Gaussian graphical modeling for spectrometric data analysis

We propose Bayesian methods for Gaussian graphical models that lead to sparse and adaptively shrunk estimators of the precision (inverse covariance) matrix. Our methods are based on lasso-type regularization priors leading to parsimonious parameterization of the precision matrix, which is essential in several applications involving learning relationships among the variables. In this context, we introduce a novel type of selection prior that develops a sparse structure on the precision matrix by making most of the elements exactly zero, in addition to ensuring positive definiteness - thus conducting model selection and estimation simultaneously. More importantly, we extend these methods to analyze clustered data using finite mixtures of Gaussian graphical model and infinite mixtures of Gaussian graphical models. We discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalizing constants that are functions of parameters of interest, which result from the restriction of positive definiteness on the correlation matrix. We evaluate the operating characteristics of our method via several simulations and demonstrate the application to real data examples in genomics.

Large-scale precision matrix estimation is of fundamental importance yet challenging in many contemporary applications for recovering Gaussian graphical models. In this paper, we suggest a new approach of innovated scalable efficient estimation (ISEE) for estimating large precision matrix. Motivated by the innovated transformation, we convert the original problem into that of large covariance matrix estimation. The suggested method combines the strengths of recent advances in high-dimensional sparse modeling and large covariance matrix estimation. Compared to existing approaches, our method is scalable and can deal with much larger precision matrices with simple tuning. Under mild regularity conditions, we establish that this procedure can recover the underlying graphical structure with significant probability and provide efficient estimation of link strengths. Both computational and theoretical advantages of the procedure are evidenced through simulation and real data examples.

The covariance matrix (or its inverse, the precision matrix) is central to many chemometric techniques. Traditional sample estimators perform poorly for high‐dimensional data such as metabolomics data. Because of this, many traditional inference techniques break down or produce unreliable results. In this paper, we selectively review several modern estimators of the covariance and precision matrix that improve upon the traditional sample estimator. We focus on 3 general techniques: eigenvalue‐shrinkage estimation, ridge‐type estimation, and structured estimation. These methods rely on different assumptions regarding the structure of the covariance or precision matrix. Various examples, in particular using metabolomics data, are used to compare these techniques and to demonstrate that in concert with, eg, principal component analysis, multivariate analysis of variance, and Gaussian graphical models, better results are obtained.

Differential network analysis investigates how the network of connected genes changes from one condition to another and has become a prevalent tool to provide a deeper and more comprehensive understanding of the molecular etiology of complex diseases. Based on the asymptotically normal estimation of large Gaussian graphical model (GGM) in the high-dimensional setting, we developed a computationally efficient test for differential network analysis through testing the equality of two precision matrices, which summarize the conditional dependence network structures of the genes. Additionally, we applied a multiple testing procedure to infer the differential network structure with false discovery rate (FDR) control. Through extensive simulation studies with different combinations of parameters including sample size, number of vertices, level of heterogeneity and graph structure, we demonstrated that our method performed much better than the current available methods in terms of accuracy and computational time. In real data analysis on lung adenocarcinoma, we revealed a differential network with 3503 nodes and 2550 edges, which consisted of 50 clusters with an FDR threshold at 0.05. Many of the top gene pairs in the differential network have been reported relevant to human cancers. Our method represents a powerful tool of network analysis for high-dimensional biological data.

In this paper, we consider the problem of fitting a sparse precision matrix to multivariate Gaussian data. The zero elements in the precision matrix correspond to conditional independencies between variables. We focus on the estimation of a class of sparse precision matrix which represents the scale-free networks. It has been demonstrated that some of the important networks display features similar to scale-free graphs. We propose a new log-likelihood formulation, which promotes the sparseness of the precision matrix as well as the topological structure of scale-free networks. To optimize this new energy formulation, the alternating direction method of multipliers form is used with the general \(L_1\)-regularized loss optimization. We tested our proposed method on various databases. Our proposed method exhibits better estimation performance with various number of samples, N, and different selection of sparsity parameter, \(\rho \).

