rqPen: A package for estimating quantile regression models using penalized objective functions.

Description

The package estimates a quantile regression model using LASSO, Adaptive LASSO, SCAD, MCP, elastic net, and their group counterparts, with the exception of elastic net for which there is no group penalty implementation.

rqPen functions

The most important functions are rq.pen(), rq.group.pen(), rq.pen.cv() and rq.group.pen.cv(). These functions fit quantile regression models with individual or group penalties. The cv functions automate the cross-validation process for selection of tuning parameters.

Author(s)

Maintainer: Ben Sherwood ben.sherwood@ku.edu

Authors:

• Shaobo Li

• Adam Maidman

See Also

Useful links:

• https://github.com/bssherwood/rqpen/blob/master/ignore/rqPenArticle.pdf

Plots of coefficients by lambda for cv.rq.group.pen and cv.rq.pen

Description

Warning: this function is no longer exported.

Usage

beta_plots(model, voi = NULL, logLambda = TRUE, loi = NULL, ...)

Arguments

Value

Plot of how beta estimates change with lambda.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

## Not run: 
  set.seed(1)
  x <- matrix(rnorm(800),nrow=100)
  y <- 1 + x[,1] - 3*x[,5] + rnorm(100)
  lassoModels <- cv.rq.pen(x,y)
  b_plot <- beta_plots(lassoModels)

## End(Not run)

Plot of how coefficients change with tau

Description

Plot of how coefficients change with tau

Usage

bytau.plot(x, ...)

Arguments

Value

Returns the plot of how coefficients change with tau.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Plot of how coefficients change with tau.

Description

Plot of how coefficients change with tau.

Usage

## S3 method for class 'rq.pen.seq'
bytau.plot(x, a = NULL, lambda = NULL, lambdaIndex = NULL, vars = NULL, ...)

Arguments

Value

A plot of coefficient values by tau.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

  set.seed(1)
  x <- matrix(rnorm(800),nrow=100)
  y <- 1 + x[,1] - 3*x[,5] + rnorm(100)
  lassoModels <- rq.pen(x,y,tau=seq(.1,.9,.1))
  bytau.plot(lassoModels,lambda=lassoModels$lambda[5])

Plot of coefficients varying by quantiles for rq.pen.seq.cv object

Description

Produces plots of how coefficient estimates vary by quantile for models selected by using cross validation.

Usage

## S3 method for class 'rq.pen.seq.cv'
bytau.plot(
  x,
  septau = ifelse(x$fit$penalty != "gq", TRUE, FALSE),
  cvmin = TRUE,
  useDefaults = TRUE,
  vars = NULL,
  ...
)

Arguments

Value

Returns plots of coefficient estimates varying by quantile.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

 set.seed(1)
 x <- matrix(runif(800),nrow=100)
 y <- 1 + x[,1] - 3*x[,5] + (1+x[,4])*rnorm(100)
 lmcv <- rq.pen.cv(x,y,tau=seq(.1,.9,.1))
 bytau.plot(lmcv)

Coefficients from a cv.rq.group.pen object

Description

Coefficients from a cv.rq.group.pen object

Usage

## S3 method for class 'cv.rq.group.pen'
coef(object, lambda = "min", ...)

Arguments

Value

Vector of coefficients.

Returns Coefficients of a cv.rq.pen object

Description

Warning: this function is no longer exported, due to the switch from cv.rq.pen() to rq.pen.cv().

Usage

## S3 method for class 'cv.rq.pen'
coef(object, lambda = "min", ...)

Arguments

Value

Coefficients for a given lambda, or the lambda associated with the minimum cv value.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Returns coefficients of a rq.pen.seq object

Description

Returns coefficients of a rq.pen.seq object

Usage

## S3 method for class 'rq.pen.seq'
coef(
  object,
  tau = NULL,
  a = NULL,
  lambda = NULL,
  modelsIndex = NULL,
  lambdaIndex = NULL,
  ...
)

Arguments

Value

A list of a matrix of coefficients for each tau and a combination

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

x <- matrix(runif(800),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(100)
m1 <- rq.pen(x,y,penalty="ENet",a=c(0,.5,1),tau=c(.25,.75),lambda=c(.1,.05,.01))
allCoefs <- coef(m1)
targetCoefs <- coef(m1,tau=.25,a=.5,lambda=.1)
idxApproach <- coef(m1,modelsIndex=2)
bothIdxApproach <- coef(m1,modelsIndex=2,lambdaIndex=1)

Returns coefficients from a rq.pen.seq.cv object.

Description

Returns coefficients from a rq.pen.seq.cv object.

Usage

## S3 method for class 'rq.pen.seq.cv'
coef(
  object,
  septau = ifelse(object$fit$penalty != "gq", TRUE, FALSE),
  cvmin = TRUE,
  useDefaults = TRUE,
  tau = NULL,
  ...
)

Arguments

Value

Returns coefficients

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

 ## Not run: 
 set.seed(1)
 x <- matrix(rnorm(800),nrow=100)
 y <- 1 + x[,1] - 3*x[,5] + rnorm(100)
 lassoModels <- rq.pen.cv(x,y,tau=seq(.1,.9,.1))
 coefficients(lassoModels,septau=FALSE)
 coefficients(lassoModels,cvmin=FALSE)

## End(Not run)

Old cross validation function for group penalty

Description

This function is no longer exported. Recommend using rq.group.pen.cv() instead.

Usage

cv.rq.group.pen(
  x,
  y,
  groups,
  tau = 0.5,
  lambda = NULL,
  penalty = "SCAD",
  intercept = TRUE,
  criteria = "CV",
  cvFunc = "check",
  nfolds = 10,
  foldid = NULL,
  nlambda = 100,
  eps = 1e-04,
  init.lambda = 1,
  alg = "huber",
  penGroups = NULL,
  ...
)

Arguments

Value

Returns the following:

beta

Matrix of coefficients for different values of lambda

residuals

Matrix of residuals for different values of lambda.

rho

Vector of rho, unpenalized portion of the objective function, for different values of lambda.

cv

Data frame with "lambda" and second column is the evaluation based on the criteria selected.

lambda.min

Lambda which provides the smallest statistic for the selected criteria.

penalty

Penalty selected.

intercept

Whether intercept was included in model.

groups

Group structure for penalty function.

