Package 'fastQR' reference manual

Title:	Fast QR Decomposition and Update
Description:	Efficient algorithms for performing, updating, and downdating the QR decomposition, R decomposition, or the inverse of the R decomposition of a matrix as rows or columns are added or removed. It also includes functions for solving linear systems of equations, normal equations for linear regression models, and normal equations for linear regression with a RIDGE penalty. For a detailed introduction to these methods, see the book by Golub and Van Loan (2013, <doi:10.1007/978-3-319-05089-8>) for complete introduction to the methods.
Authors:	Mauro Bernardi [aut, cre], Claudio Busatto [aut], Manuela Cattelan [aut]
Maintainer:	Mauro Bernardi <[email protected]>
License:	GPL (>= 2)
Version:	1.0.0
Built:	2025-02-18 11:19:08 UTC
Source:	https://github.com/cranhaven/cranhaven.r-universe.dev

The QR factorization of a matrix

Description

qr provides the QR factorization of the matrix $X\in\mathbb{R}^{n\times p}$ with $n>p$ . The QR factorization of the matrix $X$ returns the matrices $Q\in\mathbb{R}^{n\times n}$ and $R\in\mathbb{R}^{n\times p}$ such that $X=QR$ . See Golub and Van Loan (2013) for further details on the method.

Arguments

`X`	a $n\times p$ matrix.
`type`	either "givens" or "householder".
`nb`	integer. Defines the number of block in the block recursive QR decomposition. See Golud and van Loan (2013).
`complete`	logical expression of length 1. Indicates whether an arbitrary orthogonal completion of the $Q$ matrix is to be made, or whether the $R$ matrix is to be completed by binding zero-value rows beneath the square upper triangle.

Value

A named list containing

Q: the Q matrix.
R: the R matrix.

References

Golub GH, Van Loan CF (2013). Matrix computations, Johns Hopkins Studies in the Mathematical Sciences, Fourth edition. Johns Hopkins University Press, Baltimore, MD. ISBN 978-1-4214-0794-4; 1-4214-0794-9; 978-1-4214-0859-0.

Björck Å (2015). Numerical methods in matrix computations, volume 59 of Texts in Applied Mathematics. Springer, Cham. ISBN 978-3-319-05088-1; 978-3-319-05098-8, doi:10.1007/978-3-319-05089-8.

Björck Å (2024). Numerical Methods for Least Squares Problems: Second Edition. Society for Industrial and Applied Mathematics, Philadelphia, PA. doi:10.1137/1.9781611977950, https://doi.org/10.1137/1.9781611977950.

Bernardi M, Busatto C, Cattelan M (2024). “Fast QR updating methods for statistical applications.” 2412.05905, https://arxiv.org/abs/2412.05905.

Examples

## generate sample data
set.seed(1234)
n <- 10
p <- 6
X <- matrix(rnorm(n * p, 1), n, p)

## QR factorization via Givens rotation
output <- qr(X, type = "givens", complete = TRUE)
Q      <- output$Q
R      <- output$R

## check
round(Q %*% R - X, 5)
max(abs(Q %*% R - X))

## QR factorization via Householder rotation
output <- qr(X, type = "householder", complete = TRUE)
Q      <- output$Q
R      <- output$R

## check
round(Q %*% R - X, 5)
max(abs(Q %*% R - X))

## generate sample data
set.seed(1234)
n <- 10
p <- 6
X <- matrix(rnorm(n * p, 1), n, p)

## QR factorization via Givens rotation
output <- qr(X, type = "givens", complete = TRUE)
Q      <- output$Q
R      <- output$R

## check
round(Q %*% R - X, 5)
max(abs(Q %*% R - X))

## QR factorization via Householder rotation
output <- qr(X, type = "householder", complete = TRUE)
Q      <- output$Q
R      <- output$R

## check
round(Q %*% R - X, 5)
max(abs(Q %*% R - X))

Cholesky decomposition via QR factorization.

Description

qrchol, provides the Cholesky decomposition of the symmetric and positive definite matrix $X^\top X\in\mathbb{R}^{p\times p}$ , where $X\in\mathbb{R}^{n\times p}$ is the input matrix.

Usage

qrchol(X, nb = NULL)
qrchol(X, nb = NULL)

Arguments

`X`	an $(n\times p)$ matrix.
`nb`	number of blocks for the recursive block QR decomposition, default is NULL.

