kergp (version 0.5.7)

covRadial: 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",
             ...)

Value

An object with class "covRadial".

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.

References

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

See Also

k1Fun1Exp, k1Fun1Matern3_2, k1Fun1Matern5_2 or k1Fun1Gauss for examples of functions that can be used as values for the k1Fun1 formal.

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))