R: OSCAR Penalty

oscar {lqa}

R Documentation

OSCAR Penalty

Description

Object of the penalty class to handle the OSCAR penalty (Bondell \& Reich, 2008)

Usage

oscar (lambda = NULL, ...)

Arguments

lambda

two-dimensional tuning parameter. The first component corresponds to the regularization parameter \lambda that drives the relevance of the OSCAR penalty for likelihood inference. The second component corresponds to c (see details below) Both must be nonnegative.

...

further arguments

Details

Bondell \& Reich (2008) propose a shrinkage method for linear models called OSCAR that simultaneously select variables while grouping them into predictive clusters. The OSCAR penalty is defined as

P_{\tilde{\lambda}}^{osc}(\boldsymbol{\beta}) = \lambda\left( \sum_{k=1}^p |\beta_k| + c \sum_{j < k} \max\{|\beta_j|, |\beta_k|\} \right), \quad \tilde{\lambda} = (\lambda, c)

where c \geq 0 and \lambda > 0 are tuning parameters with c controlling the relative weighting of the L_\infty-norms and \lambda controlling the magnitude of penalization. The L_1-norm entails sparsity, while the pairwise maximum (L_\infty-)norm encourages equality of coefficients.

Due to equation (3) in Bondell \& Reich (2008), we use the alternative formulation

P_{\tilde{\lambda}}^{osc}(\boldsymbol{\beta}) = \lambda \sum_{j=1}^p \{c(j-1) + 1\}|\beta|_{(j)},

where |\beta|_{(1)} \leq |\beta|_{(2)} \leq \ldots \leq |\beta|_{(p)} denote the ordered absolute values of the coefficients. However, there could be some difficulties in the LQA algorithm since we need an ordering of regressors which can differ between two adjacent iterations. In the worst case, this can lead to oscillations and hence to no convergence of the algorithm. Hence, for the OSCAR penalty it is recommend to use \gamma < 1, e.g. \gamma = 0.01 when to apply lqa.update2 for fitting the GLM in order to facilitate convergence.

Value

An object of the class penalty. This is a list with elements

penalty

character: the penalty name.

lambda

double: the (nonnegative) regularization parameter.

first.derivative

function: This returns the J-dimensional vector of the first derivative of the J penalty terms with respect to |\mathbf{a}^\top_j\boldsymbol{\beta|}.

a.coefs

function: This returns the p-dimensional coefficient vector \mathbf{a}_j of the J penalty terms.

Author(s)

Jan Ulbricht

References

Bondell, H. D. \& B. J. Reich (2008) Simultaneous regression shrinkage, variable selection and clustering of predictors with oscar. Biometrics 64, 115–123.