Eigenvalues of autocovariance matrix: A practical method to identify the Koopman eigenfrequencies

Area of Science:

• Dynamical Systems Theory
• Spectral Analysis
• Time Series Analysis

Background:

• The Koopman operator provides a powerful framework for analyzing nonlinear dynamical systems by linearizing their dynamics in an infinite-dimensional function space.
• Inferring the spectral properties of the infinite-dimensional Koopman operator from finite time series data is a significant challenge in data-driven analysis.
• Existing methods like Singular Spectrum Analysis (SSA) and Dynamic Mode Decomposition (DMD) offer empirical approaches but lack a unified theoretical foundation.

Purpose of the Study:

• To establish a theoretical link between the leading eigenvalues of the autocovariance matrix of an observable and the eigenvalues of the Koopman operator.
• To provide a rigorous mathematical foundation for data-driven methods used in dynamical systems analysis.
• To develop a practical, theorem-based methodology for identifying Koopman operator eigenfrequencies from time series data.

Main Methods:

• Construction of a Hilbert space H_f and a Koopman-like operator K acting on it for any observable f with existing time-delayed autocovariances.
• Proof of a one-to-one correspondence between the leading eigenvalues of the autocovariance matrix and the energy of f (eigenvectors of K).
• Utilizing representation theorems of isometric operators and the weak-mixing property of observables for theoretical proofs.

Main Results:

• Demonstrated that leading eigenvalues of the autocovariance matrix directly correspond to Koopman operator eigenvalues (energy of f).
• Showed that for ergodic systems with finite invariant measures, H_f and K align with the Krylov subspace and classical Koopman operator, respectively.
• Established that leading temporal empirical orthogonal functions correspond to Koopman eigenfrequencies in such systems.
• Proposed a practical methodology based on the convergence of renormalized Gram matrix eigenvalues for identifying Koopman eigenfrequencies.

Conclusions:

• The study provides a strong theoretical foundation for empirical methods like SSA, DAHD, Hankel DMD, and HAVOK by linking them to Koopman operator spectral theory.
• A novel, theorem-based approach is presented for extracting Koopman eigenfrequencies from time series, validated with numerical examples and real-world oceanographic data.
• The findings enhance our understanding of spectral analysis in dynamical systems and offer a robust tool for time series decomposition and prediction.