R: Class of tensor-product spatial covariances

covTensorProduct-class {DiceKriging} | R Documentation

## 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.

[Package DiceKriging version 1.5.8 Index]