References

• Yuan, M. and Lin, Y. (2006). Model selection and estimation in regression with grouped variables. J. R. Statist. Soc. B, 68, 49-67.

• Peng, B. and Wang, L. (2015). An Iterative Coordinate Descent Algorithm for High-Dimensional Nonconvex Penalized Quantile Regression. Journal of Computational and Graphical Statistics, 24, 676-694.

Examples

## Not run: 
x <- matrix(rnorm(800),nrow=100)
y <- 1 + x[,1] - 3*x[,5] + rnorm(100)
cv_model <- cv.rq.group.pen(x,y,groups=c(rep(1,4),rep(2,4)),criteria="BIC")

## End(Not run)

Plots of cross validation results as a function of lambda.

Description

Plots of cross validation results as a function of lambda.

Usage

cv_plots(model, logLambda = TRUE, loi = NULL, ...)

Arguments

Value

returns a cross validation plot

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Cross validation plot for cv.rq.group.pen object

Description

Cross validation plot for cv.rq.group.pen object

Usage

## S3 method for class 'cv.rq.group.pen'
plot(x, ...)

Arguments

Value

A cross validation plot.

Plot of coefficients of rq.pen.seq object as a function of lambda

Description

Plot of coefficients of rq.pen.seq object as a function of lambda

Usage

## S3 method for class 'rq.pen.seq'
plot(
  x,
  vars = NULL,
  logLambda = TRUE,
  tau = NULL,
  a = NULL,
  lambda = NULL,
  modelsIndex = NULL,
  lambdaIndex = NULL,
  main = NULL,
  ...
)

Arguments

Value

Returns plot(s) of coefficients as they change with lambda.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

set.seed(1)
x <- matrix(rnorm(100*8,sd=10),ncol=8)
y <- 1 + x[,1] + 3*x[,3] - x[,8] + rt(100,3)
m1 <- rq.pen(x,y,tau=c(.1,.5,.7),penalty="SCAD",a=c(3,4))
plot(m1,a=3,tau=.7)
plot(m1)
mlist <- list()
for(i in 1:6){
mlist[[i]] <- paste("Plot",i)
}
plot(m1,main=mlist)

Plots cross validation results from a rq.pen.seq.cv object

Description

Provides plots of cross-validation results by lambda. If septau is set to TRUE then plots the cross-validation results for each quantile. If septau is set to FALSE then provides one plot for cross-validation results across all quantiles.

Usage

## S3 method for class 'rq.pen.seq.cv'
plot(
  x,
  septau = ifelse(x$fit$penalty != "gq", TRUE, FALSE),
  tau = NULL,
  logLambda = TRUE,
  main = NULL,
  ...
)

Arguments

Value

Plots of the cross validation results by lambda.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

set.seed(1)
x <- matrix(rnorm(100*8,sd=1),ncol=8)
y <- 1 + x[,1] + 3*x[,3] - x[,8] + rt(100,3)
m1 <- rq.pen.cv(x,y,tau=c(.1,.3,.7))
plot(m1)
plot(m1,septau=FALSE)

Prediction for a cv.rq.pen object

Description

This function is no longer exported.

Usage

## S3 method for class 'cv.rq.pen'
predict(object, newx, lambda = "lambda.min", ...)

Arguments

Value

A vector of predictions.

Predictions from a qic.select object

Description

Predictions from a qic.select object

Usage

## S3 method for class 'qic.select'
predict(object, newx, sort = FALSE, ...)

Arguments

Value

A matrix of predicted values.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

x <- matrix(runif(800),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(100)
m1 <- rq.pen(x,y,tau=c(.25,.75))
q1 <- qic.select(m1)
newx <- matrix(runif(80),ncol=8)
preds <- predict(q1,newx)

Prediction for a rq.pen object

Description

This function is no longer exported.

Usage

## S3 method for class 'rq.pen'
predict(object, newx, ...)

Arguments

Value

A vector of predictions.

Predictions from rq.pen.seq object

Description

Predictions from rq.pen.seq object

Usage

## S3 method for class 'rq.pen.seq'
predict(
  object,
  newx,
  tau = NULL,
  a = NULL,
  lambda = NULL,
  modelsIndex = NULL,
  lambdaIndex = NULL,
  sort = FALSE,
  ...
)

Arguments

Value

A matrix of predictions for each tau and a combination

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

x <- matrix(runif(800),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(100)
m1 <- rq.pen(x,y,penalty="ENet",a=c(0,.5,1),tau=c(.25,.75),lambda=c(.1,.05,.01))
newx <- matrix(runif(80),ncol=8)
allCoefs <- predict(m1,newx)
targetCoefs <- predict(m1,newx,tau=.25,a=.5,lambda=.1)
idxApproach <- predict(m1,newx,modelsIndex=2)
bothIdxApproach <- predict(m1,newx,modelsIndex=2,lambdaIndex=1)

Predictions from rq.pen.seq.cv object

Description

Predictions from rq.pen.seq.cv object

Usage

## S3 method for class 'rq.pen.seq.cv'
predict(
  object,
  newx,
  tau = NULL,
  septau = ifelse(object$fit$penalty != "gq", TRUE, FALSE),
  cvmin = TRUE,
  useDefaults = TRUE,
  sort = FALSE,
  lambda = NULL,
  lambdaIndex = NULL,
  ...
)

Arguments

Value

A matrix of predictions for each tau and a combination

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

x <- matrix(runif(1600),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(200)
m1 <- rq.pen.cv(x,y,penalty="ENet",a=c(0,.5,1),tau=c(.25,.75),lambda=c(.1,.05,.01))
newx <- matrix(runif(80),ncol=8)
cvpreds <- predict(m1,newx)

Print a qic.select object

Description

Print a qic.select object

Usage

## S3 method for class 'qic.select'
print(x, ...)

Arguments

Value

Prints the coefficients of the qic.select object

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Print a rq.pen.seq object

Description

Print a rq.pen.seq object

Usage

## S3 method for class 'rq.pen.seq'
print(x, ...)

Arguments

Value

If only one model, prints a data.frame of the number of nonzero coefficients and lambda. Otherwise prints information about the quantiles being modeled and choices for a.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Prints a rq.pen.seq.cv object

Description

Prints a rq.pen.seq.cv object

Usage

## S3 method for class 'rq.pen.seq.cv'
print(x, ...)

Arguments

Value

Print of btr and gtr from a rq.pen.seq.cv object. If only one quantile is modeled then only btr is returned.

