Sparse-Coded Time-Delay Graph DMD for Nonlinear State-Space Modeling on Graphs

This study proposes a method for predicting river water-level distribution using time-delay dynamic mode decomposition (tdDMD) and sparse modeling using a graph filter bank (GFB). In a previous study, sparse-coded dynamic mode decomposition on graph (SC-DMD-G) proposed by Arai et al. (2021) enabled the derivation of the time-evolution equation of water-level changes on a graph using a data-driven approach. However, SC-DMD-G has limitations in accurately representing nonlinear and complex variations in river water levels, making it challenging to improve its prediction accuracy. Therefore, tdDMD was integrated into SC-DMD-G to achieve more precise linear modeling of nonlinear dynamical systems. To validate the effectiveness of the proposed method, two types of experiments were conducted: one using artificial data and the other using real observed data. In the artificial data experiment, time-series data generated using a process-driven model, i.e., a tank model, are used to compare the reproduction performances among dynamic mode decomposition (DMD), tdDMD, SC-DMD-G, and the proposed method. In the real observed data experiment, water-level observation data collected via web scraping were used to evaluate river water-level distribution prediction during non-rainfall periods to compare the prediction performance of the proposed method with existing methods. These experimental results demonstrate that the proposed method outperforms DMD, tdDMD, and SC-DMD-G in terms of reproduction and prediction performance.