Package {fastkqr}

Type: Package
Title: A Fast Algorithm for Kernel Quantile Regression
Version: 1.0.1
Date: 2026-06-30
Author: Qian Tang [aut, cre], Yuwen Gu [aut], Boxiang Wang [aut]
Maintainer: Qian Tang <qian-tang@uiowa.edu>
Description: Implements fast algorithms for kernel quantile regression and related models, including non-crossing kernel quantile regression and regularized linear quantile regression. The methods are described in Tang, Gu and Wang (2026) <doi:10.1080/10618600.2025.2541004>.
Depends: R (≥ 3.5.0)
Imports: stats, parallel
License: GPL-2
NeedsCompilation: yes
Encoding: UTF-8
Suggests: knitr, rmarkdown
VignetteBuilder: knitr
Config/roxygen2/version: 8.0.0
RoxygenNote: 7.2.3
Packaged: 2026-06-30 17:22:29 UTC; qtang7
Repository: CRAN
Date/Publication: 2026-07-01 22:30:22 UTC

Extract Coefficients from a cv.fastlqr Object

Description

Extract Coefficients from a cv.fastlqr Object

Usage

## S3 method for class 'cv.fastlqr'
coef(object, s = c("lambda.1se", "lambda.min"), ...)

Arguments

object

A fitted cv.fastlqr object.

s

"lambda.1se", "lambda.min", or numeric lambda value(s).

...

Passed to coef.fastlqr().

Value

A coefficient matrix.

Extract Coefficients from a fastlqr Object

Description

Extract Coefficients from a fastlqr Object

Usage

## S3 method for class 'fastlqr'
coef(object, s = NULL, ...)

Arguments

object

A fitted fastlqr object.

s

Optional lambda value(s). If omitted, all coefficients are returned.

...

Not used.

Value

A coefficient matrix with rows (Intercept) and predictors.

Extract model coefficients from a kqr object.

Description

Computes the coefficients at the requested value(s) for lambda from a kqr() object.

Usage

## S3 method for class 'kqr'
coef(object, s = NULL, ...)

Arguments

object

Fitted kqr() object.

s

Value(s) of the penalty parameter lambda at which coefficients are required. Default is the entire sequence.

...

Not used.

Details

s is the new vector of lambda values at which predictions are requested. If s is not in the lambda sequence used for fitting the model, the coef function will use linear interpolation to make predictions. The new values are interpolated using a fraction of coefficients from both left and right lambda indices.

Value

The coefficients at the requested values for lambda.

See Also

kqr() and predict.kqr().

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
lambda <- 10^seq(0, -2, length.out = 3)
fit <- kqr(x, y, lambda = lambda, tau = 0.5)
coef(fit)

Extract model coefficients from a nckqr object.

Description

Computes the coefficients at the requested value(s) for lambda1 for a given lambda2 from a nckqr() object.

Usage

## S3 method for class 'nckqr'
coef(object, s1 = NULL, s2, ...)

Arguments

object

A fitted nckqr object.

s1

Value(s) of the penalty parameter lambda1 at which coefficients are required. Default is the entire sequence used to create the model.

s2

Value of the penalty parameter lambda2 at which coefficients are required.

...

Not used.

Details

s1 is the new vector of lambda1 values at which predictions are requested. If s1 is not in the lambda sequence used for fitting the model, the coef function will use linear interpolation to make predictions. The new values are interpolated using a fraction of coefficients from both left and right lambda indices.

Value

The coefficients for the non-crossing kernel quantile regression model.

See Also

nckqr() and predict.nckqr().

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
tau <- c(0.25, 0.5, 0.75)
lambda1 <- 1
lambda2 <- 10^seq(0, -2, length.out = 3)
fit <- nckqr(x, y, lambda1 = lambda1, lambda2 = lambda2, tau = tau)
coef(fit, s1 = lambda1, s2 = lambda2[1])

cross-validation for selecting the tuning parameter of kernel quantile regression

Description

Performs k-fold cross-validation for kqr().

Usage

cv.kqr(
  x,
  y,
  tau,
  lambda = NULL,
  sigma = NULL,
  kernel = "rbfdot",
  nfolds = 5L,
  foldid,
  parallel = FALSE,
  ncores = max(1L, parallel::detectCores() - 1L),
  Kfull = NULL,
  ...
)

Arguments

x

A numerical input matrix. The dimension is n rows and p columns.

y

Response variable.

tau

A user-supplied tau value for a quantile level.

lambda

A user-supplied lambda sequence.

sigma

Kernel bandwidth parameter used by "rbfdot" and "laplacedot".

kernel

Kernel function. Supported values are "rbfdot", and "laplacedot". Default is "rbfdot".

nfolds

The number of folds in cross-validation. Default is 5.

foldid

An optional vector which indexed the observations into each cross-validation fold. If supplied, nfolds is overridden.

parallel

If TRUE, fit cross-validation folds in parallel. Default is FALSE.

ncores

Number of worker processes used when parallel=TRUE.