Value

an upper triangular matrix of dimension $p\times p$ which represents the Cholesky decomposition of $X^\top X$ .

Fast downdating of the QR factorization

Description

qrdowndate provides the update of the QR factorization after the deletion of $m>1$ rows or columns to the matrix $X\in\mathbb{R}^{n\times p}$ with $n>p$ . The QR factorization of the matrix $X\in\mathbb{R}^{n\times p}$ returns the matrices $Q\in\mathbb{R}^{n\times n}$ and $R\in\mathbb{R}^{n\times p}$ such that $X=QR$ . The $Q$ and $R$ matrices are factorized as $Q=\begin{bmatrix}Q_1&Q_2\end{bmatrix}$ and $R=\begin{bmatrix}R_1\\R_2\end{bmatrix}$ , with $Q_1\in\mathbb{R}^{n\times p}$ , $Q_2\in\mathbb{R}^{n\times (n-p)}$ such that $Q_1^{\top}Q_2=Q_2^\top Q_1=0$ and $R_1\in\mathbb{R}^{p\times p}$ upper triangular matrix and $R_2\in\mathbb{R}^{(n-p)\times p}$ . qrupdate accepts in input the matrices $Q$ and either the complete matrix $R$ or the reduced one, $R_1$ . See Golub and Van Loan (2013) for further details on the method.

Usage

qrdowndate(Q, R, k, m = NULL, type = NULL, fast = NULL, complete = NULL)
qrdowndate(Q, R, k, m = NULL, type = NULL, fast = NULL, complete = NULL)

Arguments

`Q`	a $n\times n$ matrix.
`R`	a $n\times p$ upper triangular matrix.
`k`	position where the columns or the rows are removed.
`m`	number of columns or rows to be removed. Default is $m=1$ .
`type`	either 'row' of 'column', for deleting rows or columns. Default is 'column'.
`fast`	fast mode: disable to check whether the provided matrices are valid inputs. Default is FALSE.
`complete`	logical expression of length 1. Indicates whether an arbitrary orthogonal completion of the $Q$ matrix is to be made, or whether the $R$ matrix is to be completed by binding zero-value rows beneath the square upper triangle.

Value

A named list containing

Q: the updated Q matrix.
R: the updated R matrix.

References

Björck Å (2015). Numerical methods in matrix computations, volume 59 of Texts in Applied Mathematics. Springer, Cham. ISBN 978-3-319-05088-1; 978-3-319-05098-8, doi:10.1007/978-3-319-05089-8.

Bernardi M, Busatto C, Cattelan M (2024). “Fast QR updating methods for statistical applications.” 2412.05905, https://arxiv.org/abs/2412.05905.

Examples

## Remove one column
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R

## select the column to be deleted
## from X and update X
k  <- 2
X1 <- X[, -k]

## downdate the QR decomposition
out <- fastQR::qrdowndate(Q = Q, R = R,
                          k = k, m = 1,
                          type = "column",
                          fast = FALSE,
                          complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Remove m columns
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R

## select the column to be deleted from X
## and update X
m  <- 2
k  <- 2
X1 <- X[, -c(k,k+m-1)]

## downdate the QR decomposition
out <- fastQR::qrdowndate(Q = Q, R = R,
                          k = k, m = 2,
                          type = "column",
                          fast = TRUE,
                          complete = FALSE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Remove one row
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R

## select the row to be deleted from X and update X
k  <- 5
X1 <- X[-k,]

## downdate the QR decomposition
out <- fastQR::qrdowndate(Q = Q, R = R,
                          k = k, m = 1,
                          type = "row",
                          fast = FALSE,
                          complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Remove m rows
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R

## select the rows to be deleted from X and update X
k  <- 5
m  <- 2
X1 <- X[-c(k,k+1),]

## downdate the QR decomposition
out <- fastQR::qrdowndate(Q = Q, R = R,
                          k = k, m = m,
                          type = "row",
                          fast = FALSE,
                          complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Remove one column
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R

## select the column to be deleted
## from X and update X
k  <- 2
X1 <- X[, -k]

## downdate the QR decomposition
out <- fastQR::qrdowndate(Q = Q, R = R,
                          k = k, m = 1,
                          type = "column",
                          fast = FALSE,
                          complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Remove m columns
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R

