Though Gaussian graphical models have been widely used in many scientific fields, relatively limited progress has been made to link graph structures to external covariates. We propose a Gaussian graphical regression model, which regresses both the mean and the precision matrix of a Gaussian graphical model on covariates. In the context of co-expression quantitative trait locus (QTL) studies, our method can determine how genetic variants and clinical conditions modulate the subject-level network structures, and recover both the population-level and subject-level gene networks. Our framework encourages sparsity of covariate effects on both the mean and the precision matrix. In particular for the precision matrix, we stipulate simultaneous sparsity, i.e., group sparsity and element-wise sparsity, on effective covariates and their effects on network edges, respectively. We establish variable selection consistency first under the case with known mean parameters and then a more challenging case with unknown means depending on external covariates, and establish in both cases the $\ell_2$ convergence rates and the selection consistency of the estimated precision parameters. The utility and efficacy of our proposed method is demonstrated through simulation studies and an application to a co-expression QTL study with brain cancer patients.
翻译:虽然Gausian图形模型在许多科学领域被广泛使用,但在将图形结构与外部共变体联系起来方面进展相对有限。 我们提议了Gausian图形回归模型,该模型将Gausian图形模型的平均值和精确矩阵都倒退到共同变数上。 在共同表达定量特征位置(QTL)研究中,我们的方法可以确定基因变量和临床条件如何调节主题层次网络结构,并恢复人口层次和主题层次的基因网络。我们的框架鼓励对平均值和精确矩阵产生共变效应的宽度。特别是对于精确矩阵,我们规定了同时的宽度,即群体宽度和元素偏度,分别规定了有效的共变数及其对网络边缘的影响。我们首先在案例下与已知的中值参数建立变量选择一致性,然后根据外部变数确定一个更具有挑战性的案例。我们的框架鼓励对平均值和精确矩阵的共变数效应的宽度效应。我们提出的方法的实用性和有效性和有效性通过实验室的模拟研究来显示对估计精确参数的精确度应用。