covmodel {constrainedKriging}

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 covmodel() without any function argument displays a table with all available models and their parameters, see Examples.

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 nugget, spherical or gauss, etc, see Details. If a model has several extra parameters, say a, b, ... then they must be given as c(a, b, ...).

add.covmodel

an object of the class covmodel that is added to the covariance model defined by modelname (see examples)

x

a covariance model generated by covmodel

...

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 2-dimensional 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+(1-b/3)h^a)(1+h^a)^{(-b/a-1)}

  The parameter a must be in (0,2] and b 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{1-h^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^3-3.5h^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 (for a=0.5) and the stable model (for a = 1).

• gauss

  C(h)=e^{-h^2}

  This model is a special case of the stable model (for a=2). See gneiting for an alternative model that does not have the disadvantages of the Gaussian model.

• gencauchy (generalised cauchy)

  C(h)= \left(1+h^a\right)^{(-b/a)}

  The parameter a must be in (0,2] and b positive. This model allows for random fields where fractal dimension and Hurst coefficient can be chosen independently.

• gengneiting (generalised gneiting) If a=1 and let \beta = b+1 then

  C(h)=\left(1+\beta h\right) (1-h)^{\beta} 1_{[0,1]}(h)

  If a=2 and let \beta = b+2 then

  C(h)=\left(1+\beta h+\left(\beta ^2-1\right)h^2/3\right) (1-h)^{\beta} 1_{[0,1]}(h)

  If a=3 and let \beta = b+3 then

  C(h)=\left(1+\beta h+\left(2\beta ^2-3\right)\frac{h^2}{5} +\left(\beta ^2-4\right)\beta \frac{h^3}{15}\right)(1-h)^{\beta} 1_{[0,1]}(h)

  The parameter a is a positive integer; here only the cases a=1, 2, 3 are implemented. For two dimensional regions the parameter b must greater than or equal to (2 + 2a +1)/2.

• gneiting

  C(h)=\left(1 + 8 sh + 25 (sh)^2 + 32 (sh)^3\right)(1-sh)^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 the 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.

• 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 and b>0,\quad or
  c>0, a>0 and b=0,\quad or
  c>0, a\ge0, and b<0.
  Note that this class is over-parametrised; always one of the three parameters a, c, and scale can be eliminated in the formula.

• lgd1 (local-global 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 and a msut be in (0,0.5]. The random field has for 2-dimensional regions fractal dimension 3 - a/2 and Hurst coefficient 1 -b/2 for b \in (0,1]

• matern

  C(h)= 2^{1-a} \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: if a > m then the graph is m 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)= (1-h)^a 1_{[0,1]}(h)

  This covariance function is valid for 2 dimensions iff a \ge 1.5. For a=1 we get the well-known 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]. See exponential and gaussian 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 (for a=0.5).

• whittle

  C(h) = 2^{1-a} \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: if a > m then the graph is m 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, [e-mail]

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)