## select the column to be deleted from X
## and update X
m  <- 2
k  <- 2
X1 <- X[, -c(k,k+m-1)]

## downdate the QR decomposition
out <- fastQR::qrdowndate(Q = Q, R = R,
                          k = k, m = 2,
                          type = "column",
                          fast = TRUE,
                          complete = FALSE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Remove one row
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R

## select the row to be deleted from X and update X
k  <- 5
X1 <- X[-k,]

## downdate the QR decomposition
out <- fastQR::qrdowndate(Q = Q, R = R,
                          k = k, m = 1,
                          type = "row",
                          fast = FALSE,
                          complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Remove m rows
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R

## select the rows to be deleted from X and update X
k  <- 5
m  <- 2
X1 <- X[-c(k,k+1),]

## downdate the QR decomposition
out <- fastQR::qrdowndate(Q = Q, R = R,
                          k = k, m = m,
                          type = "row",
                          fast = FALSE,
                          complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

Ordinary least squares for the linear regression model

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.

Usage

qrls(y, X, X_test = NULL, type = NULL)
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".

Value

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.

Examples


## 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

## 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

Ordinary least squares for the linear multivariate regression model

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.

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".

Value

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

Examples


## 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

## 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

RIDGE estimator for the linear multivariate regression model

Description

qrmridge, or LS for linear multivariate regression models, solves the following optimization problem

$\textrm{min}_\beta ~ \frac{1}{2}\|Y-XB\|_2^2,$

Arguments

`Y`	a matrix of dimension $(n\times q$ response variables.
`X`	an $(n\times p)$ full column rank matrix of predictors.
`lambda`	a vector of lambdas.
`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".

Value

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

Examples

## 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::qrmridge(Y = Y, X = X, lambda = 1, X_test = X_test, type = "QR")
output$coeff

## 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::qrmridge(Y = Y, X = X, lambda = 1, X_test = X_test, type = "QR")
output$coeff

Cross-validation of the RIDGE estimator for the linear multivariate regression model

Description

qrmridge_cv, or LS for linear multivariate regression models, solves the following optimization problem

$\textrm{min}_\beta ~ \frac{1}{2}\|Y-XB\|_2^2,$

Arguments

`Y`	a matrix of dimension $(n\times q$ response variables.
`X`	an $(n\times p)$ full column rank matrix of predictors.
`lambda`	a vector of lambdas.
`k`	an integer vector defining the number of groups for CV.
`seed`	ad integer number defining the seed for random number generation.
`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".

Value

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

Examples

## 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::qrmridge_cv(Y = Y, X = X, lambda = c(1,2),
                                 k = 5, seed = 12, X_test = X_test, type = "QR")
output$coeff

## 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::qrmridge_cv(Y = Y, X = X, lambda = c(1,2),
                                 k = 5, seed = 12, X_test = X_test, type = "QR")
output$coeff

RIDGE estimation for the linear regression model

Description

lmridge, or RIDGE for linear regression models, solves the following penalized optimization problem

$\textrm{min}_\beta ~ \frac{1}{n}\|y-X\beta\|_2^2+\lambda\Vert\beta\Vert_2^2,$

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.

Usage

qrridge(y, X, lambda, X_test = NULL, type = NULL)
qrridge(y, X, lambda, X_test = NULL, type = NULL)

Arguments

`y`	a vector of length- $n$ response vector.
`X`	an $(n\times p)$ matrix of predictors.
`lambda`	a vector of lambdas.
`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".

Value

A named list containing

mean_y: mean of the response variable.
mean_X: a length- $p$ vector containing the mean of each column of the design matrix.
path: the whole path of estimated regression coefficients.
ess: explained sum of squares for the whole path of estimated coefficients.
GCV: generalized cross-validation for the whole path of lambdas.
GCV_min: minimum value of GCV.
GCV_idx: inded corresponding to the minimum values of GCV.
coeff: a length- $p$ vector containing the solution for the parameters $\beta$ which corresponds to the minimum of GCV.
lambda: the vector of lambdas.
scales: the vector of standard deviations of each column of the design matrix.

