Real-Time Forecasting of Metro Origin-Destination Matrices with High-Order Weighted Dynamic Mode Decomposition

Forecasting short-term ridership of different origin-destination pairs (i.e., OD matrix) is crucial to the real-time operation of a metro system. However, this problem is notoriously difficult due to the large-scale, high-dimensional, noisy, and highly skewed nature of OD matrices. In this paper, we address the short-term OD matrix forecasting problem by estimating a low-rank high-order vector autoregression (VAR) model. We reconstruct this problem as a data-driven reduced-order regression model and estimate it using dynamic mode decomposition (DMD). The VAR coefficients estimated by DMD are the best-fit (in terms of Frobenius norm) linear operator for the rank-reduced full-size data. To address the practical issue that metro OD matrices cannot be observed in real time, we use the boarding demand to replace the unavailable OD matrices. Moreover, we consider the time-evolving feature of metro systems and improve the forecast by exponentially reducing the weights for historical data. A tailored online update algorithm is then developed for the high-order weighted DMD model (HW-DMD) to update the model coefficients at a daily level, without storing historical data or retraining. Experiments on data from two large-scale metro systems show that the proposed HW-DMD is robust to noisy and sparse data, and significantly outperforms baseline models in forecasting both OD matrices and boarding flow. The online update algorithm also shows consistent accuracy over a long time, allowing us to maintain an HW-DMD model at much low costs.