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Online Time Series Forecasting with Theoretical Guarantees
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Proposes TOT, a theoretical framework for online time series forecasting under latent-driven distribution shifts, proving latent variables reduce Bayes risk and identifying latents from four adjacent observations, with a plug-in architecture evaluated on synthetic and six real benchmarks.
Problem
Online time series forecasting where unobserved latent variables cause distribution shifts that change the mapping from historical to future observations; existing methods lack theoretical guarantees on risk reduction and identifiability.
Approach
Proves that conditioning a forecaster on identifiable latent variables tightens Bayes risk, shows latent states and their causal dynamics are identifiable from four consecutive observations, and builds a VAE-style plug-in blueprint with independent noise estimators for latent and observed transitions plus an L1 sparsity penalty on the mixing procedure.
Key results
- On synthetic data, TOT achieves MCC of 0.9258 (Dataset A) and 0.9324 (Dataset B), versus IDOL at 0.3788 and 0.8593.
- Plugging TOT into OneNet reduces MSE on ETTh2 horizon 24 from 0.532 to 0.525 and on Exchange horizon 24 from 0.047 to 0.030.
- Online-T+TOT lowers ECL horizon-24 MSE from 11.339 to 4.381 and Exchange horizon-1 MSE from 0.113 to 0.010.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add TOT to the forecasting backbone | Online time-series forecasting backbones · ETTh2, ETTm1, WTH, ECL, Traffic, Exchange | MSE | Helped |
| Add estimated latent variables alongside observations | Synthetic forecasting · Synthetic A | MSE: 0.7005(0.0088) -> 0.5027(0.0045) | Helped |
| Add estimated latent variables alongside observations | Synthetic forecasting · Synthetic B | MSE: 0.6532(0.0062) -> 0.4395(0.0097) | Helped |
| Add TOT for latent representation identification | Synthetic data latent-variable identification · Synthetic A | MCC: 0.3788(0.0245) -> 0.9258(0.0034) | Helped |
| Remove the sparsity constraint L_s (ablation) | TOT · Exchange and ETTh2 | forecasting performance | Hurt |
| Remove the KL divergence L_KL (ablation) | TOT · Exchange and ETTh2 | forecasting performance | Hurt |
| Remove the reconstruction loss L_r (ablation) | TOT · Exchange and ETTh2 | forecasting performance | Hurt |
Surprising findings
- Forecasting with estimated latents approaches but does not match the upper bound using ground-truth latents on synthetic data, while the baseline using only observations is clearly worse.
- Gains are larger on OneNet than on LSTD or Proceed, which the authors attribute to LSTD and Proceed already modeling latent-driven shifts.
Limitations
- Theory assumes fully observed, discretely sampled data.
- Identifiability assumptions (injectivity, spectral uniqueness, bounded densities) are strong and hard to verify in practice.
- Real-world improvements are modest on some datasets, e.g. ECL versus LSTD, and some results are mixed.
Experimental setup
Datasets: ETTh2 ETTm1 Weather ECL Traffic Exchange synthetic data (Dataset A and B)
Benchmarks: ETTh2 ETTm1 Weather ECL Traffic Exchange synthetic simulation
Metrics: MSE MAE MCC
Baselines: LSTD proceed-T OneNet OneNet-T Online-T IDOL TDRL
Builds on: IDOL TDRL LSTD OneNet Proceed
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