CovarianceFct returns the values of an isotropic covariance function
Variogram returns the values of an isotropic variogram model PrintModelList prints the list of currently implemented models
GetModelNames returns a list of currently implemented models
CovarianceFct(x,model,param,dim=1)
Variogram(x,model,param,dim=1)
PrintModelList()
GetModelNames()PrintModelList() for all
optionsparam=c(NA,variance,nugget,scale,...), in this order;
The dots ... stand for additional parameters of the
modelCovarianceFct returns a vector of values of the covariance function.
Variogram returns a vector of values of the variogram model.
PrintModelList prints a table of the currently implemented covariance
functions and the matching methods.
PrintModelList returns NULL. GetModelNames returns a list of implemented models
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. hyperbolic, see below),
in the order $\kappa_1$, $\kappa_2$, $\kappa_3$.
The implemented models are in standard notation of a covariance
function (variance 1,
nugget 0, scale=1) and for positive real arguments $x$:
bessel$$C(x)=2^a \Gamma(a+1)x^{-a} J_a(x)$$The parameter$\kappa$is greater than or equal to$\frac{d-2}2$, where$d$is the
dimension of the random field.cauchy$$C(x)=\left(1+x^2\right)^{-\kappa}$$The parameter$\kappa$is positive.
The model possesses two generalisations, thegencauchymodel and thehyperbolicmodel.cauchytbm$$C(x)=\left(1+\left(1-\frac{\kappa_2}{\kappa_3}
\right)x^{\kappa_1}\right)
\left(1+x^{\kappa_1}\right)^{-\frac{\kappa_2}{\kappa_1}-1}$$The parameter$\kappa_1$is in (0,2],$\kappa_2$is positive, and$\kappa_3$is an integer.
The model is valid for dimensions$d\le\kappa_3$.
It allows for simulating random fields where
fractal dimension and Hurst coefficient can be chosen
independently.
It has negative correlations for$\kappa_2>1$and large$x$.circular$$C(x)=
\left(1-\frac 2\pi
\left(x \sqrt{1-x^2} +
\arcsin(x)\right)\right)
1_{[0,1]}(x)$$This isotropic covariance function is valid only for dimensions
less than or equal to 2.cone
This model is used only for methods based on marked point processes
(seeRFMethods); it is defined only in two dimensions.
The corresponding (boolean)
function is a truncated cone with socle. The base has radius$\frac12$. The model has three parameters,$\kappa_1$,$\kappa_2$, and$\kappa_3$:
$kappa_1$gives the radius of the top circle of the cone, given
as part of the socle radius;$kappa_1\in[0,1)$.
$kappa_2$gives the height of the socle.
$kappa_3$gives the height of the truncated cone.
cubic$$C(x)=(1- 7x^2+8.75x^3-3.5x^5+0.75 x^7)1_{[0,1]}(x)$$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)exponential$$C(x)=e^{-x}, \quad x\ge0$$This model is a special case of thewhittlematernmodel
(for$\kappa=\frac12$there)
and thestableclass (for$\kappa=1$).
%% \item\code{expPLUScirc}
%% \deqn{\kappa_1 e^{-x} +
%% (1-\kappa_1)
%% \left(1-\frac2\pi
%% \left(\frac xb\sqrt{1-(x/\kappa_2)^2}+\arcsin(x/\kappa_2)\right)
%% \right) 1_{[0,1]}(x)}{
%% a exp(-x) +
%% (1-a)(1-2/pi(x/b*sqrt(1-x^2/b^2)+asin(x/b))) if
%% 0<=x<=1, 0="" otherwise}="" %%="" the="" parameter="" \eqn{\kappa_1}{a}="" is="" in="" \eqn{[0,1]}="" and="" \eqn{\kappa_2}{b}="" positive.="" this="" isotropic="" covariance="" function="" valid="" only="" for="" dimensions="" less="" than="" or="" equal="" to="" 2.<="" li="">gaussian$$C(x)=e^{-x^2}$$This model is a special case of thestableclass
(for$\kappa=2$there).
Note that the corresponding function for the random coins
method (cf. the methods based on marked point processes inRFMethods) is$$e^{- 2 x^2}.$$Seegneitingfor an alternative model that does not have
the disadvantages of the Gaussian model.gencauchy(generalisedcauchy)
$$C(x)=(1+x^{\kappa_1})^{-\kappa_2/\kappa_1}$$The parameter$\kappa_1$is in (0,2], and$\kappa_2$is positive.
This model allows for simulating random fields where
fractal dimension and Hurst coefficient can be chosen
independently.gengneiting(generalisedgneiting)
if$\kappa_1=1$then$$C(x)=\left(1+(\kappa_2+1)x\right) * (1-x)^{\kappa_2+1}
1_{[0,1]}(x)$$if$\kappa_1=2$then$$C(x)=\left(1+(\kappa_2+2)x+\left((\kappa_2+2)^2-1\right)x^2/3\right)
(1-x)^{\kappa_2+2} 1_{[0,1]}(x)$$if$\kappa_1=3$then$$C(x)=\left(1+(\kappa_2+3)x+\left(2(\kappa_2+3)^2-3\right)x^2/5
+\left((\kappa_2+3)^2-4\right)(\kappa_2+3)x^3/15\right)(1-x)^{\kappa_2+3}
1_{[0,1]}(x)$$The parameter$\kappa_1$is a positive integer; here only the
cases$\kappa_1=1, 2, 3$are implemented.
