| ao {lqa} | R Documentation |
Approximated Octagon Penalty
Description
Object of the penalty class to handle the AO penalty (Ulbricht, 2010).
Usage
ao (lambda = NULL, ...)
Arguments
lambda |
two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter |
... |
further arguments. |
Details
The basic idea of the AO penalty is
to use a linear combination of L_1-norm and the bridge penalty with \gamma > 1 where the amount of
the bridge penalty part is driven by empirical
correlation. So, consider the penalty
P_{\tilde{\lambda}}^{ao}(\boldsymbol{\beta}) = \sum_{i = 2}^p \sum_{j< i} p_{\tilde{\lambda},ij}
(\boldsymbol{\beta}), \quad \tilde{\lambda} = (\lambda, \gamma)
where
p_{\tilde{\lambda},ij} = \lambda[(1 - |\varrho_{ij}|) (|\beta_i| + |\beta_j|) + |\varrho_{ij}|(|\beta_i|^\gamma + |\beta_j|^\gamma)],
and \varrho_{ij} denotes the value of the (empirical) correlation of the i-th and j-th regressor. Since we are going to
approximate an octagonal polytope in two dimensions, we will refer to this penalty as approximated octagon
(AO) penalty. Note that P_{\tilde{\lambda}}^{ao}(\boldsymbol{\beta}) leads to a dominating lasso term if the regressors are uncorrelated and to a
dominating bridge term if they are nearly perfectly correlated.
The penalty can be rearranged as
P_{\tilde{\lambda}}^{ao}(\boldsymbol{\beta}) = \sum_{i=1}^p p_{\tilde{\lambda},i}^{ao}(\beta_i),
where
p_{\tilde{\lambda},i}^{ao}(\beta_i) = \lambda \left\{|\beta_i|\sum_{j \neq i} (1 - |\varrho_{ij}|) + |\beta_i|^\gamma \sum_{j \neq i} |\varrho_{ij}|\right\}.
It uses two tuning parameters \tilde{\lambda} = (\lambda, \gamma), where \lambda controls the penalty amount and \gamma
manages the approximation of the pairwise L_\infty-norm.
Value
An object of the class penalty. This is a list with elements
penalty |
character: the penalty name. |
lambda |
double: the (nonnegative) regularization parameter. |
getpenmat |
function: computes the diagonal penalty matrix. |
Author(s)
Jan Ulbricht
References
Ulbricht, Jan (2010) Variable Selection in Generalized Linear Models. Ph.D. Thesis. LMU Munich.