Kfull

Optional precomputed full n by n kernel matrix. If supplied, fold-specific training and validation kernels are sliced from this matrix.

...

Additional arguments passed into kqr

Details

The function computes the average cross-validation error and reports the standard error.

Value

An object of class cv.kqr is returned, which is a list with the components describing the cross-validation error.

lambda

The lambda candidate values.

cvm

Mean cross-validation error.

cvsd

Estimates of standard error of cross-validation error.

cvup

The upper curve: cvm + cvsd.

cvlo

The lower curve: cvm - cvsd.

lambda.min

The lambda incurring the minimum cross-validation error.

lambda.1se

The largest lambda whose error is within one standard error of the minimum.

cv.min

The cross-validation error at lambda.min.

cv.1se

The cross-validation error at lambda.1se.

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
lambda <- 10^seq(0, -2, length.out = 3)
foldid <- rep(1:3, length.out = nrow(x))
cv.fit <- cv.kqr(x, y, lambda = lambda, tau = 0.5, foldid = foldid)

cross-validation for selecting the tuning parameter 'lambda2' of non-crossing kernel quantile regression

Description

Performs k-fold cross-validation for nckqr().

Usage

cv.nckqr(
  x,
  y,
  tau,
  lambda1 = NULL,
  lambda2 = NULL,
  sigma = NULL,
  kernel = "rbfdot",
  nfolds = 5L,
  foldid,
  parallel = FALSE,
  ncores = max(1L, parallel::detectCores() - 1L),
  Kfull = NULL,
  ...
)

Arguments

x

A numerical input matrix. The dimension is n rows and p columns.

y

Response variable.

tau

A user-supplied tau sequence.

lambda1

A user-supplied lambda1 value.

lambda2

A user-supplied lambda2 sequence.

sigma

Kernel bandwidth parameter used by "rbfdot" and "laplacedot".

kernel

Kernel function. Supported values are "rbfdot", and "laplacedot". Default is "rbfdot".

nfolds

The number of folds in cross-validation. Default is 5.

foldid

An optional vector which indexed the observations into each cross-validation fold. If supplied, nfolds is overridden.

parallel

If TRUE, fit cross-validation folds in parallel. Default is FALSE.

ncores

Number of worker processes used when parallel=TRUE.

Kfull

Optional precomputed full n by n kernel matrix. If supplied, fold-specific training and validation kernels are sliced from this matrix.

...

Additional arguments passed into nckqr

Details

The function computes the average cross-validation error and reports the standard error.

Value

An object of class cv.nckqr is returned, which is a list with the components describing the cross-validation error.

lambda2

The lambda2 candidate values.

cvm

Mean cross-validation error.

cvsd

Estimates of standard error of cross-validation error.

cvup

The upper curve: cvm + cvsd.

cvlo

The lower curve: cvm - cvsd.

lambda.min

The lambda2 incurring the minimum cross-validation error.

lambda.1se

The largest lambda whose error is within one standard error of the minimum.

cv.min

The cross-validation error at lambda.min.

cv.1se

The cross-validation error at lambda.1se.

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
lambda <- 10^seq(0, -2, length.out = 3)
foldid <- rep(1:3, length.out = nrow(x))
cvres <- cv.nckqr(x, y, tau = c(0.25, 0.5, 0.75),
  lambda1 = 1, lambda2 = lambda, foldid = foldid)

Cross-Validation for Regularized Linear Quantile Regression

Description

Selects the ridge penalty for qr() by K-fold cross-validation.

Usage

cv.qr(
  x,
  y,
  lambda,
  tau = 0.5,
  nfolds = 5L,
  foldid = NULL,
  delta = 0.125,
  eps = 1e-05,
  maxit = 1e+06,
  gam = 1e-07,
  is_exact = FALSE,
  standardize = FALSE,
  keep.x = FALSE,
  parallel = FALSE,
  ncores = max(1L, parallel::detectCores() - 1L),
  ...
)

Arguments

x

Numeric predictor matrix.

y

Numeric response vector.

lambda

Positive ridge penalty sequence.

tau

Single quantile level in (0, 1).

nfolds

Number of folds.

foldid

Optional fold assignments.

delta

Smoothing parameter passed to qr().

eps

Stopping criterion passed to qr().

maxit

Maximum iterations passed to qr().

gam

Numerical ridge passed to qr().

is_exact

Whether to use exact projection in qr().

standardize

Standardization is not implemented; must be FALSE.

keep.x

Whether to store training x in the final model.

parallel

Parallel CV is not implemented for this first version.

ncores

Reserved for future parallel CV support.

