# Direct KRR Models Direct kernel ridge regression (DKRR), following the same approcah as for DKPLSR (Bennett & Embrechts 2003). The method builds kernel Gram matrices and then runs a RR algorithm on them. This is not equivalent to the "true" KRR (= LS-SVM) algorithm. latin1 ```r dkrr(X, Y, weights = NULL, lb = 1e-2, kern = "krbf", ...) ## S3 method for class 'Dkrr' coef(object, ..., lb = NULL) ## S3 method for class 'Dkrr' predict(object, X, ..., lb = NULL) ``` ## Arguments - `X`: For the main function: Training X-data ($n, p$). --- For the auxiliary functions: New X-data ($m, p$) to consider. - `Y`: Training Y-data ($n, q$). - `weights`: Weights ($n, 1$) to apply to the training observations. Internally, weights are "normalized" to sum to 1. Default to `NULL` (weights are set to $1 / n$). - `lb`: A value of regularization parameter $lambda$. - `kern`: Name of the function defining the considered kernel for building the Gram matrix. See `krbf` for syntax, and other available kernel functions. - `...`: Optional arguments to pass in the kernel function defined in `kern` (e.g. `gamma` for `krbf`). - `object`: For the auxiliary functions: A fitted model, output of a call to the main function. ## Returns For `dkrr`: - **X**: Matrix with the training X-data ($n, p$). - **fm**: List with the outputs of the RR ((`V`): eigenvector matrix of the correlation matrix (n,n); (`TtDY`): intermediate output; (`sv`): singular values of the matrix (1,n); (`lb`): value of regularization parameter $lambda$; (`xmeans`): the centering vector of X (p,1); (`ymeans`): the centering vector of Y (q,1); (`weights`): the weights vector of X-variables (p,1) - **K**: kernel Gram matrix - **kern**: kernel function - **dots**: Optional arguments passed in the kernel function For `predict.Dkrr`: - **pred**: A list of matrices ($m, q$) with the Y predicted values for the new X-data - **K**: kernel Gram matrix ($m, nlv$), with values for the new X-data For `coef.Dkrr`: - **int**: matrix (1,nlv) with the intercepts - **B**: matrix (n,nlv) with the coefficients - **df**: model complexity (number of degrees of freedom) ## References Bennett, K.P., Embrechts, M.J., 2003. An optimization perspective on kernel partial least squares regression, in: Advances in Learning Theory: Methods, Models and Applications, NATO Science Series III: Computer & Systems Sciences. IOS Press Amsterdam, pp. 227-250. Rosipal, R., Trejo, L.J., 2001. Kernel Partial Least Squares Regression in Reproducing Kernel Hilbert Space. Journal of Machine Learning Research 2, 97-123. ## Note The second example concerns the fitting of the function sinc(x) described in Rosipal & Trejo 2001 p. 105-106 ## Examples ```r ## EXAMPLE 1 n <- 6 ; p <- 4 Xtrain <- matrix(rnorm(n * p), ncol = p) ytrain <- rnorm(n) Ytrain <- cbind(y1 = ytrain, y2 = 100 * ytrain) m <- 3 Xtest <- Xtrain[1:m, , drop = FALSE] Ytest <- Ytrain[1:m, , drop = FALSE] ; ytest <- Ytest[1:m, 1] lb <- 2 fm <- dkrr(Xtrain, Ytrain, lb = lb, kern = "krbf", gamma = .8) coef(fm) coef(fm, lb = .6) predict(fm, Xtest) predict(fm, Xtest, lb = c(0.1, .8)) pred <- predict(fm, Xtest)$pred msep(pred, Ytest) lb <- 2 fm <- dkrr(Xtrain, Ytrain, lb = lb, kern = "kpol", degree = 2, coef0 = 10) predict(fm, Xtest) ## EXAMPLE 1 x <- seq(-10, 10, by = .2) x[x == 0] <- 1e-5 n <- length(x) zy <- sin(abs(x)) / abs(x) y <- zy + rnorm(n, 0, .2) plot(x, y, type = "p") lines(x, zy, lty = 2) X <- matrix(x, ncol = 1) fm <- dkrr(X, y, lb = .01, gamma = .5) pred <- predict(fm, X)$pred plot(X, y, type = "p") lines(X, zy, lty = 2) lines(X, pred, col = "red") ```

dkrr function