Traffic Graph Convolutional Recurrent Neural Network: A Deep Learning Framework for Network-Scale Traffic Learning and Forecasting

Zhiyong Cui; Kristian Henrickson; Ruimin Ke; Ziyuan Pu; Yinhai Wang

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Traffic Graph Convolutional Recurrent Neural Network: A Deep Learning Framework for Network-Scale Traffic Learning and Forecasting

Zhiyong Cui, Kristian Henrickson, Ruimin Ke, Ziyuan Pu, Yinhai Wang

arXiv:1802.07007v3Feb 20, 2018cs.LGstat.ML

TL;DR

Traffic forecasting requires modeling time-varying patterns and complex spatial dependencies on road networks. The paper proposes TGC-LSTM, which combines traffic graph convolution based on physical network structure with LSTM and interpretability-oriented regularization. On two real-world traffic datasets, it outperforms baseline methods and identifies influential roadway segments.

Problem
Traffic forecasting must handle time-varying traffic patterns and complicated spatial dependencies, while existing RNN-based methods can capture spurious relationships that are difficult to interpret.
Method
The paper learns the traffic network as a graph, uses traffic graph convolution within TGC-LSTM, and optionally regularizes graph-convolution weights and features.
Results
The proposed TGC-LSTM outperforms multiple baseline methods on two real-world traffic-speed datasets and identifies influential roadway segments.
Takeaways & Limitations
The learned graph-convolution weights provide a way to interpret roadway interactions and recognize influential segments in real-world traffic networks.

Abstract

from arXiv · show

Traffic forecasting is a particularly challenging application of spatiotemporal forecasting, due to the time-varying traffic patterns and the complicated spatial dependencies on road networks. To address this challenge, we learn the traffic network as a graph and propose a novel deep learning framework, Traffic Graph Convolutional Long Short-Term Memory Neural Network (TGC-LSTM), to learn the interactions between roadways in the traffic network and forecast the network-wide traffic state. We define the traffic graph convolution based on the physical network topology. The relationship between the proposed traffic graph convolution and the spectral graph convolution is also discussed. An L1-norm on graph convolution weights and an L2-norm on graph convolution features are added to the model's loss function to enhance the interpretability of the proposed model. Experimental results show that the proposed model outperforms baseline methods on two real-world traffic state datasets. The visualization of the graph convolution weights indicates that the proposed framework can recognize the most influential road segments in real-world traffic networks.

I. INTRODUCTION

Traffic forecasting must model time-varying states and complex spatial dependencies across physical road networks. The paper addresses these challenges with TGC-LSTM, graph-based convolution, regularization for interpretability, and evaluation on public real-world data.

Traffic forecasting predicts future network states from historical states and the physical roadway network.
Statistical methods have limited capacity for high-dimensional traffic time series, motivating machine learning approaches.
RNN-based methods can capture spatial dependencies but may learn noise and spurious relationships that are difficult to interpret physically.
Existing graph convolutions may use receptive fields that are not confined by the physical traffic-network structure.
TGC-LSTM combines a traffic graph convolution operator with LSTM to model spatial and dynamic temporal dependencies.
The framework adds optional L1 regularization on graph-convolution weights and L2 regularization on features, and uses publicly available traffic-speed data.

II. LITERATURE REVIEW

Prior traffic-forecasting research spans neural architectures and graph-based methods, but traffic networks require representations that respect roadway structure, physical characteristics, and directional impact propagation.

Deep neural models, including DBNs, auto-encoders, RNNs, LSTMs, and GRUs, have been applied to traffic forecasting.
CNN-based approaches can introduce spurious spatial relationships when traffic networks are converted into images.
Graph neural methods represent traffic networks with adjacency or Laplacian matrices and extend convolution to graph-structured data.
Traffic-network graphs differ from other graph types because road states vary over time and roads have meaningful physical characteristics.
The paper treats congestion impacts as bidirectionally propagated between upstream and downstream roads, using an undirected graph.
K-hop neighborhoods include each node itself, while clipped powers of A + I encode neighbor existence without hop-count weighting.
The free-flow reachable matrix encodes whether vehicles can traverse between nodes within m time-steps at free-flow speed.

B. Traffic Forecasting Problem

The forecasting problem maps historical graph signals from a traffic network to a subsequent traffic state while learning how traffic impacts transmit between neighboring nodes.

