PrintModelList()
x <- 0:100

# the following five model definitions are the same!
## (1) very traditional form
(cv <- CovarianceFct(x, model="bessel", c(NA,2,1,5,0.5)))

## (2) traditional form in list notation
model <- list(model="bessel", param=c(NA,2,1,5,0.5))
cv - CovarianceFct(x, model=model)

## (3) nested model definition
cv - CovarianceFct(x, model="bessel",
param=cbind(c(2, 5, 0.5), c(1, 0, 0)))

#### most general notation in form of lists
## (4) isotropic notation
model <- list(list(model="bessel", var=2, kappa=0.5, scale=5),
"+",
list(model="nugget", var=1))
cv - CovarianceFct(x, model=model)

## (5) anisotropic notation
model <- list(list(model="bessel", var=2, kappa=0.5, aniso=0.2),
"+",
list(model="nugget", var=1, aniso=1))
cv - CovarianceFct(as.matrix(x), model=model)

# The model gneitingdiff was defined in RandomFields v1.0.
# This isotropic covariance function is valid for dimensions less
# than or equal to 3 and has two positive parameters.
# It is a class of models with compact support that allows for
# smooth parametrisation of the differentiability up to order 6.
# The former model gneitingdiff' must now be coded as
gneitingdiff <-  function(p){
list(list(m="gneiting", v=p[2], s=p[6]*p[4]*10*sqrt(2)/47),
"*",
list(m="whittle", k=p[5], v=1.0, s=p[4]),
"+",
list(m="nugget", v=p[3]))
}
# and then
param <- c(NA, runif(5,max=10)) ## as usual, here an example
CovarianceFct(x,gneitingdiff(param))
## instead of formerly CovarianceFct(x,"gneitingdiff",param)
}
\keyword{spatial}

`

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