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Non-stationary Diffusion For Probabilistic Time Series Forecasting
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Proposes NsDiff, a diffusion model for probabilistic time series forecasting that uses a location-scale noise model endpoint and an uncertainty-aware noise schedule to capture time-varying uncertainty.
Problem
Existing diffusion models for probabilistic time series forecasting assume constant (stationary) noise variance from the additive noise model, so they fail to capture non-stationary, time-varying uncertainty.
Approach
Replaces the fixed-variance additive noise endpoint N(0,I) or N(f(X),I) with a location-scale noise model endpoint N(f(X), g(X)) using a pretrained conditional mean and variance estimator, and derives an uncertainty-aware forward noise schedule that integrates time-varying variance into the diffusion process, with a quadratic-solved estimate of the data variance at inference.
Key results
- NsDiff achieves CRPS 0.392 on ETTh1 and QICE 1.470 vs. TMDM's 2.821, and best CRPS on 8 of 9 real datasets (Solar excepted).
- On synthetic quadratic-variance data, NsDiff reduces QICE from 11.404 (TMDM) to 1.336, an 88.3% improvement.
- Compared to TMDM, QICE is reduced by 66.3% on Traffic and 47.9% on ETTh1 per the paper.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NsDiff instead of TMDM | probabilistic forecasting · ETTh1 | QICE: 2.821 -> 1.470 | Helped |
| Use NsDiff instead of TMDM | probabilistic forecasting · Traffic | QICE: 10.676 -> 3.601 | Helped |
| Use NsDiff instead of the strongest baseline | probabilistic forecasting · Solar | CRPS: 0.293 -> 0.300 | Hurt |
| Add the LSNM assumption (variable conditional variance) (ablation) | NsDiff · ETTh1 | QICE: 2.821 -> 1.470 | Helped |
| Add the LSNM assumption (variable conditional variance) (ablation) | NsDiff · ETTh1 | CRPS: 0.452 -> 0.392 | Helped |
| Add the uncertainty-aware noise schedule (ablation) | NsDiff · ETTh1 | QICE: 3.184 -> 1.470 | Helped |
| Add the uncertainty-aware noise schedule (ablation) | NsDiff · ETTh1 | CRPS: 0.413 -> 0.392 | Helped |
| Train end-to-end rather than pretraining the estimators (ablation) | NsDiff · ETTh1 | CRPS: 0.3795 -> 0.3868 | No clear effect |
Surprising findings
- Assuming a perfect variance estimator (w/o UANS) improves CRPS over the unit-variance variant but yields worse QICE and higher variance, suggesting overfitting of the variance estimator.
- End-to-end training without pretraining reaches comparable CRPS to pretrained training on ETTh1, with only a slight degradation.
Limitations
- Memory cost is higher than some baselines (e.g., TMDM 221 MB training vs NsDiff 68 MB; NsDiff relies on heavy Non-stationary Transformer mean estimator).
- Solar dataset with low uncertainty variation shows NsDiff not best on CRPS.
- Requires pretraining of mean and variance estimators in a two-stage pipeline; end-to-end training is slightly harder to converge.
Experimental setup
Datasets: Electricity ILI ETTh1 ETTh2 ETTm1 ETTm2 ExchangeRate Traffic SolarEnergy synthetic LSNM datasets
Benchmarks: ETTh1 ETTh2 ETTm1 ETTm2 ECL EXG ILI Solar Traffic synthetic linear/quadratic variance datasets
Metrics: CRPS QICE MSE MAE
Baselines: TimeGrad CSDI TimeDiff TMDM DiffusionTS
Builds on: DDPM TMDM TimeGrad Non-stationary Transformer DLinear
Closest papers by meaning
- DiffPTS: Rethinking Diffusion ELBO for Probabilistic Time Series Forecasting (NeurIPS 2026 · similarity 0.93)
- Elucidated Rolling Diffusion Models for Probabilistic Forecasting of Complex Dynamics (NeurIPS 2025 · similarity 0.90)
- Probabilistic Time Series Modeling with Decomposable Denoising Diffusion Model (ICML 2024 · similarity 0.90)
- Auto-Regressive Moving Diffusion Models for Time Series Forecasting (AAAI 2025 · similarity 0.90)
- Latent Diffusion Transformer for Probabilistic Time Series Forecasting (AAAI 2024 · similarity 0.89)
- Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes (ICML 2024 · similarity 0.89)
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