# Ordinary least squares for the linear multivariate regression model qrmls, or LS for linear multivariate regression models, solves the following optimization problem [REMOVE_ME]$$ \textrm{min}_\beta ~ \frac{1}{2}\|Y-XB\|_2^2, [REMOVE_ME_2]$$ for $Y\in\mathbb{R}^{n \times q}$ and $X\in\mathbb{R}^{n\times p}$, to obtain a coefficient matrix $\widehat{B}\in\mathbb{R}^{p\times q}$. The design matrix $X\in\mathbb{R}^{n\times p}$ contains the observations for each regressor. ## Arguments - `Y`: a matrix of dimension $(n\times q$ response variables. - `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 matrix of dimension $p\times q$ containing the solution for the parameters $B$. - **fitted**: a matrix of dimension $n\times q$ of fitted values, $\widehat{Y}=X\widehat{B}$. - **residuals**: a matrix of dimension $n\times q$ of residuals, $\varepsilon=Y-\widehat{Y}$. - **XTX**: the matrix $X^\top X$. - **Sigma_hat**: a matrix of dimension $q\times q$ containing the estimated residual variance-covariance matrix. - **df**: degrees of freedom. - **R**: $R$ matrix of the QR decomposition of the matrix $X^\top X$. - **XTy**: $X^\top y$. - **R2**: $R^2$, coefficient of determination, measure of goodness-of-fit of the model. - **predicted**: predicted values for the test set, $X_{\text{test}}\widehat{B}$. It is only available if X_test is not NULL. - **PMSE**: ## Description qrmls, or LS for linear multivariate regression models, solves the following optimization problem $$ \textrm{min}_\beta ~ \frac{1}{2}\|Y-XB\|_2^2, $$ for $Y\in\mathbb{R}^{n \times q}$ and $X\in\mathbb{R}^{n\times p}$, to obtain a coefficient matrix $\widehat{B}\in\mathbb{R}^{p\times q}$. 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 q <- 3 X <- matrix(rnorm(n * p, 1), n, p) X[,1] <- 1 eps <- matrix(rnorm(n*q), n, q) B <- matrix(0, p, q) B[,1] <- rep(1, p) B[,2] <- rep(2, p) B[,3] <- rep(-1, p) Y <- X %*% B + eps X_test <- matrix(rnorm(5 * p, 1), 5, p) output <- fastQR::qrmls(Y = Y, X = X, X_test = X_test, type = "QR") output$coeff ```

qrmls function