Distributed Traffic State Estimation

• Distributed traffic state estimation frameworks are systems that utilize autonomous sensors and vehicles to locally measure and collaboratively fuse traffic density, flow, and speed data.
• They integrate techniques like Kalman filtering, operator-learning surrogates, and vertical federated architectures to enhance estimation accuracy, scalability, and privacy.
• Simulation studies indicate that consensus-driven filtering and redundancy protocols significantly reduce estimation errors and improve computational efficiency under sparse, intermittent connectivity.

A distributed traffic state estimation framework refers to any architectural paradigm wherein spatiotemporal traffic state (e.g., density, flow, speed) is estimated not centrally, but by a collection of autonomous, cooperating nodes—such as infrastructure sensors, connected vehicles, or local processing agents—each making partial observations and fusing information through networked communication protocols. These frameworks address the inherent scalability, reliability, and privacy demands of future transportation systems, especially under dense mixed traffic and intermittent connectivity. State-of-the-art implementations exploit model-based filters (finite-dimensional state-space), operator-learning surrogates, vertical federated architectures, and consensus or redundancy protocols to ensure robust, accurate, and privacy-preserving traffic estimation.

1. Macroscopic and Microscopic State Modeling

Distributed frameworks typically require traffic state representations amenable to local observation and network fusion. The dominant formulations include:

• Second-Order Macroscopic Models: The Aw–Rascle–Zhang (ARZ) model is widely adopted, comprising density $\rho(t,d)$ and relative flow $\psi(t,d)$. Its state-space discretization is:

$$x_{i,k} = \begin{bmatrix}\rho_{i,k}\\psi_{i,k}\end{bmatrix}, \quad x_k=\begin{bmatrix}x_{1,k}^T&\cdots&x_{N,k}^T\end{bmatrix}^T\in\mathbb R^{2N}$$

This yields nonlinear discrete-time dynamics governing evolution and control inputs (e.g., entering demand, boundary conditions) (Heij et al., 7 Dec 2025).

• Cell Transmission Models and SMM: Under switching-mode assumptions, the scalar conservation law is discretized for each spatial section, supporting piecewise-linear or mode-dependent dynamics (Sun et al., 2016).
• Operator Learning Frameworks: Distributed surrogates such as ON-Traffic utilize neural operators mapping probe data and boundary inputs to spatiotemporal state fields $(\rho(x,t),v(x,t))$, facilitating direct inference and uncertainty quantification in high-dimensional domains (Rap et al., 18 Mar 2025).
• Mixed-Traffic State-Space Models: For human-driven and autonomous vehicles, collective system states $x(t)\in\mathbb R^{N\cdot m}$ evolve per block matrices, with distributed observation via local agent measurements (Doostmohammadian et al., 10 Nov 2025).

The design of state variables and discretizations is critical for distributed estimation performance and determines the information structure available for network fusion and consensus.

2. Distributed Filtering and Consensus Mechanisms

The core estimation methodologies for distributed frameworks center on local model-based filtering with network-informed consensus steps.

• Distributed Kalman Filters: Each node (e.g., RSU or connected vehicle) maintains a local estimate of the global state via information form Kalman filtering, linearized prediction steps, and local corrections conditioned on cell-local measurements. Information is fused network-wide through multi-round consensus updates employing doubly-stochastic weighting (often Metropolis rule) over dynamically connected V2X graphs (Heij et al., 7 Dec 2025).
• Consensus-Augmented Estimation: In spatially partitioned networks, agents incorporate consensus terms in the Kalman filter update, promoting agreement on overlapping regions. The Distributed Local Kalman Consensus Filter (DLKCF) derives consensus gains based on filter error covariances and neighbor overlap structures, maintaining globally asymptotically stable (GAS) error dynamics in observable modes and boundedness under unobservable dynamics or arbitrary switching (Sun et al., 2016).
• Redundant Observability and Fault Tolerance: Network observability can be structurally guaranteed via $q$-node/link-connected topologies and multiple independent sensors per dynamical subsystem, enabling tolerance to up to $q-1$ faulty nodes or links (Doostmohammadian et al., 10 Nov 2025).

Consensus-based fusion not only ensures global consistency but enables resilience against intermittent connectivity, sensor dropout, and localized data unreliability.

3. Privacy and Data Integration: Vertical Federated Architectures

Distributed estimation frameworks increasingly incorporate privacy-preserving machine learning paradigms to address the partitioning of data ownership and compliance with secure data-sharing.

