covRadial: Creator for the Class '"covRadial"'
In kergp: Gaussian Process Laboratory

Description Usage Arguments Details Value Note References See Also Examples

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

   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",
             ...)

`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`.

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

K(x, x') = sigma^2 * k1(r)

where k1(r) is a suitable correlation kernel for a one-dimensional input, and r is given by

r = sigma2 * sqrt(sum((x[l] - x'[l])^2 / theta[l]^2))

In this default form, the radial kernel depends on d + 1 parameters: the ranges theta[l] > 0 and the variance sigma2.

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

A correlation version uses sigma2 = 1. This is obtained by using cov = "corr".

Finally, the correlation kernel k1(r) can depend on a "shape" parameter, e.g. have the form k1(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.

An object with class "covRadial".

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 k1(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.

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

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

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))