Large-scale microarray gene expression data provide the possibility of constructing genetic networks or biological pathways. Gaussian graphical models have been suggested to provide an effective method for constructing such genetic networks. However, most of the available methods for constructing Gaussian graphs do not account for the sparsity of the networks and are computationally more demanding or infeasible, especially in the settings of high dimension and low sample size. We introduce a threshold gradient descent (TGD) regularization procedure for estimating the sparse precision matrix in the setting of Gaussian graphical models and demonstrate its application to identifying genetic networks. Such a procedure is computationally feasible and can easily incorporate prior biological knowledge about the network structure. Simulation results indicate that the proposed method yields a better estimate of the precision matrix than the procedures that fail to account for the sparsity of the graphs. We also present the results on inference of a gene network for isoprenoid biosynthesis in Arabidopsis thaliana. These results demonstrate that the proposed procedure can indeed identify biologically meaningful genetic networks based on microarray gene expression data.

Gene regulatory networks (GRNs) are often inferred based on Gaussian graphical models that could identify the conditional dependence among genes by estimating the corresponding precision matrix. Classical Gaussian graphical models are usually designed for single network estimation and ignore existing knowledge such as pathway information. Therefore, they can neither make use of the common information shared by multiple networks, nor can they utilize useful prior information to guide the estimation. In this paper, we propose a new weighted fused pathway graphical lasso (WFPGL) to jointly estimate multiple networks by incorporating prior knowledge derived from known pathways and gene interactions. Based on the assumption that two genes are less likely to be connected if they do not participate together in any pathways, a pathway-based constraint is considered in our model. Moreover, we introduce a weighted fused lasso penalty in our model to take into account prior gene interaction data and common information shared by multiple networks. Our model is optimized based on the alternating direction method of multipliers (ADMM). Experiments on synthetic data demonstrate that our method outperforms other five state-of-the-art graphical models. We then apply our model to two real datasets. Hub genes in our identified state-specific networks show some shared and specific patterns, which indicates the efficiency of our model in revealing the underlying mechanisms of complex diseases.

Loss function, unbiasedness, and optimality of Gaussian graphical model selection

Biological networks are important for the analysis of human diseases, which summarize the regulatory interactions and other relationships between different molecules. Understanding and constructing networks for molecules, such as DNA, RNA and proteins, can help elucidate the mechanisms of complex biological systems. The Gaussian Graphical Models (GGMs) are popular tools for the estimation of biological networks. Nonetheless, reconstructing GGMs from high-dimensional datasets is still challenging. The current methods cannot handle the sparsity and high-dimensionality issues arising from datasets very well. Here, we developed a new GGM, called the GR2D2 (Graphical $R^2$-induced Dirichlet Decomposition) model, based on the R2D2 priors for linear models. Besides, we provided a data-augmented block Gibbs sampler algorithm. The R code is available at https://github.com/RavenGan/GR2D2. The GR2D2 estimator shows superior performance in estimating the precision matrices compared with the existing techniques in various simulation settings. When the true precision matrix is sparse and of high dimension, the GR2D2 provides the estimates with smallest information divergence from the underlying truth. We also compare the GR2D2 estimator with the graphical horseshoe estimator in five cancer RNA-seq gene expression datasets grouped by three cancer types. Our results show that GR2D2 successfully identifies common cancer pathways and cancer-specific pathways for each dataset.

The multivariate adaptive regression spline (MARS) is one of the alternative nonparametric approaches of the Gaussian graphical model (GGM) in order to construct the undirected networks of complex biological systems. In GGM, it is assumed that the binary form of the estimated precision matrix, computed from the systems' elements under their conditional independencies, represents the undirected interactions of these elements by using the lasso regression. From previous studies, it has been shown that if MARS is estimated by discarding all interactions and nonlinear components, resulting in a sole inference of the main effects, it can be a strong alternative of GGM. We call this type of the summarized MARS model the lasso-based MARS model. Furthermore, in the construction of the networks via the optimal MARS model, it is assumed that the estimated regression coefficients, i.e., the important terms of the estimated model, can be the indication of the pairwise links between the species of the systems. Hereby, in this study, we investigate whether the inference of the networks via the Pearson, Spearman, and Kendall's tau correlation coefficients has better performance than the networks constructed by the important regression coefficients in the lasso-based MARS model. In the analyses, we use both simulated and real datasets and compare the estimated networks by various accuracy measures from specificity and F-measure to Matthews correlation coefficient. From the analyses, we find that the suggested approaches improve all the listed criteria significantly without any computational cost.