Calculate information criterion for penalized quantile regression models. Currently not exported.

Description

Calculate information criterion for penalized quantile regression models. Currently not exported.

Usage

qic(model, n, method = c("BIC", "AIC", "PBIC"))

Arguments

Value

Let \hat{\beta} be the coefficient vectors for the estimated model. Function returns the value

\log(\sum_{i=1}^n w_i \rho_\tau(y_i-x_i^\top\hat{\beta})) + d*b/(2n),

where d is the number of nonzero coefficients and b depends on the method used. For AIC b=2, for BIC b=log(n) and for PBIC d=log(n)*log(p) where p is the dimension of \hat{\beta}. The values of w_i default to one and are set using weights when fitting the models. Returns this value for each coefficient vector in the model, so one for every value of \lambda.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

References

Lee ER, Noh H, Park BU (2014). “Model Selection via Bayesian Information Criterion for Quantile Regression Models.” Journal of the American Statistical Association, 109(505), 216–229. ISSN 01621459.

Examples

## Not run: 
set.seed(1)
x <- matrix(runif(800),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(100)
m1 <- rq.pen(x,y,tau=c(.25,.75))
# returns the IC values for tau=.25
qic(m1$models[[1]],m1$n) 
# returns the IC values for tau=.75
qic(m1$models[[2]],m1$n)

## End(Not run) 

Select tuning parameters using IC

Description

Selects tuning parameter \lambda and a according to information criterion of choice. For a given \hat{\beta} the information criterion is calculated as

\log(\sum_{i=1}^n w_i \rho_\tau(y_i-x_i^\top\hat{\beta})) + d*b/(2n),

where d is the number of nonzero coefficients and b depends on the method used. For AIC b=2, for BIC b=log(n) and for PBIC d=log(n)*log(p) where p is the dimension of \hat{\beta}. If septau set to FALSE then calculations are made across the quantiles. Let \hat{\beta}^q be the coefficient vector for the qth quantile of Q quantiles. In addition let d_q and b_q be d and b values from the qth quantile model. Note, for all of these we are assuming eqn and a are the same. Then the summary across all quantiles is

\sum_{q=1}^Q w_q[ \log(\sum_{i=1}^n m_i \rho_\tau(y_i-x_i^\top\hat{\beta}^q)) + d_q*b_q/(2n)],

where w_q is the weight assigned for the qth quantile model.

Usage

qic.select(obj, ...)

Arguments

Value

Returns a qic.select object.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

References

Lee ER, Noh H, Park BU (2014). “Model Selection via Bayesian Information Criterion for Quantile Regression Models.” Journal of the American Statistical Association, 109(505), 216–229. ISSN 01621459.

Examples

set.seed(1)
x <- matrix(runif(800),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(100)
m1 <- rq.pen(x,y,penalty="ENet",a=c(0,.5,1),tau=c(.25,.75))
qic.select(m1)

Select tuning parameters using IC

Description

Selects tuning parameter \lambda and a according to information criterion of choice. For a given \hat{\beta} the information criterion is calculated as

\log(\sum_{i=1}^n w_i \rho_\tau(y_i-x_i^\top\hat{\beta})) + d*b/(2n),

where d is the number of nonzero coefficients and b depends on the method used. For AIC b=2, for BIC b=log(n) and for PBIC d=log(n)*log(p) where p is the dimension of \hat{\beta}. If septau set to FALSE then calculations are made across the quantiles. Let \hat{\beta}^q be the coefficient vector for the qth quantile of Q quantiles. In addition let d_q and b_q be d and b values from the qth quantile model. Note, for all of these we are assuming eqn and a are the same. Then the summary across all quantiles is

\sum_{q=1}^Q w_q[ \log(\sum_{i=1}^n m_i \rho_\tau(y_i-x_i^\top\hat{\beta}^q)) + d_q*b_q/(2n)],

where w_q is the weight assigned for the qth quantile model.

Usage

## S3 method for class 'rq.pen.seq'
qic.select(
  obj,
  method = c("BIC", "AIC", "PBIC"),
  septau = ifelse(obj$penalty != "gq", TRUE, FALSE),
  tauWeights = NULL,
  ...
)

Arguments

Value

coefficients

Coefficients of the selected models.

ic

Information criterion values for all considered models.

modelsInfo

Model info for the selected models related to the original object obj.

gic

Information criterion summarized across all quantiles. Only returned if septau set to FALSE

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

References

Lee ER, Noh H, Park BU (2014). “Model Selection via Bayesian Information Criterion for Quantile Regression Models.” Journal of the American Statistical Association, 109(505), 216–229. ISSN 01621459.

Examples

set.seed(1)
x <- matrix(runif(800),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(100)
m1 <- rq.pen(x,y,penalty="ENet",a=c(0,.5,1),tau=c(.25,.75))
qic.select(m1)

Select tuning parameters using IC

Description

Selects tuning parameter \lambda and a according to information criterion of choice. For a given \hat{\beta} the information criterion is calculated as

\log(\sum_{i=1}^n w_i \rho_\tau(y_i-x_i^\top\hat{\beta})) + d*b/(2n),

where d is the number of nonzero coefficients and b depends on the method used. For AIC b=2, for BIC b=log(n) and for PBIC d=log(n)*log(p) where p is the dimension of \hat{\beta}. If septau set to FALSE then calculations are made across the quantiles. Let \hat{\beta}^q be the coefficient vector for the qth quantile of Q quantiles. In addition let d_q and b_q be d and b values from the qth quantile model. Note, for all of these we are assuming eqn and a are the same. Then the summary across all quantiles is

\sum_{q=1}^Q w_q[ \log(\sum_{i=1}^n \rho_\tau(y_i-x_i^\top\hat{\beta}^q)) + d_q*b_q/(2n)],

where w_q is the weight assigned for the qth quantile model.

