# hrqglas:
An R Package for Group Variable Selection for Quantile and Robust Mean
Regression

## Overview

This R package provides a program to conduct group-wise variable
selection and estimation for quantile and robust mean regression. The
group lasso penalty \(p_{\lambda_j}(\beta_j)=\lambda_j\|\beta_j\|_2\),
where \(j\) is the group index, is used
for group-wise variable selection (Yuan and Lin, 2006). For quantile
regression, the check loss is approximated by the Huber loss for the
median and the tilted version of Huber loss at other quantiles. This
approximation overcomes the nondifferentiability at the origin of check
loss, which may otherwise cause instable estimation. Statistical
consistency has been shown for this approximated quantile regression
estimates (Sherwood and Li, 2021). The estimation algorithm follows Yang
and Zou (2015), and it is computational efficient and stable. A robust
estimation of mean regression is a byproduct of this implementation as
Huber loss, with appropriate choices of the tuning parameter, is
intrinsically a robust loss function that is insensitive to
outliers.

## Installation

Currently not available on CRAN. User can install from github with
following code.

`devtools::install_github("shaobo-li/hrqglas")`

## Example

```
library(hrqglas)
n<- 200
p<- 30
x0<- matrix(rnorm(n*p),n,p)
X<- cbind(x0, x0^2, x0^3)[,order(rep(1:p,3))]
y<- -2+X[,1]+0.5*X[,2]-X[,3]-0.5*X[,7]+X[,8]-0.2*X[,9]+rt(n,2)
group<- rep(1:p, each=3)
# quantile regression
fit<- hrq_glasso(x=X, y=y, group.index=group, method="quantile", tau=0.3)
fit.cv<- cv.hrq_glasso(x=X, y=y, group.index=group, method="quantile", tau=0.3, loss="check")
plot(fit.cv)
# mean regression
fit1<- hrq_glasso(x=X, y=y, group.index=group, method="mean")
fit.cv1<- cv.hrq_glasso(x=X, y=y, group.index=group, method="mean", loss="se")
plot(fit.cv1)
```

## References

Sherwood, B., & Li, S. (2022). Quantile regression feature
selection and estimation with grouped variables using Huber
approximation. *Statistics and Computing*, 32(5), 75. https://doi.org/10.1007/s11222-022-10135-w

Yang, Y. and Zou, H., (2015) A Fast Unified Algorithm for Solving
Group-lasso Penalize Learning Problems, *Statistics and
Computing*, 25 1129-1141. https://doi.org/10.1007/s11222-014-9498-5.

Yuan, M. and Lin, Y., (2005) Model Selection and Estimation in
Regression with Grouped Variables, , 68 49-67. https://doi.org/10.1111/j.1467-9868.2005.00532.x.