Anti-circulant dynamic mode decomposition with sparsity-promoting for highway traffic dynamics analysis

TL;DR

This paper introduces circDMDsp, a data-driven framework for analyzing highway traffic dynamics by combining anti-circulant data augmentation with sparsity-promoting dynamic mode decomposition. By enlarging observables via an anti-circulant operator and enforcing sparsity on dynamic amplitudes, circDMDsp yields stable, interpretable dynamic modes that capture mean, daily, and weekly patterns in traffic speed while robustly denoising data and enabling long-term prediction. The Seattle traffic case study demonstrates superior reconstruction and forecast accuracy over standard DMD variants, reveals oscillatory modes with periods tied to daily and weekly cycles, and provides insights into short-term predictability limits and sensor-level variability. These results suggest circDMDsp as a practical tool for ITS applications, capable of informing capacity planning, real-time management, and downstream forecasting while offering a pathway to online and missing-data extensions.

Abstract

Highway traffic states data collected from a network of sensors can be considered a high-dimensional nonlinear dynamical system. In this paper, we develop a novel data-driven method -- anti-circulant dynamic mode decomposition with sparsity-promoting (circDMDsp) -- to study the dynamics of highway traffic speed data. Particularly, circDMDsp addresses several issues that hinder the application of existing DMD models: limited spatial dimension, presence of both recurrent and non-recurrent patterns, high level of noise, and known mode stability. The proposed circDMDsp framework allows us to numerically extract spatial-temporal coherent structures with physical meanings/interpretations: the dynamic modes reflect coherent spatial bases, and the corresponding temporal patterns capture the temporal oscillation/evolution of these dynamic modes. Our result based on Seattle highway loop detector data showcases that traffic speed data is governed by a set of periodic components, e.g., mean pattern, daily pattern, and weekly pattern, and each of them has a unique spatial structure. The spatiotemporal patterns can also be used to recover/denoise observed data and predict future values at any timestamp by extrapolating the temporal Vandermonde matrix. Our experiments also demonstrate that the proposed circDMDsp framework is more accurate and robust in data reconstruction and prediction than other DMD-based models.

Figures (15)

• Figure 1: A: Illustration of the Koopman operator for nonlinear dynamical system; B: an example of applying Koopman operator on a nonlinear dynamical system.
• Figure 2: An illustration of the proposed circDMDsp when $\tau = 3$ for a $4\times 5$ matrix $\boldsymbol{X}$. The model includes four steps: (1) applying anti-circulant operator to enlarge the number of observables; (2) using DMD to decompose the anti-circulant matrix into dynamic modes, amplitude and temporal evolution; (3) employing a sparsity-promoting method (DMDSP) to obtain a concise model; (4) utilizing the anti-circulant structure to conduct historical data reconstruction and future values prediction.
• Figure 3: (a): The highway traffic speed data collected from Seattle, USA with 5-minutes resolution; (b): the cumulative eigenvalue percentage (CEP) of training data calculated by singular value decomposition. The red dots represent the first eigenvalue that reaches 80% and 90% of CEP, respectively. (top: SB, bottom: NB).
• Figure 4: (a): The average traffic speed with standard deviation in SB (top) and NB (bottom). The blue line is for weekdays, and the red line is for weekends, and the gray area denotes the average traffic speed $\pm$ standard deviation; (b): The average traffic speed of each sensor during the training period in SB (left) and NB (right). The sensors are installed on four highways: I-5, I-90, I-405, and SR-520 (North up).
• Figure 5: The eigenvalues $\boldsymbol{\lambda}$ with the unit circle on the complex plane in SB. The dot in blue denotes steady dynamic modes and the cross in red denotes varying dynamic modes. The sum of absolute differences between $|\boldsymbol{\lambda}|$ and $\boldsymbol{1}$ is shown in the title.