Usage

## S3 method for class 'rq.pen.seq.cv'
qic.select(
  obj,
  method = c("BIC", "AIC", "PBIC"),
  septau = ifelse(obj$fit$penalty != "gq", TRUE, FALSE),
  weights = NULL,
  ...
)

Arguments

Value

coefficients

Coefficients of the selected models.

ic

Information criterion values for all considered models.

modelsInfo

Model info for the selected models related to the original object obj.

gic

Information criterion summarized across all quantiles. Only returned if septau set to FALSE

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

References

Lee ER, Noh H, Park BU (2014). “Model Selection via Bayesian Information Criterion for Quantile Regression Models.” Journal of the American Statistical Association, 109(505), 216–229. ISSN 01621459.

Examples

set.seed(1)
x <- matrix(runif(800),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(100)
m1 <- rq.pen.cv(x,y,penalty="ENet",a=c(0,.5,1),tau=c(.25,.75))
qic.select(m1)

Title Quantile regression estimation and consistent variable selection across multiple quantiles

Description

Uses the group lasso penalty across the quantiles to provide consistent selection across all, K, modeled quantiles. Let \beta^q be the coefficients for the kth quantiles, \beta_j be the Q-dimensional vector of the jth coefficient for each quantile, and \rho_\tau(u) is the quantile loss function. The method minimizes

\sum_{q=1}^Q \frac{1}{n} \sum_{i=1}^n m_i \rho_\tau(y_i-x_i^\top\beta^q) + \lambda \sum_{j=1}^p ||\beta_j||_{2,w} .

Uses a Huber approximation in the fitting of model, as presented in Sherwood and Li (2022). Where,

||\beta_j||_{2,w} = \sqrt{\sum_{k=1}^K w_kv_j\beta_{kj}^2},

where w_k is a quantile weight that can be specified by tau.penalty.factor, v_j is a predictor weight that can be assigned by penalty.factor, and m_i is an observation weight that can be set by weights.

Usage

rq.gq.pen(
  x,
  y,
  tau,
  lambda = NULL,
  nlambda = 100,
  eps = ifelse(nrow(x) < ncol(x), 0.01, 0.001),
  weights = NULL,
  penalty.factor = NULL,
  scalex = TRUE,
  tau.penalty.factor = NULL,
  gmma = 0.2,
  max.iter = 200,
  lambda.discard = TRUE,
  converge.eps = 1e-04,
  beta0 = NULL
)

Arguments

Value

An rq.pen.seq object.

models:

A list of each model fit for each tau and a combination.

n:

Sample size.

p:

Number of predictors.

alg:

Algorithm used. Options are "huber" or any method implemented in rq(), such as "br".

tau:

Quantiles modeled.

a:

Tuning parameters a used.

modelsInfo:

Information about the quantile and a value for each model.

lambda:

Lambda values used for all models. If a model has fewer coefficients than lambda, say k. Then it used the first k values of lambda. Setting lambda.discard to TRUE will gurantee all values use the same lambdas, but may increase computational time noticeably and for little gain.

penalty:

Penalty used.

call:

Original call.

Each model in the models list has the following values.

coefficients:

Coefficients for each value of lambda.

rho:

The unpenalized objective function for each value of lambda.

PenRho:

The penalized objective function for each value of lambda.

nzero:

The number of nonzero coefficients for each value of lambda.

tau:

Quantile of the model.

a:

Value of a for the penalized loss function.

Author(s)

Shaobo Li and Ben Sherwood, ben.sherwood@ku.edu

References

Wang M, Kang X, Liang J, Wang K, Wu Y (2024). “Heteroscedasticity identification and variable selection via multiple quantile regression.” Journal of Statistical Computation and Simulation, 94(2), 297-314.

Sherwood B, Li S (2022). “Quantile regression feature selection and estimation with grouped variables using Huber approximation.” Statistics and Computing, 32(5), 75.

Examples

## Not run:  
n<- 200
p<- 10
X<- matrix(rnorm(n*p),n,p)
y<- -2+X[,1]+0.5*X[,2]-X[,3]-0.5*X[,7]+X[,8]-0.2*X[,9]+rt(n,2)
taus <- seq(0.1, 0.9, 0.2)
fit<- rq.gq.pen(X, y, taus)
#use IC to select best model, see rq.gq.pen.cv() for a cross-validation approach
qfit <- qic.select(fit)

## End(Not run)

Title Cross validation for consistent variable selection across multiple quantiles.

Description

Title Cross validation for consistent variable selection across multiple quantiles.

Usage

rq.gq.pen.cv(
  x = NULL,
  y = NULL,
  tau = NULL,
  lambda = NULL,
  nfolds = 10,
  cvFunc = c("rq", "se"),
  tauWeights = NULL,
  foldid = NULL,
  printProgress = FALSE,
  weights = NULL,
  ...
)

Arguments

Details

Let y_{b,i} and x_{b,i} index the observations in fold b. Let \hat{\beta}_{\tau,a,\lambda}^{-b} be the estimator for a given quantile and tuning parameters that did not use the bth fold. Let n_b be the number of observations in fold b. Then the cross validation error for fold b is

\mbox{CV}(b,\tau) = \sum_{q=1}^Q \frac{1}{n_b} \sum_{i=1}^{n_b} m_{b,i}v_q \rho_\tau(y_{b,i}-x_{b,i}^\top\hat{\beta}_{\tau_q,a,\lambda}^{-b}).

Where, m_{b,i} is the weight for the ith observation in fold b and v_q is a quantile specific weight. Note that \rho_\tau() can be replaced squared error loss. Provides results about how the average of the cross-validation error changes with \lambda. Uses a Huber approximation in the fitting of model, as presented in Sherwood and Li (2022).

