Package 'wflsa'

Determine the Diagonal of Matrix $A$

Description

The wFLSA algorithm requires the choice of a matrix $A$ such that $A - D'D$ is positive semidefinite. We choose the matrix $A$ to be diagonal with a fixed value $a$. This function determines the smallest value of $a$ such that $A - D'D$ is indeed positive semidefinite. We do this be determining the largest eigenvalue

Usage

calculate_diagonal_matrix_A(W, eta1, eta2)

Arguments

W	Weight matrix $W$
eta1, eta2	The values $\eta_1 = \lambda_1 / \rho$ and $\eta_2 = \lambda_2 / \rho$

Value

Value of $a$

References

Zhu, Y. (2017). An Augmented ADMM Algorithm With Application to the Generalized Lasso Problem. Journal of Computational and Graphical Statistics, 26(1), 195–204. https://doi.org/10.1080/10618600.2015.1114491

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Solving Generalized LASSO with fixed $\lambda = 1$

Description

Solves efficiently the generalized LASSO problem of the form

$\hat{\beta} = \operatorname{argmin} \frac{1}{2} || y - \beta ||_2^2 + ||D\beta||_1$

where $\beta$ and $y$ are $m$-dimensional vectors and $D$ is a $(c \times m)$-matrix where $c \geq m$. We solve this optimization problem using an adaption of the ADMM algorithm presented in Zhu (2017).

Usage

genlassoRcpp(Y, W, m, eta1, eta2, a, rho, max_iter, eps, truncate)

Arguments

Y	The $y$ vector of length $m$
W	The weight matrix $W$ of dimensions $m \times m$
m	The number of graphs
eta1	Equals $\lambda_1 / rho$
eta2	Equals $\lambda_2 / rho$
a	Value added to the diagonal of $-D'D$ so that the matrix is positive definite, see matrix_A_inner_ADMM in package CVN
rho	The ADMM's parameter
max_iter	Maximum number of iterations
eps	Stopping criterion. If differences are smaller than $\epsilon$, algorithm is halted
truncate	Values below truncate are set to 0

Value

The estimated vector $\hat{\beta}$

References

Zhu, Y. (2017). An Augmented ADMM Algorithm With Application to the Generalized Lasso Problem. Journal of Computational and Graphical Statistics, 26(1), 195–204. https://doi.org/10.1080/10618600.2015.1114491

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Print Function for the Weighted Fused LASSO Signal Approximator (wFLSA) fit object

Description

This function prints information about the fitted wFLSA object, including the number of variables, the number of lambda pairs, and the estimated beta coefficients.

Usage

## S3 method for class 'wflsa.fit'
print(x, ...)

Arguments

x	An object of class wflsa.fit.
...	Additional arguments to be passed to the print function.

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The Weighted Fused Lasso Signal Approximator (wFLSA) Algorithm

Description

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.

See Also

genlassoRcpp()

Examples

# Example usage of the wflsa function
y <- c(1, 2, 3)
W <- matrix(c(1, 0, 0, 0, 1, 0, 0, 0, 1), ncol = 3)
lambda1 <- c(0.1, 0.2)
lambda2 <- c(0.1, 0.2)
result <- wflsa(y, W, lambda1, lambda2)

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