find_lambda_by_sparsity {Glarmadillo}

Find Optimal Lambda by Sparsity Level

Description

This function performs a grid search over a range of lambda values to identify the lambda that achieves a desired level of sparsity in the precision matrix estimated by Graphical Lasso. Sparsity is defined as the proportion of zero elements (excluding the diagonal) in the precision matrix.

Usage

find_lambda_by_sparsity(
  s,
  lambda_grid,
  desired_sparsity,
  mtol = 1e-04,
  maxIterations = 10000,
  ltol = 1e-06
)

Arguments

s

The sample covariance matrix of the data.

lambda_grid

A numeric vector of lambda values to be tested in the grid search.

desired_sparsity

The target sparsity level as a proportion of zero elements in the precision matrix. This should be a value between 0 and 1.

mtol

The convergence threshold for Graphical Lasso optimization.

maxIterations

The maximum number of iterations for Graphical Lasso optimization.

ltol

The tolerance for determining whether elements are considered zero when calculating sparsity.

Value

A list containing the following components: - best_lambda: the lambda value that results in sparsity closest to the desired level. - best_sparsity_difference: the smallest difference between achieved and desired sparsity. - actual_sparsity: a numeric vector of actual sparsity levels for each lambda tested. - lambda_grid: the vector of lambda values tested.

Examples

# Generate a sparse covariance matrix
values <- c(160, 50)
n <- values[1]
p <- values[2]
s <- generate_sparse_cov_matrix(n, p, standardize = TRUE, sparse_rho = 0, scale_power = 0)

# Define a sequence of lambda values for the grid search
lambda_find <- c(0.1, 0.2, 0.3, 0.4)

# Perform a grid search to find the lambda value
# that results in a precision matrix with approximately 80% sparsity
lambda_results <- find_lambda_by_sparsity(s, lambda_find, desired_sparsity = 0.8)

# Inspect the optimal lambda value
optimal_lambda <- lambda_results$best_lambda

# Inspect the sparsity levels for each lambda tested
sparsity_levels <- lambda_results$actual_sparsity