It is well known that regression methods designed for clean data will lead to erroneous results if directly applied to corrupted data. Despite the recent methodological and algorithmic advances in Gaussian graphical model estimation, how to achieve efficient and scalable estimation under contaminated covariates is unclear. Here a new methodology called convex conditioned innovative scalable efficient estimation (COCOISEE) for Gaussian graphical model under both additive and multiplicative measurement errors is developed. It combines the strengths of the innovative scalable efficient estimation in Gaussian graphical model and the nearest positive semi-definite matrix projection, thus enjoying stepwise convexity and scalability. Comprehensive theoretical guarantees are provided and the effectiveness of the proposed methodology is demonstrated through numerical studies.

This doctoral thesis contributes to the advancement of statistical analyses of gene expression profiles using Gaussian graphical models (GGMs). The main focus has been on studying the properties of GGMs obtained with the LW shrinkage. The premise is that, despite its advantages, the shrinkage approach introduces some biases that have not been studied in sufficient detail. These biases have the potential of obscuring the interpretation of the network structure and impeding the validation of earlier analyses. In this sense, this thesis may be considered a continuation of the works of Schäfer and Strimmer, where the LW-shrinkage was originally used to model GRNs with GGMs. The thesis is organized as follows; Chapter 1 presents a test of statistical significance for GGMs based on the LW-shrinkage. Here the probability density of the ‘shrunk’ partial correlation is derived by means of geometric arguments. In Chapter 2 a network analysis of (matched) nasal and bronchial expression profiles is presented. The method developed in the previous chapter is employed to explore whether expression profiles from nasal epithelial cells can be used as a proxy for bronchial epithelial cells. Chapter 3 shows the existence of a non-linear bias on the partial correlations obtained with the LW-shrinkage. This bias is removed via ‘un-shrinking’; a new concept that de-regularizes the partial correlation. In Chapter 4 the LW-shrinkage is revisited from a data-level perspective. The goal is to explore the correspondence between the shrinkage-based covariance matrix and the dataset.

Nitrogen is an essential nutrient for all life forms. Like most unicellular organisms, the yeast Saccharomyces cerevisiae transports and catabolizes good nitrogen sources in preference to poor ones. Nitrogen catabolite repression (NCR) refers to this selection mechanism. We propose an approach based on Gaussian graphical models (GGMs), which enable to distinguish direct from indirect interactions between genes, to identify putative NCR genes from putative NCR regulatory motifs and over-represented motifs in the upstream noncoding sequences of annotated NCR genes. Because of the high-dimensionality of the data, we use a shrinkage estimator of the covariance matrix to infer the GGMs. We show that our approach makes significant and biologically valid predictions. We also show that GGMs are more effective than models that rely on measures of direct interactions between genes.

Functional data capture a wide range of processes, including growth curves and spectral absorption patterns. In this study, we analyse air pollution data from the In-service Aircraft for a Global Observing System, focusing on the spatial interactions among atmospheric chemicals and their dependence on meteorological conditions. This analysis necessitates functional regression, where both response and covariates are functional objects evolving throughout the troposphere. Quantifying both the functional dependencies between the response and covariates and the interdependencies among multivariate response functions poses significant challenges. To address these challenges, we introduce a functional Gaussian graphical regression model, which extends conditional Gaussian graphical models to partially separable functional data. We propose a doubly penalized estimator for model inference and develop a novel adaptation of Kullback–Leibler cross-validation, specifically tailored for graphical estimators. This criterion, named joint Kullback–Leibler cross-validation, simultaneously accounts for both precision and regression matrices, particularly in scenarios where the population comprises multiple sub-groups. Model performance is evaluated in terms of Kullback–Leibler divergence and graph recovery power.

This paper concerns the specification, and performance, of scale-free prior distributions with a view toward large-scale network inference from small-sample data sets. We devise three scale-free priors and implement them in the framework of Gaussian graphical models. Gaussian graphical models are used in gene network inference where high-throughput data describing a large number of variables with comparatively few samples are frequently analyzed by practitioners. And, although there is a consensus that many such networks are scale-free, the modus operandi is to assign a random network prior. Simulations demonstrate that the scale-free priors outperform the random network prior at recovering scale-free trees with degree exponents near 2, such as are characteristic of many real-world systems. On the other hand, the random network prior compares favorably at recovering scale-free trees characterized by larger degree exponents.