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Bayesian sparse covariance decomposition with a graphical structure

机译：具有图形结构的贝叶斯稀疏协方差分解

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摘要

We consider the problem of estimating covariance matrices of a particular structure that is a summation of a low-rank component and a sparse component. This is a general covariance structure encountered in multiple statistical models including factor analysis and random effects models, where the low-rank component relates to the correlations among variables coming from the latent factors or random effects and the sparse component displays the correlations of the remaining residuals. We propose a Bayesian method for estimating the covariance matrices of such structures by representing the covariance model in the form of a factor model with an unknown number of latent factors. We introduce binary indicators for factor selection and rank estimation for the low-rank component, combined with a Bayesian lasso method for the estimation of the sparse component. Simulation studies show that our method can recover the rank as well as the sparsity of the two respective components. We further extend our method to a latent-factor Markov graphical model, with a focus on the sparse conditional graphical model of the residuals as well as selecting the number of factors. We show through simulations that our Bayesian model can successfully recover both the number of latent factors and the Markov graphical model of the residuals.
机译：我们考虑估计特定结构的协方差矩阵的问题，该协方差矩阵是低秩分量和稀疏分量的总和。这是在包括因子分析和随机效应模型在内的多个统计模型中遇到的一般协方差结构，其中低秩成分与来自潜在因子或随机效应的变量之间的相关性相关，而稀疏成分显示其余残差的相关性。我们提出了一种贝叶斯方法，用于通过以未知数量的潜在因子的因子模型形式表示协方差模型来估计此类结构的协方差矩阵。我们引入了用于低阶成分的因子选择和秩估计的二元指标，并结合了贝叶斯套索方法来估计稀疏成分。仿真研究表明，我们的方法可以恢复两个分量的等级以及稀疏性。我们进一步将我们的方法扩展到潜在因子马尔可夫图形模型，重点是残差的稀疏条件图形模型以及选择因子的数量。通过仿真显示，我们的贝叶斯模型可以成功地恢复潜在因子的数量和残差的马尔可夫图形模型。

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