Description Usage Arguments Details Value Author(s) References See Also Examples
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.
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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 |
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. |
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.
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 |
the actual sequence of |
df |
The number of nonzero coefficients for each value of |
dim |
dimension of coefficient matrix (ices) |
Bo Peng
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
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