Identifiability Analysis of Linear ODE Systems with Hidden Confounders

TL;DR

This paper investigates the conditions under which linear ODE systems with hidden confounders are identifiable, extending existing theory to include systems with causal dependencies among latent variables and validating results through simulations.

Contribution

It provides a systematic analysis of identifiability for linear ODE systems with hidden confounders, including causal dependencies, under various observation scenarios, filling a key gap in the literature.

Findings

Identifiability conditions established for systems with non-causal hidden confounders.

Extended analysis to systems with causal dependencies among latent variables.

Simulation results support theoretical identifiability conditions.

Abstract

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.