Object of the penalty to handle the fused lasso penalty (Tibshirani et al., 2005)
Usage
fused.lasso (lambda = NULL, ...)
Arguments
lambda
two-dimensional tuning parameter. The first component corresponds to the regularization parameter $\lambda_1$ of the lasso penalty,
the second one to the regularization parameter $\lambda_2$ of the fusion penalty. Both must be nonnegative.
...
further arguments
Value
An object of the class penalty. This is a list with elements
penaltycharacter: the penalty name.
lambdadouble: the (nonnegative) regularization parameter.
first.derivativefunction: 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.coefsfunction: This returns the p-dimensional coefficient vector $\mathbf{a}_j$ of the J penalty terms.
Details
The fused lasso penalty is defined as
$$P_{\tilde{\lambda}}^{fl} (\boldsymbol{\beta}) = \lambda_1 \sum_{i=1}^p |\beta_i| + \lambda_2 \sum_{i=2}^p |\beta_{i} - \beta_{i-1}|,$$
where $\tilde{\lambda} = (\lambda_1, \lambda_2)$ contains two regularization parameters.
The main idea of the fused lasso penalty is to encourage sparsity in the coefficients by using the $L_1$-norm lasso penalty, and additionally to
force sparsity in the differences of the coefficients by the $L_1$-norm of their differences as reflected in the second penalty term. As a
result,
the fused lasso penalty conveys the estimated coefficients to behave in a smooth manner, with only a small number of big jumps.
See Tibshirani et al. (2005) for further details.
References
Tibshirani, R., M. Saunders, S. Rosset, J. Zhu and K. Knight (2005) Sparsity and smoothness via the fused lasso.
Journal of the Royal Statistical Society B67, 91--108.