S4 class of tensor-product (or separable) covariances.
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.
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) |
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.
signature(x = "covTensorProduct") Print covariance function. See show,km-method.
signature(x = "covTensorProduct") Get the coefficients of the covariance function.
% \item{coef<-}{\code{signature(x = "covTensorProduct")} Set the coefficients of the covariance function. }
O. Roustant, D. Ginsbourger
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.
covStruct.create to construct a covariance structure.