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Inductive Graph Neural Networks for Spatiotemporal Kriging
Yuankai Wu, Dingyi Zhuang, Aurelie Labbe, Lijun Sun
TL;DR
Spatiotemporal kriging needs to recover signals for unsampled sensors, while existing scalable approaches are often transductive and require retraining for new sensors. IGNNK uses inductive GNN message passing with sampled subgraphs and reconstructed adjacency matrices, achieving strong performance across datasets and competitive transfer to an unseen dataset. The paper therefore supports GNN-based kriging for new graph structures and virtual sensors, while noting scope for richer temporal and multivariate modeling.
Problem
Spatiotemporal kriging must recover signals at unsampled locations, but existing scalable methods are often transductive and require full retraining for new sensors.
Method
IGNNK generates random subgraphs, reconstructs each sample's adjacency matrix, and reconstructs signals to learn inductive spatial message passing for dynamic network kriging.
Results
IGNNK consistently outperforms baseline models across five real-world spatiotemporal datasets and transfers competitively from METR-LA and SeData to unseen PeMS-Bay data.
Takeaways & Limitations
A trained IGNNK model can perform kriging at new locations without retraining, transfer to new graph structures, and support virtual sensors.
Takeaways & Limitations
The current approach assumes all time points in the short recovery window are correlated and represents the window as h features; future work targets multivariate data and richer temporal models.
Abstract
from arXiv · showhide
Time series forecasting and spatiotemporal kriging are the two most important tasks in spatiotemporal data analysis. Recent research on graph neural networks has made substantial progress in time series forecasting, while little attention has been paid to the kriging problem -- recovering signals for unsampled locations/sensors. Most existing scalable kriging methods (e.g., matrix/tensor completion) are transductive, and thus full retraining is required when we have a new sensor to interpolate. In this paper, we develop an Inductive Graph Neural Network Kriging (IGNNK) model to recover data for unsampled sensors on a network/graph structure. To generalize the effect of distance and reachability, we generate random subgraphs as samples and reconstruct the corresponding adjacency matrix for each sample. By reconstructing all signals on each sample subgraph, IGNNK can effectively learn the spatial message passing mechanism. Empirical results on several real-world spatiotemporal datasets demonstrate the effectiveness of our model. In addition, we also find that the learned model can be successfully transferred to the same type of kriging tasks on an unseen dataset. Our results show that: 1) GNN is an efficient and effective tool for spatial kriging; 2) inductive GNNs can be trained using dynamic adjacency matrices; 3) a trained model can be transferred to new graph structures and 4) IGNNK can be used to generate virtual sensors.
1 Introduction
Spatiotemporal kriging must recover signals at unsampled locations, but scalable network methods are often transductive and GNN research has focused more on forecasting. IGNNK addresses this gap with inductive graph message passing for dynamic networks and shows strong performance and transferability.
- Motivation: Spatiotemporal kriging interpolates signals at unsampled locations from observations at sampled locations during the same period.The resulting high-resolution data can support travel-time estimation and disaster evaluation while reducing sensor-network costs.
- Limitations of existing methods: Gaussian-process regression is computationally expensive at scale and difficult to adapt to networked systems with existing graph kernels.Graph-regularized matrix and tensor completion provide alternatives for large-scale network kriging, but the broader motivation highlights the need for more flexible methods.
- IGNNK motivation: GNNs characterize complex spatial dependencies through message passing and can generalize representations to unseen nodes or entirely new subgraphs.IGNNK applies this inductive capability to real-time spatiotemporal kriging on dynamic network structures, using location information to capture distance and reachability.
- Empirical results: IGNNK achieves the best performance on almost all datasets when compared with state-of-the-art kriging methods.The result suggests that the model generalizes spatiotemporal dynamics across sensor networks.
- Empirical results: Models trained on METR-LA and SeData provide very competitive performance on PeMS-Bay, despite never seeing that dataset during training.This experiment evaluates transferability across traffic-speed kriging datasets.
2 Related Work
Prior work spans matrix and tensor completion, graph convolutional models for spatiotemporal data, and inductive GNNs. However, existing inductive approaches largely focus on binary graph structures and do not fully incorporate distance effects in spatiotemporal settings.
- Matrix and tensor completion: Network spatiotemporal kriging can be formulated as matrix completion with several completely missing rows, using network structure as side information.Related methods include low-rank tensor models that capture variable dependencies or incorporate spatial autoregressive dynamics.
- GNNs for spatiotemporal data: Graph convolutional networks have been combined with recurrent and temporal-convolutional models to represent temporal dynamics on graphs.GCNs derive graph convolution operations through spectral filters and their first-order approximations.
- Inductive GNNs: Inductive GNNs learn node representations that generalize beyond the nodes used during training.Subsequent work applies masking, local subgraphs, and graph sampling to support reconstruction, unseen entities, transferability, and scalable training.
