R: L1-Norm based Improved Correlation-based Penalty

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 \lambda_1 for the lasso penalty term, the second one \lambda_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_{\lambda}^{licb}(\boldsymbol{\beta}) = \lambda_1 \sum_{i=1}^p |\beta_i| + \lambda_2 \sum_{i=1}^{p-1} \sum_{j > i} \left\{\frac{|\beta_i - \beta_j|}{1 - \varrho_{ij}} + \frac{|\beta_i + \beta_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{\beta|}.

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.