Estimates a sparse precision matrix from ordinal item response data via the Graphical Lasso (Friedman, Hastie, Tibshirani 2008). The polychoric correlation matrix is used as input. The optimal regularization parameter lambda is selected by the Extended Bayesian Information Criterion (Foygel and Drton 2010).
Usage
Glasso(
U,
na = NULL,
Z = NULL,
w = NULL,
gamma = 0.5,
n_lambda = 50,
lambda_ratio = 0.01,
penalize_diagonal = TRUE,
max_iter = 100,
eps = 1e-06,
edge_tol = 1e-08,
verbose = FALSE,
...
)Arguments
- U
U is either a data class of exametrika, or raw data. When raw data is given, it is converted to the exametrika class with the dataFormat function.
- na
na argument specifies the numbers or characters to be treated as missing values.
- Z
Z is a missing indicator matrix of the type matrix or data.frame
- w
w is item weight vector
- gamma
EBIC tuning parameter in
[0, 1]. Default is 0.5.- n_lambda
Number of lambda values in the search grid. Default is 50.
- lambda_ratio
Ratio of lambda_min to lambda_max. Default is 0.01.
- penalize_diagonal
Logical. If TRUE, the L1 penalty is applied to the diagonal of Theta. Default is TRUE.
- max_iter
Maximum number of outer iterations for the block coordinate descent algorithm. Default is 100.
- eps
Convergence tolerance. Default is 1e-6.
- edge_tol
Threshold below which an off-diagonal element of Theta is considered zero (no edge). Default is 1e-8.
- verbose
Logical. If TRUE, progress messages are displayed.
- ...
Additional arguments (currently unused; reserved for future extensions).
Value
- theta
Estimated precision matrix at the optimal lambda.
- W
Working covariance matrix at the optimal lambda.
- lambda_opt
Selected lambda value.
- gamma
EBIC tuning parameter used.
- ebic_opt
Minimum EBIC value.
- n_edge
Number of edges in the selected model.
- path
Data frame with columns lambda, ebic, n_edge over the search grid.
Details
The Graphical Lasso estimates the precision matrix Theta by maximizing the penalized log-likelihood:
log det(Theta) - tr(S Theta) - lambda * ||Theta||_1
subject to Theta being positive definite. Optimization is performed by block coordinate descent (Algorithm 17.2 of Hastie, Tibshirani, Friedman 2009) with cyclical coordinate descent for the inner lasso step.
Lambda is searched on a log-scale grid from lambda_max (the maximum absolute off-diagonal of S) down to lambda_max * lambda_ratio. For each lambda, the EBIC is computed and the lambda minimizing EBIC is returned. Warm-starting (W and beta cache from the previous lambda) is used to accelerate the search.
References
Friedman, J., Hastie, T., and Tibshirani, R. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3), 432-441.
Foygel, R., and Drton, M. (2010). Extended Bayesian Information Criteria for Gaussian Graphical Models. Advances in Neural Information Processing Systems 23.