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Modeling spatio-temporal locality in multi-step forecasting of geo-referenced time series

Annunziata D'Aversa, Gianvito Pio, Michelangelo Ceci

arXiv:2608.25698v1cs.LG

TL;DR

Forecasting from distributed sensors must account for spatial dependencies, yet existing methods model spatial information globally rather than capturing spatio-temporal locality. SPALT uses linear model trees to model that locality, and across datasets and horizons it almost always outperformed the considered competitors.

  • Problem

    For multi-step forecasting from distributed sensors, existing methods model spatial information globally rather than capturing spatio-temporal locality.

    Source: §2.2 · p. 6

  • Method

    SPALT extends linear model trees to selectively add spatial features for similar-trend sensor series and applies locality-aware reduced-error pruning.

    Source: §3.2 · p. 8, §5 · p. 29

  • Results

    Across datasets and forecast horizons, SPALT almost always outperformed the considered competitors, with comparable results in a few exceptions.

    Source: §4.3 · p. 17

  • Takeaways & Limitations

    The results support SPALT as a state-of-the-art approach for multi-step predictions from geodistributed sensors that exploit spatio-temporal locality.

    Source: §4.3 · p. 29

    Evaluation is confined to the energy domain; effectiveness in other application domains remains for future evaluation.

    Source: §5 · p. 29

Abstract

from arXiv · show

Forecasting future measurements from geographically distributed sensors is essential across many domains. However, the spatial distribution of these sensors raises multiple challenges, primarily due to spatial autocorrelation phenomena, that introduce inter-dependencies among nearby locations, that cannot therefore be treated independently. While some existing approaches can capture such phenomena, they generally model the spatial dimension globally across all locations. On the other hand, the method we propose in this paper, called SPALT, focuses on capturing spatial relationships among time series with similar trends, even if they occur at different times, thus modeling the spatio-temporal locality. SPALT leverages linear model trees, which allow us to consider the spatial autocorrelation locally: during the tree-building process, the adopted heuristics group time series exhibiting similar trends into the same node, on which additional features considering the spatial dimension are selectively injected. Additionally, we propose a new pruning strategy, based on Reduced Error Pruning, that also considers the spatio-temporal locality during the tree simplification. Designed for a multi-step setting, SPALT provides forecasts for multiple future time steps across multiple sensors simultaneously. The characteristics exhibited by SPALT can provide significant benefits in different domains, where measurements come from distributed sensors. In this paper, we focus on data produced by sensors located in multiple renewable power plants measuring their energy production at regular, short intervals. Experiments on 3 real-world datasets demonstrate the effectiveness of SPALT in forecasting the production of energy at different time horizons, and its superior performance in comparison with tree-based models and state-of-the-art neural networks that incorporate both temporal and spatial dimensions.

1 Introduction

The paper proposes a linear model tree for multi-step forecasting that captures spatial relationships among time series with similar trends and adds a pruning strategy based on spatio-temporal locality. Its evaluation focuses on forecasting renewable power production at fine temporal resolutions.

  • The proposed model captures spatial relationships among instances with similar trends while representing both linear and non-linear dependencies, rather than modeling spatial information globally.Its approach is designed to capture spatio-temporal locality within a linear model tree.Source: §1 · pp. 2–3
  • The proposed pruning strategy adapts Reduced Error Pruning to spatio-temporal locality to reduce model complexity and help avoid overfitting.The strategy is motivated by preferring simpler models when they achieve similar generalization error.Source: §1 · p. 3
  • The target application is fine-resolution forecasting of renewable power production for 6, 12, or 18 future time steps, using intervals from 5 to 15 minutes.The forecasts are intended to support both network planning and short-term actions such as real-time load balancing.Source: §1 · p. 3
  • The paper reports a time-complexity analysis showing that modeling spatio-temporal locality does not alter the underlying learning algorithms’ time complexity.Source: §1 · p. 3
  • The evaluation covers three real-world datasets and includes an ablation study of spatio-temporal locality and the proposed pruning strategy.The ablation assesses forecasting errors and model complexity.Source: §1 · pp. 3–4

2 Related work

Related work spans spatio-temporal forecasting across domains and energy-sector applications, but the reviewed methods generally do not model spatio-temporal locality across locations.

