weighted.fusion: Weighted Fusion Penalty
In lqa: Penalized Likelihood Inference for GLMs

Description Usage Arguments Details Value Author(s) References See Also

Object of the penalty class to handle the weighted fusion penalty (Daye \& Jeng, 2009)

1	weighted.fusion(lambda = NULL, ...)

`lambda`	three-dimensional tuning parameter. The first component corresponds to the regularization parameter λ_1 of the lasso penalty. The second component corresponds to the regularization parameter λ_2 of the fusion penalty. Both components must be nonnegative. The third component corresponds to γ > 0 that determines the fusion penalty.
`...`	further arguments.

Another extension of correlation-based penalization has been proposed by Daye \& Jeng (2009). They introduce the weighted fusion penalty to utilize the correlation information from the data by penalizing the pairwise differences of coefficients via correlation-driven weights. As a consequence, highly correlated regressors are allowed to be treated similarly in regression. The weighted fusion penalty is defined as

P_{λ}^{wf}(\boldsymbol{β})= λ_1 ∑_{j=1}^p|β_j| + P_{λ_2}^{cd} (\boldsymbol{β}),

where

P_{λ_2}^{cd}(\boldsymbol{β}) = \frac{λ_2}{p}∑_{i < j} ω_{ij} \{β_i - \textrm{sign} (\varrho_{ij})β_j\}^2

is referred to as correlation-driven penalty function. Daye \& Jeng (2009) propose to use

ω_{ij} = \frac{|\varrho_{ij}|^γ}{1 - |\varrho_{ij}|},

where γ > 0 is an additional tuning parameter. Consequently, the weighted fusion penalty consists of three tuning parameters λ = (λ_1, λ_2, γ). The effect is that ω_{ij} \rightarrow ∞ as |\varrho_{ij}| \rightarrow 1 so that the correlation-driven penalty function tends to equate the magnitude of the coefficients of the corresponding regressors x_i and x_j. Note that the lasso penalty term in the weighted fusion penalty is responsible for variable selection.

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

Daye, Z. J. \& X. J. Jeng (2009) Shrinkage and model selection with correlated variabeles via weighted fusion. Computational Statistics and Data Analysis 53, 1284–1298.

penalty, penalreg, icb, licb, ForwardBoost