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PGCN: Progressive Graph Convolutional Networks for Spatial-Temporal Traffic Forecasting

Yuyol Shin, Yoonjin Yoon

arXiv:2202.08982v3cs.LG

TL;DR

Traffic forecasting is difficult because spatial-temporal correlations change, while adaptive graphs learned during training may not reflect test-time inputs. PGCN builds progressive graphs from online traffic trends and combines graph convolution with temporal convolutions. Across seven real-world datasets, it consistently achieves state-of-the-art performance; the authors identify information dilution across graph structures as a limitation of the current graph-convolution module.

  • Problem

    Traffic correlations change over time, while many adaptive-graph methods finalize their graphs during training and do not adapt to online test data.

    Source: §I · pp. 1–2

  • Method

    PGCN uses parameterized cosine similarity of online node signals to update progressive graphs and combines graph convolution with dilated causal convolution for traffic forecasting.

    Source: §I · p. 2

  • Results

    PGCN consistently achieves state-of-the-art performance on all seven datasets used in the study.

    Source: §I · p. 2, §VI · p. 11

  • Takeaways & Limitations

    The authors conclude that progressive adaptation to online inputs supports generalization across single-feature and multi-feature datasets and different study sites.

    Source: §VI · p. 11

    The current diffusion-convolution module sums representations from different graphs, which may dilute information and potentially compromise performance.

    Source: §VI · p. 11

Abstract

from arXiv · show

The complex spatial-temporal correlations in transportation networks make the traffic forecasting problem challenging. Since transportation system inherently possesses graph structures, many research efforts have been put with graph neural networks. Recently, constructing adaptive graphs to the data has shown promising results over the models relying on a single static graph structure. However, the graph adaptations are applied during the training phases and do not reflect the data used during the testing phases. Such shortcomings can be problematic especially in traffic forecasting since the traffic data often suffer from unexpected changes and irregularities in the time series. In this study, we propose a novel traffic forecasting framework called Progressive Graph Convolutional Network (PGCN). PGCN constructs a set of graphs by progressively adapting to online input data during the training and testing phases. Specifically, we implemented the model to construct progressive adjacency matrices by learning trend similarities among graph nodes. Then, the model is combined with the dilated causal convolution and gated activation unit to extract temporal features. With residual and skip connections, PGCN performs the traffic prediction. When applied to seven real-world traffic datasets of diverse geometric nature, the proposed model achieves state-of-the-art performance with consistency in all datasets. We conclude that the ability of PGCN to progressively adapt to input data enables the model to generalize in different study sites with robustness.

I. INTRODUCTION

Traffic forecasting is challenged by complex spatial-temporal correlations that can change over time, while existing adaptive graphs may not reflect online data at test time. PGCN updates its graph from online traffic trends and achieves consistently strong results across seven datasets.

I. INTRODUCTION

  • Earlier adaptive-graph methods use graphs finalized during training, so they do not adapt to online traffic data during testing.Source: §I · pp. 1–2
  • Traffic correlations can shift between nodes and across instances of the same time of day, including for sensors that are not adjacent.The passage notes that day-of-week, holidays, weather, accidents, and sports events may contribute to changing trends.Source: §I · p. 2, Figure 1 · p. 2
  • PGCN combines a progressive graph convolution module for time-varying spatial correlations with dilated causal convolution and gated activation for temporal features.The graph updates edges and weights using online input data.Source: §I · p. 2
  • PGCN consistently achieves state-of-the-art performance across seven real-world datasets, remaining within 1.5% of the best model except for a few cases.It is also competitive when structural-network information is unavailable.Source: §I · p. 2

II. RELATED WORKS

Related work covers graph neural network families, graph construction for traffic forecasting, and temporal feature extraction. It identifies the need for graph updates from online inputs during both training and testing.

A. Graph Neural Networks

  • Graph neural networks learn graph-structured data through convolutional, attentional, or message-passing approaches that differ in how they aggregate neighboring-node information.Source: A. Graph Neural Networks · pp. 2–3

B. GNNs for Spatial-Temporal Traffic Forecasting

  • Traffic graph methods incorporate static structural information, semantic or input-derived graphs, and adaptive graphs, but many finalize graphs before validation and testing.The section motivates updating edge connections and weights from online inputs in both training and testing.Source: B. GNNs for Spatial-Temporal Traffic… · p. 3

C. Temporal Feature Extraction

  • Temporal feature extraction has used recurrent networks, temporal and dilated convolutions, self-attention, and graphs with temporal connectivity.Source: C. Temporal Feature Extraction · p. 3

III. PRELIMINARIES

The preliminaries define transportation-network and graph-signal notation, then formulate forecasting as predicting future traffic states from historical observations and a graph.

