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 l_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)^(1/4), 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 l_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 1.2.0 Index]