# Ordinary least squares for the linear regression model qrls, or LS for linear regression models, solves the following optimization problem [REMOVE_ME]$$ \textrm{min}_\beta ~ \frac{1}{2}\|y-X\beta\|_2^2, [REMOVE_ME_2]$$ for $y\in\mathbb{R}^n$ and $X\in\mathbb{R}^{n\times p}$, to obtain a coefficient vector $\widehat{\beta}\in\mathbb{R}^p$. The design matrix $X\in\mathbb{R}^{n\times p}$ contains the observations for each regressor. ```r qrls(y, X, X_test = NULL, type = NULL) ``` ## Arguments - `y`: a vector of length-$n$ response vector. - `X`: an $(n\times p)$ full column rank matrix of predictors. - `X_test`: an $(q\times p)$ full column rank matrix. Test set. By default it set to NULL. - `type`: either "QR" or "R". Specifies the type of decomposition to use: "QR" for the QR decomposition or "R" for the Cholesky factorization of $A^\top A$. The default is "QR". ## Returns A named list containing - **coeff**: a length-$p$ vector containing the solution for the parameters $\beta$. - **fitted**: a length-$n$ vector of fitted values, $\widehat{y}=X\widehat{\beta}$. - **residuals**: a length-$n$ vector of residuals, $\varepsilon=y-\widehat{y}$. - **residuals_norm2**: the L2-norm of the residuals, $\Vert\varepsilon\Vert_2^2.$ - **y_norm2**: the L2-norm of the response variable. $\Vert y\Vert_2^2.$ - **XTX_Qmat**: $Q$ matrix of the QR decomposition of the matrix $X^\top X$. - **XTX_Rmat**: $R$ matrix of the QR decomposition of the matrix $X^\top X$. - **QXTy**: $QX^\top y$, where $Q$ matrix of the QR decomposition of the matrix $X^\top X$. - **R2**: $R^2$, coefficient of determination, measure of goodness-of-fit of the model. - **predicted**: predicted values for the test set, $X_{\text{test}}\widehat{\beta}$. It is only available if X_test is not NULL. ## Description qrls, or LS for linear regression models, solves the following optimization problem $$ \textrm{min}_\beta ~ \frac{1}{2}\|y-X\beta\|_2^2, $$ for $y\in\mathbb{R}^n$ and $X\in\mathbb{R}^{n\times p}$, to obtain a coefficient vector $\widehat{\beta}\in\mathbb{R}^p$. The design matrix $X\in\mathbb{R}^{n\times p}$ contains the observations for each regressor. ## Examples ```r ## generate sample data set.seed(10) n <- 30 p <- 6 X <- matrix(rnorm(n * p, 1), n, p) X[,1] <- 1 eps <- rnorm(n, sd = 0.5) beta <- rep(0, p) beta[1:3] <- 1 beta[4:5] <- 2 y <- X %*% beta + eps X_test <- matrix(rnorm(5 * p, 1), 5, p) output <- fastQR::qrls(y = y, X = X, X_test = X_test) output$coeff ```

qrls function