# Direct KPLSR Models Direct kernel PLSR (DKPLSR) (Bennett & Embrechts 2003). The method builds kernel Gram matrices and then runs a usual PLSR algorithm on them. This is faster (but not equivalent) to the "true" NIPALS KPLSR algorithm such as described in Rosipal & Trejo (2001). latin1 ```r dkplsr(X, Y, weights = NULL, nlv, kern = "krbf", ...) ## S3 method for class 'Dkpls' transform(object, X, ..., nlv = NULL) ## S3 method for class 'Dkpls' coef(object, ..., nlv = NULL) ## S3 method for class 'Dkplsr' predict(object, X, ..., nlv = NULL) ``` ## Arguments - `X`: For the main function: Matrix with the training X-data ($n, p$). --- For auxiliary functions: A matrix with new X-data ($m, p$) to consider. - `Y`: Matrix with the training Y-data ($n, q$). - `weights`: vector of weights ($n, 1$) to apply to the training observations. Internally, weights are "normalized" to sum to 1. Default to `NULL` (weights are set to $1 / n$). - `nlv`: For the main function: The number(s) of LVs to calculate. --- For auxiliary functions: The number(s) of LVs to consider. - `kern`: Name of the function defining the considered kernel for building the Gram matrix. See `krbf` for syntax, and other available kernel functions (`krbf`, `kpol`, `ktanh`). - `...`: Optional arguments to pass in the kernel function defined in `kern` (e.g. `gamma` for `krbf`, `gamma` and `coef0` for `ktanh`, `gamma` and `coef0` and `degree` for `kpol`). - `object`: For auxiliary functions: A fitted model, output of a call to the main function. ## Returns For `dkplsr`: - **X**: Matrix with the training X-data ($n, p$). - **fm**: List with the outputs of the PLSR ((`T`): the X-score matrix (n,nlv); (`P`): the X-loadings matrix (p,nlv); (`R`): The PLS projection matrix (p,nlv); (`W`): The X-loading weights matrix (p,nlv); (`C`): The Y-loading weights matrix; (`TT`): the X-score normalization factor; (`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); (`U`): intermediate output. - **K**: kernel Gram matrix - **kern**: kernel function - **dots**: Optional arguments passed in the kernel function For `transform.Dkplsr` : A matrix ($m, nlv$) with the projection of the new X-data on the X-scores For `predict.Dkplsr`: - **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.Dkplsr`: - **int**: matrix (1,nlv) with the intercepts - **B**: matrix (n,nlv) with the coefficients ## 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] nlv <- 2 fm <- dkplsr(Xtrain, Ytrain, nlv = nlv, kern = "krbf", gamma = .8) transform(fm, Xtest) transform(fm, Xtest, nlv = 1) coef(fm) coef(fm, nlv = 1) predict(fm, Xtest) predict(fm, Xtest, nlv = 0:nlv)$pred pred <- predict(fm, Xtest)$pred msep(pred, Ytest) nlv <- 2 fm <- dkplsr(Xtrain, Ytrain, nlv = nlv, kern = "kpol", degree = 2, coef0 = 10) predict(fm, Xtest, nlv = nlv) ## EXAMPLE 2 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) nlv <- 3 fm <- dkplsr(X, y, nlv = nlv) pred <- predict(fm, X)$pred plot(X, y, type = "p") lines(X, zy, lty = 2) lines(X, pred, col = "red") ```

dkplsr function