The recently introduced graphical continuous Lyapunov models provide a new approach to statistical modeling of correlated multivariate data. The models view each observation as a one-time cross-sectional snapshot of a multivariate dynamic process in equilibrium. The covariance matrix for the data is obtained by solving a continuous Lyapunov equation that is parametrized by the drift matrix of the dynamic process. In this context, different statistical models postulate different sparsity patterns in the drift matrix, and it becomes a crucial problem to clarify whether a given sparsity assumption allows one to uniquely recover the drift matrix parameters from the covariance matrix of the data. We study this identifiability problem by representing sparsity patterns by directed graphs. Our main result proves that the drift matrix is globally identifiable if and only if the graph for the sparsity pattern is simple (i.e., does not contain directed two-cycles). Moreover, we present a necessary condition for generic identifiability and provide a computational classification of small graphs with up to 5 nodes.

翻译：近期提出的图形化连续Lyapunov模型为相关多元数据的统计建模提供了一种新方法。该模型将每个观测视为处于平衡态的多元动态过程的一次横截面快照。数据协方差矩阵通过求解由动态过程漂移矩阵参数化的连续Lyapunov方程获得。在此框架下，不同统计模型对漂移矩阵设置不同的稀疏模式，因此关键问题在于明确给定的稀疏假设是否允许从数据协方差矩阵唯一恢复漂移矩阵参数。我们通过有向图表示稀疏模式来研究这一可辨识性问题。主要结果表明：当且仅当稀疏模式对应的图为简单图（即不包含双向二环）时，漂移矩阵具有全局可辨识性。此外，我们给出了通用可辨识性的必要条件，并提供了包含至多5个节点的小型图的计算分类结果。