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Selection of the Regularization Parameter in Graphical Models Using Network Characteristics
Authors:Adria Caballe Mestres  Natalia Bochkina  Claus Mayer
Institution:1. University of Edinburgh &2. Maxwell Institute, Scotland, United Kingdom;3. Biomathematics &4. Statistics Scotland, Scotland, United Kingdom;5. The Alan Turing Institute for Data Science, British Library, London, United Kingdom;6. Biomathematics &
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 article, 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 case study of gene expression data with the aim to discover the genes relevant to colon cancer. Using these data, we find graph structures, which are verified to display significant biological gene associations. Supplementary material is available online.
Keywords:Clustering  Gene expression  Graphical lasso  High dimension  Hyperparameter estimation  Sparse precision matrix
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