2.1 Spatio-temporal forecasting of multiple time series

  • Earlier approaches combine CNNs and LSTMs or graph networks to model spatial and temporal dependencies, including DMVST-Net for taxi demand and Graph WaveNet for traffic forecasting.DMVST-Net also represents location similarity through a semantic graph view, while Graph WaveNet uses dilated temporal convolutions and a self-adaptive adjacency matrix.Source: §2.1 · p. 4
  • D2STGNN (Shao et al., 2022), STAEformer, and HSTGNN are identified as approaches aiming to model spatio-temporal locality.The paper names these methods as conceptually close state-of-the-art competitors.Source: §2.1 · p. 5

2.2 Forecasting methods in the energy sector

  • Ceci et al. (2017) and D’Aversa et al. (2022) jointly forecast 24 hourly photovoltaic-production values or 12 monthly energy-consumption values, augmenting features with LISA or PCNM spatial indicators.Both works use multi-step settings that return the future values simultaneously.Source: §2.2 · p. 5
  • Energy methods include sparse spatially linked VARs, distance-filtered mutual-information graphs with LSTMs, spatio-temporal GCNs, MTGNN, and GAP-LSTM, but the reviewed methods model spatial information globally rather than locally.The cited applications include wind power or speed forecasting and other domains; VAR models cannot capture complex nonlinear dependencies.Source: §2.2 · pp. 5–6

3 The proposed method

SPALT uses linear model trees to capture linear and nonlinear dependencies while modeling spatio-temporal locality, and applies locality-aware pruning to simplify the learned trees.

3.1 Background

  • For multiple geo-referenced series, SPALT uses the MIMO forecasting strategy to predict a vector of future values jointly from historical observations.The method targets multiple future time steps across locations, rather than applying a single-step model repeatedly.Source: §3.1 · pp. 6–7, §3.2 · p. 8

3.2 SPALT

  • SPALT’s leaves use multivariate linear functions to predict h future values from w historical values, combining linear models’ strengths with trees’ ability to capture nonlinear dependencies.Leaf models are fitted with Ordinary Least Squares, subject to a minimum training-instance portion set by msp.Source: §3.2 · p. 8
  • Quantile binning limits candidate split thresholds, while SPALT selects splits by minimizing a weighted child-node MSE that considers all h future time steps.For each input feature, b quantiles provide candidate thresholds instead of evaluating every possible threshold.Source: §3.2 · pp. 8–9

3.3 Capturing spatio-temporal locality in SPALT

  • Within each leaf, SPALT augments historical features with proximity-weighted observations from other locations at the same time, retaining the augmented model only when validation error improves.If a leaf contains instances from only one location, no locality features are added; this selective choice avoids unnecessary complexity and can reduce overfitting.Source: §3.3 · pp. 10–11

3.4 Reduced error pruning based on spatio-temporal locality

  • SPALT’s locality-aware Reduced Error Pruning compares unpruned and pruned trees, including a pruned version with locality features, and prunes when validation performance improves.Pruning proceeds bottom-up and reuses the validation set used to assess locality features.Source: §3.4 · pp. 11–12

3.5 Time complexity analysis

  • SPALT’s locality and pruning computations preserve the asymptotic time complexity of building a linear model tree without locality features or pruning.The stated overall bound is O((1/msp) · m · n · max(m, log n)).Source: §3.5 · p. 14

4 Experiments

Experiments evaluate SPALT across three georeferenced energy datasets, multiple forecast horizons, and predictive-error and runtime measures. Overall, SPALT generally outperforms the competitors, while ablations indicate that spatio-temporal locality contributes to accuracy and pruning reduces model complexity.

