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Learning Optimal Projection for Forecast Reconciliation of Hierarchical Time Series
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Learns an oblique projection matrix for coherent hierarchical forecast reconciliation end-to-end inside a neural forecaster (TimesNet/Autoformer), comparing generalized Euclidean and general idempotent projections against classical reconciliation baselines.
Problem
Forecasting hierarchical time series requires coherent forecasts that add up across the hierarchy; existing projection-based reconciliation uses pre-specified projection matrices (orthogonal, OLS, MinT) that either ignore error structure or are hard to estimate.
Approach
Model the reconciliation projection as a learnable structured dense layer inside the neural forecaster, trained end-to-end: a generalized Euclidean projection with a symmetric positive-definite W, or a general oblique projection enforced to be idempotent (PS=I) via a Lagrange penalty.
Key results
- On Labour, the AF-GenProj RMSE is 21.96 vs. 24.61 for the best AF MinT-var baseline (10 runs).
- Across datasets, the proposed end-to-end methods (Eucl, GenProj) achieve the lowest RMSE within each backbone, e.g., TimesNet-GenProj 10.12 on Labour vs. 10.95 for TimesNet-MinT-var.
- Learned oblique projections have higher spectral norms on harder-to-predict datasets, meaning they assign larger weights to difficult series.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use learnable general oblique projection instead of the strongest listed baseline | TimesNet · Labour | RMSE: 10.51±1.42 -> 10.12±1.35 | Helped |
| use learnable general oblique projection instead of the strongest listed baseline | Autoformer · Traffic | RMSE: 0.79±0.25 -> 0.71±0.24 | Helped |
| use learnable general oblique projection instead of the strongest listed baseline | TimesNet · Wiki | RMSE: 617.94±51.42 -> 613.95±49.37 | Helped |
| use learnable general oblique projection instead of the strongest listed baseline | Autoformer · Tourism | RMSE: 416.49±18.99 -> 413.18±17.04 | Helped |
| use learnable general oblique projection instead of the strongest listed baseline | TimesNet · TourismLarge | RMSE: 112.26±3.76 -> 110.82±3.71 | Helped |
| use general oblique projection instead of generalized Euclidean projection (ablation) | TimesNet · Labour | RMSE: 10.22±1.67 -> 10.12±1.35 | Helped |
| add the idempotence constraint to the projection (ablation) | TimesNet · Labour | RMSE: 10.39±1.37 -> 10.12±1.35 | Helped |
| use general oblique projection instead of generalized Euclidean projection (ablation) | Autoformer · Traffic | RMSE: 0.71±0.11 -> 0.71±0.24 | No clear effect |
| add the idempotence constraint to the projection (ablation) | Autoformer · Traffic | RMSE: 0.73±0.18 -> 0.71±0.24 | Helped |
| use general oblique projection instead of generalized Euclidean projection (ablation) | TimesNet · Wiki | RMSE: 615.41±48.95 -> 613.95±49.37 | Helped |
| add the idempotence constraint to the projection (ablation) | TimesNet · Wiki | RMSE: 622.95±46.87 -> 613.95±49.37 | Helped |
| use general oblique projection instead of generalized Euclidean projection (ablation) | Autoformer · Tourism | RMSE: 415.37±14.78 -> 413.18±17.04 | Helped |
Surprising findings
- Relaxing the idempotence constraint (Gen, lambda=0) still performs competitively, though it loses the unbiasedness guarantee.
- Naive and SeasonalNaive baselines are competitive on Traffic and Labour datasets.
Limitations
- Generalized Euclidean projection imposes a specific structure on P that may not always be required.
- General oblique projection enforces idempotence only approximately via a Lagrange penalty (lambda=10^4), so projection certification depends on that choice.
- Unconstrained relaxed variant (Gen) loses unbiasedness guarantees.
Experimental setup
Datasets: Labour Traffic Wiki Tourism TourismLarge
Benchmarks: Labour Traffic Wiki Tourism TourismLarge
Metrics: RMSE wMAPE CRPS
Baselines: Naive SeasonalNaive Bottom-Up Top-Down MinT-ols MinT-shr MinT-var ERM Hier-E2E
Builds on: Panagiotelis et al. (2021) geometric projection view of reconciliation MinT Rangapuram et al. (2021) end-to-end hierarchical forecasting TimesNet Autoformer
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