The identifiability analysis of linear Ordinary Differential Equation (ODE) systems is a necessary prerequisite for making reliable causal inferences about these systems. While identifiability has been well studied in scenarios where the system is fully observable, the conditions for identifiability remain unexplored when latent variables interact with the system. This paper aims to address this gap by presenting a systematic analysis of identifiability in linear ODE systems incorporating hidden confounders. Specifically, we investigate two cases of such systems. In the first case, latent confounders exhibit no causal relationships, yet their evolution adheres to specific functional forms, such as polynomial functions of time $t$. Subsequently, we extend this analysis to encompass scenarios where hidden confounders exhibit causal dependencies, with the causal structure of latent variables described by a Directed Acyclic Graph (DAG). The second case represents a more intricate variation of the first case, prompting a more comprehensive identifiability analysis. Accordingly, we conduct detailed identifiability analyses of the second system under various observation conditions, including both continuous and discrete observations from single or multiple trajectories. To validate our theoretical results, we perform a series of simulations, which support and substantiate our findings.

翻译：线性常微分方程（ODE）系统的可辨识性分析是确保对这些系统做出可靠因果推断的必要前提。尽管在系统完全可观测的场景下，可辨识性已得到充分研究，但当潜在变量与系统交互时，其可辨识性条件仍未被探索。本文旨在填补这一空白，对包含隐藏混杂因素的线性ODE系统进行系统的可辨识性分析。具体而言，我们研究了此类系统的两种情况。在第一种情况下，潜在混杂因素之间不存在因果关系，但其演化遵循特定的函数形式，例如时间 $t$ 的多项式函数。随后，我们将此分析扩展到隐藏混杂因素存在因果依赖性的场景，其中潜在变量的因果结构由有向无环图（DAG）描述。第二种情况是第一种情况的更复杂变体，因此需要进行更全面的可辨识性分析。相应地，我们在多种观测条件下对第二种系统进行了详细的可辨识性分析，包括来自单条或多条轨迹的连续和离散观测。为了验证我们的理论结果，我们进行了一系列仿真实验，这些实验支持并证实了我们的发现。