Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability

Area of Science:

• Complex Systems Analysis
• Nonlinear Dynamics
• Geophysical Fluid Dynamics

Background:

• Many scientific and engineering processes exhibit multiscale spatio-temporal patterns driven by complex dynamics and external forcings.
• Extracting salient modes of variability from incomplete observations is crucial for understanding and predicting these phenomena.
• Classical singular spectrum analysis (SSA) struggles with strongly nonlinear dynamics, such as intermittency and rare events.

Purpose of the Study:

• To develop a novel technique for analyzing high-dimensional, complex time series.
• To recover features characteristic of strongly nonlinear dynamics missed by classical methods.
• To provide a robust method for empirical mode extraction from complex datasets.

Main Methods:

• The proposed technique utilizes Laplacian eigenmaps for time-lagged embedded data.
• This approach creates a reduced data representation preserving nonlinear geometrical structure.
• Truncated singular-value decomposition is then applied to this reduced representation.

Main Results:

• The method successfully captures intermittent modes in a North Pacific general circulation model (Kuroshio current).
• It achieves dimensional reduction in a low-order atmospheric model exhibiting chaotic intermittent regime transitions.
• Demonstrates superior performance over classical SSA in challenging nonlinear scenarios.

Conclusions:

• The developed technique effectively analyzes complex time series with strong nonlinearities.
• It offers a powerful alternative for mode extraction when classical SSA fails.
• This method enhances the understanding of complex phenomena in climate and atmospheric science.