- Inductive GNNs: Existing inductive approaches focus on binary graph structures, while the effects of distance in spatiotemporal data are not fully considered.This distinction motivates methods that use richer spatial information than connectivity alone.
3 Methodology
IGNNK formulates network spatiotemporal kriging as inductive reconstruction of unsampled sensor signals. It trains on randomly sampled subgraphs with reconstructed adjacency matrices and uses a multi-layer DGCN to recover masked signals, including virtual sensors.
- 3.1 Problem description: Spatiotemporal kriging interpolates time-series signals at unsampled sensor locations from sampled sensors on an irregular network.The recovered signals represent traffic states at locations without sensors and can function as virtual sensors.
- 3.1 Problem description: IGNNK learns to generalize the message-passing mechanism from historical sampled-sensor data to estimate signals for new or unsampled nodes.The formulation allows sampled sensors to change because sensors may retire or new sensors may be introduced.
- 3.2 Subgraph signals and random masks: Random subgraph sampling selects observed and missing nodes, extracts a temporal window, constructs its adjacency matrix, and generates a mask for training.The sampling scheme is random rather than local, helping characterize distance effects in spatially connected graphs.
- 3.3 GNN architecture: IGNNK reconstructs the full subgraph signal matrix from masked inputs, treating the recovery window as h correlated features rather than modeling long-term temporal dependencies.This simplification is motivated by the relatively short real-time kriging window.
- 3.3 GNN architecture: The architecture uses a three-layer DGCN suited to directed networks with asymmetric distance and reachability relationships.An additional DGCN layer is added because masked nodes initially pass zero-valued signals to their neighbors, while the residual H1 connection preserves missing-sensor information.
- 3.4 Loss function and prediction: Training minimizes total reconstruction error on both observed and unseen nodes, then applies the trained model to a current network adjacency matrix with virtual-sensor inputs masked to zero.The resulting estimates provide the kriging outputs for virtual sensors.
4 Numerical Experiments
Experiments evaluate IGNNK across diverse spatiotemporal datasets, compare distance-based and binary adjacency representations, and test transferability and virtual-sensor generation. IGNNK generally performs strongly, while results show that spatial distance information and network-specific structure affect kriging quality.
- Experimental setup: Experiments use five real-world datasets spanning traffic speed, solar power, and precipitation, with approximately 25% of nodes held out as unsampled locations.The evaluation includes METR-LA, NREL, USHCN, SeData, and PeMS-Bay; training uses the first 70% of time points and testing the following 30%.
- Adjacency construction: The distance-based adjacency matrix weights sensor pairs by their distance, whereas SeData uses a binary matrix indicating neighboring sensors.The distance formulation uses pairwise sensor distance and a normalization factor; the binary formulation assigns 1 to neighbors and 0 otherwise.
- Kriging performance: IGNNK consistently outperforms baseline models on four datasets, providing the lowest RMSE and MAE for almost all datasets.The compared methods include kNN, OKriging, KPMF, and other kriging or matrix/tensor factorization baselines.
- Kriging performance: On USHCN, kNN and OKriging are comparable to IGNNK, with OKriging achieving lower MAE because dense sensors and smooth precipitation favor local spatial consistency.For SeData, binary connectivity omits pairwise distance, which is important for highway traffic dynamics; consequently, IGNNK is expected to be less powerful there.
- Kriging performance: IGNNK reconstructions are closer to ground truth than kNN and GLTL in a METR-LA evening-peak spatial visualization.The figure compares ground truth, IGNNK, kNN, and GLTL speed values at the same crowded time point.
- Transfer learning: IGNNK transfer from METR-LA to PeMS-Bay outperforms other baselines except a model trained on PeMS-Bay, whereas transfer from SeData performs worse than the mean predictor.The SeData transfer model has R2 below 0; Gaussian adjacency generally outperforms binary adjacency, especially outside peak hours.
- Virtual sensors: IGNNK generates virtual sensors whose outputs generally vary smoothly between observed sensors, but it misses abrupt changes caused by unmodeled on-ramp or off-ramp effects.The METR-LA dynamic graph uses road distance and does not incorporate those additional traffic-structure factors.
5 Conclusion
IGNNK provides an inductive framework for spatiotemporal kriging that can be applied to new locations without retraining. Experiments report consistent baseline improvements, transferability, virtual-sensor generation, and future extensions.
- IGNNK applies a trained model directly to new locations of interest without retraining.
- IGNNK consistently outperforms other baseline models across five real-world spatiotemporal datasets.
- IGNNK demonstrates transferability in a traffic-data kriging task and can generate virtual sensors from surrounding sensor data.
- Future work includes multivariate spatiotemporal tensors, stronger temporal models, and integrated forecasting for existing and virtual sensors.