rqPen (version 2.3)

QICD.group: Group Penalized Quantile Regression with QICD Algorithm

Description

QICD.group produces group penalized quantile regression estimates using the QICD algorithm. If no initial values are given, LASSO estimates will be used. This function can handle the LASSO, SCAD, and MCP penalties.

Usage

QICD.group(y, x, groups, tau=.5, lambda, intercept=TRUE, penalty="SCAD", 
                 initial_beta=NULL, maxin=100, maxout=20, eps = 1e-05, 
				 coef.cutoff=1e-08, a=3.7, scalex, ...)

Value

Returns a vector containing the intercept (if intercept=TRUE) and the estimated coefficients for each column in x.

Arguments

y

Vector of response values.

x

n x p matrix of observations with each row corresponding to one observation. Do not include column of 1's; set intercept=TRUE, if intercept is desired.

groups

Vector of length p with the group number of the corresponding coefficient. Coefficients in the same group will either all be 0, or none will be 0.

tau

Conditional quantile being modelled.

lambda

Tuning parameter for LASSO, SCAD, and MCP penalties. Must be positive.

intercept

If TRUE, an intercept is included in the model. If FALSE, no intercept is included.

penalty

Penalty function for the coefficients. Either "SCAD", "MCP", or "LASSO". See details for description of penalties.

initial_beta

Vector of initial values for QICD algorithm. The vector should contain the intercept first (if intercept=TRUE) and then the p coefficients. If NULL, exact LASSO estimates will be computed and used as initial values.

maxin

Maximum number of iterations on the minimization step of the QICD algorithm.

maxout

Maximum number of iterations on the majorization step of the QICD algorithm.

eps

Threshold for convergence of algorithm.

coef.cutoff

Coefficients with magnitude less than this value are set to 0.

a

Second tuning parameter for SCAD and MCP penalties. Must be greater than 2 for SCAD and greater than 1 for MCP.

scalex

If set to true the predictors will be scaled to have mean zero and standard deviation of one before fitting the model. The output returned will be on the original scale of the data.

...

The extra arguments will not be used.

Author

Adam Maidman

Details

The QICD algorithm should only be used for the LASSO penalty if initial_beta can be set to LASSO estimates with a similar lambda (similar to a "warm start"). Otherwise, exact LASSO estimates will be used as initial values for the QICD algorithm: this will cause unnecessary computations and could lead to less accurate estimates.

References

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

[2] Wu, Y. and Liu, Y. (2009). Variable selection in quantile regression. Statistica Sinica, 19, 801--817.

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

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

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

Examples

library(splines)
n = 50
p = 100
x <- matrix(rnorm(n*p),nrow=n)
z1 <- runif(n)
z2 <- runif(n)
x <- cbind(x, bs(z1), bs(z2))
groups <- c( 1:p, rep(101,3), rep(102,3) )
y <- 0 + x[,1] - 3*x[,5] + z1^3 + rnorm(n)
fit1 <- QICD.group(y,x, groups, tau=.5, lambda=1, intercept=TRUE, penalty="SCAD")
fit2 <- QICD.group(y,x, groups, tau=.7, lambda=1, intercept=TRUE, penalty="SCAD")