4.1 Datasets

  • The three datasets cover UK photovoltaic plants, a wind farm, and commercial wind plants, with 12 historical measurements as input and geographic coordinates available for spatial modeling.UKPV uses 332 plants at 5-minute intervals; SDWPF covers 134 turbines at 10-minute intervals; WPP covers 60 plants at 15-minute intervals.Source: §4.1 · pp. 15–16

4.2 Competitors

  • Competitors comprise default-parameter scikit-learn regression trees, random forests, and XGBoost, with PCNM spatial-feature and differenced-data variants, plus four multi-step spatio-temporal neural architectures.The neural models are MTGNN (Wu et al., 2020), D2STGNN (Shao et al., 2022), STAEformer, and HSTGNN; validation data monitored loss and early stopping.Source: §4.2 · p. 16

4.3 Results and discussion

  • Higher horizons worsen average results and increase standard deviations, but SPALT’s RSE remains clearly below 1.0 at h = 18, unlike several competitors.RT, RT+PCNM, MTGNN, and D2STGNN approach RSE values of 1.0; HSTGNN and STAEformer do so on SDWPF.Source: §4.3 · p. 17
  • SPALT almost always outperforms the competitors; on WPP, STAEformer has significantly lower MAE and RMSE, while SPALT and HSTGNN achieve the best RSE.The WPP RSE comparison indicates that SPALT performs better than STAEformer relative to the location-specific mean baseline.Source: §4.3 · p. 17, Table 2 · p. 18, Table 3 · p. 20
  • SPALT ranks first by average RMSE across all horizons, with its advantage over XGB and XGB+PCNM statistically significant at α = 0.001 for every horizon.Its average-rank gap is close to 3 at h = 6 and close to 2 at h = 12 and h = 18.Source: §4.3 · pp. 17, 19, Figure 8 · p. 23, Figure 9 · p. 23
  • SPALT runs faster than RF and the neural architectures but slower than RT and XGB, whose predictive errors are significantly worse; its runtime scales nearly linearly with instance count.The comparison covers the considered datasets and forecast horizons.Source: §4.3 · p. 19, Table 6 · p. 24
  • Removing spatio-temporal locality significantly worsens every error measure and increases node counts, while pruning reduces nodes by about 23%–35% but does not significantly improve MAE, RMSE, or RSE.The locality comparison is significant at α = 0.001 across all measures; locality features were introduced in roughly 64%–77% of leaf nodes.Source: §4.3 · p. 21, Table 7 · p. 25, Table 10 · p. 28
  • Changing the distance weighting or quantile-bin count does not significantly affect results, suggesting the default configuration balances computational effort and predictive performance.The weighting comparison and tests of b = 10 and b = 50 against the default b = 25 use h = 12.Source: §4.3 · pp. 23, 28, 29, Table 11 · p. 28, Table 12 · p. 28

5 Conclusion

SPALT extends linear model trees to capture spatio-temporal locality in multi-step forecasting, and experiments on three renewable-energy datasets showed its effectiveness and benefits from locality-aware modeling and pruning. Future work will test other domains and ensemble extensions.

5 Conclusion

  • SPALT uses linear model trees for multi-step forecasting and captures spatio-temporal locality by enriching leaf models with historical trends from other locations in the same node.Its Reduced Error Pruning strategy incorporates spatio-temporal locality into tree simplification.Source: §5 · p. 29
  • Across three renewable-energy datasets, SPALT was effective against tree-based and spatio-temporal neural models; locality significantly improved results, while pruning reduced complexity without hurting predictive accuracy.The ablation study found statistically significant advantages from considering spatio-temporal locality and a significant complexity reduction from the pruning strategy.Source: §5 · p. 29
  • Future work will evaluate SPALT beyond energy applications and explore ensembles such as forests and extremely randomized linear model trees.Source: §5 · p. 29

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