R: Weighted Fusion Penalty

weighted.fusion {lqa}

R Documentation

Weighted Fusion Penalty

Description

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

Usage

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

Arguments

lambda

three-dimensional tuning parameter. The first component corresponds to the regularization parameter \lambda_1 of the lasso penalty. The second component corresponds to the regularization parameter \lambda_2 of the fusion penalty. Both components must be nonnegative. The third component corresponds to \gamma > 0 that determines the fusion penalty.

...

further arguments.

Details

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_{\lambda}^{wf}(\boldsymbol{\beta})= \lambda_1 \sum_{j=1}^p|\beta_j| + P_{\lambda_2}^{cd} (\boldsymbol{\beta}),

where

P_{\lambda_2}^{cd}(\boldsymbol{\beta}) = \frac{\lambda_2}{p}\sum_{i < j} \omega_{ij} \{\beta_i - \textrm{sign} (\varrho_{ij})\beta_j\}^2

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

\omega_{ij} = \frac{|\varrho_{ij}|^\gamma}{1 - |\varrho_{ij}|},

where \gamma > 0 is an additional tuning parameter. Consequently, the weighted fusion penalty consists of three tuning parameters \lambda = (\lambda_1, \lambda_2, \gamma). The effect is that \omega_{ij} \rightarrow \infty 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.

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.

Author(s)

Jan Ulbricht

References

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.