lqa (version 1.0-3)

licb: 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 nonne

...

further arguments

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.

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.

References

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

See Also

penalty, penalreg, icb, weighted.fusion