covTensorProduct-class {DiceKriging}

Class of tensor-product spatial covariances

Description

S4 class of tensor-product (or separable) covariances.

Value

covTensorProduct

separable covariances depending on 1 set of parameters, such as Gaussian, exponential, Matern with fixed nu... or on 2 sets of parameters, such as power-exponential.

Objects from the Class

A d-dimensional tensor product (or separable) covariance kernel C(x,y) is the tensor product of 1-dimensional covariance kernels : C(x,y) = C(x1,y1)C(x2,y2)...C(xd,yd).

In 1-dimension, the covariance kernels are parameterized as in (Rasmussen, Williams, 2006). Denote by theta the range parameter, p the exponent parameter (for power-exponential covariance), s the standard deviation, and h=|x-y|. Then we have C(x,y) = s^2 * k(x,y), with:

Gauss	k(x,y) = exp(-1/2*(h/theta)^2)
Exponential	k(x,y) = exp(-h/theta)
Matern(3/2)	k(x,y) = (1+sqrt(3)*h/theta)*exp(-sqrt(3)*h/theta)
Matern(5/2)	k(x,y) = (1+sqrt(5)*h/theta+(1/3)*5*(h/theta)^2)
	*exp(-sqrt(5)*h/theta)
Power-exponential	k(x,y) = exp(-(h/theta)^p)

Slots

d:

Object of class "integer". The spatial dimension.

name:

Object of class "character". The covariance function name. To be chosen between "gauss", "matern5_2", "matern3_2", "exp", and "powexp"

paramset.n:

Object of class "integer". 1 for covariance depending only on the ranges parameters, 2 for "powexp" which also depends on exponent parameters.

var.names:

Object of class "character". The variable names.

sd2:

Object of class "numeric". The variance of the stationary part of the process.

known.covparam:

Object of class "character". Internal use. One of: "None", "All".

nugget.flag:

Object of class "logical". Is there a nugget effect?

nugget.estim:

Object of class "logical". Is the nugget effect estimated or known?

nugget:

Object of class "numeric". If there is a nugget effect, its value (homogeneous to a variance).

param.n:

Object of class "integer". The total number of parameters.

range.n:

Object of class "integer". The number of range parameters.

range.names:

Object of class "character". Names of range parameters, for printing purpose. Default is "theta".

range.val:

Object of class "numeric". Values of range parameters.

shape.n:

Object of class "integer". The number of shape parameters (exponent parameters in "powexp").

shape.names:

Object of class "character". Names of shape parameters, for printing purpose. Default is "p".

shape.val:

Object of class "numeric". Values of shape parameters.

Methods

show

signature(x = "covTensorProduct") Print covariance function. See show,km-method.

coef

signature(x = "covTensorProduct") Get the coefficients of the covariance function.

Author(s)

O. Roustant, D. Ginsbourger

References

N.A.C. Cressie (1993), Statistics for spatial data, Wiley series in probability and mathematical statistics.

C.E. Rasmussen and C.K.I. Williams (2006), Gaussian Processes for Machine Learning, the MIT Press, http://www.gaussianprocess.org/gpml/

M.L. Stein (1999), Interpolation of spatial data, some theory for kriging, Springer.

See Also

covStruct.create to construct a covariance structure.