`Glarmadillo` is an R package designed for solving the Graphical Lasso (GLasso) problem using the Armadillo library. It provides an efficient implementation for estimating sparse inverse covariance matrices from observed data, ideal for applications in statistical learning and network analysis. The package includes functionality to generate random sparse covariance matrices and specific shape sparse covariance matrices, facilitating simulations, statistical method testing, and educational purposes.

## Installation

To install the latest version of `Glarmadillo` from GitHub, run the following commands in R:

``````# Install the devtools package if it's not already installed
if (!require(devtools)) install.packages("devtools")

# Install the Glarmadillo package from GitHub

## Usage

Here’s an example demonstrating how to generate a sparse covariance matrix, solve the GLasso problem, and find an optimal lambda by sparsity level:

``````# Load the Glarmadillo package

# Define the dimension and rank for the covariance matrix
n <- 160
p <- 50

# Generate a sparse covariance matrix
s <- generate_sparse_cov_matrix(n, p, standardize = TRUE, sparse_rho = 0, scale_power = 0)

# Solve the Graphical Lasso problem
# Set the regularization parameter
rho <- 0.1
gl_result <- glarma(s, rho, mtol = 1e-4, maxIterations = 10000, ltol = 1e-6)

# Define a sequence of lambda values for the grid search
lambda_grid <- 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_grid, desired_sparsity = 0.8)

# Inspect the optimal lambda value and the sparsity levels for each lambda tested
optimal_lambda <- lambda_results\$best_lambda
sparsity_levels <- lambda_results\$actual_sparsity``````

## Parameter Selection Tips

When selecting parameters for `generate_sparse_cov_matrix`, consider the following guidelines based on the matrix dimension (`n`):

• For smaller dimensions (e.g., `n` around 50), the `scale_power` parameter can be set lower, such as around 0.8.
• For larger dimensions (e.g., `n` equals 1000), `scale_power` can be increased to 1.2. Since the fitting changes with the expansion of `n`, 1.5 is the upper limit and should not be exceeded.
• Regarding sparsity, the sparsity coefficient `sparse_rho` should vary inversely with `scale_power`. For example, if `scale_power` is 0.8, `sparse_rho` can be set to 0.4; if `scale_power` is 1.2, `sparse_rho` can be reduced to 0.1. Adjust according to the specific matrix form.

For `glarma`, the `mtol` parameter, which controls the overall matrix convergence difference, typically does not need adjustment. The number of iterations usually does not reach the maximum, and convergence generally occurs within 20 iterations. The critical aspect is the adjustment of `ltol`. It is recommended to decrease `ltol` as the matrix size increases. For instance, when `n` is 20, `ltol` can be set to 1e-5; when `n` is 1000, it should be set to 1e-7. This is because `ltol` is the convergence condition for each column; if `n` is too large and `ltol` is not sufficiently reduced, the final results can vary significantly.