# icb: Improved Correlation-based Penalty In lqa: Penalized Likelihood Inference for GLMs

## Description

Object of the penalty class to handle the Improved Correlation-Based (ICB) Penalty (Ulbricht, 2010).

## Usage

 1 icb(lambda = NULL, ...) 

## Arguments

 lambda two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter λ_1 for the lasso penalty term, the second one λ_2 for the correlation-based penalty. Both parameters must be nonnegative. ... further arguments

## Details

The improved correlation-based (ICB) penalty is defined as

P_{λ}^{icb}(\boldsymbol{β}) = λ_1 |\boldsymbol{β}|_1 + \frac{1}{2}λ_2 \boldsymbol{β}^\top \mathbf{M}^{cb} \boldsymbol{β},

with tuning parameter λ = (λ_1, λ_2), where \mathbf{M}^{cb} = (m_{ij}) is determined by m_{ij} = 2∑_{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.

## 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.

Jan Ulbricht

## References

Ulbricht, Jan (2010) Variable Selection in Generalized Linear Models. Ph.D. Thesis. LMU Munich.

penalty, penalreg, licb, weighted.fusion