Lee Wonyul, Liu Yufeng
Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599-3260, USA.
Department of Statistics and Operations Research, Department of Genetics, Department of Biostatistics, Carolina Center for Genome Sciences, University of North Carolina, Chapel Hill, NC 27599-3260, USA.
J Mach Learn Res. 2015;16:1035-1062.
Estimation of inverse covariance matrices, known as precision matrices, is important in various areas of statistical analysis. In this article, we consider estimation of multiple precision matrices sharing some common structures. In this setting, estimating each precision matrix separately can be suboptimal as it ignores potential common structures. This article proposes a new approach to parameterize each precision matrix as a sum of common and unique components and estimate multiple precision matrices in a constrained minimization framework. We establish both estimation and selection consistency of the proposed estimator in the high dimensional setting. The proposed estimator achieves a faster convergence rate for the common structure in certain cases. Our numerical examples demonstrate that our new estimator can perform better than several existing methods in terms of the entropy loss and Frobenius loss. An application to a glioblastoma cancer data set reveals some interesting gene networks across multiple cancer subtypes.
估计逆协方差矩阵（即精度矩阵）在统计分析的各个领域都很重要。在本文中，我们考虑估计具有一些共同结构的多个精度矩阵。在这种情况下，单独估计每个精度矩阵可能不是最优的，因为它忽略了潜在的共同结构。本文提出了一种新方法，将每个精度矩阵参数化为共同分量和独特分量的和，并在约束最小化框架中估计多个精度矩阵。我们在高维情况下建立了所提出估计器的估计一致性和选择一致性。在某些情况下，所提出的估计器对于共同结构实现了更快的收敛速度。我们的数值例子表明，在熵损失和弗罗贝尼乌斯损失方面，我们的新估计器比几种现有方法表现更好。对胶质母细胞瘤癌症数据集的应用揭示了跨多个癌症亚型的一些有趣的基因网络。
