Graphical continuous Lyapunov models offer a new perspective on modeling causally interpretable dependence structure in multivariate data by treating each independent observation as a one-time cross-sectional snapshot of a temporal process. Specifically, the models assume that the observations are cross-sections of independent multivariate Ornstein-Uhlenbeck processes in equilibrium. The Gaussian equilibrium exists under a stability assumption on the drift matrix, and the equilibrium covariance matrix is determined by the continuous Lyapunov equation. Each graphical continuous Lyapunov model assumes the drift matrix to be sparse, with a support determined by a directed graph. A natural approach to model selection in this setting is to use an $\ell_1$-regularization technique that, based on a given sample covariance matrix, seeks to find a sparse approximate solution to the Lyapunov equation. We study the model selection properties of the resulting lasso technique to arrive at a consistency result. Our detailed analysis reveals that the involved irrepresentability condition is surprisingly difficult to satisfy. While this may prevent asymptotic consistency in model selection, our numerical experiments indicate that even if the theoretical requirements for consistency are not met, the lasso approach is able to recover relevant structure of the drift matrix and is robust to aspects of model misspecification.

翻译：图连续李雅普诺夫模型通过将每个独立观测视为时间过程的一次性截面快照，为多变量数据中因果可解释的依赖结构建模提供了新视角。具体而言，该模型假设观测数据是处于平衡态的独立多元奥恩斯坦-乌伦贝克过程的截面。在漂移矩阵满足稳定性假设的前提下，高斯平衡存在，且平衡协方差矩阵由连续李雅普诺夫方程确定。每个图连续李雅普诺夫模型假设漂移矩阵具有稀疏性，其支撑集由有向图决定。在此框架中进行模型选择的自然方法是采用ℓ1正则化技术，该技术基于给定样本协方差矩阵，旨在寻找李雅普诺夫方程的稀疏近似解。我们研究了所得Lasso技术的模型选择特性，并得出一致性结论。详细分析表明，所涉及的不可表示性条件出乎意料地难以满足。尽管这可能导致模型选择渐近一致性的缺失，但数值实验显示，即使未满足一致性的理论要求，Lasso方法仍能有效恢复漂移矩阵的相关结构，并对模型误设具有鲁棒性。