Uncertainty Intervals for Graph-based Spatio-Temporal Traffic Prediction

• Mean-only traffic predictions are harder to interpret, whereas confidence intervals support road-maintenance scheduling and low-delay route planning.Source: §1 · p. 1
• Monte Carlo Dropout requires B stochastic forward passes and assumes multivariate Gaussian uncertainty, despite traffic distributions often being skewed.At maximum measured speed, for example, decreases may be more likely than increases.Source: §1 · p. 1
• Quantile Graph Wavenet estimates non-Gaussian, asymmetric uncertainty by conditioning a Graph Wavenet model on requested quantiles.It replaces mean-prediction optimization with quantile-loss training to estimate density at specific points.Source: §1 · p. 1
• The proposed method models asymmetric and skewed densities with one forward pass for each requested quantile instead of estimating variance through repeated sampling.Source: §1 · p. 1

• Quantile regression estimates the quantile function at chosen points, equivalent to evaluating the distribution’s inverse cumulative distribution function.Source: §2 · p. 2
• A neural network approximation of the quantile function reparameterizes a random sample from a uniform distribution into an approximation of the target distribution.Source: §2 · p. 2
• For scalar distribution Z, the inverse-CDF value F^-1_Z(τ) minimizes expected quantile loss for quantile τ.Source: §2 · p. 2
• The 1-Wasserstein projection Q is illustrated as minimizing transport error onto uniformly weighted Dirac distributions.Source: Figure 1 · p. 2, §2 · p. 2

3 Graph WaveNet

• Graph WaveNet applies temporal causal convolutions and graph diffusion convolutions throughout a spatio-temporal architecture for traffic sensors on a road network.Source: §3 · p. 2

3.1 Temporal Convolutions

• The temporal module gates two parallel TCN layers, using tanh and sigmoid activations to control information flow.Source: §3.1 · p. 2
• Dilated causal convolutions select inputs at every k-step interval, while receptive fields grow exponentially with layer count and dilation factor.Source: Figure 2 · p. 2, §3.1 · p. 2

3.2 Spatial Graph Diffusion Convolution

• Graph diffusion convolution uses forward and reverse transition matrices and sparse graph neighborhoods rather than a full graph Laplacian.Its update has complexity O(K|E|) and accommodates non-symmetric edges.Source: §3.2 · p. 3

3.3 Loss Function

• The training objective is mean absolute error over Q prediction timesteps, N locations, and C measurements.Source: §3.3 · p. 3

• Quantile Graph WaveNet conditions its input on quantile τ and trains with quantile loss to implicitly predict the CDF instead of the mean.Source: §4 · p. 3
• The Huber quantile loss scales with error magnitude within threshold κ, addressing the original loss’s sign-and-quantile-only gradient scaling.The experiments report κ = 0.05 as effective.Source: §4 · p. 3

• METR-LA contains 207 loop detectors and is split into training, validation, and test sections with ratios 7:1:2.Source: §5 · p. 3
• Traffic4cast provides Berlin, Istanbul, and Moscow grids at 495x436 pixels, with 181 training days and 18 validation days.Source: §5 · p. 3
• Because Traffic4cast grids contain many infrequently reporting pixels, the method samples sufficiently observed pixels before constructing the traffic graph.Source: §5 · pp. 3–4
• Equidistant sampling from the remaining pixels uses d = 1200m and yields approximately 1300 nodes in Berlin, with average speeds measured every 5 minutes.Source: §5 · p. 4, Figure 3 · p. 4

6 Results

• The evaluation compares uncertainty estimates and calibration with MC Dropout, while also reporting mean prediction accuracy against previous methods.Source: §6 · p. 4
• Quantile Graph WaveNet generates 0.9 confidence intervals whose boundaries or means are conditioned on distinct τ values, enabling efficient skewed and asymmetric densities.Source: §6 · p. 4, Figure 4 · p. 4
• Both methods require calibration, but Quantile Graph WaveNet can remap every τ value; its recalibration is not perfect outside the learned density region.Source: §6 · p. 5, Figure 5 · p. 5
• Mean-accuracy comparison is presented as supplementary evidence rather than the primary contribution of the research.Source: §6 · p. 5

D Model 15 min 30 min 60 min MAE MSE MAE MSE MAE MSE

• Graph Wavenet has higher mean prediction accuracy than its Quantile estimation counterpart across all datasets.Source: D Model 15 min 30 min 60 min MAE MSE… · p. 6

• The method is a step toward reliable traffic-prediction uncertainty estimates, but its initial results are described only as promising.Source: §7 · p. 6
• The approach has three limitations: quantile crossing, undefined density estimates for missing values, and the need for post-training calibration.Source: §7 · p. 6
• Future work should consider non-Gaussian densities and develop more descriptive confidence metrics for traffic-prediction uncertainty applications.Source: §7 · p. 6

• Traffic uncertainty estimates support better planning for scheduled road maintenance.Source: Broader Impact · p. 6
• More informative uncertainty estimates can improve routing decisions for critical trips.Source: Broader Impact · p. 6
• Improved traffic prediction may reduce travel delays by facilitating better use of available capacity.Source: Broader Impact · p. 6