Object of the penalty
class to handle the AO penalty (Ulbricht, 2010).
1 |
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. |
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.
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. |
Jan Ulbricht
Ulbricht, Jan (2010) Variable Selection in Generalized Linear Models. Ph.D. Thesis. LMU Munich.
Questions? Problems? Suggestions? Tweet to @rdrrHQ or email at ian@mutexlabs.com.
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