Vessel Traffic Flow Prediction on Sparse Data via Spatio-Temporal Graph Neural Networks with a Learnable Tweedie Head

Abstract: Accurate vessel traffic flow prediction is crucial for smart port operations and navigational safety. However, maritime traffic flow data are often highly sparse with intermittent bursts, making robust forecasting challenging. Under such conditions, conventional spatio-temporal graph neural networks (ST-GNNs) can degrade toward conservative near-zero predictions and fail to capture non-zero activity. Although zero-inflated negative binomial (ZINB) models partially address excess zeros, their two-part formulation can still remain conservative around abrupt transitions. To address these issues, we propose a model-agnostic learnable Tweedie head that can be attached as a plug-and-play output module to arbitrary ST-GNN backbones. Instead of likelihood-based Tweedie training, which typically requires surrogate objectives, our approach optimizes the closed-form Tweedie unit deviance and predicts the mean for point forecasting while learning a node-level variance power to capture heterogeneous variability across port areas. Experiments on a maritime traffic graph constructed from real-world AIS data in the Port of Los Angeles and Long Beach show that the proposed head consistently improves RMSE across multiple ST-GNN backbones, especially on non-zero events, leading to more reliable forecasts for practical maritime traffic control.

• The paper introduces a model-agnostic learnable Tweedie head trained with closed-form unit deviance, allowing spatio-temporal graph neural networks to model zero-inflated and overdispersed vessel counts without likelihood approximations.
• Across STGCN, DCRNN, and Graph WaveNet on LA/LB AIS data, the Tweedie head achieved the lowest overall and non-zero RMSE, with non-zero RMSE values of 1.4570, 1.4831, and 1.4634, respectively.
• The method improves sensitivity to vessel arrivals and bursts over standard regression models, but remains conservative at extreme peaks and requires validation across ports, seasons, dynamic graphs, and uncertainty-calibrated forecasts.

Problem setting and motivation

Vessel traffic flow prediction supports smart-port logistics and navigational safety, but maritime traffic differs from urban road traffic in ways that stress standard forecasting pipelines. Marine areas are open spaces with stochastic arrival patterns, producing count series that are severely zero-inflated (over 70% zeros in the dataset used here) and heavy-tailed, with sudden bursts of vessel arrivals. Under such targets, conventional spatio-temporal graph neural networks (ST-GNNs) trained with standard regression losses tend to converge to conservative near-zero predictions: they achieve deceptively low overall error while systematically missing non-zero activity.

The paper positions itself against two existing remedies. Zero-inflated negative binomial (ZINB) heads model structural zeros and counts via a two-part mixture, but their predictive mean is discounted as $(1-\pi)\mu$; when the zero gate $\pi$ does not drop quickly at an onset of activity, forecasts remain conservative around abrupt transitions, and the two-part formulation is fragile when its zero-generation assumptions are violated (2606.07694). Tweedie-based ST-GNNs avoid an explicit zero-gate by using the compound Poisson–Gamma distribution, but because the exact Tweedie log-likelihood for $1

Proposed method

The core contribution is a lightweight, model-agnostic learnable Tweedie head that attaches to any ST-GNN backbone and is trained on the closed-form Tweedie unit deviance rather than a likelihood surrogate. The head splits backbone features into two streams predicting, per node and horizon, the mean $\mu = \mathrm{Softplus}(H_\mu) + \epsilon$ and a node-level variance power $p = 1.01 + 0.98\,\sigma(H_p)$, constraining $p \in (1, 2)$ so that the distribution corresponds to compound Poisson–Gamma. The learnable $p$ spans near-Poisson behavior ($p \approx 1.1$) to overdispersed Gamma-like behavior ($p \approx 1.9$), capturing spatial heterogeneity across port areas without manual tuning.

The training objective is the unit deviance

$$d(y,\mu,p)=2\left(\frac{y^{2-p}}{(1-p)(2-p)}-\frac{y\,\mu^{1-p}}{1-p}+\frac{\mu^{2-p}}{2-p}\right),$$

with the first term set to zero when $\pi$0. This objective is non-negative, equals zero if and only if $\pi$1, requires no dispersion parameter (implicitly $\pi$2), and backpropagates stably without evaluating the normalizing term $\pi$3 that makes likelihood-based training intractable. At inference, the point forecast is simply the Tweedie mean $\pi$4, avoiding ZINB's gate-discounted expectation.

