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Uncertainty Quantification of Spatiotemporal Travel Demand with Probabilistic Graph Neural Networks

Qingyi Wang, Shenhao Wang, Dingyi Zhuang, Haris Koutsopoulos, Jinhua Zhao

TL;DR

The paper introduces Prob-GNN, a framework that incorporates probabilistic distribution assumptions into graph neural networks to quantify spatiotemporal uncertainty in travel-demand forecasting. By evaluating twelve architectures formed from six probabilistic assumptions and two deterministic encoders (GCN, GAT), the work demonstrates that distributional choices largely drive uncertainty calibration and predictive intervals, while architectural tweaks have smaller effects. Key findings show two-parameter distributions, such as truncated Gaussian and Laplace, provide superior uncertainty estimation and stability under domain shifts (e.g., COVID-19), with uncertainty patterns revealing higher variability in afternoon peaks and busy areas. The study advances resilient urban planning by linking uncertainty quantification to actionable patterns in ridership, and suggests broader exploration of probabilistic assumptions for diverse transportation contexts.

Abstract

Recent studies have significantly improved the prediction accuracy of travel demand using graph neural networks. However, these studies largely ignored uncertainty that inevitably exists in travel demand prediction. To fill this gap, this study proposes a framework of probabilistic graph neural networks (Prob-GNN) to quantify the spatiotemporal uncertainty of travel demand. This Prob-GNN framework is substantiated by deterministic and probabilistic assumptions, and empirically applied to the task of predicting the transit and ridesharing demand in Chicago. We found that the probabilistic assumptions (e.g. distribution tail, support) have a greater impact on uncertainty prediction than the deterministic ones (e.g. deep modules, depth). Among the family of Prob-GNNs, the GNNs with truncated Gaussian and Laplace distributions achieve the highest performance in transit and ridesharing data. Even under significant domain shifts, Prob-GNNs can predict the ridership uncertainty in a stable manner, when the models are trained on pre-COVID data and tested across multiple periods during and after the COVID-19 pandemic. Prob-GNNs also reveal the spatiotemporal pattern of uncertainty, which is concentrated on the afternoon peak hours and the areas with large travel volumes. Overall, our findings highlight the importance of incorporating randomness into deep learning for spatiotemporal ridership prediction. Future research should continue to investigate versatile probabilistic assumptions to capture behavioral randomness, and further develop methods to quantify uncertainty to build resilient cities.

Uncertainty Quantification of Spatiotemporal Travel Demand with Probabilistic Graph Neural Networks

TL;DR

The paper introduces Prob-GNN, a framework that incorporates probabilistic distribution assumptions into graph neural networks to quantify spatiotemporal uncertainty in travel-demand forecasting. By evaluating twelve architectures formed from six probabilistic assumptions and two deterministic encoders (GCN, GAT), the work demonstrates that distributional choices largely drive uncertainty calibration and predictive intervals, while architectural tweaks have smaller effects. Key findings show two-parameter distributions, such as truncated Gaussian and Laplace, provide superior uncertainty estimation and stability under domain shifts (e.g., COVID-19), with uncertainty patterns revealing higher variability in afternoon peaks and busy areas. The study advances resilient urban planning by linking uncertainty quantification to actionable patterns in ridership, and suggests broader exploration of probabilistic assumptions for diverse transportation contexts.

Abstract

Recent studies have significantly improved the prediction accuracy of travel demand using graph neural networks. However, these studies largely ignored uncertainty that inevitably exists in travel demand prediction. To fill this gap, this study proposes a framework of probabilistic graph neural networks (Prob-GNN) to quantify the spatiotemporal uncertainty of travel demand. This Prob-GNN framework is substantiated by deterministic and probabilistic assumptions, and empirically applied to the task of predicting the transit and ridesharing demand in Chicago. We found that the probabilistic assumptions (e.g. distribution tail, support) have a greater impact on uncertainty prediction than the deterministic ones (e.g. deep modules, depth). Among the family of Prob-GNNs, the GNNs with truncated Gaussian and Laplace distributions achieve the highest performance in transit and ridesharing data. Even under significant domain shifts, Prob-GNNs can predict the ridership uncertainty in a stable manner, when the models are trained on pre-COVID data and tested across multiple periods during and after the COVID-19 pandemic. Prob-GNNs also reveal the spatiotemporal pattern of uncertainty, which is concentrated on the afternoon peak hours and the areas with large travel volumes. Overall, our findings highlight the importance of incorporating randomness into deep learning for spatiotemporal ridership prediction. Future research should continue to investigate versatile probabilistic assumptions to capture behavioral randomness, and further develop methods to quantify uncertainty to build resilient cities.
Paper Structure (25 sections, 3 equations, 6 figures, 4 tables)

This paper contains 25 sections, 3 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Travel Demand Prediction with Deep Learning
  4. Uncertainty Quantification Methods
  5. Theoretical Framework
  6. Probabilistic Assumptions $\mathcal{G}$
  7. Deterministic Assumptions in $\mathcal{F}$
  8. GCN and GAT for Spatial Encoding
  9. LSTM for Temporal Encoding
  10. Specific Examples: Gaussian Distributions and Mean-Variance Estimation
  11. Evaluation
  12. Case Study
  13. Data
  14. Experiment Setup
  15. Results and Discussion
  16. ...and 10 more sections

Figures (6)

  • Figure 1: Spatial Autocorrelation (Moran's I)
  • Figure 2: Average Daily Ridership (T1: Immediately Before; T2: Stay-at-home; T3: Initial Recovery; T4: Steady Recovery)
  • Figure 3: Proposed model architecture. The model processes past demand information using spatial and temporal layers, with the addition of auxiliary information to produce estimates of distributions parameters
  • Figure 4: Training Curves. Top: GCN; Bottom: GAT
  • Figure 5: Calibration Plot of GCN Models (a) CTA rail and (b) Ridesharing. The line $y=x$ represents a perfectly-calibrated model. The calibration error (CE) corresponds to the areas between the $y=x$ line and the empirical quantiles.
  • ...and 1 more figures