R: Fused Lasso Penalty

fused.lasso {lqa}

R Documentation

Fused Lasso Penalty

Description

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

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.

Value

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{\beta|}.

a.coefs

function: This returns the p-dimensional coefficient vector \mathbf{a}_j of the J penalty terms.

Author(s)

Jan Ulbricht

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 B 67, 91–108.