# Linear Ridge Regression Fitting linear ridge regression models (RR) (Hoerl & Kennard 1970, Hastie & Tibshirani 2004, Hastie et al 2009, Cule & De Iorio 2012) by SVD factorization. latin1 ```r rr(X, Y, weights = NULL, lb = 1e-2) ## S3 method for class 'Rr' coef(object, ..., lb = NULL) ## S3 method for class 'Rr' 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 $lambda$. If `lb = 0`, a pseudo-inverse is used. - `object`: For the auxiliary functions: A fitted model, output of a call to the main function. - `...`: --- For the auxiliary functions: Optional arguments. Not used. ## Returns For `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 $(p,1)$. For `coef.Rr`: - **int**: matrix (1,nlv) with the intercepts - **B**: matrix (n,nlv) with the coefficients - **df**: model complexity (number of degrees of freedom) For `predict.Rr`: - **pred**: A list of matrices ($m, q$) with the Y predicted values for the new X-data ## References Cule, E., De Iorio, M., 2012. A semi-automatic method to guide the choice of ridge parameter in ridge regression. arXiv:1205.0686. Hastie, T., Tibshirani, R., 2004. Efficient quadratic regularization for expression arrays. Biostatistics 5, 329-340. https://doi.org/10.1093/biostatistics/kxh010 Hastie, T., Tibshirani, R., Friedman, J., 2009. The elements of statistical learning: data mining, inference, and prediction, 2nd ed. Springer, New York. Hoerl, A.E., Kennard, R.W., 1970. Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics 12, 55-67. https://doi.org/10.1080/00401706.1970.10488634 Wu, W., Massart, D.L., de Jong, S., 1997. The kernel PCA algorithms for wide data. Part I: Theory and algorithms. Chemometrics and Intelligent Laboratory Systems 36, 165-172. https://doi.org/10.1016/S0169-7439(97)00010-5 ## Examples ```r 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 <- .1 fm <- rr(Xtrain, Ytrain, lb = lb) coef(fm) coef(fm, lb = .8) predict(fm, Xtest) predict(fm, Xtest, lb = c(0.1, .8)) pred <- predict(fm, Xtest)$pred msep(pred, Ytest) ```

rr function