HIGHER ORDER EXTENDED DYNAMIC MODE DECOMPOSITION BASED ON THE STRUCTURED TOTAL LEAST SQUARES

We develop a data-driven approach for analyzing the underlying dynamics from snapshots, which is called the higher order extended dynamic mode decomposition (HOEDMD) in this paper. The HOEDMD method, generalizing the extended dynamic mode decomposition, can handle the case when the spectral complexity of the dynamical system exceeds its spatial complexity. Moreover, the proposed method is capable of analyzing the snapshots taken from multiple trajectories by a mode-frequency-individual decomposition. We also introduce the structured total least squares technique for denoising and debiasing purposes and discuss efficient implementation details. The ability of our proposed method to accurately retrieve the modes with frequencies in linear dynamical systems is proved, which further provides an empirical choice for an optimal order. Finally, we evaluate the proposed structured total least squares based HOEDMD algorithm and apply it to four kinds of dynamical systems: a synthetic linear system to show that the proposed algorithm is less sensitive to the noises; a nonlinear dynamical system of iterates from a multilinear PageRank model to illustrate the necessity of introducing higher order cases; real-world signals for time series classification to indicate individual coefficients could parameterize trajectories and kernel tricks can be employed to enhance its performance on nonlinear real-world systems; and a real-world dynamical system of fMRI data to show the proposed algorithm retrieves modes more stably over several other dynamic mode decomposition variants.