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Structured Matrix Basis for Multivariate Time Series Forecasting with Interpretable Dynamics
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Proposes Sumba, a multivariate time series forecasting model that generates dynamic spatial structures as convex combinations of a learnable structured (SVD-factored, shared orthogonal factors) matrix basis, yielding lower-variance and interpretable graphs.
Problem
Forecasting multivariate time series whose spatial correlations evolve over time; existing two-stage dynamic graph methods produce unbounded, high-variance spatial structures that are sensitive to short input windows and lack interpretability.
Approach
Directly parameterize the spatial structure generation function with a learnable matrix basis of SVD-factored matrices sharing orthogonal factors (U, V via Clay map on SO(N)), combined by input-adaptive convex coefficients, with a bounded-norm guarantee on the output and low-rank approximation for efficiency.
Key results
- Sumba achieves best results in most settings on six datasets for horizons 3 and 6, with up to 8.5% improvement over the best baseline on Weather.
- Ablation: removing the dynamic GCN increases errors by up to 13.83% (MAE on Electricity/Weather/PEMS), and removing structure regularization (shared U,V) or orthogonal parameterization yields average drops of 4.95% and 3.05%.
- On Solar-Energy at horizons 12 and 24, improvements exceed 20% in MAE and 25% in MAPE over existing methods as claimed by the authors.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace Dynamic GCN with vanilla GCN (ablation) | Sumba · Electricity, Weather, and PEMS (F=3, 6) | MAE: 0.148 -> 0.153; 0.0587 -> 0.0693; 0.137 -> 0.145; 0.163 -> 0.169; 0.0704 -> 0.0817; 0.147 -> 0.158 | Hurt |
| use distinct Um, Vm for each basis matrix (ablation) | Sumba · Electricity, Weather, and PEMS (F=3, 6) | MAE: 0.148 -> 0.148; 0.0587 -> 0.0659; 0.137 -> 0.138; 0.163 -> 0.164; 0.0704 -> 0.0722; 0.147 -> 0.149 | Hurt |
| remove orthogonal parameterization on U and V (ablation) | Sumba · Electricity, Weather, and PEMS (F=3, 6) | MAE: 0.148 -> 0.151; 0.0587 -> 0.0633; 0.137 -> 0.142; 0.163 -> 0.167; 0.0704 -> 0.0706; 0.147 -> 0.151 | Hurt |
Surprising findings
- Sumba's alpha coefficients over time reveal two dominant spatial structures alternating with day and night on Solar-Energy, and a midnight-only structure on Electricity.
Limitations
- Number of basis matrices M is empirically tuned rather than set in a data-driven way.
- Long-term forecasting capability needs further enhancement, per the authors.
- Single-run variance is reported over only three random seeds.
Experimental setup
Datasets: Electricity Weather PEMS ETTh2 Traffic Solar-Energy
Benchmarks: Electricity Weather PEMS ETTh2 Traffic Solar-Energy
Metrics: MAE RMSE MAPE
Baselines: MICN ModernTCN PatchTST FEDformer Autoformer Reformer S4 MTGNN MegaCRN iTransformer CrossFormer Card
Builds on: TPGNN MTGNN Clay map iTransformer
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