An Explicit Mean-Covariance Parameterization for Multivariate Response Linear Regression

We develop a new method to fit the multivariate response linear regression model that exploits a parametric link between the regression coefficient matrix and the error covariance matrix. Specifically, we assume that the correlations between entries in the multivariate error random vector are proportional to the cosines of the angles between their corresponding regression coefficient matrix columns, so as the angle between two regression coefficient matrix columns decreases, the correlation between the corresponding errors increases. We highlight two models under which this parameterization arises: a latent variable reduced-rank regression model and the errors-in-variables regression model. We propose a novel nonconvex weighted residual sum of squares criterion which exploits this parameterization and admits a new class of penalized estimators. The optimization is solved with an accelerated proximal gradient descent algorithm. Our method is used to study the association between microRNA expression and cancer drug activity measured on the NCI-60 cell lines. An R package implementing our method, MCMVR, is available online.我们提出了一种全新方法，用于拟合多响应线性回归模型（multivariate response linear regression model），该方法利用了回归系数矩阵（regression coefficient matrix）与误差协方差矩阵（error covariance matrix）之间的参数关联。具体而言，我们假设多元误差随机向量（multivariate error random vector）中各元素间的相关性，与它们各自对应的回归系数矩阵列向量之间夹角的余弦值成正比：即两个回归系数矩阵列向量的夹角越小，对应误差项的相关性越强。我们重点阐述了该参数化形式所适配的两类模型：潜变量降秩回归模型（latent variable reduced-rank regression model）与变量误差回归模型（errors-in-variables regression model）。我们提出了一种新颖的非凸加权残差平方和准则（nonconvex weighted residual sum of squares criterion），该准则利用上述参数化形式，并支持一类全新的惩罚估计量（penalized estimators）。其优化问题可通过加速近端梯度下降算法（accelerated proximal gradient descent algorithm）求解。我们将所提方法应用于NCI-60细胞系（NCI-60 cell lines）中测得的微RNA（microRNA）表达与癌症药物活性之间的关联分析。实现该方法的R包（R package）MCMVR已在线公开。