Value

An rq.pen.seq.cv object.

cverr:

Matrix of cvSummary function, default is average, cross-validation error for each model, tau and a combination, and lambda.

cvse:

Matrix of the standard error of cverr foreach model, tau and a combination, and lambda.

fit:

The rq.pen.seq object fit to the full data.

btr:

Let blank, unlike rq.pen.seq.cv() or rq.group.pen.cv(), because optmizes the quantiles individually does not make sense with this penalty.

gtr:

A data.table for the combination of a and lambda that minimize the cross validation error across all tau.

gcve:

Group, across all quantiles, cross-validation error results for each value of a and lambda.

call:

Original call to the function.

Author(s)

Shaobo Li and Ben Sherwood, ben.sherwood@ku.edu

References

Wang M, Kang X, Liang J, Wang K, Wu Y (2024). “Heteroscedasticity identification and variable selection via multiple quantile regression.” Journal of Statistical Computation and Simulation, 94(2), 297-314.

Sherwood B, Li S (2022). “Quantile regression feature selection and estimation with grouped variables using Huber approximation.” Statistics and Computing, 32(5), 75.

Examples

## Not run:  
n<- 200
p<- 10
X<- matrix(rnorm(n*p),n,p)
y<- -2+X[,1]+0.5*X[,2]-X[,3]-0.5*X[,7]+X[,8]-0.2*X[,9]+rt(n,2)
taus <- seq(0.1, 0.9, 0.2)
cvfit<- rq.gq.pen.cv(x=X, y=y, tau=taus)
cvCoefs <- coefficients(cvfit)

## End(Not run)

Estimates a quantile regression model with a group penalized objective function.

Description

Warning: function is no longer exported. Recommend using rq.group.pen() instead. Similar to cv.rq.pen function, but uses group penalty. Group penalties use the L1 norm instead of L2 for computational convenience. As a result of this the group lasso penalty is the same as the typical lasso penalty and thus you should only use a SCAD or MCP penalty. Only the SCAD and MCP penalties incorporate the group structure into the penalty. The group lasso penalty is implemented because it is needed for the SCAD and MCP algorithm.

Usage

rq.group.fit(
  x,
  y,
  groups,
  tau = 0.5,
  lambda,
  intercept = TRUE,
  penalty = "SCAD",
  alg = "LP",
  a = 3.7,
  penGroups = NULL,
  ...
)

Arguments

Value

Returns the following:

coefficients

Coefficients of the model.

residuals

Residuals from the fitted model.

rho

Unpenalized portion of the objective function.

tau

Quantile being modeled.

n

Sample size.

intercept

Whether intercept was included in model.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu and Adam Maidman

References

• Yuan, M. and Lin, Y. (2006). Model selection and estimation in regression with grouped variables. J. R. Statist. Soc. B, 68, 49-67.

• Peng, B. and Wang, L. (2015). An Iterative Coordinate Descent Algorithm for High-Dimensional Nonconvex Penalized Quantile Regression. Journal of Computational and Graphical Statistics, 24, 676-694.

Fits quantile regression models using a group penalized objective function.

Description

Let the predictors be divided into G groups with G corresponding vectors of coefficients, \beta_1,\ldots,\beta_G. Let \rho_\tau(a) = a[\tau-I(a<0)]. Fits quantile regression models for Q quantiles by minimizing the penalized objective function of

\sum_{q=1}^Q \frac{1}{n} \sum_{i=1}^n m_i \rho_\tau(y_i-x_i^\top\beta^q) + \sum_{q=1}^Q \sum_{g=1}^G P(||\beta^q_g||_k,w_q*v_j*\lambda,a).

Where w_q and v_j are designated by penalty.factor and tau.penalty.factor respectively and m_i can be set by weights. The value of k is chosen by norm. Value of P() depends on the penalty. Briefly, but see references or vignette for more details,

Group LASSO (gLASSO)

P(||\beta||_k,\lambda,a)=\lambda||\beta||_k

Group SCAD

P(||\beta||_k,\lambda,a)=SCAD(||\beta||_k,\lambda,a)

Group MCP

P(||\beta||_k,\lambda,a)=MCP(||\beta||_k,\lambda,a)

Group Adaptive LASSO

P(||\beta||_k,\lambda,a)=\frac{\lambda ||\beta||_k}{|\beta_0|^a}

Note if k=1 and the group lasso penalty is used then this is identical to the regular lasso and thus function will stop and suggest that you use rq.pen() instead. For Adaptive LASSO the values of \beta_0 come from a Ridge solution with the same value of \lambda. If the Huber algorithm is used than \rho_\tau(y_i-x_i^\top\beta) is replaced by a Huber-type approximation. Specifically, it is replaced by h^\tau_\gamma(y_i-x_i^\top\beta)/2 where

h^\tau_\gamma(a) = a^2/(2\gamma)I(|a| \leq \gamma) + (|a|-\gamma/2)I(|a|>\gamma)+(2\tau-1)a.

Where if \tau=.5, we get the usual Huber loss function.

Usage

rq.group.pen(
  x,
  y,
  tau = 0.5,
  groups = 1:ncol(x),
  penalty = c("gLASSO", "gAdLASSO", "gSCAD", "gMCP"),
  lambda = NULL,
  nlambda = 100,
  eps = ifelse(nrow(x) < ncol(x), 0.05, 0.01),
  alg = c("huber", "br"),
  a = NULL,
  norm = 2,
  group.pen.factor = NULL,
  tau.penalty.factor = rep(1, length(tau)),
  scalex = TRUE,
  coef.cutoff = 1e-08,
  max.iter = 5000,
  converge.eps = 1e-04,
  gamma = IQR(y)/10,
  lambda.discard = TRUE,
  weights = NULL,
  ...
)

Arguments

Value

An rq.pen.seq object.

models

A list of each model fit for each tau and a combination.

n

Sample size.

p

Number of predictors.

alg

Algorithm used.

tau

Quantiles modeled.

penalty

Penalty used.

a

Tuning parameters a used.

lambda

Lambda values used for all models. If a model has fewer coefficients than lambda, say k. Then it used the first k values of lambda. Setting lambda.discard to TRUE will gurantee all values use the same lambdas, but may increase computational time noticeably and for little gain.

modelsInfo

Information about the quantile and a value for each model.

call

Original call.

