licb {lqa}R Documentation

L1-Norm based Improved Correlation-based Penalty

Description

Object of the penalty class to handle the L1-Norm based Improved Correlation-Based (LICB) Penalty (Ulbricht, 2010).

Usage

licb (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 L_1-norm based correlation penalty term. Both parameters must be nonnegative.

...

further arguments

Details

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

P_{λ}^{licb}(\boldsymbol{β}) = λ_1 ∑_{i=1}^p |β_i| + λ_2 ∑_{i=1}^{p-1} ∑_{j > i} ≤ft\{\frac{|β_i - β_j|}{1 - \varrho_{ij}} + \frac{|β_i + β_j|}{1 + \varrho_{ij}}\right\}.

The LICB 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.

first.derivative

function: This returns the J-dimensional vector of the first derivative of the J penalty terms with respect to |\mathbf{a}^\top_j\boldsymbol{β|}.

a.coefs

function: This returns the p-dimensional coefficient vector \mathbf{a}_j of the J penalty terms.

Author(s)

Jan Ulbricht

References

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

See Also

penalty, penalreg, icb, weighted.fusion


[Package lqa version 1.0-3 Index]