A. Notations and Definitions

  • A transportation network is represented as a directed graph with nodes, edges, and an adjacency matrix whose entries weight pairwise connections.A graph signal records node features; the notation also specifies signals observed over the last T time steps.Source: A. Notations and Definitions · p. 3

B. Problem Definition

  • Traffic forecasting maps historical graph signals and the transportation graph to predicted traffic states over T′ future time steps.The inputs may include historical speed, flow, and occupancy observations.Source: B. Problem Definition · p. 3

IV. PROGRESSIVE GRAPH CONVOLUTIONAL NETWORKS

PGCN constructs progressive graphs from online node-signal trends, then uses these graphs to model changing spatial relationships. Its graph construction compares normalized signals so similar trends can be linked beyond physical adjacency.

A. Progressive Graph Construction

  • Parameterized cosine similarity is used to adapt progressive graphs to traffic changes using online signals, extending relationships beyond simple spatial adjacency.The approach aims to weight nodes with similar signals more highly regardless of proximity.Source: A. Progressive Graph Construction · p. 4
  • A progressive graph is a time-indexed set of graphs whose adjacency weights encode pairwise node-signal similarities.Source: A. Progressive Graph Construction · p. 4
  • Minmax-normalized node signals are compared by trend rather than absolute magnitude, so differently scaled signals with matching trends can have cosine similarity 1.The passage gives [40] and [50, 60, 50, 70, 50] as an example.Source: A. Progressive Graph Construction · p. 4
  • The five-step profiles illustrate changing graph relationships: v1 is most similar to v4 at time t, but that connection disappears at t + h.Source: Figure 2 · p. 4, A. Progressive Graph Construction · p. 4

B. Progressive Graph Convolution Module

The module uses diffusion convolution to aggregate spatial information, then extends it with a progressive adjacency matrix derived from graph signals.

B. Progressive Graph Convolution Module

C. Dilated Causal Convolution

PGCN extracts temporal features with dilated causal convolutions and gated activation units. Causality excludes future information, while dilation expands the receptive field.

C. Dilated Causal Convolution

D. Overall Architecture of PGCN

PGCN stacks spatial-temporal layers with residual and skip connections, sharing the progressive graph-constructor parameters across layers. The evaluation datasets span highway and urban settings, with varied regions, measures, and temporal coverage.

D. Overall Architecture of PGCN

  • Each spatial-temporal layer combines dilated causal convolution, a gated activation unit, progressive graph convolution, and a residual connection.The output uses skip connections from each layer and two fully connected layers with ReLU activation.Source: D. Overall Architecture of PGCN · p. 5
  • The progressive graph-constructor parameter is shared across stacked layers, which increase the temporal receptive field.Source: D. Overall Architecture of PGCN · p. 5

A. Datasets

  • The datasets include highway-speed and flow measurements from Bay Area, Los Angeles, Seattle, and San Bernardino locations, plus Seoul urban speed data.Reported examples include 5-minute PeMS-Bay data from 325 detectors and METR-LA data from 207 detectors; PEMS04 and PEMS08 include speed, flow, and occupancy.Source: A. Datasets · pp. 5–6
  • PEMS07 adds highway-flow data from 883 loop detectors across Los Angeles and Ventura counties; the dataset summary is presented in Table I.Its observations cover four months at 5-minute resolution.Source: A. Datasets · p. 6, Table I · p. 6

B. Experiment Settings

The experiments use MAE, RMSE, and MAPE, with an eight-layer PGCN trained under specified convolution and optimization settings. Comparisons cover historical, recurrent, graph-convolutional, and adaptive-graph baselines.

B. Experiment Settings

  • Evaluation metrics are Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE).Source: B. Experiment Settings · p. 6
  • PGCN uses eight spatial-temporal layers with 32 hidden dimensions, kernel size 2, and alternating dilation factors 1 and 2.Training uses Adam with an initial learning rate of 0.001 for 100 epochs, selecting the validation-best model.Source: B. Experiment Settings · p. 6

C. Methods for Comparison

  • Baselines include HA and FC-LSTM, alongside DCRNN and Graph WaveNet.HA uses the most recent traffic state; FC-LSTM combines LSTM units with a fully connected layer.Source: C. Methods for Comparison · p. 6
  • Other comparisons include TGC-GRU, DMSTGCN, STSGCN, AGCRN, and Z-GCNETs.Z-GCNETs was not evaluated on Seattle-Loop because computing zigzag persistence images was too time-consuming.Source: C. Methods for Comparison · p. 6

D. Experimental Results

PGCN is consistently among the strongest models across seven datasets, prediction horizons, and metrics, with only a few lower rankings. Its performance gap from the best model is generally small, though baseline leaders vary by dataset.

