# Creator for the Class `"covANOVA"` Creator for the class `"covANOVA"`. ```r covANOVA(k1Fun1 = k1Fun1Gauss, cov = c("corr", "homo"), iso = 0, iso1 = 1L, hasGrad = TRUE, inputs = NULL, d = NULL, parNames, par = NULL, parLower = NULL, parUpper = NULL, label = "ANOVA kernel", ...) ``` ## Arguments - `k1Fun1`: A kernel function of a **scalar** numeric variable, and possibly of an extra "shape" parameter. This function can also 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 can 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 value of the variance parameter $\delta$ for the covariance kernel. Contrarily to other kernel classes, that parameter is not equal to the variance. Thus, mind that choosing (`"corr"`) corresponds to $\delta=1$ but **does not** correspond to a correlation kernel, see details below. Partial matching is allowed. - `iso`: Integer. The value `1L` corresponds to an isotropic covariance, with all the inputs sharing the same range value. - `iso1`: Integer. This applies only when `k1Fun1` contains one or more parameters that can be called 'shape' parameters. At now, only one such parameter can be found in `k1Fun1` and consequently `iso1` must be of length one. With `iso1 = 0` the shape parameter in `k1Fun1` will generate `d` parameters in the `covANOVA` object with their name suffixed by the dimension. When `iso1` is `1` only one shape parameter will be created in the `covANOVA` object. - `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. - `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. - `par`: Numeric values for the parameters. Can be `NA`. - `parLower`: Numeric values for the lower bounds on the parameters. Can be `-Inf`. - `parUpper`: Numeric values for the upper bounds on the parameters. Can be `Inf`. - `label`: A short description of the kernel object. - `...`: Other arguments passed to the method `new`. ## Returns An object with class `"covANOVA"`. ## Details A ANOVA kernel on the $d$-dimensional Euclidean space takes the form $$ K(\mathbf{x},\,\mathbf{x}') = \delta^2\prod_{\ell = 1}^d (1 + \tau_\ell^2 \kappa(r_\ell))K(x1, x2) = \delta^2 * prod_l (1 + \tau[l]^2 k1(r[l]) $$ where $k1(r)$ is a suitable correlation kernel for a one-dimensional input, and $r[l]$ is given by c("$r[l] = (x2[l] -\n$", "$ x2[l]) / \\theta[l]$") for $l =1$ to $d$. In this default form, the ANOVA kernel depends on $2d + 1$ parameters: the **ranges** $\theta[l] > 0$, the **variance ratios** $\tau[l]^2$, and the variance parameter $\delta^2$. 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 c("$\\delta^2 =\n$", "$ 1$"). This is obtained by using `cov = "corr"`. Mind that it does not correspond to a correlation kernel. Indeed, in general, the variance is equal to $$ K(\mathbf{x},\,\mathbf{x}) = \delta^2\prod_{\ell = 1}^d (1 + \tau_\ell^2).K(x, x) = \delta^2 * prod_l (1 + \tau[l]^2). $$ 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 ANOVA kernel, making it possible to estimate it by ML along with the range(s) and the variance. ## Examples ```r ## Not run: if (require(DiceKriging)) { ## a 16-points factorial design and the corresponding response d <- 2; n <- 16; x <- seq(from = 0.0, to = 1.0, length.out = 4) X <- expand.grid(x1 = x, x2 = x) y <- apply(X, 1, DiceKriging::branin) ## kriging model with matern5_2 covariance structure, constant ## trend. A crucial point is to set the upper bounds! mycov <- covANOVA(k1Fun1 = k1Fun1Matern5_2, d = 2, cov = "homo") coefUpper(mycov) <- c(2.0, 2.0, 5.0, 5.0, 1e10) mygp <- gp(y ~ 1, data = data.frame(X, y), cov = mycov, multistart = 100, noise = TRUE) nGrid <- 50; xGrid <- seq(from = 0, to = 1, length.out = nGrid) XGrid <- expand.grid(x1 = xGrid, x2 = xGrid) yGrid <- apply(XGrid, 1, DiceKriging::branin) pgp <- predict(mygp, XGrid)$mean mykm <- km(design = X, response = y) pkm <- predict(mykm, XGrid, "UK")$mean c("km" = sqrt(mean((yGrid - pkm)^2)), "gp" = sqrt(mean((yGrid - pgp)^2))) } ## End(Not run) ```

covANOVA function