Urban traffic analysis and forecasting through shared Koopman eigenmodes

TL;DR

This work addresses data scarcity in city-scale traffic forecasting by introducing cross-city knowledge transfer of Koopman eigenmodes through constrained Hankelized DMD (TrHDMD). By extracting shared city heartbeats—invariant spectral features across data-rich cities—and transferring them to data-scarce targets using a constrained DMD framework, the method yields interpretable, time-invariant modes and improved forecasting performance. The approach leverages time-delay Hankel embeddings (HDMD) to linearize nonlinear city dynamics and identifies ε-maximally shared eigenvalues to guide transfer, validated on multi-city loop-detector data with results showing competitive prediction accuracy and clear advantages over HDMD. The work demonstrates practical impact for traffic management in data-poor cities, providing a principled, low-parameter-tuning mechanism to incorporate transferable temporal patterns and detect changes in patterns such as hysteresis effects across dates.

Abstract

Predicting traffic flow in data-scarce cities is challenging due to limited historical data. To address this, we leverage transfer learning by identifying periodic patterns common to data-rich cities using a customized variant of Dynamic Mode Decomposition (DMD): constrained Hankelized DMD (TrHDMD). This method uncovers common eigenmodes (urban heartbeats) in traffic patterns and transfers them to data-scarce cities, significantly enhancing prediction performance. TrHDMD reduces the need for extensive training datasets by utilizing prior knowledge from other cities. By applying Koopman operator theory to multi-city loop detector data, we identify stable, interpretable, and time-invariant traffic modes. Injecting ``urban heartbeats'' into forecasting tasks improves prediction accuracy and has the potential to enhance traffic management strategies for cities with varying data infrastructures. Our work introduces cross-city knowledge transfer via shared Koopman eigenmodes, offering actionable insights and reliable forecasts for data-scarce urban environments.

Figures (7)

• Figure 1: Our work examines city traffic flow as an amalgam of spectral components hereafter called 'city heartbeats:' We map city dynamics into a space where the nonlinear city traffic dynamics become linear. We perform DMD and break down city traffic into a spectrum of modes characterized by specific frequencies and growth/decay rates, then we identify the shared patterns that could be transferred to downstream tasks in the target city, see Section \ref{['sec:methods']}. This approach differs from traditional time series decomposition methods, which typically categorize data into trend, seasonal, and residual components. In our framework, all DMD modes can be analogously considered as seasonal components, with modes exhibiting infinite periods akin to trend components, and noisy modes potentially truncated (see Fig. \ref{['fig:eigenvalueDisplay']}) serving as residual components. The advantages of our framework can be summarized as follows: (1) Minimal dependency on parameter tuning: We do not rely heavily on precise parameter tuning or extensive training data, the process of delay tuning is described in Section \ref{['sec:exp-linearization']}. (2) Invariant spectral features: The spectral features extracted are also interpretable in the form of characteristic cycle times instead of high-level features, allowing for clearer explanations of the interface between cities. See Section \ref{['sec:exp-eigen-extract']} (3) Transferability validation: We evaluate the validity of the transfer by assessing the performance improvement of both Koopman methods and additional time series forecasting methods that require prior information as discussed in Section \ref{['sec:exp-transfer-beyond']}.
• Figure 2: Visualization of Averaged Temporal Dynamics in Stuttgart Traffic via Koopman Operator Analysis: this figure presents the top three eigenmodes extracted from right-singular eigenvectors of Stuttgart traffic data, spanning several days and depicted in both the original observational space and the transformed high-dimensional Koopman space. Each subplot (a through d) displays the normalized singular vectors corresponding to varied backcasting intervals: 5 minutes, 3 hours, 12 hours, and 24 hours, respectively. These eigenmodes are scaled with their eigenvalues and are maximally normalized within the range of [-1, 1]. They exhibit periodic behaviors that underscore the traffic patterns across the Stuttgart region. The chosen delay or backtest period influences the complexity of the observed patterns: initially, the traffic dynamics demonstrate noisy, complex, high-frequency, non-linear characteristics. Yet, within the expansive Koopman space, these dynamics are simplified into basic quasi-sinusoidal waveforms. The selection of the delay/backcast window balances between promoting linearity and preserving the informative nature of the eigenmodes.
• Figure 3: Polar comparison of Koopman eigenvalues among source cities: (a) Displays all eigenvalues calculated by HDMD with a delay of 300 for Bordeaux, Graz, and Madrid, based on data trained over three consecutive days. (b) For enhanced clarity, only eigenvalues exceeding an amplitude threshold of $10^{-2}$ are shown, determined using an optimal singular value hard thresholding (SVHT) procedure dl2014optimal. There is significant overlap among the eigenvalues on the unit circle, indicating consistent, shared patterns across the cities. Notably, the symmetry along the $\angle \lambda=0$ axis illustrates that the eigenvalues are pairwise conjugated, as they originate from the diagonalization of a real matrix.
• Figure 4: Subpatterns with shared frequencies in Graz and Madrid. We observe that they still differ in both amplitude and phase. Within the Koopman framework, the cross-city shared, and transferrable information is identified from the shared Koopman eigenvalues, while each city's unique characteristics are preserved in both the city-specific Koopman modes and the Koopman eigenvalues.
• Figure 5: Comparison of HDMD and TrHDMD prediction results. The graphs show flow readings of 30 detectors in the target city (Stuttgart) on May 07, 2012. The left graph is the ground truth, the middle graph shows predicted flows using HDMD, and the right graph illustrates the TrHDMD predictions. We observe that TrHDMD has a better prediction performance, with the congested periods and free-flow periods more accurately captured than with HDMD.

Theorems & Definitions (1)

• Definition 1: $\epsilon$-maximally shared eigenvalues