# Improved Correlation-based Penalty

### Description

Object of the `penalty`

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

### Usage

1 |

### Arguments

`lambda` |
two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter |

`...` |
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. |

### Author(s)

Jan Ulbricht

### References

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

### See Also

`penalty`

, `penalreg`

, `licb`

, `weighted.fusion`

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