Data-adaptive harmonic spectra and multilayer Stuart-Landau models.

Area of Science:

• Dynamical systems analysis
• Time series decomposition
• Spectral analysis

Background:

• Multivariate time series analysis often requires complex models.
• Extracting meaningful dynamics from complex data remains challenging.
• Existing methods may not fully capture data-adaptive features.

Purpose of the Study:

• To develop a novel harmonic decomposition method for multivariate time series.
• To introduce data-adaptive harmonic (DAH) modes and associated spectra.
• To demonstrate the effectiveness of DAH decomposition in modeling complex systems.

Main Methods:

• Integral operator approach with periodic semigroup kernels.
• Derivation of spectral decomposition theorems for mixing invariant measures.
• Definition of multidimensional power and phase spectra based on data-adaptive eigenmodes.
• Application of DAH decomposition to multilayer stochastic models (MSMs).

Main Results:

• Eigenvalues correspond to singular values of cross-spectral matrices, grouped by Fourier frequency.
• DAH modes exhibit data-adaptive phases, enabling multidimensional phase spectrum definition.
• DAH decomposition simplifies modeling to frequency-stacked elemental models.
• Successful application to Lorenz 96 and stochastic heat equation models.

Conclusions:

• DAH decomposition effectively extracts spatio-temporal modes and reveals dynamical features.
• Multilayer Stuart-Landau models (MSLMs) accurately capture time-evolving field patterns and statistics.
• The DAH framework offers a powerful tool for analyzing and modeling complex time series data.