矩阵图形模型：基于偏相关的联合估计
Matrix Graphical Model Via Joint Estimation of Partial Correlations
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- Sungshin Women’s University（诚信女子大学）
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中文总结 AI 辅助
针对矩阵图形模型中现有回归方法导致图不对称且调参困难的问题，提出统一优化框架下联合估计所有偏相关的方法，保持对称性并提升图恢复性能，且成功应用于肺结核蛋白质网络分析。
中文摘要 AI 辅助
矩阵图形模型旨在可分离协方差假设下刻画矩阵型数据中的条件依赖结构。在该框架中，精度矩阵被分解为Kronecker积，从而能够分别对行域和列域上的无向图进行建模。现有方法已针对该问题被开发，包括基于似然的方法和基于回归的图估计程序。基于似然的方法直接估计精度矩阵并间接恢复图结构，而基于回归的方法直接针对变量间的边进行估计，因此优于前者。然而，现有的基于回归的方法基于多个惩罚回归问题，这自然导致估计图的不对称性以及在选择调谐参数时的计算困难。为解决这些局限，我们提出在矩阵图形模型中对偏相关进行联合估计。所提方法在一个统一的优化框架内同时估计所有偏相关，从而保持对称性并减轻选择最优模型的负担。数值研究表明，与现有方法相比，所提方法改善了图恢复性能。我们还分析了从肺结核患者收集的、在多个时间点重复测量的蛋白质表达数据，其中所提方法比较了两组患者之间的蛋白质网络并恢复了时间依赖结构。
英文摘要
Matrix graphical models aim to characterize conditional dependence structures in matrix-variate data under a separable covariance assumption. In this framework, the precision matrix is decomposed as a Kronecker product, enabling separate modeling of undirected graphs across row and column domains. Existing methods have been developed for this problem, including likelihood-based approaches and regression-based procedures for graph estimation. Likelihood-based methods estimate precision matrices directly and recover graph structures indirectly, whereas regression-based approaches directly target estimating edges among variables, thus outperforming the former. However, existing regression-based methods are based on multiple penalized regression problems, which naturally yields asymmetry in estimated graphs and computational difficulty in selecting tuning parameters. To address the limitations, we propose a joint estimation of partial correlations in matrix graphical models. The proposed method estimates all partial correlations simultaneously within a unified optimization framework, thereby preserving symmetry and easing the pain of selecting the best models. Numerical studies demonstrate that the proposed method improves graph recovery performance compared to existing approaches. We also analyze protein expression data collected from patients with pulmonary tuberculosis, measured repeatedly at multiple time points, where the proposed method compares protein networks between two groups of patients and recovers the temporal dependence structure.
