conquer.reg {conquer} | R Documentation |
Penalized Convolution-Type Smoothed Quantile Regression
Description
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.
Usage
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,
weights = NULL,
para.scad = 3.7,
para.mcp = 3,
epsilon = 0.001,
iteMax = 500,
phi0 = 0.01,
gamma = 1.2,
iteTight = 3
)
Arguments
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 |
weights |
(optional) A vector specifying groups weights for group Lasso and sparse group Lasso. The length must be equal to the number of groups. If not specified, the default weights are square roots of group sizes.
For example , if |
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 |
Value
An object containing the following items will be returned:
coeff
If the input
lambda
is a scalar, thencoeff
returns a(p + 1)
vector of estimated coefficients, including the intercept. If the inputlambda
is a sequence, thencoeff
returns a(p + 1)
bynlambda
matrix, wherenlambda
refers to the length oflambda
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.
References
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 Also
See conquer.cv.reg
for regularized quantile regression with cross-validation.
Examples
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