...

Compatibility arguments. method="fast_lqr" is accepted; removed linear QR methods are rejected.

Value

An object of class cv.fastlqr.

Solve the kernel quantile regression. The solution path is computed at a grid of values of tuning parameter lambda.

Description

Solve the kernel quantile regression. The solution path is computed at a grid of values of tuning parameter lambda.

Usage

kqr(
  x,
  y,
  lambda,
  tau,
  delta = 0.125,
  eps = 1e-05,
  maxit = 1e+06,
  gam = 1e-07,
  sigma = NULL,
  kernel = "rbfdot",
  Kmat = NULL,
  is_exact = FALSE
)

Arguments

x

A numerical input matrix. The dimension is n rows and p columns.

y

Response variable. The length is n.

lambda

A user-supplied lambda sequence.

tau

A user-supplied tau value for a quantile level.

delta

The smoothing index for method='huber'. Default is 0.125.

eps

Stopping criterion.

maxit

Maximum number of iterates.

gam

A small number for numerical stability.

sigma

Kernel bandwidth parameter used by "rbfdot" and "laplacedot".

kernel

Kernel function. Supported values are "rbfdot", and "laplacedot". Default is "rbfdot".

Kmat

Optional precomputed training kernel matrix. If supplied, it is reused instead of recomputing kernelMat(x, x, sigma, kernel).

is_exact

Exact or approximated solutions. Default is FALSE.

Details

The function implements an accelerated proximal gradient descent to solve kernel quantile regression.

Value

An object with S3 class kqr

alpha

An n+1 by L matrix of coefficients, where n is the number of observations and L is the number of tuning parameters. The first row of alpha contains the intercepts.

lambda

The lambda sequence that was actually used.

delta

The smoothing index.

npass

The total number of iterates used to train the classifier.

jerr

Warnings and errors; 0 if none.

info

A list includes some settings used to fit this object: eps, maxit

.

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
lambda <- 10^seq(0, -2, length.out = 3)
fit <- kqr(x, y, lambda = lambda, tau = 0.5)

Solve the non-crossing kernel quantile regression

Description

Trains the kernel quantile regression

Usage

nckqr(
  x,
  y,
  lambda1,
  lambda2,
  tau,
  delta = 0.125,
  eps = 1e-08,
  maxit = 5e+06,
  gam = 1e-07,
  sigma = NULL,
  kernel = "rbfdot",
  Kmat = NULL,
  is_exact = FALSE
)

Arguments

x

A numerical input matrix with n rows and p columns.

y

Response variable. The length is n.

lambda1

A user-supplied lambda1 sequence. The length is L1.

lambda2

A user-supplied lambda2 sequence. The length is L2.

tau

A user-supplied tau sequence for quantile levels. The length is ntau.

delta

The smoothing index for method='huber'. Default is 0.125.

eps

Stopping criterion.

maxit

Maximum number of iterates.

gam

A small number for numerical stability.

sigma

Kernel bandwidth parameter used by "rbfdot" and "laplacedot".

kernel

Kernel function. Supported values are "rbfdot", and "laplacedot". Default is "rbfdot".

Kmat

Optional precomputed training kernel matrix. If supplied, it is reused instead of recomputing kernelMat(x, x, sigma, kernel).

is_exact

Exact or approximated solutions.

Details

The function implements the majorization-minimization method to solve non-crossing kernel quantile regression.

Value

An object with S3 class nckqr

alpha

An n+1 by L matrix of coefficients, where n represents the number of observations, ntau represents the number of quantile levels, and L denotes the number of tuning parameters.

tau

The tau sequence that was actually used.

lambda1

The lambda1 sequence that was actually used.

lambda2

The lambda2 sequence that was actually used.

delta

The smoothing index.

npass

The total number of iterates used to train the classifier.

jerr

Warnings and errors; 0 if none.

info

A list includes some settings used to fit this object: eps, maxit

.

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
tau <- c(0.25, 0.5, 0.75)
lambda1 <- 1
lambda2 <- 10^seq(0, -2, length.out = 3)
fit <- nckqr(x, y, lambda1 = lambda1, lambda2 = lambda2, tau = tau)

Predict from a cv.fastlqr Object

Description

Predict from a cv.fastlqr Object

Usage

## S3 method for class 'cv.fastlqr'
predict(object, newx, s = c("lambda.1se", "lambda.min"), ...)

Arguments

object

A fitted cv.fastlqr object.

newx

New predictor matrix.

s

"lambda.1se", "lambda.min", or numeric lambda value(s).

