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Identifiability Analysis of Linear ODE Systems with Hidden Confounders

Yuanyuan Wang, Biwei Huang, Wei Huang, Xi Geng, Mingming Gong

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.

Identifiability Analysis of Linear ODE Systems with Hidden Confounders

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 . 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.

Paper Structure

This paper contains 34 sections, 12 theorems, 86 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Causal interpretation of the ODE system
  4. Identifiability analysis of the linear ODE system (\ref{['eq:ODE']})
  5. Identifiability analysis of the linear ODE system (\ref{['eq:ODE1']})
  6. Identifiability analysis of the linear ODE system (\ref{['eq:ODE2']})
  7. Identifiability condition from a single whole trajectory
  8. Identifiability condition from discrete observations sampled from a single trajectory
  9. Identifiability condition from p controllable whole trajectories
  10. Identifiability condition from discrete observations sampled from p controllable trajectories
  11. Simulations
  12. Related work
  13. Conclusion
  14. Appendix for "Identifiability Analysis of Linear ODE Systems with Hidden Confounders"
  15. Summary of notations and proposed identifiability conditions
  16. ...and 19 more sections

Key Result

Lemma 2.1

For $\boldsymbol{x}_0 \in \mathbb{R}^d, A\in \mathbb{R}^{d\times d}$, the ODE system eq:ODE is $(\boldsymbol{x}_0, A)$-identifiable if and only if condition A0 is satisfied.

Figures (4)

  • Figure 1: Example causal structures of the ODE system \ref{['eq:ODE1']} and \ref{['eq:ODE2']}.
  • Figure 2: Causal structures of the ODE system \ref{['eq:ODE2']} with parameter matrix $M$ and $M'$.
  • Figure 3: Example causal structure of the damped harmonic oscillators model with $3$ oscillators and $2$ latent variables.
  • Figure 4: Causal structure of the population model.

Theorems & Definitions (31)

  • Definition 2.1
  • Lemma 2.1
  • Definition 3.1
  • Theorem 3.1
  • Definition 4.1
  • Theorem 4.1
  • Definition 4.2
  • Theorem 4.2
  • Definition 4.3
  • Theorem 4.3
  • ...and 21 more