\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{constant}\cr
    Identically constant. Any scale parameter is ignored.
    
    \item \code{cone}\cr
    This model is used only for methods based on marked point processes
    (see \command{\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{cutoff} (hypermodel, see also below)
    \deqn{C(x)=\phi(x), 0\le x \le \kappa_2}{C(x) = phi(x),   0 <= x <= b}
    
    \deqn{C(x)=b_1 (r^{\kappa_3} - x)^{2 \kappa_3}, \kappa_2\le x \le
      \kappa_2\kappa_3}{C(x) = b_0 (r - x)^{2 c},   b <= x <= bc}
    
    \deqn{C(x)=0,  \kappa_2\kappa_3\le x}{C(x) = 0,   bc <= x}
   
    The cutoff model is a functional of the covariance function
    \eqn{\phi}{phi};
    see below for general comments on hypermodels.
    
    The first parameter, \eqn{\kappa_1}{a}, gives the number of
    subsequent models that build \eqn{\phi}{phi};
    \eqn{\kappa_2\ge0}{b>0}, \eqn{\kappa_3>0}{c>0}.
    The parameters \eqn{r} and \eqn{b_0}
    are chosen internally such that \eqn{C} is a smooth function.

    The cutoff-model is not a valid covariance function for any
    choice of phi and the parameters. Only some few
    necessary conditions can be checked.
    
    For certain models \eqn{\phi}{phi}, i.e. \code{stable},
    \code{whittle} and \code{gencauchy}, some sufficient conditions
    are known.
    
    \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}
    \deqn{C(x)=e^{-x},  \quad x\ge0}{C(x)=exp(-x)}
    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
    \command{\link{RFMethods}}) is
    \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 
    \eqn{\kappa_2>0,\quad}{b>0,   }
    or\cr
    \eqn{\kappa_3>0}{c>0 },  \eqn{\kappa_1>0}{a>0 }  and 
    \eqn{\kappa_2=0,\quad}{b=0,   }
    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)=
      1-\frac\beta{\alpha+\beta}|x|^{\alpha}, |x|\le 1 \qquad \hbox{and} \qquad
    \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 \code{Stein} (hypermodel, see also below)
    \deqn{C(x)=a_0 + a_2 x^2 + \phi(x), 0\le x \le \kappa_2}{C(x) = a_0 + a_2
      x^2 + phi(x),   0 <= x <= b}
    
    \deqn{C(x)=b_1 (\kappa_3 - x)^3/x, \kappa_2\le x \le
      \kappa_2\kappa_3}{C(x) = b_0 (c - x)^3/x,   b <= x <= bc}
    
    \deqn{C(x)=0,  \kappa_2\kappa_3\le x}{C(x) = 0,   bc <= x}
   
    The Stein model is a functional of the covariance function
    \eqn{\phi}{phi};
    see below for for general comments on hypermodels.
    
    The first parameter, \eqn{\kappa_1}{a}, gives the number of
    subsequent models that build \eqn{\phi}{phi};
    \eqn{\kappa_2\ge0}{b>0}, \eqn{\kappa_3\ge1}{c>=1}.
    The parameters \eqn{a_0}, \eqn{a_2} and \eqn{b_0}
    are chosen internally such that \eqn{C} is a smooth function.

    The Stein-model is not a valid covariance function for any
    choice of phi and the parameters. Only some few
    necessary conditions can be checked.
    
    For certain models \eqn{\phi}{phi}, i.e. \code{stable},
    \code{whittle}, \code{gencauchy}, and the variogram
    model \code{fractalB}
    some sufficient conditions are known.
    
    \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
  \command{\link{RFparameters}}\code{()$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
    \command{\link{GaussRF}}.

    \item Hypermodels\cr
    hypermodels are functions or functionals of covariance functions
    or variograms. The first parameter is always the number of the following
    covariance models included.
    
    The hypermodel inherits the anisotropy parameters (or the scale
    parameter) from the first submodel. The given anisotropy parameters
    are ignored.
    
    \bold{Important!} Hyper models are in an experimental stage:\cr
    (i) the (current) algorithm does not allow for a complete
    check whether the parameters for a hypermodel are well chosen.
    So, only use parameter combinations for which you are sure they lead to a
    positive definite function.
    \cr
    (ii) behaviour and parameters may change in future version!
  }
}
\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.
    \emph{SIAM review} % see also cauchy
  }

  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})}
\seealso{\command{\link{EmpiricalVariogram}},
  \command{\link{GetPracticalRange}},
  \command{\link{parameter.range}},
  \code{\link{RandomFields}},
  \command{\link{RFparameters}},
  \command{\link{ShowModels}}.}

\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]),
       "*",
       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}