1) Overall Performance Evaluation:

  • Baseline leaders vary: AGCRN leads on several Urban-core, Seattle-Loop, and PEMS07 tasks, while DMSTGCN is more competitive on multi-feature PEMS04 and PEMS08.Source: 1) Overall Performance Evaluation · p. 6
  • PGCN ranks best or runner-up across metrics, horizons, and datasets, except for four specified MAPE or RMSE cases.It ranks third for 15-min MAPE on PEMS08 and 60-min RMSE on Urban-core, and fourth for 60-min MAPE on PEMS04 and 15-min MAPE on PEMS07.Source: Figure 5 · p. 8, 1) Overall Performance Evaluation · p. 8
  • PGCN's performance gap from the best model stays below 1.5%, except DCRNN's 2.44% MAPE edge on Seattle-Loop at 15 minutes and AGCRN's 2.05% edge on PEMS04 at 60 minutes.Source: 1) Overall Performance Evaluation · p. 9

2) Evaluation of Progressive Graph Convolution:

The progressive adjacency combination provides consistent results across datasets, while the strongest graph configuration varies by task. The feature-selection results favor single-feature input overall, with a PEMS08 exception at shorter horizons.

2) Evaluation of Progressive Graph Convolution:

  • Transition-plus-progressive performs best at every horizon on PeMS-Bay, Urban-core, PEMS04, and PEMS07 except 15-min PEMS07; other datasets favor different combinations.The all-three-matrix combination leads on Seattle-Loop and PEMS08 except at 30 minutes on PEMS08, where progressive plus self-adaptive leads.Source: 2) Evaluation of Progressive Graph… · p. 9
  • Progressive adjacency alone is usually not competitive, but adds complementary information; progressive and self-adaptive matrices remain alternatives when the structural graph is unavailable.Source: 2) Evaluation of Progressive Graph… · p. 9
  • Transition-plus-progressive adjacency is the most consistent combination across seven datasets, although the best configuration varies by dataset and horizon.Source: Table III · p. 9, 2) Evaluation of Progressive Graph… · p. 9
  • In PeMS-Bay, weights rise when sensor speed trends align and fall as their rates diverge, even when absolute speeds differ.When many nodes share similar free-flow trends, the softmax can keep an individual pair's weight low.Source: 2) Evaluation of Progressive Graph… · pp. 9–10, Figure 6 · p. 10

3) Feature Selection for Multi-Feature Data:

  • The single-feature model is most accurate at all horizons on PEMS04 and at 60 minutes on both datasets; on PEMS08, flow plus occupancy wins at 15 and 30 minutes.The authors report that single-feature input offers robust forecasting with lower computational requirements and avoids time-intensive feature selection.Source: 3) Feature Selection for Multi-Feature… · p. 10

4) Computation Efficiency:

PGCN reports state-of-the-art performance across seven datasets alongside computational efficiency, and identifies temporal weighting, graph aggregation, and external features as future directions.

4) Computation Efficiency:

  • PGCN's progressive adjacency uses only Wadj ∈ R^T×T for similarity calculation, with fewer parameters than the compared adaptive-graph constructions.The paper also reports better generalization than Graph WaveNet and DMSTGCN; Table V covers parameter counts and computation cost on PeMS-Bay.Source: 4) Computation Efficiency · p. 10, Table V · p. 11
  • STSGCN's localized spatial-temporal adjacency requires substantial computation, while Z-GCNETs adds about 10 seconds per observation to generate persistence images.The paper also identifies weather, road characteristics, and POI as external features for future study across diverse sites.Source: 4) Computation Efficiency · p. 11, §VI · p. 11

VI. CONCLUSION AND FUTURE STUDY

  • PGCN achieves state-of-the-art performance on all seven datasets and generalizes across single-feature and multi-feature datasets.The conclusion also describes this generalization as supporting robustness.Source: §VI · p. 11
  • The temporal feature module keeps its weights fixed across inputs; future work will assign weights dynamically while preserving sequential information.The authors note that temporal correlations between time steps may change.Source: §VI · p. 11
  • The current diffusion convolution sums representations from different graphs, which may dilute information and compromise performance.The authors propose developing a graph convolution module that aggregates representations without information dilution.Source: §VI · p. 11
  • PGCN's robust performance without structural-network information is reported, while incorporating weather, road characteristics, and POI remains future work.The authors plan to explore how these external features affect forecasting across diverse study sites.Source: §VI · p. 11

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