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Graph-based Time Series Clustering for End-to-End Hierarchical Forecasting
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Proposes a graph-based pyramidal architecture that learns hierarchical time series clusters end-to-end via trainable graph pooling and a differentiable forecast reconciliation layer, improving multistep forecasting on sensor/energy benchmarks.
Problem
Hierarchical time series forecasting when the hierarchy is not given a priori, combining relational (graph) and hierarchical (aggregation) inductive biases in deep spatiotemporal forecasting.
Approach
Model hierarchy levels as a pyramidal graph whose cluster assignments are learned end-to-end via Gumbel-softmax straight-through selection matrices, trained to forecast both bottom series and learned aggregates, with a min-cut regularizer on the graph and a differentiable projection onto coherent forecasts.
Key results
- On METR-LA, the GRU-based HGP-TTS (G) variant reaches MAE 3.007 vs. 3.071 for GConv-TTS and 3.012 for Diff-TTS baseline (5 runs).
- On AQI, HGP-TTS (G) achieves MAE 12.02 vs. 12.07 for the best baseline Gated-TTS, a small margin.
- The tuned variant HGP-TTS (T) achieves the best MAE at all horizons on METR-LA (2.68/3.02/3.40) and at 60 min on PeMS-BAY (1.87), while SGP is best on PeMS-BAY at 15 and 30 min.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use learned hierarchical forecasting with graph convolution | HIGT-TTS versus Diff-TTS · Metr-LA | MAE: 3.012 { $\pm$ .005} -> 3.007 { $\pm$ .009} | Helped |
| Use learned hierarchical forecasting with diffusion convolution | HIGT-TTS versus Diff-TTS · PeMS-Bay | MAE: 1.569 { $\pm$ .004} -> 1.566 { $\pm$ .005} | Helped |
| Use learned hierarchical forecasting with gated message passing | HIGT-TTS versus GUNet-TTS · CER | MAE: 4.08 { $\pm$ .02} -> 4.05 { $\pm$ .01} | Helped |
| Use learned hierarchical forecasting with gated message passing | HIGT-TTS versus Gated-TTS · AQI | MAE: \underline{12.07 { $\pm$ .02}} -> 12.02 { $\pm$ .04} | Helped |
| Use tuned HIGT architecture | Traffic forecasting · LA, 15 min | MAE: \underline{2.69 { $\pm$ .00}} -> 2.68 { $\pm$ .01} | Helped |
| Use tuned HIGT architecture | Traffic forecasting · LA, 30 min | MAE: \underline{3.05 { $\pm$ .00}} -> 3.02 { $\pm$ .01} | Helped |
| Use tuned HIGT architecture | Traffic forecasting · LA, 60 min | MAE: \underline{3.44 { $\pm$ .01}} -> 3.40 { $\pm$ .01} | Helped |
| Use tuned HIGT architecture | Traffic forecasting · PeMS-Bay, 15 min | MAE: \textbf{1.30 { $\pm$ .00}} -> \underline{1.31 { $\pm$ .00}} | Hurt |
| Use tuned HIGT architecture | Traffic forecasting · PeMS-Bay, 30 min | MAE: \textbf{1.60 { $\pm$ .00}} -> \underline{1.61 { $\pm$ .00}} | Hurt |
| Use tuned HIGT architecture | Traffic forecasting · PeMS-Bay, 60 min | MAE: \underline{1.88 { $\pm$ .00}} -> 1.87 { $\pm$ .00} | Helped |
| Add relational message passing (ablation) | HIGT (T) · LA, 15/30/60 min | MAE: 2.80 { $\pm$ .01} -> 2.68 { $\pm$ .01}; 3.14 { $\pm$ .01} -> 3.02 { $\pm$ .01}; 3.47 { $\pm$ .02} -> 3.40 { $\pm$ .01} | Helped |
| Add relational message passing (ablation) | HIGT (T) · PeMS-Bay, 15/30/60 min | MAE: 1.32 { $\pm$ .00} -> 1.31 { $\pm$ .00}; 1.63 { $\pm$ .00} -> 1.61 { $\pm$ .00}; 1.88 { $\pm$ .01} -> 1.87 { $\pm$ .00} | Helped |
Surprising findings
- Removing the hierarchical propagation on METR-LA gives MAE 2.68 at 15 min, matching the full tuned model, suggesting the hierarchical propagation gain is small in that setting.
Limitations
- Reconciliation inverse of QQ^T scales as O(M^3) memory and O(M^2) time, limiting very large collections.
- Evaluated only on univariate, homogeneous sensor datasets with static graphs; extensions to multivariate or irregular series are left to future work.
- Gains over strong baselines are often small (e.g., within about 0.05 MAE on several settings).
Experimental setup
Datasets: METR-LA PeMS-BAY CER AQI
Benchmarks: METR-LA PeMS-BAY CER AQI
Metrics: MAE MRE
Baselines: RNN FC-RNN GConv-TTS Diff-TTS Gated-TTS GUNet-TTS DCRNN GWNet Gated-GN SGP
Builds on: Select-Reduce-Connect framework min-cut pooling Gumbel softmax Graph WaveNet-style spatiotemporal GNNs MinT forecast reconciliation
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