Description Usage Arguments Details Value Author(s) References See Also
Object of the penalty
class to handle the OSCAR penalty (Bondell \& Reich, 2008)
1 |
lambda |
two-dimensional tuning parameter. The first component corresponds to the regularization parameter λ 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 |
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{λ}}^{osc}(\boldsymbol{β}) = λ≤ft( ∑_{k=1}^p |β_k| + c ∑_{j < k} \max\{|β_j|, |β_k|\} \right), \quad \tilde{λ} = (λ, c)
where c ≥q 0 and λ > 0 are tuning parameters with c controlling the relative weighting of the L_∞-norms and λ controlling the magnitude of penalization. The L_1-norm entails sparsity, while the pairwise maximum (L_∞-)norm encourages equality of coefficients.
Due to equation (3) in Bondell \& Reich (2008), we use the alternative formulation
P_{\tilde{λ}}^{osc}(\boldsymbol{β}) = λ ∑_{j=1}^p \{c(j-1) + 1\}|β|_{(j)},
where |β|_{(1)} ≤q |β|_{(2)} ≤q … ≤q |β|_{(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 γ < 1, e.g. γ = 0.01 when to apply lqa.update2
for fitting the GLM in order to facilitate convergence.
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{β|}. |
a.coefs |
function: This returns the p-dimensional coefficient vector \mathbf{a}_j of the J penalty terms. |
Jan Ulbricht
Bondell, H. D. \& B. J. Reich (2008) Simultaneous regression shrinkage, variable selection and clustering of predictors with oscar. Biometrics 64, 115–123.
penalty
, lasso
, fused.lasso
, weighted.fusion
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