Selection of the Regularization Parameter in Graphical Models using Network Characteristics

Abstract

Gaussian graphical models represent the underlying graph structure of conditional dependence between random variables which can be determined using their partial correlation or precision matrix. In a high-dimensional setting, the precision matrix is estimated using penalized likelihood by adding a penalization term which controls the amount of sparsity in the precision matrix and totally characterizes the complexity and structure of the graph. The most commonly used penalization term is the L1 norm of the precision matrix scaled by the regularization parameter which determines the trade-off between sparsity of the graph and fit to the data. In this paper we propose several procedures to select the regularization parameter in the estimation of graphical models that focus on recovering reliably the appropriate network structure of the graph. We conduct an extensive simulation study to show that the proposed methods produce useful results for different network topologies. The approaches are also applied in a high-dimensional real case study of gene expression data with the aim to discover the genes relevant to colon cancer. Using this data, we find graph structures which are verified to display significant biological gene associations.