lqa (version 1.0-3)

genet: Generalized Elastic Net Penalty

Description

Object of the penalty to handle the Generalized Elastic Net (GENET) penalty (Ulbricht, 2010).

Usage

genet (lambda = NULL, ...)

Arguments

lambda

three-dimensional tuning parameter. The first component corresponds to the regularization parameter $\lambda$. This must be a nonnegative real number. The second component $0 \leq \alpha \leq 1$ drives the linear combination of $L_1$ penalty and the bri

...

further arguments.

Value

• An object of the class penalty. This is a list with elements
• penalty
  character: the penalty name.
• lambda
  double: the (nonnegative) tuning parameter.
• getpenmat
  function: computes the diagonal penalty matrix.

Details

The GENET penalty can be defined as $$P_{\bar{\lambda}}^{genet}(\boldsymbol{\beta}) = \lambda \left{\alpha \sum_{i=1}^p |\beta_i| + (1-\alpha) \sum_{i=1}^p |\beta_i|^\gamma \right}, \quad 0 \leq \alpha \leq 1, \: \gamma > 1$$ with tuning parameter vector $\bar{\lambda} = (\lambda, \alpha, \gamma)$.

The regularization parameter $\lambda$ determines the overall relevance of the GENET penalty. The balance between $L_1$-norm penalization, and hence variable selection, and bridge penalization for incorporating the grouping effect is managed by an overall tuning parameter $\alpha$. For motivation and further details on the GENET penalty see Ulbricht (2010).

References

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

See Also

penalty, ao