Examples


## 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::qrridge(y = y, X = X,
                              lambda = 0.2,
                              X_test = X_test)
output$coeff

## 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::qrridge(y = y, X = X,
                              lambda = 0.2,
                              X_test = X_test)
output$coeff

Cross-validation of the RIDGE estimator for the linear regression model

Description

qrridge_cv, or LS for linear multivariate regression models, solves the following optimization problem

$\textrm{min}_\beta ~ \frac{1}{2}\|Y-XB\|_2^2,$

Arguments

`y`	a vector of length- $n$ response vector.
`X`	an $(n\times p)$ full column rank matrix of predictors.
`lambda`	a vector of lambdas.
`k`	an integer vector defining the number of groups for CV.
`seed`	ad integer number defining the seed for random number generation.
`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".

Value

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: the matrix $X^\top X$ .
XTy: $X^\top y$ .
sigma_hat: estimated residual variance.
df: degrees of freedom.
Q: $Q$ matrix of the QR decomposition of the matrix $X^\top X$ .
R: $R$ 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.

Examples


## generate sample data
set.seed(10)
n         <- 30
p         <- 6
X         <- matrix(rnorm(n * p, 1), n, p)
X[,1]     <- 1
eps       <- rnorm(n)
beta      <- rep(1, p)
y         <- X %*% beta + eps
X_test    <- matrix(rnorm(5 * p, 1), 5, p)
output    <- fastQR::qrridge_cv(y = y, X = X, lambda = c(1,2), 
                                k = 5, seed = 12, X_test = X_test, type = "QR")
output$coeff

## generate sample data
set.seed(10)
n         <- 30
p         <- 6
X         <- matrix(rnorm(n * p, 1), n, p)
X[,1]     <- 1
eps       <- rnorm(n)
beta      <- rep(1, p)
y         <- X %*% beta + eps
X_test    <- matrix(rnorm(5 * p, 1), 5, p)
output    <- fastQR::qrridge_cv(y = y, X = X, lambda = c(1,2), 
                                k = 5, seed = 12, X_test = X_test, type = "QR")
output$coeff

Solution of linear system of equations, via the QR decomposition.

Description

solves systems of equations $Ax=b$ , for $A\in\mathbb{R}^{n\times p}$ and $b\in\mathbb{R}^n$ , via the QR decomposition.

Usage

qrsolve(A, b, type = NULL, nb = NULL)
qrsolve(A, b, type = NULL, nb = NULL)

Arguments

`A`	an $(n\times p)$ full column rank matrix.
`b`	a vector of dimension $n$ .
`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".
`nb`	number of blocks for the recursive block QR decomposition, default is NULL.

Value

x a vector of dimension $p$ that satisfies $Ax=b$ .

References

Björck Å (2015). Numerical methods in matrix computations, volume 59 of Texts in Applied Mathematics. Springer, Cham. ISBN 978-3-319-05088-1; 978-3-319-05098-8, doi:10.1007/978-3-319-05089-8.

Bernardi M, Busatto C, Cattelan M (2024). “Fast QR updating methods for statistical applications.” 2412.05905, https://arxiv.org/abs/2412.05905.