Each model in the models list has the following values.

coefficients

Coefficients for each value of lambda.

rho

The unpenalized objective function for each value of lambda.

PenRho

The penalized objective function for each value of lambda. If scalex=TRUE then this is the value for the scaled version of x.

nzero

The number of nonzero coefficients for each value of lambda.

tau

Quantile of the model.

a

Value of a for the penalized loss function.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu, Shaobo Li shaobo.li@ku.edu and Adam Maidman

References

Peng B, Wang L (2015). “An iterative coordinate descent algorithm for high-dimensional nonconvex penalized quantile regression.” J. Comput. Graph. Statist., 24(3), 676-694.

Examples

## Not run:  
set.seed(1)
x <- matrix(rnorm(200*8,sd=1),ncol=8)
y <- 1 + x[,1] + 3*x[,3] - x[,8] + rt(200,3)
g <- c(1,1,1,2,2,2,3,3)
tvals <- c(.25,.75)
r1 <- rq.group.pen(x,y,groups=g)
r5 <- rq.group.pen(x,y,groups=g,tau=tvals)
#Linear programming approach with group SCAD penalty and L1-norm
m2 <- rq.group.pen(x,y,groups=g,alg="br",penalty="gSCAD",norm=1,a=seq(3,4))
# No penalty for the first group
m3 <- rq.group.pen(x,y,groups=g,group.pen.factor=c(0,rep(1,2)))
# Smaller penalty for the median
m4 <- rq.group.pen(x,y,groups=g,tau=c(.25,.5,.75),tau.penalty.factor=c(1,.25,1))

## End(Not run)

Performs cross validation for a group penalty.

Description

Performs cross validation for a group penalty.

Usage

rq.group.pen.cv(
  x,
  y,
  tau = 0.5,
  groups = 1:ncol(x),
  lambda = NULL,
  a = NULL,
  cvFunc = NULL,
  nfolds = 10,
  foldid = NULL,
  groupError = TRUE,
  cvSummary = mean,
  tauWeights = rep(1, length(tau)),
  printProgress = FALSE,
  weights = NULL,
  ...
)

Arguments

Details

Two cross validation results are returned. One that considers the best combination of a and lambda for each quantile. The second considers the best combination of the tuning parameters for all quantiles. Let y_{b,i}, x_{b,i}, and m_{b,i} index the response, predictors, and weights of observations in fold b. Let \hat{\beta}_{\tau,a,\lambda}^{-b} be the estimator for a given quantile and tuning parameters that did not use the bth fold. Let n_b be the number of observations in fold b. Then the cross validation error for fold b is

\mbox{CV}(b,\tau) = \frac{1}{n_b} \sum_{i=1}^{n_b} m_{b,i} \rho_\tau(y_{b,i}-x_{b,i}^\top\hat{\beta}_{\tau,a,\lambda}^{-b}).

Note that \rho_\tau() can be replaced by a different function by setting the cvFunc parameter. The function returns two different cross-validation summaries. The first is btr, by tau results. It provides the values of lambda and a that minimize the average, or whatever function is used for cvSummary, of \mbox{CV}(b). In addition it provides the sparsest solution that is within one standard error of the minimum results.

The other approach is the group tau results, gtr. Consider the case of estimating Q quantiles of \tau_1,\ldots,\tau_Q with quantile (tauWeights) of v_q. The gtr returns the values of lambda and a that minimizes the average, or again whatever function is used for cvSummary, of

\sum_{q=1}^Q v_q\mbox{CV}(b,\tau_q).

If only one quantile is modeled then the gtr results can be ignored as they provide the same minimum solution as btr.

Value

An rq.pen.seq.cv object.

cverr

Matrix of cvSummary function, default is average, cross-validation error for each model, tau and a combination, and lambda.

cvse

Matrix of the standard error of cverr foreach model, tau and a combination, and lambda.

fit

The rq.pen.seq object fit to the full data.

btr

A data.table of the values of a and lambda that are best as determined by the minimum cross validation error and the one standard error rule, which fixes a. In btr the values of lambda and a are selected seperately for each quantile.

gtr

A data.table for the combination of a and lambda that minimize the cross validation error across all tau.

gcve

Group, across all quantiles, cross-validation error results for each value of a and lambda.

call

Original call to the function.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu and Shaobo Li shaobo.li@ku.edu

Examples

set.seed(1)
x <- matrix(rnorm(100*8,sd=1),ncol=8)
y <- 1 + x[,1] + 3*x[,3] - x[,8] + rt(100,3)
g <- c(1,1,1,1,2,2,3,3)
tvals <- c(.25,.75)
## Not run: 
m1 <- rq.group.pen.cv(x,y,tau=c(.1,.3,.7),groups=g)
m2 <- rq.group.pen.cv(x,y,penalty="gAdLASSO",tau=c(.1,.3,.7),groups=g)
m3 <- rq.group.pen.cv(x,y,penalty="gSCAD",tau=c(.1,.3,.7),a=c(3,4,5),groups=g)
m4 <- rq.group.pen.cv(x,y,penalty="gMCP",tau=c(.1,.3,.7),a=c(3,4,5),groups=g)

## End(Not run)

Estimates a quantile regression model with a lasso penalized quanitle loss function.

Description

Fits a quantile regression model with the LASSO penalty. Uses the augmented data approach similar to the proposal in Sherwood and Wang (2016).

Usage

rq.lasso.fit(
  x,
  y,
  tau = 0.5,
  lambda = NULL,
  weights = NULL,
  intercept = TRUE,
  coef.cutoff = 1e-08,
  method = "br",
  penVars = NULL,
  scalex = TRUE,
  lambda.discard = TRUE,
  ...
)

Arguments

Value

Returns the following:

coefficients

Coefficients from the penalized model.

PenRho

Penalized objective function value. If scalex=TRUE then this is the value for the scaled version of x.

residuals

Residuals from the model.

rho

Objective function evaluation without the penalty.

tau

Conditional quantile being modeled.

n

Sample size.

References

• Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B, 58, 267–288.

• Wu, Y. and Liu, Y. (2009). Variable selection in quantile regression. Statistica Sinica, 19, 801–817.

