# Creator for the Class `"covRadial"` Creator for the class `"covRadial"`, which describes **radial kernels**. ```r 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`. ## Returns An object with class `"covRadial"`. ## Details A radial kernel on the $d$-dimensional Euclidean space takes the form $$ K(\mathbf{x},\,\mathbf{x}') = \sigma^2 k_1(r)K(x, x') =sigma^2 * k1(r) $$ where $k1(r)$ is a suitable correlation kernel for a one-dimensional input, and $r$ is given by $$ r = \left\{\sum_{\ell = 1}^d [x_\ell - x'_\ell]^2 / \theta_\ell^2\right\}^{1/2}.r = sigma2 * sqrt(sum((x[l] - x'[l])^2 / theta[l]^2)) $$ 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 c("$k1(r,\n$", "$ 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. ## 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 $k1(r)$ 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. ## See Also `k1Fun1Exp`, `k1Fun1Matern3_2`, `k1Fun1Matern5_2` or `k1Fun1Gauss` for examples of functions that can be used as values for the `k1Fun1` formal. ## References Gregory Fassauher and Michael McCourt (2016) **Kernel-based Approximation Methods using MATLAB**. World Scientific. ## Examples ```r 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)) ```

covRadial function