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Identifiability Analysis of Linear ODE Systems with Hidden Confounders
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Gives necessary-and-sufficient identifiability conditions for linear ODE systems with hidden confounders (polynomial-driven latent variables and DAG-structured latents) from continuous or discrete observations of single or multiple trajectories, validated by simulations.
Problem
Identifiability of linear ODE systems when latent confounders interact with the observed system, so that parameters cannot be fully observed and causal inference may be misleading.
Approach
Augment the partially observed linear ODE with polynomial basis functions of time (or nilpotent latent-DAG dynamics giving polynomial terms) to obtain a fully observable homogeneous linear ODE, then reuse the known identifiability criterion (linear independence of Krylov vectors) to derive conditions for the observed-variable dynamics.
Key results
- In identifiable simulated cases, parameter MSE decreases toward zero as sample size grows (e.g., A MSE 2.65E-05 at n=500 for the single-trajectory case), while unidentifiable cases stay high (e.g., A MSE 0.0461 at n=500).
- For the p-controllable-trajectory setting, identifiable cases reach MSE on the order of 1e-22 to 1e-23 for A, B, G, versus about 0.59 for A in the unidentifiable case at n=50.
- Necessary and sufficient identifiability conditions are derived for the independent-latent system (rank condition A1) and the single-trajectory latent-DAG system (condition B1).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Satisfy B1 rather than violate it | ODE (3), single trajectory · η-identifiability simulation, n=10 | MSE (A): 0.0994 -> 6.00E-05 | Helped |
| Satisfy B1 rather than violate it | ODE (3), single trajectory · η-identifiability simulation, n=10 | MSE (BG²z₀): 0.6482 -> 0.0007 | Helped |
| Satisfy B1 rather than violate it | ODE (3), single trajectory · η-identifiability simulation, n=500 | MSE (A): 0.0461 -> 2.65E-05 | Helped |
| Satisfy B1 rather than violate it | ODE (3), single trajectory · η-identifiability simulation, n=500 | MSE (BG²z₀): 0.3648 -> 0.0002 | Helped |
| Satisfy B2, B3 and B4 rather than violate B2 | ODE (3), p controllable trajectories · {ηᵢ}₁ᵖ-identifiability simulation, n=10 | MSE (A): 0.6349 -> 5.83E-22 | Helped |
| Satisfy B2, B3 and B4 rather than violate B2 | ODE (3), p controllable trajectories · {ηᵢ}₁ᵖ-identifiability simulation, n=10 | MSE (G): 0.0044 -> 2.27E-21 | Helped |
| Satisfy B2, B3 and B4 rather than violate B2 | ODE (3), p controllable trajectories · {ηᵢ}₁ᵖ-identifiability simulation, n=50 | MSE (A): 0.5876 -> 5.16E-23 | Helped |
| Satisfy B2, B3 and B4 rather than violate B2 | ODE (3), p controllable trajectories · {ηᵢ}₁ᵖ-identifiability simulation, n=50 | MSE (G): 0.0045 -> 2.39E-22 | Helped |
Surprising findings
- Under a single trajectory with latent confounders, matrix B and latent initial conditions z0 cannot be identified; only the composite terms Bz0, BGz0, etc. are identifiable, yet these suffice for causal inference on observed variables.
Limitations
- Verifying the identifiability conditions in practice is hard because true parameters are unknown.
- Analysis restricted to linear ODEs with polynomial or specific (exponential, trigonometric) latent evolutions and latent DAG structure, with latent variables not influenced by observables.
Experimental setup
Datasets: simulated linear ODE trajectories
Metrics: mean squared error of parameter estimates variance of squared error over 100 replications
Builds on: Stanhope et al. identifiability of linear ODEs from a single trajectory Wang et al. identifiability from discrete observations of linear ODEs Wang et al. identifiability of SDEs
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