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
...
further arguments.
Value
An object of the class penalty. This is a list with elements
penaltycharacter: the penalty name.
lambdadouble: the (nonnegative) regularization parameter.
getpenmatfunction: computes the diagonal penalty matrix.
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.
References
Daye, Z. J. & X. J. Jeng (2009) Shrinkage and model selection with correlated variabeles via weighted fusion. Computational Statistics
and Data Analysis53, 1284--1298.