Examples


## generate sample data
set.seed(1234)
n <- 10
p <- 4
A <- matrix(rnorm(n * p, 1), n, p)
b <- rnorm(n)

## solve the system of linear equations using qr
x1 <- fastQR::qrsolve(A = A, b = b)
x1

## solve the system of linear equations using rb qr
x2 <- fastQR::qrsolve(A = A, b = b, nb = 2)
x2

## check
round(x1 - solve(crossprod(A)) %*% crossprod(A, b), 5)
round(x2 - solve(crossprod(A)) %*% crossprod(A, b), 5)

## generate sample data
set.seed(1234)
n <- 10
p <- 4
A <- matrix(rnorm(n * p, 1), n, p)
b <- rnorm(n)

## solve the system of linear equations using qr
x1 <- fastQR::qrsolve(A = A, b = b)
x1

## solve the system of linear equations using rb qr
x2 <- fastQR::qrsolve(A = A, b = b, nb = 2)
x2

## check
round(x1 - solve(crossprod(A)) %*% crossprod(A, b), 5)
round(x2 - solve(crossprod(A)) %*% crossprod(A, b), 5)

Fast updating of the QR factorization

Description

qrupdate provides the update of the QR factorization after the addition of $m>1$ rows or columns to the matrix $X\in\mathbb{R}^{n\times p}$ with $n>p$ . The QR factorization of the matrix $X$ returns the matrices $Q\in\mathbb{R}^{n\times n}$ and $R\in\mathbb{R}^{n\times p}$ such that $X=QR$ . The $Q$ and $R$ matrices are factorized as $Q=\begin{bmatrix}Q_1&Q_2\end{bmatrix}$ and $R=\begin{bmatrix}R_1\\R_2\end{bmatrix}$ , with $Q_1\in\mathbb{R}^{n\times p}$ , $Q_2\in\mathbb{R}^{n\times (n-p)}$ such that $Q_1^{\top}Q_2=Q_2^\top Q_1=0$ and $R_1\in\mathbb{R}^{p\times p}$ upper triangular matrix and $R_2\in\mathbb{R}^{(n-p)\times p}$ . qrupdate accepts in input the matrices $Q$ and either the complete matrix $R$ or the reduced one, $R_1$ . See Golub and Van Loan (2013) for further details on the method.

Usage

qrupdate(Q, R, k, U, type = NULL, fast = NULL, complete = NULL)
qrupdate(Q, R, k, U, type = NULL, fast = NULL, complete = NULL)

Arguments

`Q`	a $n\times p$ matrix.
`R`	a $p\times p$ upper triangular matrix.
`k`	position where the columns or the rows are added.
`U`	either a $n\times m$ matrix or a $p\times m$ matrix of columns or rows to be added.
`type`	either 'row' of 'column', for adding rows or columns. Default is 'column'.
`fast`	fast mode: disable to check whether the provided matrices are valid inputs. Default is FALSE.
`complete`	logical expression of length 1. Indicates whether an arbitrary orthogonal completion of the $Q$ matrix is to be made, or whether the $R$ matrix is to be completed by binding zero-value rows beneath the square upper triangle.

Value

A named list containing

Q: the updated Q matrix.
R: the updated R matrix.

References

Björck Å (2015). Numerical methods in matrix computations, volume 59 of Texts in Applied Mathematics. Springer, Cham. ISBN 978-3-319-05088-1; 978-3-319-05098-8, doi:10.1007/978-3-319-05089-8.

Bernardi M, Busatto C, Cattelan M (2024). “Fast QR updating methods for statistical applications.” 2412.05905, https://arxiv.org/abs/2412.05905.

Examples

## Add one column
## generate sample data
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R

## create column u to be added
k  <- p+1
u  <- matrix(rnorm(n), n, 1)
X1 <- cbind(X, u)

## update the QR decomposition
out <- fastQR::qrupdate(Q = Q, R = R,
                       k = k, U = u,
                       type = "column",
                       fast = FALSE,
                       complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Add m columns
## create data: n > p
set.seed(1234)
n <- 10
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R

## create the matrix of two columns to be added
## in position 2
k  <- 2
m  <- 2
U  <- matrix(rnorm(n*m), n, m)
X1 <- cbind(X[,1:(k-1)], U, X[,k:p])

# update the QR decomposition
out <- fastQR::qrupdate(Q = Q, R = R,
                        k = k, U = U, type = "column",
                       fast = FALSE, complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Add one row
## create data: n > p
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the row u to be added
u  <- matrix(data = rnorm(p), p, 1)
k  <- n+1
if (k<=n) {
  X1 <- rbind(rbind(X[1:(k-1), ], t(u)), X[k:n, ])
} else {
  X1 <- rbind(rbind(X, t(u)))
}

## update the QR decomposition
out <- fastQR::qrupdate(Q = Q, R = R,
                        k = k, U = u,
                        type = "row",
                        complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Add m rows
## create data: n > p
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the matrix of rows U to be added:
## two rows in position 5
m  <- 2
U  <- matrix(data = rnorm(p*m), p, m)
k  <- 5
if (k<=n) {
  X1 <- rbind(rbind(X[1:(k-1), ], t(U)), X[k:n, ])
} else {
  X1 <- rbind(rbind(X, t(U)))
}

## update the QR decomposition
out <- fastQR::qrupdate(Q = Q, R = R,
                        k = k, U = U,
                        type = "row",
                        complete = FALSE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Add one column
## generate sample data
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R

## create column u to be added
k  <- p+1
u  <- matrix(rnorm(n), n, 1)
X1 <- cbind(X, u)

## update the QR decomposition
out <- fastQR::qrupdate(Q = Q, R = R,
                       k = k, U = u,
                       type = "column",
                       fast = FALSE,
                       complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Add m columns
## create data: n > p
set.seed(1234)
n <- 10
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R

