lqa (version 1.0-3)

enet: Elastic Net Penalty

Description

Object of the penalty to handle the elastic net (enet) penalty (Zou & Hastie, 2005)

Usage

enet (lambda = NULL, ...)

Arguments

lambda

two-dimensional tuning parameter. The first component corresponds to the regularization parameter $\lambda_1$ of the lasso penalty, the second one to the regularization parameter $\lambda_2$ of the ridge penalty. Both must be nonnegative.

...

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 elastic net penalty has been introduced in the linear model context by Zou & Hastie (2005). The elastic net enables simultaneous automatic variable selection and continuous shrinkage. Furthermore, contrary to the lasso it can select groups of correlated variables. This is related to the so called grouping effect, where strongly correlated regressors tend to be in or out of the model together.

The elastic net penalty $$P_{\lambda}^{en} (\boldsymbol{\beta}) = \lambda_1 \sum_{i=1}^p |\beta_i| + \lambda_2 \sum_{i=1}^p \beta_i^2, \quad \lambda = (\lambda_1, \lambda_2)$$ is a linear combination of the lasso penalty and the ridge penalty. Therefore the penalty covers these both as extreme cases.

References

Zou, H. & T. Hastie (2005) Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society B 67, 301--320.

See Also

penalty, genet, lasso, ridge