R: Penalized Convolution-Type Smoothed Quantile Regression

conquer.reg {conquer}

R Documentation

Penalized Convolution-Type Smoothed Quantile Regression

Description

Fit sparse quantile regression models in high dimensions via regularized conquer methods with "lasso", "scad" and "mcp" penalties. For "scad" and "mcp", the iteratively reweighted \ell_1-penalized algorithm is complemented with a local adpative majorize-minimize algorithm.

Usage

conquer.reg(
  X,
  Y,
  lambda = 0.2,
  tau = 0.5,
  kernel = c("Gaussian", "logistic", "uniform", "parabolic", "triangular"),
  h = 0,
  penalty = c("lasso", "scad", "mcp"),
  para = NULL,
  epsilon = 0.001,
  iteMax = 500,
  phi0 = 0.01,
  gamma = 1.2,
  iteTight = 3
)

Arguments

`X`	A n by p design matrix. Each row is a vector of observation with p covariates.
`Y`	An n-dimensional response vector.
`lambda`	(optional) Regularization parameter. Default is 0.2.
`tau`	(optional) Quantile level (between 0 and 1). Default is 0.5.
`kernel`	(optional) A character string specifying the choice of kernel function. Default is "Gaussian". Choices are "Gaussian", "logistic", "uniform", "parabolic" and "triangular".
`h`	(optional) Bandwidth/smoothing parameter. Default is \max\{0.5 (log(p) / n)^{0.25}, 0.05\}*.
`penalty`	(optional) A character string specifying the penalty. Default is "lasso". The other two options are "scad" and "mcp".
`para`	(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".
`epsilon`	(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 `epsilon`. Default is 0.001.
`iteMax`	(optional) Maximum number of iterations. Default is 500.
`phi0`	(optional) The initial quadratic coefficient parameter in the local adaptive majorize-minimize algorithm. Default is 0.01.
`gamma`	(optional) The adaptive search parameter (greater than 1) in the local adaptive majorize-minimize algorithm. Default is 1.2.
`iteTight`	(optional) Maximum number of tightening iterations in the iteratively reweighted \ell_1-penalized algorithm. Do not need to specify if the penalty is lasso. Default is 3.

Value

An object containing the following items will be returned:

coeff: A (p + 1) vector of estimated coefficients, including the intercept.
bandwidth: Bandwidth value.
tau: Quantile level.
kernel: Kernel function.
penalty: Penalty type.
lambda: Regularization parameter.
n: Sample size.
p: Number of the covariates.

Author(s)

Xuming He <xmhe@umich.edu>, Xiaoou Pan <xip024@ucsd.edu>, Kean Ming Tan <keanming@umich.edu>, and Wen-Xin Zhou <wez243@ucsd.edu>

References

Fan, 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.

Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33-50.

Tan, 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.

Examples

n = 200; p = 500; s = 10
beta = c(rep(1.5, s), rep(0, p - s))
X = matrix(rnorm(n * p), n, p)
Y = X %*% beta + rt(n, 2)

## Regularized conquer with lasso penalty at tau = 0.8
fit.lasso = conquer.reg(X, Y, lambda = 0.05, tau = 0.8, kernel = "Gaussian", penalty = "lasso")
beta.lasso = fit.lasso$coeff

#' ## Regularized conquer with scad penalty at tau = 0.8
fit.scad = conquer.reg(X, Y, lambda = 0.13, tau = 0.8, kernel = "Gaussian", penalty = "scad")
beta.scad = fit.scad$coeff

[Package conquer version 1.2.1 Index]