## create the matrix of two columns to be added
## in position 2
k  <- 2
m  <- 2
U  <- matrix(rnorm(n*m), n, m)
X1 <- cbind(X[,1:(k-1)], U, X[,k:p])

# update the QR decomposition
out <- fastQR::qrupdate(Q = Q, R = R,
                        k = k, U = U, type = "column",
                       fast = FALSE, complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Add one row
## create data: n > p
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the row u to be added
u  <- matrix(data = rnorm(p), p, 1)
k  <- n+1
if (k<=n) {
  X1 <- rbind(rbind(X[1:(k-1), ], t(u)), X[k:n, ])
} else {
  X1 <- rbind(rbind(X, t(u)))
}

## update the QR decomposition
out <- fastQR::qrupdate(Q = Q, R = R,
                        k = k, U = u,
                        type = "row",
                        complete = TRUE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

## Add m rows
## create data: n > p
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the matrix of rows U to be added:
## two rows in position 5
m  <- 2
U  <- matrix(data = rnorm(p*m), p, m)
k  <- 5
if (k<=n) {
  X1 <- rbind(rbind(X[1:(k-1), ], t(U)), X[k:n, ])
} else {
  X1 <- rbind(rbind(X, t(U)))
}

## update the QR decomposition
out <- fastQR::qrupdate(Q = Q, R = R,
                        k = k, U = U,
                        type = "row",
                        complete = FALSE)

## check
round(out$Q %*% out$R - X1, 5)
max(abs(out$Q %*% out$R - X1))

Cholesky decomposition via R factorization.

Description

rchol, provides the Cholesky decomposition of the symmetric and positive definite matrix $X^\top X\in\mathbb{R}^{p\times p}$ , where $X\in\mathbb{R}^{n\times p}$ is the input matrix.

Arguments

`X`	an $(n\times p)$ matrix, with $n\geq p$ . If $n< p$ an error message is returned.

Value

an upper triangular matrix of dimension $p\times p$ which represents the Cholesky decomposition of $X^\top X$ .

References

Björck Å (2015). Numerical methods in matrix computations, volume 59 of Texts in Applied Mathematics. Springer, Cham. ISBN 978-3-319-05088-1; 978-3-319-05098-8, doi:10.1007/978-3-319-05089-8.

Bernardi M, Busatto C, Cattelan M (2024). “Fast QR updating methods for statistical applications.” 2412.05905, https://arxiv.org/abs/2412.05905.

Examples


set.seed(1234)
n <- 10
p <- 6
X <- matrix(rnorm(n * p, 1), n, p)

## compute the Cholesky decomposition of X^TX
S <- fastQR::rchol(X = X)
S

## check
round(S - chol(crossprod(X)), 5)

set.seed(1234)
n <- 10
p <- 6
X <- matrix(rnorm(n * p, 1), n, p)

## compute the Cholesky decomposition of X^TX
S <- fastQR::rchol(X = X)
S

## check
round(S - chol(crossprod(X)), 5)

Fast downdating of the R matrix

Description

rdowndate provides the update of the thin R matrix of the QR factorization after the deletion of $m\geq 1$ rows or columns to the matrix $X\in\mathbb{R}^{n\times p}$ with $n>p$ . The R factorization of the matrix $X$ returns the upper triangular matrix $R\in\mathbb{R}^{p\times p}$ such that $X^\top X=R^\top R$ . See Golub and Van Loan (2013) for further details on the method.

Usage

rdowndate(R, k = NULL, m = NULL, U = NULL, fast = NULL, type = NULL)
rdowndate(R, k = NULL, m = NULL, U = NULL, fast = NULL, type = NULL)

Arguments

`R`	a $p\times p$ upper triangular matrix.
`k`	position where the columns or the rows are removed.
`m`	number of columns or rows to be removed.
`U`	a $p\times m$ matrix of rows to be removed. It should only be provided when rows are being removed.
`fast`	fast mode: disable to check whether the provided matrices are valid inputs. Default is FALSE.
`type`	either 'row' of 'column', for removing rows or columns.

Value

R the updated R matrix.

References

Björck Å (2015). Numerical methods in matrix computations, volume 59 of Texts in Applied Mathematics. Springer, Cham. ISBN 978-3-319-05088-1; 978-3-319-05098-8, doi:10.1007/978-3-319-05089-8.

Bernardi M, Busatto C, Cattelan M (2024). “Fast QR updating methods for statistical applications.” 2412.05905, https://arxiv.org/abs/2412.05905.

