conquer.reg {conquer} | R Documentation |
Fit sparse quantile regression models in high dimensions via regularized conquer methods with "lasso", "elastic-net", "group lasso", "sparse group lasso", "scad" and "mcp" penalties.
For "scad" and "mcp", the iteratively reweighted \ell_1
-penalized algorithm is complemented with a local adpative majorize-minimize algorithm.
conquer.reg(
X,
Y,
lambda = 0.2,
tau = 0.5,
kernel = c("Gaussian", "logistic", "uniform", "parabolic", "triangular"),
h = 0,
penalty = c("lasso", "elastic", "group", "sparse-group", "scad", "mcp"),
para.elastic = 0.5,
group = NULL,
para.scad = 3.7,
para.mcp = 3,
epsilon = 0.001,
iteMax = 500,
phi0 = 0.01,
gamma = 1.2,
iteTight = 3
)
X |
An |
Y |
An |
lambda |
(optional) Regularization parameter. Can be a scalar or a sequence. If the input is a sequence, the function will sort it in ascending order, and run the regression accordingly. Default is 0.2. |
tau |
(optional) Quantile level (between 0 and 1). Default is 0.5. |
kernel |
(optional) A character string specifying the choice of kernel function. Default is "Gaussian". Choices are "Gaussian", "logistic", "uniform", "parabolic" and "triangular". |
h |
(optional) Bandwidth/smoothing parameter. Default is |
penalty |
(optional) A character string specifying the penalty. Default is "lasso" (Tibshirani, 1996). The other options are "elastic" for elastic-net (Zou and Hastie, 2005), "group" for group lasso (Yuan and Lin, 2006), "sparse-group" for sparse group lasso (Simon et al., 2013), "scad" (Fan and Li, 2001) and "mcp" (Zhang, 2010). |
para.elastic |
(optional) The mixing parameter between 0 and 1 (usually noted as |
group |
(optional) A |
para.scad |
(optional) The constant parameter for "scad". Default value is 3.7. Only specify it if |
para.mcp |
(optional) The constant parameter for "mcp". Default value is 3. Only specify it if |
epsilon |
(optional) A tolerance level for the stopping rule. The iteration will stop when the maximum magnitude of the change of coefficient updates is less than |
iteMax |
(optional) Maximum number of iterations. Default is 500. |
phi0 |
(optional) The initial quadratic coefficient parameter in the local adaptive majorize-minimize algorithm. Default is 0.01. |
gamma |
(optional) The adaptive search parameter (greater than 1) in the local adaptive majorize-minimize algorithm. Default is 1.2. |
iteTight |
(optional) Maximum number of tightening iterations in the iteratively reweighted |
An object containing the following items will be returned:
coeff
If the input lambda
is a scalar, then coeff
returns a (p + 1)
vector of estimated coefficients, including the intercept. If the input lambda
is a sequence, then coeff
returns a (p + 1)
by nlambda
matrix, where nlambda
refers to the length of lambda
sequence.
bandwidth
Bandwidth value.
tau
Quantile level.
kernel
Kernel function.
penalty
Penalty type.
lambda
Regularization parameter(s).
n
Sample size.
p
Number of the covariates.
Belloni, A. and Chernozhukov, V. (2011). \ell_1
penalized quantile regression in high-dimensional sparse models. Ann. Statist., 39, 82-130.
Fan, J. and Li, R. (2001). Variable selection via nonconcave regularized likelihood and its oracle properties. J. Amer. Statist. Assoc., 96, 1348-1360.
Fan, J., Liu, H., Sun, Q. and Zhang, T. (2018). I-LAMM for sparse learning: Simultaneous control of algorithmic complexity and statistical error. Ann. Statist., 46, 814-841.
Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33-50.
Simon, N., Friedman, J., Hastie, T. and Tibshirani, R. (2013). A sparse-group lasso. J. Comp. Graph. Statist., 22, 231-245.
Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. R. Statist. Soc. Ser. B, 58, 267–288.
Tan, K. M., Wang, L. and Zhou, W.-X. (2022). High-dimensional quantile regression: convolution smoothing and concave regularization. J. Roy. Statist. Soc. Ser. B, 84, 205-233.
Yuan, M. and Lin, Y. (2006). Model selection and estimation in regression with grouped variables., J. Roy. Statist. Soc. Ser. B, 68, 49-67.
Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Ann. Statist., 38, 894-942.
Zou, H. and Hastie, T. (2005). Regularization and variable selection via the elastic net. J. R. Statist. Soc. Ser. B, 67, 301-320.
See conquer.cv.reg
for regularized quantile regression with cross-validation.
n = 200; p = 500; s = 10
beta = c(rep(1.5, s), rep(0, p - s))
X = matrix(rnorm(n * p), n, p)
Y = X %*% beta + rt(n, 2)
## Regularized conquer with lasso penalty at tau = 0.7
fit.lasso = conquer.reg(X, Y, lambda = 0.05, tau = 0.7, penalty = "lasso")
beta.lasso = fit.lasso$coeff
## Regularized conquer with elastic-net penalty at tau = 0.7
fit.elastic = conquer.reg(X, Y, lambda = 0.1, tau = 0.7, penalty = "elastic", para.elastic = 0.7)
beta.elastic = fit.elastic$coeff
## Regularized conquer with scad penalty at tau = 0.7
fit.scad = conquer.reg(X, Y, lambda = 0.13, tau = 0.7, penalty = "scad")
beta.scad = fit.scad$coeff
## Regularized conquer with group lasso at tau = 0.7
beta = c(rep(1.3, 5), rep(1.5, 5), rep(0, p - s))
err = rt(n, 2)
Y = X %*% beta + err
group = c(rep(1, 5), rep(2, 5), rep(3, p - s))
fit.group = conquer.reg(X, Y, lambda = 0.05, tau = 0.7, penalty = "group", group = group)
beta.group = fit.group$coeff