Research map / Time series / Long-term forecasting
Rethinking Fourier Transform from A Basis Functions Perspective for Long-term Time Series Forecasting
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Proposes Fourier Basis Mapping (FBM), which embeds the DFT as cosine/sine basis expansion to create time-frequency features that plug into linear, MLP, and PatchTST-style networks for long-term time series forecasting.
Problem
Existing Fourier-based LTSF methods treat real and imaginary frequency components as plain coefficients in frequency space, which causes inconsistent starting cycles and inconsistent series length issues, so frequency meanings are ambiguous and time-frequency relations are missed.
Approach
Reinterpret DFT real and imaginary parts as cosine and sine basis coefficients, then multiply them with orthogonal cosine and sine basis matrices to form time-frequency features that keep temporal structure; a simple mapping network (linear, 3-layer MLP, or PatchTST-like) then maps these features to the forecast.
Key results
- FBM-L reduces average MSE/MAE on eight LTSF datasets from 0.333/0.349 (NLinear) to 0.323/0.339, with improvements across all datasets and horizons.
- FBM-NL and FBM-NP achieve best average MSE among compared methods (0.325 and 0.321 vs. 0.330 for PatchTST), with FBM-NP reported as best average MSE (0.321).
- On short-term PEMS forecasting (horizon 12), FBM variants achieve the best results in most cases, e.g., PEMS04 MSE 0.071 for FBM-NL and FBM-NP versus 0.075 for PatchTST.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Fourier basis expansion to a vanilla linear network | NLinear · Eight LTSF datasets, four horizons (Average) | MSE: 0.333 -> 0.323 | Helped |
| Add Fourier basis expansion to a vanilla linear network | NLinear · Eight LTSF datasets, four horizons (Average) | MAE: 0.349 -> 0.339 | Helped |
| Add Fourier basis expansion before patching | PatchTST · Eight LTSF datasets, four horizons | MSE | Helped |
| Use Fourier basis expansion with a three-layer MLP mapping | TimeMixer · Eight LTSF datasets, four horizons | MSE | Helped |
| Replace the linear mapping with a three-layer nonlinear MLP mapping (ablation) | FBM · Eight LTSF datasets, four horizons | MSE: 0.323 vs 0.325 | Mixed |
| Replace the linear mapping with a PatchTST-style nonlinear mapping (ablation) | FBM · Eight LTSF datasets, four horizons | MSE: 0.323 vs 0.321 | Mixed |
| Use a PatchTST-style mapping instead of a three-layer MLP mapping (ablation) | FBM · Eight LTSF datasets, four horizons | MSE: 0.325 vs 0.321 | No clear effect |
Surprising findings
- Look-back window of 336 yields better results than 96 for most baselines, suggesting prior work's standard window undersells baselines.
- The authors state that the frequency spectrum alone is insufficient because changes in granularity or series length alter the meaning of frequency components.
Limitations
- FBM is better suited to series with seasonality and periodicity than to monotonic trends (stated in the checklist).
- Performance differences between FBM-L and FBM-NL depend on dataset noise and granularity, so the best variant varies by dataset.
- Hyperparameters are fixed rather than tuned, and results are single-seed without error bars.
Experimental setup
Datasets: ETTh1 ETTh2 ETTm1 ETTm2 Electricity Traffic Weather Exchange PEMS03 PEMS04 PEMS07 PEMS08
Benchmarks: ETTh1 ETTh2 ETTm1 ETTm2 Electricity Traffic Weather Exchange PEMS03 PEMS04 PEMS07 PEMS08
Metrics: MSE MAE MAPE
Baselines: NLinear PatchTST iTransformer TimeMixer N-BEATS CrossGNN FEDformer FreTS FITS FiLM BasisFormer TimesNet
Builds on: NLinear PatchTST N-BEATS FreTS FITS
Closest papers by meaning
- FilterNet: Harnessing Frequency Filters for Time Series Forecasting (NeurIPS 2024 · similarity 0.87)
- Affirm: Interactive Mamba with Adaptive Fourier Filters for Long-term Time Series Forecasting (AAAI 2025 · similarity 0.87)
- CFPT: Empowering Time Series Forecasting through Cross-Frequency Interaction and Periodic-Aware Timestamp Modeling (ICML 2025 · similarity 0.87)
- Revisiting Attention for Multivariate Time Series Forecasting (AAAI 2025 · similarity 0.87)
- Towards a General Time Series Forecasting Model with Unified Representation and Adaptive Transfer (ICML 2025 · similarity 0.86)
- TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecasting (ICLR 2025 · similarity 0.86)
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