From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction (2004.00574v1)

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.

Spectral Methods for Long-term Time Series Prediction

The paper "From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction" by Henning Lange, Steven L. Brunton, and J. Nathan Kutz presents advanced spectral methods for forecasting time series data derived from both linear and nonlinear quasi-periodic dynamical systems.

Overview of Contributions

The authors introduce algorithmic techniques leveraging spectral methods to address the challenges of long-term forecasting in dynamical systems, encompassing both linear and nonlinear regimes. The paper is bifurcated into two principal methodologies:

1. Linear Approach Using Fourier-like Spectral Methods: The authors propose an algorithm that mimics the Fourier transform but abandons the periodicity constraint, thus providing the flexibility to forecast over arbitrary sampling intervals. This method computes global optima efficiently by representing the non-convex optimization objective in the frequency domain, utilizing the Fast Fourier Transform (FFT).
2. Nonlinear Approach Extending Koopman Theory: To handle nonlinearities, the research extends the above methodology using Koopman theory—an established framework positing that a nonlinear dynamic system can be linearized in an infinite-dimensional space via a set of observable functions. This extension decomposes time series data spectrally within a nonlinear, data-driven basis.

Algorithmic Innovations

These methods collectively offer robust uncertainty quantification metrics, derived as natural byproducts of spectral analysis. The paper stresses the importance of these computational techniques in global optimization without compounding errors typical in recursive forecasting models. Additionally, the paper delineates a novel algorithmic design that merges FFT-based global optimization with gradient descent, effectively refining initial guesses and subverting implicit periodicity assumptions associated with FFT.

Comparative Analysis

The effectiveness of these methods is tested against other prevalent forecasting techniques on several synthetic datasets and real-world scenarios, such as power systems and fluid dynamics. Key distinguishing attributes of the proposed methods when juxtaposed with traditional models include:

• Enhanced resilience to noise, yielding stable long-term predictions even under greater uncertainty.
• Superior scalability due to the computational efficiency of FFT in evaluating the error surface.
• Ability to extract meaningful frequencies from data phases with minimal prior training, leveraging global spectral properties.

Implications and Future Directions

On a practical front, these methodologies furnish more accurate long-term prediction capabilities across diverse scientific domains, espousing robustness and computation speed. Theoretically, the melding of Koopman theory with spectral decomposition paradigms invigorates potential future investigations into high-dimensional nonlinear systems and controlled dynamical frameworks.

For advancing AI research further, the integration of statistical spectral theories with machine learning holds promise for developing algorithmic structures that enhance interpretability and bias correction in dynamic modeling. Future explorations could involve extending the methodology to iteratively model input-output systems with external forcings, thus broadening applicability and achieving higher generalization capacities in complex environments.

By contributing novel spectral analysis tools and advancing computational methodologies, this paper presents substantive enhancements in the domain of time series prediction, reinforcing the application and theoretical scope of dynamical spectral methods.