  Forecasting the short-term ridership among origin-destination pairs (OD
matrix) of a metro system is crucial in real-time metro operation. However,
this problem is notoriously difficult due to the high-dimensional, sparse,
noisy, and skewed nature of OD matrices. This paper proposes a High-order
Weighted Dynamic Mode Decomposition (HW-DMD) model for short-term metro OD
matrices forecasting. DMD uses Singular Value Decomposition (SVD) to extract
low-rank approximation from OD data, and a high-order vector autoregression
model is estimated on the reduced space for forecasting. To address a practical
issue that metro OD matrices cannot be observed in real-time, we use the
boarding demand to replace the unavailable OD matrices. Particularly, we
consider the time-evolving feature of metro systems and improve the forecast by
exponentially reducing the weights for old data. Moreover, we develop a
tailored online update algorithm for HW-DMD to update the model coefficients
daily without storing historical data or retraining. Experiments on data from a
large-scale metro system show the proposed HW-DMD is robust to the 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 when maintaining an HW-DMD model at low
costs.
