Predictive Nonlinear Modeling by Koopman Mode Decomposition

Machine learning has countless applications in time series analysis: controlling smart grids, detecting mechanical failures, and analyzing stock prices. Fourier mode decomposition (FMD) is the most common method of analysis because it decomposes time series into finite waveform components, or modes, but its principal shortcoming is that FMD assumes every mode has a constant amplitude, an assumption that rarely holds in real-world data. In contrast, Koopman mode decomposition (KMD) can detect modes with exponentially-increasing or - decreasing amplitudes, although it has mostly been applied to diagnosing data errors, not to prediction. What has kept KMD from being applied to prediction is partly a shortcoming in a mathematical formulation. This paper seeks to remedy that shortcoming: it provides a mathematically-precise formulation of KMD as a practical tool. This formulation, in turn, allows us to develop a novel practical method for prediction of future data. We further demonstrate our method's effectiveness using both synthetic data and real plasma flow data.