Traffic states represent measurements such as speed, travel time, or volume at N sensing locations in a graph.
The short-term task learns a function F(·) that maps T historical graph signals to the subsequent one-step signal x_T+1.
The graph-based formulation includes the network graph, k-hop neighborhood matrix, and free-flow reachable matrix as structural inputs.
A further objective is learning function parameters that represent traffic-impact transmission between adjacent and neighboring nodes.

C. Traffic Graph Convolution

Traffic Graph Convolution extends localized graph filtering to high-order neighborhoods while enforcing roadway physical reachability. Its trainable weights can represent node interactions, with a trade-off between feature richness, accuracy, and computational cost.

The TGC operation combines trainable weights, a k-hop neighborhood matrix, and a free-flow reachable matrix with the traffic state vector.
TGC expands the receptive field beyond one-hop neighbors and incorporates edge properties and high-order neighborhood structure.
Because the structural matrices are sparse and binary, the resulting graph-convolution operation is localized and sparse.
The learned TGC weights can measure interactive influence between graph nodes and support model interpretability.
Larger convolution orders increase receptive-field size and extracted neighborhood features, but need not reach the graph-specific maximum order.
Choosing the convolution order balances prediction accuracy and feature richness against computational cost.

D. Comparing TGC with Spectral Graph Convolution

The paper contrasts adjacency-based TGC with spectral graph convolutions defined through the graph Laplacian. TGC emphasizes physically localized, interpretable roadway relationships.

Spectral graph convolution applies a learnable Fourier-domain filter using the Laplacian eigen-decomposition.The operation is Udiag(θ)U^T x_t, with the Laplacian factorized as L=UΛU^T.
Localized spectral graph convolution reduces computation with a K-parameter polynomial filter and extracts weighted features from K-hop neighbors.It avoids eigen-decomposition and is spatially localized.
TGC is based on the adjacency matrix and incorporates the free-flow reachable matrix to extract roadway features according to physical properties.This provides spatial localization tied to traffic-network structure.
TGC uses one convolution layer, making its parameters more interpretable than those of SGC and LSGC, which normally require multiple layers.TGC also has more parameters and greater capacity to represent relationships between connected nodes.

E. Traffic Graph Convolutional LSTM

TGC-LSTM replaces the vanilla LSTM input with graph-convolution features so the recurrent model captures spatial and temporal traffic dependencies. Its cell-state gate also incorporates neighboring cell states during recurrence.

TGC-LSTM feeds reshaped graph-convolution features into an LSTM while retaining the vanilla gates and hidden-state structure.The graph-convolution feature vector has dimension K N.
The model’s gates map graph-convolution inputs and preceding hidden states through learned weights, biases, sigmoid activation, and tanh.Input-to-gate matrices are in R^(N×KN), while recurrent matrices are in R^(N×N).
A cell-state gate allows each node’s LSTM cell state to be affected by neighboring cell states.Its weight matrix is constrained by a K-hop adjacency matrix combined with the free-flow reachable matrix.
At the final time step, the hidden state is the predicted value, and the next-step label satisfies y_T=x_T+1.Training loss measures the residual between the prediction and true value, normally using MSE for continuous values.
Algorithm 1 describes generating the final output after T iterations from traffic-state data and graph-related matrices.The pseudocode omits mini-batch gradient descent and backpropagation-based parameter updates.

F. Traffic Graph Convolution Regularization

The model adds regularization to make graph-convolution weights sparse and stable and to keep features from different graph hops similar. These choices support interpretation aligned with roadway relationships.

Algorithm 1 computes graph-convolution features across hops using hop-specific weights, adjacency matrices, and the input traffic state.The hop loop runs from k=1 to K.
L1 regularization is added to graph-convolution weight matrices so they become sparse and more interpretable.Sparsity helps distinguish which neighboring nodes or node groups contribute most.
Graph-convolution weights can otherwise vary substantially because node features combine influences from multiple neighboring nodes.Large positive and negative weights may cancel while still producing informative features, obscuring node relationships.
Adjacent graph-convolution hops are regularized so their extracted features do not differ dramatically.The stated motivation is that influence between distant nodes is transmitted through intervening nodes.
The total loss combines prediction loss with penalties λ1R1 and λ2R2 controlling graph-convolution weight and feature regularization.These penalties are optional additions to the training objective.

