lqa (version 1.0-2)

ao: 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 $\lambda$. This must be a nonnegative real number. The second component indicates the exponent $\gamma$ of the bridge penalty term. See details be

...

further arguments.

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.

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 $$P_{\tilde{\lambda }}^{ao}(\mathit{\beta })=\sum _{i=2}^p\sum _{j<i}{p}_{\tilde{\lambda },ij}(\mathit{\beta }),\tilde{\lambda }=(\lambda ,\gamma )$$ 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 $$P_{\tilde{\lambda }}^{ao}(\mathit{\beta })=\sum _{i=1}^p{p}_{\tilde{\lambda },i}^{ao}({\beta }_i),$$ where $$\text{Missing or unrecognized delimiter for left}$$ 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.

References

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

See Also

penalty, genet