Dataset and graph construction

The evaluation uses AIS records from the Los Angeles/Long Beach (LA/LB) port region from September 1 to November 30, 2024, split chronologically 70:10:20 into 1,528 / 218 / 438 hours to prevent look-ahead bias; all graph construction uses training data only. Nodes are extracted data-drivenly: near-stationary records (SOG ≤ 1.0 knot) are clustered with DBSCAN under separate hyperparameters for terminals ($\pi$5 m) and anchorages ($\pi$6 m, both Minimum Points = 400), then converted to polygon boundaries via alpha shapes (radii 400 m and 2,000 m respectively). This yields $\pi$7 polygonal nodes respecting the breakwater boundary. Directed edges come from observed vessel transitions between nodes, filtered to at least 10 transitions, giving 198 edges with row-normalized transition-probability weights (average weight 0.1414). The target tensor aggregates hourly inflow/outflow counts per node.

This construction is a strength of the paper relative to grid-based maritime forecasting, but it also embeds assumptions worth noting: the static, transition-based adjacency ignores time-varying connectivity, and DBSCAN/alpha-shape hyperparameters are domain-specific choices whose sensitivity is not ablated.

Experimental results

Three representative backbones—STGCN, DCRNN, and Graph WaveNet (GWNet)—are each evaluated with three heads: Base (MAE loss), +ZINB, and +Tweedie, using a 24-hour input window, 6-hour horizon, hidden dimension 32, Adam at learning rate 0.001 for 100 epochs. Because overall metrics are misleading on zero-heavy targets, the paper reports Non-zero MAE and Non-zero RMSE restricted to positive ground-truth timesteps.

Metric	GWNet Base	GWNet +ZINB	GWNet +Tw	DCRNN Base	DCRNN +ZINB	DCRNN +Tw	STGCN Base	STGCN +ZINB	STGCN +Tw
MAE (All)	0.3970	0.4773	0.4873	0.3986	0.4823	0.4735	0.3905	0.4907	0.4827
MAE (Non-zero)	1.1969	0.9940	0.9709	1.2530	0.9681	0.9794	1.1902	0.9623	0.9723
RMSE (All)	0.9172	0.9112	0.8951	0.9255	0.8971	0.8802	0.9024	0.9164	0.8865
RMSE (Non-zero)	1.6720	1.5241	1.4570	1.7432	1.5089	1.4831	1.6682	1.4877	1.4634

Three findings emerge. First, Base models attain the best overall MAE (e.g., 0.3970 for GWNet) yet the worst non-zero errors, confirming the near-zero bias diagnosis. Second, both sparsity-aware heads substantially reduce Non-zero MAE (roughly 0.96–0.99 versus 1.19–1.25), with ZINB and Tweedie comparable on that metric. Third—and this is the paper's strongest quantitative claim—the +Tweedie variant achieves the lowest overall RMSE and Non-zero RMSE across all three backbones, indicating better capture of surge magnitudes since RMSE penalizes large errors disproportionately. The consistency across architecturally distinct backbones supports the plug-and-play claim.

The case study reinforces this: GWNet produces flatlined near-zero forecasts; GWNet+ZINB improves event sensitivity but, during spikes of 4–5 vessels, predicts barely 1 vessel—an effect the authors attribute to the slow response of the zero-inflation gate in the discounted mean $\pi$8. The Tweedie head tracks rise-and-fall patterns more sharply at both an anchorage node and an active terminal node, though it still does not match absolute burst maxima.

Limitations and open questions

The paper is candid about several constraints. Even with the Tweedie head, predictions remain conservative during the most extreme events; matching the maxima of stochastic surges is not achieved, and no tail-aware objective is explored here. Evaluation covers a single region over three months at hourly aggregation, so generalization across ports, seasons, and temporal resolutions is unverified. The framework performs point forecasting only—no calibrated uncertainty quantification despite the probabilistic head—and relies on a static transition-based graph, excluding exogenous drivers such as weather or schedules and time-varying connectivity. Finally, the deviance objective fixes $\pi$9 and optimizes mean prediction; whether the learned $1

Conclusion

This paper diagnoses the failure mode of conventional ST-GNNs on zero-inflated, bursty maritime traffic—low overall error masking missed non-zero events—and proposes a closed-form, deviance-trained, learnable Tweedie head as a plug-and-play alternative to likelihood-surrogate Tweedie models and gate-discounted ZINB heads. Across STGCN, DCRNN, and GWNet on real AIS data from LA/LB, it consistently delivers the best RMSE and Non-zero RMSE, establishing a simple, backbone-agnostic recipe for spike-sensitive forecasting in sparse spatio-temporal settings, while leaving extreme-surge fidelity, cross-region transfer, and calibrated uncertainty as open problems.