A. Dataset Description

The evaluation uses two network-scale Seattle traffic-speed datasets with five-minute observations: LOOP freeway sensors and INRIX road-segment speeds.

LOOP contains 323 sensor stations across four connected Seattle-area freeways during 2015.Measurements are collected at five-minute intervals.
INRIX contains speeds for 1014 Seattle downtown road segments during 2012.The data are aggregated from GPS probes collected through commercial vehicle fleets and mobile apps, at five-minute intervals.
Free-flow speeds are 60 mph for all LOOP segments and range from 20 mph to 60 mph across the INRIX network.Distance adjacency and free-flow reachable matrices are calculated from roadway characteristics and topology.

B. Experimental Settings

The experiments compare TGC-LSTM with statistical, neural, recurrent, convolutional, and graph-convolution baselines on two datasets, using three forecasting metrics and graph-convolution settings with K=3 hops. Results indicate that TGC-LSTM outperforms the compared approaches, while performance differences reflect the models’ ability to represent spatial-temporal dependencies.

Baselines: The study compares TGC-LSTM with ARIMA, SVR, FNN, LSTM, DiffGRU, Conv+LSTM, SGC+LSTM, and LSGC+LSTM.
Experimental configuration: K=3 is used for graph-convolution-related models in the performance comparison.
Metrics: The evaluation uses MAE, MAPE, and RMSE to assess forecasting performance.
Results: TGC-LSTM outperforms the other models on all three metrics across both datasets.
Results: SGC+LSTM improves over vanilla LSTM, whereas one-layer LSGC+LSTM does not outperform LSTM.
Dataset considerations: INRIX errors are lower than LOOP errors because missing nighttime or off-peak speeds are imputed and INRIX speeds are integer-valued.

D. Training Efficiency

TGC-LSTM converges faster and in fewer epochs than the compared graph-convolution LSTM models, but requires more time per epoch than vanilla LSTM. Regularization slightly reduces accuracy while increasing weight sparsity and feature consistency, supporting interpretability.

Convergence: TGC-LSTM converges in fewer epochs than SGC+LSTM and LSGC+LSTM, with the fastest decrease in loss.
Training cost: TGC-LSTM requires twice the training time per epoch of LSTM, while SGC+LSTM is faster and LSGC+LSTM is slightly slower.
Hop order: Increasing graph-convolution hops accelerates convergence, but values above 3 yield only marginal training and validation improvements.
Regularization: L1 regularization on graph-convolution weights and L2 regularization on features constrain learned weights and features, with a trade-off against prediction accuracy.
Regularization: With penalty rates of 0.01, MAEs increase by around 0.02 on both datasets, while weight sparsity increases and feature-regularization R_2 decreases.
Weight visualization: Weight matrices concentrate around the diagonal and form clusters corresponding to nearby or connected road segments.

F. Model Interpretation and Visualization

The proposed model’s learned graph-convolution weights align with influential physical locations in real traffic networks, while its forecasts track traffic-speed trends during peak and off-peak periods. Regularization scales affect the visualization of inferred influence.

Weight visualization: Regularization terms λ1 and λ2 change the darkness of INRIX influence lines and the sizes of LOOP influence circles.These visual encodings represent the magnitude of influence in the respective network visualizations.
Weight visualization: Dark INRIX weight regions correspond to busy, congested freeway entrance and exit ramps in downtown Seattle.LOOP highlights include major freeway intersections and a frequently congested SR-520 ramp connecting to Bellevue.
Weight visualization: The learned weight matrix captures spatial dependencies and identifies influential roadways and segments.The interpretation is supported by comparing learned weights with the physical traffic network.
Forecast visualization: Predicted traffic-speed curves track ground-truth trends at both peak and off-peak hours on randomly selected days from LOOP and INRIX.The visualization covers two locations from each dataset despite differences between the two traffic networks.
Model interpretation: The framework combines traffic graph convolution with LSTM forecasting and adds regularization to make learned weights more stable and interpretable.It is evaluated on two real-world traffic datasets and is reported to outperform compared baseline models.
Future work: Future work targets improved prediction accuracy and robustness and more interpretable convolution across spatial and temporal dimensions.The authors identify these directions as ongoing extensions of the model.