QICD: Fit a nonconvex penalized quantile regression model

In QICD: Estimate the Coefficients for Non-Convex Penalized Quantile Regression Model by using QICD Algorithm

Description

Fit a nonconvex penalized quantile model via QICD algorithm. The estimation of the coefficients will be given. The regularization path is computed for the nonconvex penalties at a grid of values for the tuning parameter lambda. High dimentional BIC for quantile regression model (QBIC) and cross validation will be used as criterion on the tuning parameter searching.

Usage

QICD(y, x, beta = NULL, tau, lambda, a = 3.7,
funname = "scad", intercept = TRUE, thresh = 1e-06, 
exclude = NULL, maxin = 100, maxout = 20)

Arguments

y	response variable. should be a numerical vector
x	input matrix, of dimension nobs x nvars; each row is an observation vector. In high dimensional cases, nvars could be larger than nobs
beta	initial value of coefficients. A numberical vector, with length of p if intercept is not included, and length of p+1 if intercept is included as the last entry, where p is the data dimention. The default value of beta is NULL, so an appropriate vector will be given automatically
tau	quantile parameter in quantile regression. A numerical value between 0 and 1.
lambda	a user supplied lambda sequence. A numerical vector.
a	an unknown parameter in SCAD and MCD penalty functions. The default value is 3.7, suggested by Fan and Li (2001).
funname	penalty function names. Character vector, which can be scad, mcp and lasso.
intercept	Should intercept(s) be fitted (default=TRUE) or set to zero (FALSE). Default value is TRUE.
thresh	Convergence threshold for QICD algorithm. Defaults value is 1E-6.
exclude	Indices of variables to be excluded from the model. Default is NULL. Can be logical vetor or numerical index specifying the exclued variables.
maxin	Maximum number of inner coordiante descent iterations; default is 100.
maxout	Maximum number of outter Majoriaztion Minimization step (MM) iterations; default is 20.

Details

The sequence of models indicated by lambda is fitted via the QICD algorithm. The QICD can substantially improves the computation speed in the p>>n setting. It combines the idea of the MM algorithm with that of the coordinate descent algorithm. For penalized quantile regression model, the objective function is

Q(β) = 1/n∑_{i=1}^nρ_{τ}(Y_i-x_i^Tβ) +∑_{j=1}^p p_{λ}(|β_j|)

where ρ_{τ} is the checkloss function for quantile regression. More specifically, we first replace the non-convex penalty function by its majorization function to create a surrogate objective function. Then we minimize the surrogate objective function with respect to a single parameter at each time and cycle through all parameters until convergence. For each univariate minimization problem, we only need to compute a one-dimensional weighted median, which ensures fast computation.

Value

An object with a list QICD.

beta_final	a matrix of coefficients. If intercept is included, diminsion is (p+1) x nlambda with the last row to be the intercepts, where nlamba is the length of lambda vector; if intercept is not included, diminsion is p x nlambda.
lambda	the actual sequence of lambda values used
df	The number of nonzero coefficients for each value of lambda
dim	dimension of coefficient matrix (ices)

Author(s)

Bo Peng

References

Peng,B and Wang,L. (2015)An Iterative Coordinate Descent Algorithm for High-dimensional Nonconvex Penalized Quantile Regression. Journal of Computational and Graphical Statistics http://amstat.tandfonline.com/doi/abs/10.1080/10618600.2014.913516 doi: 10.1080/10618600.2014.913516

Lee, E. R., Noh, H. and Park. B. (2013) Model Selection via Bayesian Information Criterion for Quantile Regression Models. Journal of the American Statistical Associa- tion, preprint. http://www.tandfonline.com/doi/pdf/10.1080/01621459.2013.836975 doi: 10.1080/01621459.2013.836975

Wang,L., Kim, Y., and Li,R. (2013+) Calibrating non-convex penalized regression in ultra-high dimension. To appear in Annals of Statistics. http://users.stat.umn.edu/~wangx346/research/nonconvex.pdf

Fan, J. and Li, R.(2001) Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties. Journal of American Statistical Association, 1348-1360. http://orfe.princeton.edu/~jqfan/papers/01/penlike.pdf

Zhang,C. (2010) Nearly Unbiase Variable Selection Under Minimax Concave Penalty. The Annals of Statistics, Vol. 38, No.2, 894-942 http://arxiv.org/pdf/1002.4734.pdf

See Also

QICD.cv, QICD.BIC

Examples

x=matrix(rnorm(10000),50)
n=dim(x)[1]
p=dim(x)[2]
intercept=1
y=x[,1]+x[,7]+x[,9]+0.1*rnorm(n)
beta1=rep(0,p+intercept)
tau=0.5
a=2.7
res=QICD(y,x,beta1,tau,lambda=10,a,"scad",intercept=intercept)

Example output

QICD documentation built on May 29, 2017, 3:04 p.m.