Examples

## Remove one column
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## select the column to be deleted from X and update X
k  <- 2
X1 <- X[, -k]

## downdate the R decomposition
R2 <- fastQR::rdowndate(R = R1, k = k,
                        m = 1, type = "column")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Remove m columns
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## select the column to be deleted from X and update X
k  <- 2
X1 <- X[, -c(k,k+1)]

## downdate the R decomposition
R2 <- fastQR::rdowndate(R = R1, k = k,
                        m = 2, type = "column")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Remove one row
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

# select the row to be deleted from X and update X
k  <- 5
X1 <- X[-k,]
U  <- as.matrix(X[k,], p, 1)

## downdate the R decomposition
R2 <-  rdowndate(R = R1, k = k, m = 1,
                 U = U, fast = FALSE, type = "row")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Remove m rows
## create data: n > p
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)
output <- fastQR::qr(X, type = "householder",
                    nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## select the rows to be deleted from X and update X
k  <- 2
m  <- 2
X1 <- X[-c(k,k+m-1),]
U  <- t(X[k:(k+m-1), ])

## downdate the R decomposition
R2 <- rdowndate(R = R1, k = k, m = m,
                U = U, fast = FALSE, type = "row")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Remove one column
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## select the column to be deleted from X and update X
k  <- 2
X1 <- X[, -k]

## downdate the R decomposition
R2 <- fastQR::rdowndate(R = R1, k = k,
                        m = 1, type = "column")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Remove m columns
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## select the column to be deleted from X and update X
k  <- 2
X1 <- X[, -c(k,k+1)]

## downdate the R decomposition
R2 <- fastQR::rdowndate(R = R1, k = k,
                        m = 2, type = "column")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Remove one row
## generate sample data
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, type = "householder",
                     nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

# select the row to be deleted from X and update X
k  <- 5
X1 <- X[-k,]
U  <- as.matrix(X[k,], p, 1)

## downdate the R decomposition
R2 <-  rdowndate(R = R1, k = k, m = 1,
                 U = U, fast = FALSE, type = "row")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Remove m rows
## create data: n > p
set.seed(10)
n      <- 10
p      <- 6
X      <- matrix(rnorm(n * p, 1), n, p)
output <- fastQR::qr(X, type = "householder",
                    nb = NULL,
                     complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## select the rows to be deleted from X and update X
k  <- 2
m  <- 2
X1 <- X[-c(k,k+m-1),]
U  <- t(X[k:(k+m-1), ])

## downdate the R decomposition
R2 <- rdowndate(R = R1, k = k, m = m,
                U = U, fast = FALSE, type = "row")

## check
max(abs(crossprod(R2) - crossprod(X1)))

Fast updating of the R matrix

Description

updates the R factorization when $m \geq 1$ rows or columns are added to the matrix $X \in \mathbb{R}^{n \times p}$ , where $n > p$ . The R factorization of $X$ produces an upper triangular matrix $R \in \mathbb{R}^{p \times p}$ such that $X^\top X = R^\top R$ . For more details on this method, refer to Golub and Van Loan (2013). Columns can only be added in positions $p+1$ through $p+m$ , while the position of added rows does not need to be specified.

Arguments

`R`	a $p\times p$ upper triangular matrix.
`U`	either a $n\times m$ matrix or a $p\times m$ matrix of columns or rows to be added.
`type`	either 'row' of 'column', for adding rows or columns.
`fast`	fast mode: disable to check whether the provided matrices are valid inputs. Default is FALSE.

Value

R the updated R matrix.

References

Björck Å (2015). Numerical methods in matrix computations, volume 59 of Texts in Applied Mathematics. Springer, Cham. ISBN 978-3-319-05088-1; 978-3-319-05098-8, doi:10.1007/978-3-319-05089-8.

Bernardi M, Busatto C, Cattelan M (2024). “Fast QR updating methods for statistical applications.” 2412.05905, https://arxiv.org/abs/2412.05905.

