covmodel {constrainedKriging}  R Documentation 
Create isotropic covariance model
Description
Function to generate isotropic covariance models, or add an isotropic covariance model to an existing isotropic model.
Usage
covmodel(modelname, mev, nugget,variance, scale, parameter, add.covmodel)
## S3 method for class 'covmodel'
print(x, ...)
Arguments
modelname 
a character scalar with the name of an isotropic
covariance model, see Details for a list of implemented models. A
call of 
mev 
a numeric scalar, variance of the measurement error. 
nugget 
a numeric scalar, variance of microstructure white noise process with range smaller than the minimal distance between any pair of support data. 
variance 
a numeric scalar, partial sill of the covariance model. 
scale 
a numeric scalar, scale ("range") parameter of the covariance model. 
parameter 
a numeric vector of further covariance parameters, missing
for some model like 
add.covmodel 
an object of the class 
x 
a covariance model generated by 
... 
further printing arguments 
Details
The name and parametrisation of the covariance models originate from the
function CovarianceFct
of the archived package RandomFields,
version 2.0.71.
The following isotropic covariance functions are implemented (equations
taken from help page of function CovarianceFct
of archived package
RandomFields, version 2.0.71, note that the variance and range
parameters are equal to 1 in the following formulae and h
is the
lag distance.):

bessel
C(h)=2^a \Gamma(a+1)h^{a} J_a(h)
For a 2dimensional region, the parameter
a
must be greater than or equal to 0. 
cauchy
C(h)=\left(1+h^2\right)^{a}
The parameter
a
must be positive. 
cauchytbm
C(h)= (1+(1b/3)h^a)(1+h^a)^{(b/a1)}
The parameter
a
must be in (0,2] andb
positive. The model is valid for 3 dimensions. It allows for simulating random fields where fractal dimension and Hurst coefficient can be chosen independently. 
circular
C(h)= \left(1\frac 2\pi \left(h \sqrt{1h^2} + \arcsin(h)\right)\right) 1_{[0,1]}(h)
This isotropic covariance function is valid only for dimensions less than or equal to 2.

constant
C(h)=1

cubic
C(h)=(1 7h^2+8.75h^33.5h^5+0.75 h^7)1_{[0,1]}(h)
This model is valid only for dimensions less than or equal to 3. It is a 2 times differentiable covariance functions with compact support.

dampedcosine
(hole effect model)C(h)= e^{a h} \cos(h)
This model is valid for 2 dimensions iff
a \ge 1
. 
exponential
C(h)=e^{h}
This model is a special case of the
whittle
model (fora=0.5
) and thestable
model (fora = 1
). 
gauss
C(h)=e^{h^2}
This model is a special case of the
stable
model (fora=2
). Seegneiting
for an alternative model that does not have the disadvantages of the Gaussian model. 
gencauchy
(generalisedcauchy
)C(h)= \left(1+h^a\right)^{(b/a)}
The parameter
a
must be in (0,2] andb
positive. This model allows for random fields where fractal dimension and Hurst coefficient can be chosen independently. 
gengneiting
(generalisedgneiting
) Ifa=1
and let\beta = b+1
thenC(h)=\left(1+\beta h\right) (1h)^{\beta} 1_{[0,1]}(h)
If
a=2
and let\beta = b+2
thenC(h)=\left(1+\beta h+\left(\beta ^21\right)h^2/3\right) (1h)^{\beta} 1_{[0,1]}(h)
If
a=3
and let\beta = b+3
thenC(h)=\left(1+\beta h+\left(2\beta ^23\right)\frac{h^2}{5} +\left(\beta ^24\right)\beta \frac{h^3}{15}\right)(1h)^{\beta} 1_{[0,1]}(h)
The parameter
a
is a positive integer; here only the casesa=1, 2, 3
are implemented. For two dimensional regions the parameterb
must greater than or equal to(2 + 2a +1)/2
. 
gneiting
C(h)=\left(1 + 8 sh + 25 (sh)^2 + 32 (sh)^3\right)(1sh)^8 1_{[0,1]}(sh)
where
s=0.301187465825
. This 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 thegaussian
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. 
hyperbolic
C(h)= c^{b}(K_{b}(a c))^{1} ( c^2 + h^2 )^{b/2} K_{b}( a [ c^2 + h^2 ]^{1/2} )
The parameters are such that
c\ge0
,a>0
andb>0,\quad
or
c>0
,a>0
andb=0,\quad
or
c>0
,a\ge0
, andb<0
.
Note that this class is overparametrised; always one of the three parametersa
,c
, and scale can be eliminated in the formula. 
lgd1
(localglobal distinguisher)C(h)= 1\frac{\beta}{a+b}h^{a}, h\le 1 \qquad \hbox{and} \qquad \frac{a}{a+b}h^{b}, h> 1
Here
b>0
anda
msut be in(0,0.5]
. The random field has for 2dimensional regions fractal dimension3  a/2
and Hurst coefficient1 b/2
forb \in (0,1]

matern
C(h)= 2^{1a} \Gamma(a)^{1} (\sqrt{2 a} h)^a K_a(\sqrt{2 a}h)
The parameter
a
must be positive. This is the model of choice if the smoothness of a random field is to be parametrised: ifa > m
then the graph ism
times differentiable. 
nugget
C(h)=1_{[0]}(h)

penta
C(h)= \left(1  \frac{22}3 h^2 +33 h^4  \frac{77}2 h^5 + \frac{33}2 h^7 \frac{11}2 h^9 + \frac 56 h^{11} \right)1_{[0,1]}(h)
valid only for dimensions less than or equal to 3. This is a 4 times differentiable covariance functions with compact support.

power
C(h)= (1h)^a 1_{[0,1]}(h)
This covariance function is valid for 2 dimensions iff
a \ge 1.5
. Fora=1
we get the wellknown triangle (or tent) model, which is valid on the real line, only. 
qexponential
C(h)= ( 2 e^{h}  a e^{2x} ) / ( 2  a )
The parameter
a
must be in[0,1]
. 
spherical
C(h)=\left(1 1.5 h+0.5 h^3\right) 1_{[0,1]}(h)
This covariance function is valid only for dimensions less than or equal to 3.

stable
C(h)=\exp\left(h^a\right)
The parameter
a
must be in(0,2]
. Seeexponential
andgaussian
for special cases. 
wave
C(h)=\frac{\sin h}{h}, \quad h>0 \qquad \hbox{and } \qquad C(0)=1
This isotropic covariance function is valid only for dimensions less than or equal to 3. It is a special case of the
bessel
model (fora=0.5
). 
whittle
C(h) = 2^{1a} \Gamma(a)^{1} h^a K_a(h)
The parameter
a
must be positive. This is the model of choice if the smoothness of a random field is to be parametrised: ifa > m
then the graph ism
times differentiable.
The default values of the arguments
mev
,
nugget
,
variance
and scale
are eq 0.
Value
an object of the class covmodel
that defines a covariance model.
Author(s)
Christoph Hofer, christoph.hofer@alumni.ethz.ch
Examples
# table with all available covariance models and their
# parameters
covmodel()
# exponential model without a measurement error and without a nugget,
# partial sill = 10, scale parameter = 15
covmodel(modelname = "exponential", variance = 10, scale = 15)
# exponential model with a measurement error ( mev = 0.5) and a
# nugget (nugget = 2.1), exponential partial sill (variance = 10)
# and scale parameter = 15
covmodel(modelname = "exponential", mev = 0.5, nugget = 2.1,
variance = 10, scale = 15)