adaptive.lasso {lqa} | R Documentation |
Object of the penalty
class to handle the adaptive lasso penalty (Zou, 2006).
adaptive.lasso (lambda = NULL, al.weights = NULL, ...)
lambda |
regularization parameter. This must be a nonnegative real number. |
al.weights |
weights used for the adaptive lasso penalty. |
... |
further arguments. |
The adaptive lasso penalty (Zou, 2006) is defined as
P_{\tilde{λ}}^{al} (\boldsymbol{β}) = λ ∑_{i=1}^p w_i |β_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{β}} of the true parameter vector \boldsymbol{β}, Zou (2006) uses the weight vector \hat\mathbf{w} = (\hat{w}_1,…,\hat{w}_p)^\top with
\hat{w}_j = \frac{1}{|\hatβ_j|^γ}, \quad j = 1, …, p
as estimates
for the adaptive weights, where γ > 0 can be chosen arbitrarily. In
the n > p case, the MLE can be used for the estimated weights. In the n << 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.
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 |β_i|. However, this function is not really required for adaptive lasso. |
Jan Ulbricht
Zou, H. (2006) The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101, 1418–1429.