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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
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Proposes Morphing-Flow (MoF), a spline-based invertible transform (Flow) plus a test-time-trained adaptation layer (Morph) that normalizes fat-tailed, shifting time series inputs for linear and Mamba forecasting backbones.
Problem
Long-term multivariate time series forecasting is destabilized by leptokurtic, fat-tailed distributions that persist even after stationarization, and by distribution shift between training and test data.
Approach
A channel-wise monotonic piecewise-linear spline with invertible forward/backward maps reduces excess kurtosis, and a test-time-trained layer modulates the spline bin parameters per instance to counter distribution drift; the module is wrapped around the backbone and inverted at the output.
Key results
- MoF-Mamba achieves the best average MSE on 8 datasets, outperforming the closest competitor by 6.3% on average and by 21.7% on Exchange.
- MoF-Linear reaches an average MSE of 0.317, matching GLinear and LiNo-level results with a linear backbone.
- Adding MoF to DLinear, Informer, and Autoformer reduces their average MSE (e.g., Informer 0.816 to 0.507, Autoformer 0.584 to 0.467).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MoF-Mamba instead of the closest baseline | Mamba time-series forecasting · Eight datasets, averaged across 4 horizons | MSE: 6.3% | Helped |
| Use MoF-Mamba instead of the closest competitor | Mamba time-series forecasting · Exchange Rate | MSE: 21.7% | Helped |
| Add the test-time-trained TTT module (ablation) | MoF · Short-term prediction windows | forecasting error: up to a 5% error reduction | Helped |
| Add the test-time-trained TTT module (ablation) | MoF · Long-term prediction windows | forecasting error | Hurt |
| Vary spline bin count B with T = 6.0 (ablation) | MoF · ETTh1, ETTm1, Weather | MSE and MAE: 0.23%; 0.67%; 1.73%; 0.22%; 0.71%; 1.32% | No clear effect |
| Vary tail-size T with B = 24 (ablation) | MoF · ETTh1, ETTm1, Weather | MSE and MAE: 1.40%; 3.54%; 3.02%; 0.69%; 2.83%; 2.48% | No clear effect |
| Add MoF to Autoformer | Autoformer · ETT (all) | MSE: 0.491 -> 0.455 | Helped |
| Add MoF to Autoformer | Autoformer · Traffic | MSE: 0.648 -> 0.677 | Hurt |
| Add MoF to Informer | Informer · Exchange Rate | MSE: 1.676 -> 0.694 | Helped |
| Add MoF to DLinear | DLinear · Exchange Rate | MSE: 0.390 -> 0.402 | Hurt |
| Add MoF to DLinear | DLinear · Average across datasets | MSE: 0.316 -> 0.310 | Helped |
| Use MoF normalization instead of instance-wise z-score, RevIN, or PatchTST normalization | PatchTST · ETTm1 | gradient variance: 1.12e-07 -> 5.38e-08 | Helped |
Surprising findings
- Adding MoF to DLinear slightly worsens Exchange-rate error (+3.16%), so gains are not uniform across datasets.
- Test-time training helps short horizons substantially but its benefit shrinks for long-term predictions.
Limitations
- Gains vary by dataset and horizon; TTT benefit is minor for long horizons.
- Fat-tail motivation relies on a synthetic clipped-Cauchy experiment and kurtosis analysis rather than a formal theory.
- Hyperparameters were unified across datasets, which may differ from original baseline reports.
Experimental setup
Datasets: ETTh1 ETTh2 ETTm1 ETTm2 Electricity Traffic Weather Exchange Rate
Benchmarks: ETTh1 ETTh2 ETTm1 ETTm2 Electricity Traffic Weather Exchange Rate
Metrics: MSE MAE excess kurtosis relative MSE
Baselines: GLinear LiNo TimeMachine CATS iTransformer PatchTST DLinear Informer Autoformer FAN SAN DishTS
Builds on: Test-Time Training (Sun et al., 2024) LTSF-Linear PatchTST Mamba RevIN
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