Geometric mode-selection scores for delay-coordinates dynamic mode decomposition

Area of Science:

• Dynamical Systems and Control Theory
• Data Analysis and Machine Learning
• Fluid Dynamics and Spatiotemporal Analysis

Background:

• Delay-coordinates dynamic mode decomposition (DC-DMD) is crucial for extracting spatiotemporal modes from complex time series.
• A significant challenge in DC-DMD is differentiating true dynamic modes from spurious ones caused by noise or incorrect model order.
• Existing methods often struggle with accurate mode selection, impacting the reliability of extracted dynamic features.

Purpose of the Study:

• To develop robust, data-driven scoring criteria for mode selection in DC-DMD.
• To address the challenge of distinguishing dynamically meaningful modes from noise-induced spurious modes.
• To provide a framework for improved accuracy in identifying coherent structures in high-dimensional time series.

Main Methods:

• Framing mode selection as a subspace geometry problem, analyzing the concentration of modes within signal subspaces.
• Developing two complementary, data-driven scoring criteria based on geometric residual and operator-theoretic analysis.
• Utilizing a block-companion formulation to reveal the Kronecker-Vandermonde structure of modes in delay embeddings.

Main Results:

• Demonstrated that true modes concentrate in a low-dimensional subspace, while spurious modes exhibit broader distribution.
• Introduced two novel scores that effectively rank modes based on geometric properties and conformity to the Kronecker-Vandermonde structure.
• Numerical experiments showed superior performance of the proposed scores over baseline methods, particularly in the small-spatial-dimension regime, as measured by PR-AUC.

Conclusions:

• Mode selection in DC-DMD is fundamentally a geometric problem related to subspace structure.
• The proposed scoring criteria offer a robust and data-driven approach to enhance the reliability of DC-DMD analysis.
• This framework provides a clearer understanding of mode behavior and improves the distinction between true and spurious modes in time series decomposition.