From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction

TL;DR

This paper introduces spectral methods for long-term forecasting of time series from linear and nonlinear dynamical systems, extending Fourier techniques with Koopman theory for improved accuracy and uncertainty quantification.

Contribution

It develops novel spectral algorithms that do not assume periodicity, handle nonlinearities via Koopman theory, and enable scalable, global optimization in forecasting.

Findings

Algorithms outperform existing methods on synthetic data

Effective in real-world power systems and fluid flow forecasting

Provides uncertainty quantification through spectral analysis

Abstract

We propose spectral methods for long-term forecasting of temporal signals stemming from linear and nonlinear quasi-periodic dynamical systems. For linear signals, we introduce an algorithm with similarities to the Fourier transform but which does not rely on periodicity assumptions, allowing for forecasting given potentially arbitrary sampling intervals. We then extend this algorithm to handle nonlinearities by leveraging Koopman theory. The resulting algorithm performs a spectral decomposition in a nonlinear, data-dependent basis. The optimization objective for both algorithms is highly non-convex. However, expressing the objective in the frequency domain allows us to compute global optima of the error surface in a scalable and efficient manner, partially by exploiting the computational properties of the Fast Fourier Transform. Because of their close relation to Bayesian Spectral Analysis, uncertainty quantification metrics are a natural byproduct of the spectral forecasting methods. We extensively benchmark these algorithms against other leading forecasting methods on a range of synthetic experiments as well as in the context of real-world power systems and fluid flows.