The parameter$\kappa_2$is greater than or equal to$(d + 2\kappa_1 +1)/2$where$d$is the
dimension of the random field.gneiting$$C(x)=\left(1 + 8 sx + 25 (sx)^2 + 32
(sx)^3\right)(1-sx)^8 1_{[0,1]}(sx)$$where$s=\frac{10\sqrt2}{47}\approx 0.3$.
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.
It is an alternative to thegaussianmodel 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.
This model is a special case ofgengneiting(for$\kappa_1=3$and$\kappa_2=5$there).gneitingdiff$$C(x)=\left(1 + 8 \frac x{\kappa_2}
+ 25 \frac {x^2}{\kappa_2^2}
+ 32 \frac {x^3}{\kappa_2^3}\right)
\left(1-\frac{x}{\kappa_2}\right)^8
\;\frac{2^{1-\kappa_1}}{\Gamma(\kappa_1)}
\,x^{\kappa_1} K_{\kappa_1}(x)1_{[0,\kappa_2]}(x)$$This isotropic covariance function is valid only for dimensions less
than or equal to 3.
The parameters$\kappa_1$and$\kappa_2$are
positive.
This class of models with compact support
allows for smooth parametrisation of the differentiability up to
order 6.holeeffect$$C(x)=e^{-\kappa x} \cos(x), \quad x\ge0$$This model is valid
for dimension 1 iff$\kappa\ge1$,
for dimension 2 iff$\kappa\ge1$,
and for dimension 3 iff$\kappa\ge \sqrt{3}$.hyperbolic$$C(x)=\frac{1}{\kappa_3^{\kappa_2}
K_{\kappa_2}(\kappa_1 \kappa_3)}
\left(\kappa_3^2 +x^2\right)^{{\kappa_2}/2}
K_{{\kappa_2}}\left(
\kappa_1 \left(\kappa_3^2 + x^2\right)^{1/2}\right), \quad
x>0$$The parameters are such that
$\kappa_3\ge0$,$\kappa_1>0$and$\kappa_2>0,\quad$or
$\kappa_3>0$,$\kappa_1>0$and$\kappa_2=0,\quad$or
$\kappa_3>0$,$\kappa_1\ge0$, and$\kappa_2<0$. note="" that="" this="" class="" is="" over-parametrised;="" always="" one="" of="" the="" three="" parameters$\kappa_1$,$\kappa_3$,="" and="" scale="" can="" be="" eliminiated="" in="" formula.="" therefore,="" these="" parameters="" should="" kept="" fixed="" any="" simulation="" study.="" model="" contains="" as="" special="" cases="" thecauchymodel, for$\kappa_3=0$and$\kappa_1=0$, respectively.power(a=1there).whittlematern.nugget$$C(x)=1_{{0}}(x)$$Here, eitherparam[2], thevariance,
orparam[3], thenugget, must be zero.pentamodel$$C(x)= \left(1 - \frac{22}3 x^2 +33 x^4 -
\frac{77}2 x^5 + \frac{33}2
x^7 -\frac{11}2 x^9 + \frac 56 x^{11}
\right)1_{[0,1]}(x)$$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)power$$C(x)= (1-x)^\kappa 1_{[0,1]}(x)$$This covariance function is valid for dimension$d$if$\kappa\ge\frac{d+1}2$.
For$\kappa=1$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, Zastavnyistable.qexponential$$C(x)=\frac{2 e^{-x}-\kappa e^{-2x}}{2-\kappa}$$The parameter$\kappa$takes values in$[0,1]$. % \item rational quadratic model\cr % See \code{cauchy} for \eqn{\kappa=1}{a=1}. % (Cressie)
spherical$$C(x)=\left(1- \frac32 x+\frac 12 x^3\right)
1_{[0,1]}(x)$$This isotropic covariance function is valid only for dimensions
less than or equal to 3.stable$$C(x)=\exp\left(-x^\kappa\right)$$The parameter$\kappa$is in$[0,2]$.
Seeexponentialandgaussianfor special cases.stable.power.power.wave$$C(x)=\frac{\sin x}x, \quad x>0$$This isotropic covariance function is valid only for dimensions less
than or equal to 3.
It is a special case of thebesselmodel
(for$\kappa=2$).whittlematern$$C(x)=2^{1-\kappa} \Gamma(\kappa)^{-1} x^\kappa
K_\kappa(x)$$The parameter$\kappa$is positive.
This is the model of choice if the smoothness of a random field is to
be parametrised. It is a special case of thehyperbolicmodel (for$\kappa_3=0$there).RFparameters()$PracticalRange. If the latter is
TRUE then the covariance function is internally
rescaled such that cov$(1)\approx 0.05$ for standard
parameters (scale==1).
Some models allow certain parameter combinations only for certain
dimensions. As any model in $d$ dimensions is also valid in 1
dimension, the default in CovarianceFct and Variogram
is dim=1.Cauchy models, generalisations and extensions
Hyperbolic model
EmpiricalVariogram,
RandomFields,
RFparameters,
ShowModels.PrintModelList()
CovarianceFct(0:100, "bessel", c(NA,2,1,5,0.5))Run the code above in your browser using DataLab