hrq_glasso {hrqglas}

Robust group variable selection for quantile and mean regression

Description

This function conducts group-wise (with known groups) variable selection for quantile and robust mean regression with the group lasso penalty. The Huber loss is used for both types of regression model, where the quantile check function is approximated by Huber loss. A full solution path is generated unless a single value of the shrinkage parameter is specified.

Usage

hrq_glasso(
  x,
  y,
  group.index,
  tau = 0.5,
  lambda = NULL,
  weights = NULL,
  w.lambda = NULL,
  gamma = 0.2,
  max_iter = 200,
  apprx = "huber",
  lambda.discard = TRUE,
  method = "quantile",
  scalex = TRUE,
  epsilon = 1e-04,
  beta0 = NULL
)

Arguments

x

Design matrix (in matrix format)

y

Response variable

group.index

A vector of group index, e.g., (1,1,1,2,2,2,3,3)

tau

Percentile

lambda

Shrinkage parameter, default is NULL so that the algorithm chooses a sequence.

weights

Observation weights, default is NULL

w.lambda

Weights for Shrinkage parameter of each group, default is NULL

gamma

Huber parameter. An initial value is 0.2, while the algorithm adaptively tunes the value in each iteration.

max_iter

Maximum number of iteration

apprx

Approximation method. Default is huber. The other option is tanh which uses the hypertangent function to approximate the first order derivative of absolute loss.

lambda.discard

Default is TRUE, meaning that the solution path stops if the relative deviance changes sufficiently small. It usually happens near the end of solution path. However, the program returns at least 70 models along the solution path.

method

Choice for mean or quantile regression. Default is quantile.

scalex

Standardize design matrix. Default is TRUE.

epsilon

The epsilon level convergence. Default is 1e-4.

beta0

Initial estimates. Default is NULL.

Value

It returns a sequence of estimated coefficients for quantile regression with group feature selection corresponding to a sequence of lambda. The estimated coefficients are in the sparse matrix format. Returned values also include the sequence of lambda, the null deviance, values of penalized loss, and unpenalized loss across the sequence of lambda.

beta

The estimated coefficients for all lambdas, stored in sparse matrix format, where each column corresponds to a lambda.

lambda

The sequence of lambdas.

null.dev

The null deviance.

pen.loss

The value of penalized loss for each lambda.

loss

The value of unpenalized loss for each lambda.

index.grp

Group indices that correspond to the estimated coefficient matrix beta.

n.grp

The number of selected groups for each lambda.

References

Sherwood, B., and Li, S. (2021) An Efficient Approach to Feature Selection and Estimation for Quantile Regression with Grouped Variables. Working paper.

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

Examples

n<- 100
p<- 10
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)
fit<- hrq_glasso(X, y, group)
fit$beta[,8]