Examples

## Add one column
## generate sample data
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create column to be added
u  <- matrix(rnorm(n), n, 1)
X1 <- cbind(X, u)

## update the R decomposition
R2 <- fastQR::rupdate(X = X, R = R1, U = u,
                      fast = FALSE, type = "column")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Add m columns
## generate sample data
set.seed(1234)
n <- 10
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the matrix of columns to be added
m  <- 2
U  <- matrix(rnorm(n*m), n, m)
X1 <- cbind(X, U)

# QR update
R2 <- fastQR::rupdate(X = X, R = R1, U = U,
                      fast = FALSE, type = "column")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Add one row
## generate sample data
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the row u to be added
u  <- matrix(data = rnorm(p), p, 1)
k  <- 5
if (k<=n) {
  X1 <- rbind(rbind(X[1:(k-1), ], t(u)), X[k:n, ])
} else {
  X1 <- rbind(rbind(X, t(u)))
}

## update the R decomposition
R2 <- fastQR::rupdate(R = R1, X = X,
                      U = u,
                      type = "row")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Add m rows
## generate sample data
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the matrix of rows to be added
m  <- 2
U  <- matrix(data = rnorm(p*m), p, m)
k  <- 5
if (k<=n) {
  X1 <- rbind(rbind(X[1:(k-1), ], t(U)), X[k:n, ])
} else {
  X1 <- rbind(rbind(X, t(U)))
}

## update the R decomposition
R2 <- fastQR::rupdate(R = R1, X = X,
                      U = U,
                      fast = FALSE,
                      type = "row")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Add one column
## generate sample data
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create column to be added
u  <- matrix(rnorm(n), n, 1)
X1 <- cbind(X, u)

## update the R decomposition
R2 <- fastQR::rupdate(X = X, R = R1, U = u,
                      fast = FALSE, type = "column")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Add m columns
## generate sample data
set.seed(1234)
n <- 10
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the matrix of columns to be added
m  <- 2
U  <- matrix(rnorm(n*m), n, m)
X1 <- cbind(X, U)

# QR update
R2 <- fastQR::rupdate(X = X, R = R1, U = U,
                      fast = FALSE, type = "column")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Add one row
## generate sample data
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the row u to be added
u  <- matrix(data = rnorm(p), p, 1)
k  <- 5
if (k<=n) {
  X1 <- rbind(rbind(X[1:(k-1), ], t(u)), X[k:n, ])
} else {
  X1 <- rbind(rbind(X, t(u)))
}

## update the R decomposition
R2 <- fastQR::rupdate(R = R1, X = X,
                      U = u,
                      type = "row")

## check
max(abs(crossprod(R2) - crossprod(X1)))

## Add m rows
## generate sample data
set.seed(1234)
n <- 12
p <- 5
X <- matrix(rnorm(n * p, 1), n, p)

## get the initial QR factorization
output <- fastQR::qr(X, complete = TRUE)
Q      <- output$Q
R      <- output$R
R1     <- R[1:p,]

## create the matrix of rows to be added
m  <- 2
U  <- matrix(data = rnorm(p*m), p, m)
k  <- 5
if (k<=n) {
  X1 <- rbind(rbind(X[1:(k-1), ], t(U)), X[k:n, ])
} else {
  X1 <- rbind(rbind(X, t(U)))
}

## update the R decomposition
R2 <- fastQR::rupdate(R = R1, X = X,
                      U = U,
                      fast = FALSE,
                      type = "row")

## check
max(abs(crossprod(R2) - crossprod(X1)))

Package 'fastQR'

Help Index

The QR factorization of a matrix

Description

Arguments

Value

References

Examples

Cholesky decomposition via QR factorization.

Description

Usage

Arguments

Value

Fast downdating of the QR factorization

Description

Usage

Arguments

Value

References

Examples

Ordinary least squares for the linear regression model

Description

Usage

Arguments

Value

Examples

Ordinary least squares for the linear multivariate regression model

Description

Arguments

Value

Examples

RIDGE estimator for the linear multivariate regression model

Description

Arguments

Value

Examples

Cross-validation of the RIDGE estimator for the linear multivariate regression model

Description

Arguments

Value

Examples

RIDGE estimation for the linear regression model

Description

Usage

Arguments

Value

Examples

Cross-validation of the RIDGE estimator for the linear regression model

Description

Arguments

Value

Examples

Solution of linear system of equations, via the QR decomposition.

Description

Usage

Arguments

Value

References

Examples

Fast updating of the QR factorization

Description

Usage

Arguments

Value

References

Examples

Cholesky decomposition via R factorization.

Description

Arguments

Value

References

Examples

Fast downdating of the R matrix

Description

Usage

Arguments

Value

References

Examples

Fast updating of the R matrix