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Modeling Latent Non-Linear Dynamical System over Time Series
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Proposes LaNoLem, which models latent non-linear polynomial dynamics from time series via an alternating minimization (extended Kalman filter inference plus sparse proximal-gradient learning) with MDL-based complexity selection.
Problem
Estimating latent states and sparse non-linear polynomial dynamical system equations directly from noisy time series, where latent states and model parameters form a circular dependency.
Approach
Introduce latent states with a polynomial non-linear transition and fit them jointly with sparse dynamics via alternating minimization (extended Kalman filter/smoother for inference, elastic-net-regularized proximal gradient for learning), with an MDL-based coding scheme to choose polynomial order.
Key results
- At 50% noise, LaNoLem obtains 42 first-place wins in coefficient error (vs. 26 for MIOSR) and 71 of 71 wins in prediction error on dysts.
- At 5% noise, MIOSR is competitive with LaNoLem on coefficient error (33 vs 26 wins), but LaNoLem wins prediction in 66 of 71 systems.
- Ablation: removing the Frobenius regularizer or l1 term reduces accuracy in some noise regimes, with the full regularizer giving the most consistent prediction wins.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| introduce latent states and model temporal dependencies | LaNoLem versus SINDy baselines · dysts, 5% noise | Prediction error first-place count: 66 vs 4 | Helped |
| introduce latent states and model temporal dependencies | LaNoLem versus SINDy baselines · dysts, 25% and 50% noise | Prediction error first-place count: 71 vs 0 | Helped |
| introduce latent states and model temporal dependencies | LaNoLem versus MIOSR · dysts, 5% noise | Coefficient error first-place count: 26 vs 33 | Mixed |
| introduce latent states and model temporal dependencies | LaNoLem versus MIOSR · dysts, 25% and 50% noise | Coefficient error first-place count: 37 vs 25; 42 vs 26 | Helped |
| add the regularizer (ablation) | LaNoLem · dysts, 5%, 25%, and 50% noise | Coefficient error | Helped |
| add the l1 norm regularizer (ablation) | LaNoLem · dysts, high noise ratios | Coefficient error | Helped |
| add the Frobenius norm regularizer (ablation) | LaNoLem · dysts, high noise ratios | Prediction error | Helped |
Surprising findings
- At 5% noise, the MIOSR baseline wins more coefficient-error comparisons than LaNoLem.
- The ablation shows the l1-only and no-regularizer variants win on coefficient error at some noise levels, so the full regularizer is not uniformly best.
Limitations
- Restricted to polynomial non-linearities, a narrower class than SR or general nonlinear models.
- Alternating minimization does not guarantee global optimum and is sensitive to initialization (SVD used for real data).
- Gaussian approximations of moments (higher-order state noise assumed negligible) are approximate.
Experimental setup
Datasets: dysts chaotic systems benchmark (71 systems) Google Trends web search volumes for outdoor activity keywords
Benchmarks: dysts (71 chaotic systems)
Metrics: normalized coefficient error mean squared error (prediction) first-place win count critical difference diagram
Baselines: MIOSR SSR STLSQ rLDS
Builds on: SINDy linear dynamical systems extended Kalman filter minimum description length
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