• Sherwood, B. and Wang, L. (2016) Partially linear additive quantile regression in ultra-high dimension. Annals of Statistics 44, 288–317.

Examples

## Not run: 
x <- matrix(rnorm(800),nrow=100)
y <- 1 + x[,1] - 3*x[,5] + rnorm(100)
lassoModel <- rq.lasso.fit(x,y,lambda=.1)

## End(Not run)

Non-convex penalized quantile regression

Description

Warning: this function is no longer exported. Produces penalized quantile regression models for a range of lambdas and penalty of choice. If lambda is unselected than an iterative algorithm is used to find a maximum lambda such that the penalty is large enough to produce an intercept only model. Then range of lambdas goes from the maximum lambda found to "eps" on the log scale. Local linear approximation approach used by Wang, Wu and Li to extend LLA as proposed by Zou and Li (2008) to the quantile regression setting.

Usage

rq.nc.fit(
  x,
  y,
  tau = 0.5,
  lambda = NULL,
  weights = NULL,
  intercept = TRUE,
  penalty = "SCAD",
  a = 3.7,
  iterations = 1,
  converge_criteria = 1e-06,
  alg = "LP",
  penVars = NULL,
  internal = FALSE,
  ...
)

Arguments

Value

Returns the following:

coefficients

Coefficients from the penalized model.

PenRho

Penalized objective function value. If scalex=TRUE then this is the value for the scaled version of x.

residuals

Residuals from the model.

rho

Objective function evaluation without the penalty.

coefficients

Coefficients from the penalized model.

tau

Conditional quantile being modeled.

n

Sample size.

penalty

Penalty used, SCAD or MCP.

penalty

Penalty selected.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu and Adam Maidman.

References

• Wang, L., Wu, Y. and Li, R. (2012). Quantile regression of analyzing heterogeneity in ultra-high dimension. J. Am. Statist. Ass, 107, 214–222.

• Wu, Y. and Liu, Y. (2009). Variable selection in quantile regression. Statistica Sinica, 19, 801–817.

• Zou, H. and Li, R. (2008). One-step sparse estimates in nonconcave penalized likelihood models. Ann. Statist., 36, 1509–1533.

• Peng, B. and Wang, L. (2015). An iterative coordinate-descent algorithm for high-dimensional nonconvex penalized quantile regression. J. Comp. Graph., 24, 676–694.

Examples

## Not run: 
x <- matrix(rnorm(800),nrow=100)
y <- 1 + x[,1] - 3*x[,5] + rnorm(100)
scadModel <- rq.nc.fit(x,y,lambda=1)

## End(Not run)

Fit a quantile regression model using a penalized quantile loss function.

Description

Let q index the Q quantiles of interest. Let \rho_\tau(a) = a[\tau-I(a<0)]. Fits quantile regression models by minimizing the penalized objective function of

\frac{1}{n} \sum_{q=1}^Q \sum_{i=1}^n m_i \rho_\tau(y_i-x_i^\top\beta^q) + \sum_{q=1}^Q \sum_{j=1}^p P(\beta^q_p,w_q*v_j*\lambda,a).

Where w_q and v_j are designated by penalty.factor and tau.penalty.factor respectively, and m_i is designated by weights. Value of P() depends on the penalty. See references or vignette for more details,

LASSO:

P(\beta,\lambda,a)=\lambda|\beta|

SCAD:

P(\beta,\lambda,a)=SCAD(\beta,\lambda,a)

MCP:

P(\beta,\lambda,a)=MCP(\beta,\lambda,a)

Ridge:

P(\beta,\lambda,a)=\lambda\beta^2

Elastic Net:

P(\beta,\lambda,a)=a*\lambda|\beta|+(1-a)*\lambda*\beta^2

Adaptive LASSO:

P(\beta,\lambda,a)=\frac{\lambda |\beta|}{|\beta_0|^a}

For Adaptive LASSO the values of \beta_0 come from a Ridge solution with the same value of \lambda. Three different algorithms are implemented

huber:

Uses a Huber approximation of the quantile loss function. See Yi and Huang 2017 for more details.

br:

Solution is found by re-formulating the problem so it can be solved with the rq() function from quantreg with the br algorithm.

The huber algorithm offers substantial speed advantages without much, if any, loss in performance. However, it should be noted that it solves an approximation of the quantile loss function.

Usage

rq.pen(
  x,
  y,
  tau = 0.5,
  lambda = NULL,
  penalty = c("LASSO", "Ridge", "ENet", "aLASSO", "SCAD", "MCP"),
  a = NULL,
  nlambda = 100,
  eps = ifelse(nrow(x) < ncol(x), 0.05, 0.01),
  penalty.factor = rep(1, ncol(x)),
  alg = c("huber", "br", "fn"),
  scalex = TRUE,
  tau.penalty.factor = rep(1, length(tau)),
  coef.cutoff = 1e-08,
  max.iter = 5000,
  converge.eps = 1e-04,
  lambda.discard = TRUE,
  weights = NULL,
  ...
)

Arguments

Value

An rq.pen.seq object.

models:

A list of each model fit for each tau and a combination.

n:

Sample size.

p:

Number of predictors.

alg:

Algorithm used. Options are "huber" or any method implemented in rq(), such as "br".

tau:

Quantiles modeled.

a:

Tuning parameters a used.

modelsInfo:

Information about the quantile and a value for each model.

lambda:

Lambda values used for all models. If a model has fewer coefficients than lambda, say k. Then it used the first k values of lambda. Setting lambda.discard to TRUE will gurantee all values use the same lambdas, but may increase computational time noticeably and for little gain.

penalty:

Penalty used.

call:

Original call.

Each model in the models list has the following values.

coefficients:

Coefficients for each value of lambda.

rho:

The unpenalized objective function for each value of lambda.

PenRho:

The penalized objective function for each value of lambda. If scalex=TRUE then this is the value for the scaled version of x.

nzero:

The number of nonzero coefficients for each value of lambda.

tau:

Quantile of the model.

a:

Value of a for the penalized loss function.

