Selection of the Regularization Parameter in Graphical Models Using Network Characteristics

<p>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.</p

Selection of the Regularization Parameter in Graphical Models Using Network Characteristics

<p>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.</p

BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips-4

<p><b>Copyright information:</b></p><p>Taken from "BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips"</p><p>http://www.biomedcentral.com/1471-2105/8/439</p><p>BMC Bioinformatics 2007;8():439-439.</p><p>Published online 12 Nov 2007</p><p>PMCID:PMC2216047.</p><p></p>l step is adapted independently for each component. The plot shows a histogram of the optimal log variance for proposals and the fixed step size used in the non-adaptive version overlaid in black, highlighting that a wide range of proposal variances are needed

BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips-2

<p><b>Copyright information:</b></p><p>Taken from "BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips"</p><p>http://www.biomedcentral.com/1471-2105/8/439</p><p>BMC Bioinformatics 2007;8():439-439.</p><p>Published online 12 Nov 2007</p><p>PMCID:PMC2216047.</p><p></p>As probe affinity categories increase, the distributions shift from left to right. The black density line is the distribution from the original BGX model and illustrates the discriminatory power of the probe affinity extension

BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips-5

<p><b>Copyright information:</b></p><p>Taken from "BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips"</p><p>http://www.biomedcentral.com/1471-2105/8/439</p><p>BMC Bioinformatics 2007;8():439-439.</p><p>Published online 12 Nov 2007</p><p>PMCID:PMC2216047.</p><p></p> on the parameter of expressed genes (left & centre). A similar improvement was observed for the IACT of the parameter, for all genes (right)

BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips-1

<p><b>Copyright information:</b></p><p>Taken from "BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips"</p><p>http://www.biomedcentral.com/1471-2105/8/439</p><p>BMC Bioinformatics 2007;8():439-439.</p><p>Published online 12 Nov 2007</p><p>PMCID:PMC2216047.</p><p></p>ted genes between two conditions using a routine incorporated in the package

BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips-3

<p><b>Copyright information:</b></p><p>Taken from "BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips"</p><p>http://www.biomedcentral.com/1471-2105/8/439</p><p>BMC Bioinformatics 2007;8():439-439.</p><p>Published online 12 Nov 2007</p><p>PMCID:PMC2216047.</p><p></p> Spike data set and the corresponding probes' affinity categories were estimated from the data. There is a positive correlation between estimated and true categories, particularly for high-affinity probes

BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips-7

<p><b>Copyright information:</b></p><p>Taken from "BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips"</p><p>http://www.biomedcentral.com/1471-2105/8/439</p><p>BMC Bioinformatics 2007;8():439-439.</p><p>Published online 12 Nov 2007</p><p>PMCID:PMC2216047.</p><p></p>ted genes between two conditions using a routine incorporated in the package

BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips-6

<p><b>Copyright information:</b></p><p>Taken from "BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips"</p><p>http://www.biomedcentral.com/1471-2105/8/439</p><p>BMC Bioinformatics 2007;8():439-439.</p><p>Published online 12 Nov 2007</p><p>PMCID:PMC2216047.</p><p></p> (left). plots the density of the difference in the posterior distributions of a given gene between two conditions (right)