| adaptive.lasso {lqa} | R Documentation |
Adaptive Lasso Penalty
Description
Object of the penalty class to handle the adaptive lasso penalty (Zou, 2006).
Usage
adaptive.lasso (lambda = NULL, al.weights = NULL, ...)
Arguments
lambda |
regularization parameter. This must be a nonnegative real number. |
al.weights |
weights used for the adaptive lasso penalty. |
... |
further arguments. |
Details
The adaptive lasso penalty (Zou, 2006) is defined as
P_{\tilde{\lambda}}^{al} (\boldsymbol{\beta}) = \lambda \sum_{i=1}^p w_i |\beta_i|,
where adaptive weights w_i are used for penalizing different coefficients in the L_1-norm penalty. Based on a root-n-consistent estimator
\hat{\boldsymbol{\beta}} of the true parameter vector \boldsymbol{\beta}, Zou (2006) uses the weight vector \hat\mathbf{w} = (\hat{w}_1,\ldots,\hat{w}_p)^\top with
\hat{w}_j = \frac{1}{|\hat\beta_j|^\gamma}, \quad j = 1, \ldots, p
as estimates
for the adaptive weights, where \gamma > 0 can be chosen arbitrarily. In
the n > p case, the MLE can be used for the estimated weights. In the n \ll p case, a ridge penalized MLE with an
optimized regularization parameter might be a good alternative. You can commit any nonnegative weights by using al.weights in the initialization of the corresponding penalty object. If you left it unspecified then al.weights = 1 will be used.
Value
An object of the class penalty. This is a list with elements
penalty |
character: the penalty name. |
lambda |
double: the (nonnegative) regularization parameter. |
getpenmat |
function: computes the diagonal penalty matrix. |
first.derivative |
function: This returns the p-dimensional vector of the first derivative of the p penalty terms with respect to |
Author(s)
Jan Ulbricht
References
Zou, H. (2006) The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101, 1418–1429.