Approximated Octagon Penalty

Description

Object of the penalty class to handle the AO penalty (Ulbricht, 2010).

Usage

1
ao (lambda = NULL, ...)

Arguments

lambda

two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter λ. This must be a nonnegative real number. The second component indicates the exponent γ of the bridge penalty term. See details below. It must hold that γ > 1.

...

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 γ > 1 where the amount of the bridge penalty part is driven by empirical correlation. So, consider the penalty

P_{\tilde{λ}}^{ao}(\boldsymbol{β}) = ∑_{i = 2}^p ∑_{j< i} p_{\tilde{λ},ij} (\boldsymbol{β}), \quad \tilde{λ} = (λ, γ)

where

p_{\tilde{λ},ij} = λ[(1 - |\varrho_{ij}|) (|β_i| + |β_j|) + |\varrho_{ij}|(|β_i|^γ + |β_j|^γ)],

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{λ}}^{ao}(\boldsymbol{β}) 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{λ}}^{ao}(\boldsymbol{β}) = ∑_{i=1}^p p_{\tilde{λ},i}^{ao}(β_i),

where

p_{\tilde{λ},i}^{ao}(β_i) = λ ≤ft\{|β_i|∑_{j \neq i} (1 - |\varrho_{ij}|) + |β_i|^γ ∑_{j \neq i} |\varrho_{ij}|\right\}.

It uses two tuning parameters \tilde{λ} = (λ, γ), where λ controls the penalty amount and γ manages the approximation of the pairwise L_∞-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.

See Also

penalty, genet