GVARCLUS Procedure

Prior to estimating the precision matrix by using the graphical lasso at each step, PROC GVARCLUS creates a threshold for the entries of the sample covariance matrix at a regularization parameter and decomposes the covariance matrix into connected components. In other words, if an entry of sample covariance is less than in its absolute value, there is no graph edge between the variables. This graph is called the thresholded sample covariance graph. The thresholded covariance matrix can be reordered and decomposed into connected components.

If you already have the solution to maximize the penalized log likelihood, you can also present the edge set via a symmetric 0-1 matrix. Let denote the solution at . The element in the matrix is 1 when and and is 0 otherwise. This graph is named the estimated concentration graph.

Mazumder and Hastie (2012) prove that the vertex partition of connected components that is obtained through thresholding the covariance matrix is exactly equal to the one that is obtained from the estimated concentration graph.