If the Huber algorithm is used than \rho_\tau(y_i-x_i^\top\beta) is replaced by a Huber-type approximation. Specifically, it is replaced by h^\tau_\gamma(y_i-x_i^\top\beta)/2 where

h^\tau_\gamma(a) = a^2/(2\gamma)I(|a| \leq \gamma) + (|a|-\gamma/2)I(|a|>\gamma)+(2\tau-1)a.

Where if \tau=.5, we get the usual Huber loss function. The Huber implementation calls the package hqreg which implements the methods of Yi and Huang (2017) for Huber loss with elastic net penalties. For non-elastic net penalties the LLA algorithm of Zou and Li (2008) is used to approximate those loss functions with a lasso penalty with different weights for each predictor.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu, Shaobo Li, and Adam Maidman

References

Zou H, Li R (2008). “One-step sparse estimates in nonconcave penalized likelihood models.” Ann. Statist., 36(4), 1509-1533.

Yi C, Huang J (2017). “Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression.” J. Comput. Graph. Statist., 26(3), 547-557.

Belloni A, Chernozhukov V (2011). “L1-Penalized quantile regression in high-dimensional sparse models.” Ann. Statist., 39(1), 82-130.

Peng B, Wang L (2015). “An iterative coordinate descent algorithm for high-dimensional nonconvex penalized quantile regression.” J. Comput. Graph. Statist., 24(3), 676-694.

Examples

n <- 200
p <- 8
x <- matrix(runif(n*p),ncol=p)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(n)
r1 <- rq.pen(x,y) #Lasso fit for median
# Lasso for multiple quantiles
r2 <- rq.pen(x,y,tau=c(.25,.5,.75))
# Elastic net fit for multiple quantiles, which must use Huber algorithm
r3 <- rq.pen(x,y,penalty="ENet",a=c(0,.5,1),alg="huber")
# First variable is not penalized
r4 <- rq.pen(x,y,penalty.factor=c(0,rep(1,7)))
tvals <- c(.1,.2,.3,.4,.5)
#Similar to penalty proposed by Belloni and Chernouzhukov. 
#To be exact you would divide the tau.penalty.factor by n. 
r5 <- rq.pen(x,y,tau=tvals, tau.penalty.factor=sqrt(tvals*(1-tvals)))

Does k-folds cross validation for rq.pen. If multiple values of a are specified then does a grid based search for best value of \lambda and a.

Description

Does k-folds cross validation for rq.pen. If multiple values of a are specified then does a grid based search for best value of \lambda and a.

Usage

rq.pen.cv(
  x,
  y,
  tau = 0.5,
  lambda = NULL,
  penalty = c("LASSO", "Ridge", "ENet", "aLASSO", "SCAD", "MCP"),
  a = NULL,
  cvFunc = NULL,
  nfolds = 10,
  foldid = NULL,
  groupError = TRUE,
  cvSummary = mean,
  tauWeights = rep(1, length(tau)),
  printProgress = FALSE,
  weights = NULL,
  ...
)

Arguments

Details

Two cross validation results are returned. One that considers the best combination of a and lambda for each quantile. The second considers the best combination of the tuning parameters for all quantiles. Let y_{b,i}, x_{b,i}, and m_{b,i} index the response, predictors, and weights of observations in fold b. Let \hat{\beta}_{\tau,a,\lambda}^{-b} be the estimator for a given quantile and tuning parameters that did not use the bth fold. Let n_b be the number of observations in fold b. Then the cross validation error for fold b is

\mbox{CV}(b,\tau) = \frac{1}{n_b} \sum_{i=1}^{n_b} m_{b,i} \rho_\tau(y_{b,i}-x_{b,i}^\top\hat{\beta}_{\tau,a,\lambda}^{-b}).

Note that \rho_\tau() can be replaced by a different function by setting the cvFunc parameter. The function returns two different cross-validation summaries. The first is btr, by tau results. It provides the values of lambda and a that minimize the average, or whatever function is used for cvSummary, of \mbox{CV}(b). In addition it provides the sparsest solution that is within one standard error of the minimum results.

The other approach is the group tau results, gtr. Consider the case of estimating Q quantiles of \tau_1,\ldots,\tau_Q with quantile (tauWeights) of v_q. The gtr returns the values of lambda and a that minimizes the average, or again whatever function is used for cvSummary, of

\sum_{q=1}^Q v_q\mbox{CV}(b,\tau_q).

If only one quantile is modeled then the gtr results can be ignored as they provide the same minimum solution as btr.

Value

An rq.pen.seq.cv object.

cverr:

Matrix of cvSummary function, default is average, cross-validation error for each model, tau and a combination, and lambda.

cvse:

Matrix of the standard error of cverr foreach model, tau and a combination, and lambda.

fit:

The rq.pen.seq object fit to the full data.

btr:

A data.table of the values of a and lambda that are best as determined by the minimum cross validation error and the one standard error rule, which fixes a. In btr the values of lambda and a are selected seperately for each quantile.

gtr:

A data.table for the combination of a and lambda that minimize the cross validation error across all tau.

gcve:

Group, across all quantiles, cross-validation error results for each value of a and lambda.

call:

Original call to the function.

Author(s)

Ben Sherwood, ben.sherwood@ku.edu

Examples

## Not run: 
x <- matrix(runif(800),ncol=8)
y <- 1 + x[,1] + x[,8] + (1+.5*x[,3])*rnorm(100)
r1 <- rq.pen.cv(x,y) #lasso fit for median
# Elastic net fit for multiple values of a and tau
r2 <- rq.pen.cv(x,y,penalty="ENet",a=c(0,.5,1),tau=c(.25,.5,.75)) 
#same as above but more weight given to median when calculating group cross validation error. 
r3 <- rq.pen.cv(x,y,penalty="ENet",a=c(0,.5,1),tau=c(.25,.5,.75),tauWeights=c(.25,.5,.25))
# uses median cross-validation error instead of mean.
r4 <- rq.pen.cv(x,y,cvSummary=median)  
#Cross-validation with no penalty on the first variable.
r5 <- rq.pen.cv(x,y,penalty.factor=c(0,rep(1,7)))

## End(Not run)