...

Passed to predict.fastlqr().

Value

A matrix of fitted linear quantiles.

Predict from a fastlqr Object

Description

Predict from a fastlqr Object

Usage

## S3 method for class 'fastlqr'
predict(object, newx, s = NULL, ...)

Arguments

object

A fitted fastlqr object.

newx

New predictor matrix. If omitted, training data are used when the model was fit with keep.x=TRUE.

s

Optional lambda value(s).

...

Not used.

Value

A matrix of fitted linear quantiles.

Predict the fitted values for a kqr object.

Description

Predict the fitted values for a kqr object.

Usage

## S3 method for class 'kqr'
predict(object, x, newx = NULL, s = NULL, ...)

Arguments

object

A fitted kqr object.

x

The predictor matrix, i.e., the x matrix used when fitting the kqr object.

newx

A matrix of new values for x at which predictions are to be made. Note that newx must be of a matrix form, predict function does not accept a vector or other formats of newx.

s

Value(s) of the penalty parameter lambda at which predictions are required. Default is the entire sequence used to create the model.

...

Not used.

Details

The result is \beta_0 + K_i' \alpha where \beta_0 and \alpha are from the kqr object and K_i is the ith row of the kernel matrix.

Value

Returns the fitted values.

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
lambda <- 10^seq(0, -2, length.out = 3)
fit <- kqr(x, y, lambda = lambda, tau = 0.5)
predict(fit, x, x[1:5, , drop = FALSE])

Predict the fitted values for a nckqr object.

Description

Predict the fitted values for a nckqr object.

Usage

## S3 method for class 'nckqr'
predict(object, x, newx = NULL, s2, s1 = NULL, ...)

Arguments

object

A fitted nckqr object.

x

The predictor matrix, i.e., the x matrix used when fitting the nckqr object.

newx

A matrix of new values for x at which predictions are to be made. Note that newx must be of a matrix form, predict function does not accept a vector or other formats of newx.

s2

Value of the penalty parameter lambda2 at which predictions are required.

s1

Value(s) of the penalty parameter lambda1 at which predictions are required. Default is the entire sequence used to create the model.

...

Not used.

Value

Returns the fitted values for the non-crossing kernel quantile regression model.

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
tau <- c(0.25, 0.5, 0.75)
lambda1 <- 1
lambda2 <- 10^seq(0, -2, length.out = 3)
fit <- nckqr(x, y, lambda1 = lambda1, lambda2 = lambda2, tau = tau)
predict(fit, x, x[1:5, , drop = FALSE], s1 = lambda1, s2 = lambda2[1])

Regularized Linear Quantile Regression

Description

Fits a parametric linear quantile regression path Q_\tau(Y | X) = \beta_0 + X \beta.

Usage

qr(
  x,
  y,
  lambda,
  tau = 0.5,
  delta = 0.125,
  eps = 1e-05,
  maxit = 1e+06,
  gam = 1e-07,
  is_exact = FALSE,
  standardize = FALSE,
  intercept = TRUE,
  keep.x = FALSE,
  ...
)

Arguments

x

Numeric predictor matrix.

y

Numeric response vector.

lambda

Positive ridge penalty sequence.

tau

Single quantile level in (0, 1).

delta

Smoothing parameter used by the Fortran solver.

eps

Stopping criterion.

maxit

Maximum number of iterations.

gam

Small ridge added internally for numerical stability.

is_exact

Whether to run the exact projection path.

standardize

Standardization is not implemented; must be FALSE.

intercept

Intercept is required by fast_lqr; must be TRUE.

keep.x

If TRUE, store the training matrix for later prediction.

...

Compatibility arguments. method="fast_lqr" is accepted; removed linear QR methods are rejected.

Details

The objective is

n^{-1}\sum_i \rho_\tau(y_i - \beta_0 - x_i^T\beta) + \lambda ||\beta||_2^2 / 2.

The intercept is not penalized. This function masks base::qr(); use base::qr() when QR decomposition is needed. This parametric linear model is the supported path for linear quantile regression; kernel methods kqr() and nckqr() no longer support kernel = "lineardot".

Value

An object of class fastlqr.

Examples

set.seed(1)
x <- matrix(rnorm(60), 30, 2)
y <- x[, 1] + rnorm(30)
lambda <- 10^seq(0, -2, length.out = 3)
fit <- qr(x, y, lambda = lambda, tau = 0.5)
predict(fit, x[1:5, , drop = FALSE])