• Vertical Federated Learning (VFL): TSE can be implemented using VFL, wherein municipal authorities (MAs) and mobility providers (MPs) train joint models over vertically partitioned features and labels without raw data exchange. Model training leverages encrypted intermediate outputs and gradient transfer using homomorphic encryption or differential privacy techniques (Zhan et al., 2 Jun 2025).
• Mutual Information (MI)-Driven Selection: Segment-level provider competence is scored using a neural MI estimator, parameterized via the Donsker–Varadhan bound, ensuring only high-quality MPs supply features for vertical training. This guarantees model accuracy while minimizing exposure to underperforming or lazy data providers.
• Penalty-Based Incentive Mechanisms: A dynamic supervision game model and double-strike penalty protocol regulate MP behavior, driving equilibrium where the lazy data provision probability vanishes and overall MA utility is maximized.

The combination of MI-guided selection and robust incentive mechanisms enables reliable distributed estimation in adversarial and heterogeneous data environments.

4. Physical Consistency and Uncertainty Quantification

Robust distributed estimation mandates adherence to physical constraints and meaningful confidence quantification.

• Projection Steps: After each prediction, local unconstrained state estimates are projected cell-wise onto feasible regions (density and flow non-negativity and upper bounds), guaranteeing physical consistency in distributed reconstructions. These projections are fed back as posterior means for subsequent information updates (Heij et al., 7 Dec 2025).
• Aleatoric Uncertainty Quantification (UQ): Operator-learning frameworks output direct predictive variances through networked mean/log-variance heads, trained jointly on NLL loss to reflect residuals. Calibration curves are empirically validated to ensure reliability in coverage bounds (Rap et al., 18 Mar 2025).
• Handling Noise and Dropout: Sensor noise and missing data are addressed by injecting Gaussian error during training and deploying models (such as VIDON branches) compatible with variable and masked input tensors, preserving inference robustness over degraded observation regimes.

Physical constraint enforcement and UQ yield interpretable, trustworthy state estimates that inform actionable traffic management directives.

5. Simulation Validation and Performance Metrics

Framework reliability is established through simulation scenarios, quantitative metric analysis, and phase transition observations.

• Scenario Design: Validation employs highway segments (e.g., 2.7 km, multi-lane), representative RSU and CV penetration rates (ranging 2–20%), real-world datasets (e.g., pNEUMA drone trajectories), and Lagrangian/microscopic simulators (SUMO, IDM).
• Performance Metrics: RMSE and SMAPE for density and flow, MAE for held-out scenarios, receding-horizon stability analysis, consensus disagreement level, and utility functions for federated settings.
• Phase Transitions and Percolation: Increasing penetration of connected vehicles reveals a sharp phase transition in network observability around 10% penetration, above which a giant connected component emerges, consensus becomes feasible, and estimation errors sharply decrease (Heij et al., 7 Dec 2025).
• Computational Efficiency: Distributed filters—DLKCF, local Kalman—offer $O(n_l^3)$ complexity per agent (vs $O(n^3)$ centralized), with order-of-magnitude speedup and maintained accuracy, especially under sensor heterogeneity (Sun et al., 2016).

These experimental results demonstrate the operational advantages—robustness, scalability, accuracy—of distributed architectures over centralized or purely local alternatives.

6. Future Directions and Practical Considerations

Distributed traffic state estimation frameworks continue to advance along several dimensions:

• Scalable Decentralized Design: Multi-round consensus and distributed filtering increasingly support large traffic networks, sparse observation regimes, and mixed-type vehicle fleets.
• Online Adaptation: Operator-learning architectures integrate receding-horizon fine-tuning, ensuring adaptation to nonstationary dynamics and traffic patterns (Rap et al., 18 Mar 2025).
• Robustness and Redundancy: Design principles rooted in graph-theoretic connectivity and redundant sensing deliver resilience against faults, dropouts, and failures.
• Privacy and Trust: Vertical federated learning protocols, MI-based selection, and penalty mechanisms underlie the trust infrastructure for cross-organizational data integration, balancing accuracy and privacy (Zhan et al., 2 Jun 2025).
• Physical-Model Hybridization: Coupling macroscopic models with data-driven surrogates enhances fidelity and interpretability.

A plausible implication is that ongoing research will focus on jointly optimizing information topology, inference robustness, incentive alignment, and computational tractability for real-time adaptive traffic management.