1% Generated by roxygen2: do not edit by hand 2% Please edit documentation in R/smqr.R 3\name{conquer.reg} 4\alias{conquer.reg} 5\title{Penalized Convolution-Type Smoothed Quantile Regression} 6\usage{ 7conquer.reg( 8 X, 9 Y, 10 lambda = 0.2, 11 tau = 0.5, 12 kernel = c("Gaussian", "logistic", "uniform", "parabolic", "triangular"), 13 h = 0, 14 penalty = c("lasso", "scad", "mcp"), 15 para = NULL, 16 epsilon = 0.001, 17 iteMax = 500, 18 phi0 = 0.01, 19 gamma = 1.2, 20 iteTight = 3 21) 22} 23\arguments{ 24\item{X}{A \eqn{n} by \eqn{p} design matrix. Each row is a vector of observation with \eqn{p} covariates.} 25 26\item{Y}{An \eqn{n}-dimensional response vector.} 27 28\item{lambda}{(\strong{optional}) Regularization parameter. Default is 0.2.} 29 30\item{tau}{(\strong{optional}) Quantile level (between 0 and 1). Default is 0.5.} 31 32\item{kernel}{(\strong{optional}) A character string specifying the choice of kernel function. Default is "Gaussian". Choices are "Gaussian", "logistic", "uniform", "parabolic" and "triangular".} 33 34\item{h}{(\strong{optional}) Bandwidth/smoothing parameter. Default is \eqn{\max\{0.5 * (log(p) / n)^{0.25}, 0.05\}}.} 35 36\item{penalty}{(\strong{optional}) A character string specifying the penalty. Default is "lasso". The other two options are "scad" and "mcp".} 37 38\item{para}{(\strong{optional}) A constant parameter for "scad" and "mcp". Do not need to specify if the penalty is lasso. The default values are 3.7 for "scad" and 3 for "mcp".} 39 40\item{epsilon}{(\strong{optional}) A tolerance level for the stopping rule. The iteration will stop when the maximum magnitude of the change of coefficient updates is less than \code{epsilon}. Default is 0.001.} 41 42\item{iteMax}{(\strong{optional}) Maximum number of iterations. Default is 500.} 43 44\item{phi0}{(\strong{optional}) The initial quadratic coefficient parameter in the local adaptive majorize-minimize algorithm. Default is 0.01.} 45 46\item{gamma}{(\strong{optional}) The adaptive search parameter (greater than 1) in the local adaptive majorize-minimize algorithm. Default is 1.2.} 47 48\item{iteTight}{(\strong{optional}) Maximum number of tightening iterations in the iteratively reweighted \eqn{\ell_1}-penalized algorithm. Do not need to specify if the penalty is lasso. Default is 3.} 49} 50\value{ 51An object containing the following items will be returned: 52\describe{ 53\item{\code{coeff}}{A \eqn{(p + 1)} vector of estimated coefficients, including the intercept.} 54\item{\code{bandwidth}}{Bandwidth value.} 55\item{\code{tau}}{Quantile level.} 56\item{\code{kernel}}{Kernel function.} 57\item{\code{penalty}}{Penalty type.} 58\item{\code{lambda}}{Regularization parameter.} 59\item{\code{n}}{Sample size.} 60\item{\code{p}}{Number of the covariates.} 61} 62} 63\description{ 64Fit sparse quantile regression models in high dimensions via regularized conquer methods with "lasso", "scad" and "mcp" penalties. For "scad" and "mcp", the iteratively reweighted \eqn{\ell_1}-penalized algorithm is complemented with a local adpative majorize-minimize algorithm. 65} 66\examples{ 67n = 200; p = 500; s = 10 68beta = c(rep(1.5, s), rep(0, p - s)) 69X = matrix(rnorm(n * p), n, p) 70Y = X \%*\% beta + rt(n, 2) 71 72## Regularized conquer with lasso penalty at tau = 0.8 73fit.lasso = conquer.reg(X, Y, lambda = 0.05, tau = 0.8, kernel = "Gaussian", penalty = "lasso") 74beta.lasso = fit.lasso$coeff 75 76#' ## Regularized conquer with scad penalty at tau = 0.8 77fit.scad = conquer.reg(X, Y, lambda = 0.13, tau = 0.8, kernel = "Gaussian", penalty = "scad") 78beta.scad = fit.scad$coeff 79} 80\references{ 81Fan, J., Liu, H., Sun, Q. and Zhang, T. (2018). I-LAMM for sparse learning: Simultaneous control of algorithmic complexity and statistical error. Ann. Statist. 46 814-841. 82 83Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33-50. 84 85Tan, K. M., Wang, L. and Zhou, W.-X. (2021). High-dimensional quantile regression: convolution smoothing and concave regularization. J. Roy. Statist. Soc. Ser. B, to appear. 86} 87\seealso{ 88See \code{\link{conquer.cv.reg}} for regularized quantile regression with cross-validation. 89} 90\author{ 91Xuming He <xmhe@umich.edu>, Xiaoou Pan <xip024@ucsd.edu>, Kean Ming Tan <keanming@umich.edu>, and Wen-Xin Zhou <wez243@ucsd.edu> 92} 93