The Weighted Fused Lasso Signal Approximator (wFLSA) Algorithm

Solves the weighted Fused LASSO Signal Approximator optimization problem using an ADMM-based approach. The problem is formulated as follows: $$ \hat{\beta} = \operatorname{argmin} \frac{1}{2} || y - \beta ||_2^2 + \lambda_1 ||\beta ||_1 + \lambda_2 \sum_{i < j} w_{ij} | \beta_i - \beta_j | $$ where:

$y$ is the response with mean $0$.
$\beta$ is the vector of coefficients to be estimated.
$|| \cdot ||_1$ and $|| \cdot ||_2$ are the $L_1$- and $L_2$-norms, respectively.
$\lambda_1 > 0$ is the regularization parameter controlling the strength of the sparsity penalty.
$\lambda_2 > 0$ is the regularization parameter controlling the smoothness.
$w_{ij} \in [0,1]$ is the weight between the $i$-th and $j$-th coefficient.

Usage

wflsa(
  y,
  W,
  lambda1 = c(0.1),
  lambda2 = c(0.1),
  rho = 1,
  max_iter = 1e+05,
  eps = 1e-10,
  truncate = 1e-04,
  offset = TRUE
)

Arguments

y: Vector of length $p$ representing the response variable (assumed to be centered).
W: Weight matrix of dimensions $p \times p$.
lambda1: Vector of positive regularization parameters for $L_1$ penalty.
lambda2: Vector of positive regularization parameters for smoothness penalty.
rho: ADMM's parameter (Default: 1).
max_iter: Maximum number of iterations (Default: 1e5).
eps: Stopping criterion. If differences are smaller than eps, the algorithm halts (Default: 1e-10).
truncate: Values below truncate are set to 0 (Default: 1e-4).
offset: Logical indicating whether to include an intercept term (Default: TRUE).

Value

A list containing:

betas: Estimated vector $\hat{\beta}$ from the Weighted Fused LASSO.
tuning_parameters: Data frame with tuning parameters. The column df contains the number of non-zero coefficients for the different lambda-values
all input variables.

Note

Important Note: The algorithm assumes $y$ to be centered, i.e., its mean is 0.

Examples