Quantifying the relationships between components of a complex system is a valuable yet challenging task, especially when the number of variables is large. The Gaussian graphical model (GGM) incorporates an undirected graph that represents the conditional dependence between variables. It has a wide variety of applications, such as biological networks, social networks, and stock networks. The precision matrix, also known as the inverse covariance matrix or concentration matrix, encodes the partial correlation between pairs of variables given the others. Nonzero entries in the precision matrix correspond to edges in the graph, making GGM a potent tool for analysis with interpretability.

Challenges in GGM include the feasibility of methods and high computational costs in the high-dimensiona...[ Read more ]

Quantifying the relationships between components of a complex system is a valuable yet challenging task, especially when the number of variables is large. The Gaussian graphical model (GGM) incorporates an undirected graph that represents the conditional dependence between variables. It has a wide variety of applications, such as biological networks, social networks, and stock networks. The precision matrix, also known as the inverse covariance matrix or concentration matrix, encodes the partial correlation between pairs of variables given the others. Nonzero entries in the precision matrix correspond to edges in the graph, making GGM a potent tool for analysis with interpretability.

Challenges in GGM include the feasibility of methods and high computational costs in the high-dimensional settings, the estimation accuracy of large precision matrix, inference on edge existence, and joint estimation of multiple graphs. To address these challenges, we propose the f̠lexible and accurate G̠aussian graphical model (FLAG), which utilizes the random effects model for pairwise conditional regression, making it feasible in the high-dimensional settings and free from sparsity assumptions on the precision matrix. The critical advantage of FLAG is its capability to perform statistical inference on each entry in the precision matrix, quantifying the uncertainty associated with each edge. The minorize-maximization (MM) algorithm and parameter-expanded expectation-maximization (PX-EM) algorithm are designed to estimate the variance components efficiently.

To handle multiple pairs, the algorithm is further accelerated by low-rank update, which significantly reduces the computational cost by eliminating repeated eigen-decompositions of large matrices. The capability of element-wise statistical inference makes FLAG method easily extended to joint estimation of multiple graphs with little additional computational cost. The FLAG method with meta-analysis applies hypothesis tests to determine whether the same pair in different groups share the same partial correlation and adjusts the shared one using meta-analysis. The FLAG with covariate adjusted (FLAG-CA) method regards the extra information that differs across groups as fixed effects and estimates the variance components based on an extended model using the revised MM and EM algorithms.

The proposed method FLAG and its extensions are evaluated using simulation data in the block-magnified precision matrix, graph with hub structure, and multiple graph settings, as well as the real data applications, including gene expression in the human brain, term association in the university websites, and stock prices in the U.S. financial market.