R: Creator for the Class '"covRadial"'

covRadial {kergp}

R Documentation

Creator for the Class `"covRadial"`

Description

Creator for the class "covRadial", which describes radial kernels.

Usage


   covRadial(k1Fun1 = k1Fun1Gauss,
             cov = c("corr", "homo"), 
             iso = 0, hasGrad = TRUE,
             inputs = NULL, d = NULL,
             parNames, par = NULL,
             parLower = NULL, parUpper = NULL,
             label = "Radial kernel",
             ...)

Arguments

`k1Fun1`	A function of a scalar numeric variable, and possibly of an extra "shape" parameter. This function should return the first-order derivative or the two-first order derivatives as an attribute with name `"der"` and with a matrix content. When an extra shape parameter exists, the gradient should also be returned as an attribute with name `"gradient"`, see Examples later. The name of the function can be given as a character string.
`cov`	A character string specifying the kind of covariance kernel: correlation kernel (`"corr"`) or kernel of a homoscedastic GP (`"homo"`). Partial matching is allowed.
`iso`	Integer. The value `1L` corresponds to an isotropic covariance, with all the inputs sharing the same range value.
`hasGrad`	Integer or logical. Tells if the value returned by the function `k1Fun1` has an attribute named `"der"` giving the derivative(s).
`inputs`	Character. Names of the inputs.
`d`	Integer. Number of inputs.
`par`, `parLower`, `parUpper`	Optional numeric values for the lower bounds on the parameters. Can be `NA` for `par`, can be `-Inf` for `parLower` and `Inf` for `parUpper`.
`parNames`	Names of the parameters. By default, ranges are prefixed `"theta_"` in the non-iso case and the range is named `"theta"` in the iso case.
`label`	A short description of the kernel object.
`...`	Other arguments passed to the method `new`.

Details

A radial kernel on the d-dimensional Euclidean space takes the form

K(\mathbf{x},\,\mathbf{x}') = \sigma^2 k_1(r)

where k_1(r) is a suitable correlation kernel for a one-dimensional input, and r is given by

r = \left\{\sum_{\ell = 1}^d [x_\ell - x'_\ell]^2 / \theta_\ell^2 \right\}^{1/2}.

In this default form, the radial kernel depends on d + 1 parameters: the ranges \theta_\ell >0 and the variance \sigma^2.

An isotropic form uses the same range \theta for all inputs, i.e. sets \theta_\ell = \theta for all \ell. This is obtained by using iso = TRUE.

A correlation version uses \sigma^2 = 1. This is obtained by using cov = "corr".

Finally, the correlation kernel k_1(r) can depend on a "shape" parameter, e.g. have the form k_1(r;\,\alpha). The extra shape parameter \alpha will be considered then as a parameter of the resulting radial kernel, making it possible to estimate it by ML along with the range(s) and the variance.

Value

An object with class "covRadial".

Note

When k1Fun1 has more than one formal argument, its arguments with position > 1 are assumed to be "shape" parameters of the model. Examples are functions with formals function(x, shape = 1.0) or function(x, alpha = 2.0, beta = 3.0), corresponding to vector of parameter names c("shape") and c("alpha", "beta"). Using more than one shape parameter has not been tested yet.

Remind that using a one-dimensional correlation kernel k_1(r) here does not warrant that a positive semi-definite kernel will result for any dimension d. This question relates to Schoenberg's theorem and the concept of completely monotone functions.

References

Gregory Fassauher and Michael McCourt (2016) Kernel-based Approximation Methods using MATLAB. World Scientific.

Examples

set.seed(123)
d <- 2; ng <- 20
xg <- seq(from = 0, to = 1, length.out = ng)
X <- as.matrix(expand.grid(x1 = xg, x2 = xg))

## ============================================================================
## A radial kernel using the power-exponential one-dimensional
## function
## ============================================================================

d <- 2
myCovRadial <- covRadial(k1Fun1 = k1Fun1PowExp, d = 2, cov = "homo", iso = 1)
coef(myCovRadial)
inputNames(myCovRadial) <- colnames(X)
coef(myCovRadial) <- c(alpha = 1.8, theta = 2.0, sigma2 = 4.0)
y <- simulate(myCovRadial, X = X, nsim = 1)
persp(x = xg, y = xg, z = matrix(y, nrow = ng))

## ============================================================================
## Define the inverse multiquadric kernel function. We return the first two
## derivatives and the gradient as attributes of the result.
## ============================================================================

myk1Fun <- function(x, beta = 2) {
    prov <- 1 + x * x
    res <- prov^(-beta)
    der <- matrix(NA, nrow = length(x), ncol = 2)
    der[ , 1] <- - beta * 2 * x * res / prov
    der[ , 2] <- -2 * beta * (1 - (1 + 2 * beta) * x * x) * res / prov / prov
    grad <- -log(prov) * res
    attr(res, "gradient") <- grad
    attr(res, "der") <- der
    res
}

myCovRadial1 <- covRadial(k1Fun1 = myk1Fun, d = 2, cov = "homo", iso = 1)
coef(myCovRadial1)
inputNames(myCovRadial1) <- colnames(X)
coef(myCovRadial1) <- c(beta = 0.2, theta = 0.4, sigma2 = 4.0)
y1 <- simulate(myCovRadial1, X = X, nsim = 1)
persp(x = xg, y = xg, z = matrix(y1, nrow = ng))

[Package kergp version 0.5.7 Index]

Creator for the Class "covRadial"