Object of the penalty class to handle the Improved Correlation-Based (ICB) Penalty (Ulbricht, 2010).
Usage
icb(lambda = NULL, ...)
Arguments
lambda
two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter $\lambda_1$ for the lasso penalty
term, the second one $\lambda_2$ for the correlation-based penalty. Both parameters must be nonnegative.
...
further arguments
Value
An object of the class penalty. This is a list with elements
penaltycharacter: the penalty name.
lambdadouble: the (nonnegative) regularization parameter.
getpenmatfunction: computes the diagonal penalty matrix.
Details
The improved correlation-based (ICB) penalty is defined as
$$P_{\lambda}^{icb}(\boldsymbol{\beta}) = \lambda_1 |\boldsymbol{\beta}|_1 + \frac{1}{2}\lambda_2 \boldsymbol{\beta}^\top \mathbf{M}^{cb} \boldsymbol{\beta},$$
with tuning parameter $\lambda = (\lambda_1, \lambda_2)$, where $\mathbf{M}^{cb} = (m_{ij})$ is determined by
$m_{ij} = 2\sum_{s\neq i}\frac{1}{1-\varrho_{is}^2}$ if $i = j$, and $m_{ij} = -2\frac{\varrho_{ij}}{1-\varrho_{ij}^2}$ otherwise.
The ICB has been introduced to overcome the major drawback of the correlation based-penalized estimator, that is its lack of sparsity.
See Ulbricht (2010) for details.
References
Ulbricht, Jan (2010) Variable Selection in Generalized Linear Models. Ph.D. Thesis. LMU Munich.