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Dimensionless embedding for nonlinear time series analysis.

Yoshito Hirata1, Kazuyuki Aihara1

  • 1Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan.

Physical Review. E
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Infinite-dimensional delay coordinates (InDDeCs) offer faster, more accurate predictions for complex systems. This study validates InDDeCs for broader nonlinear time series analysis, including chaos prediction and data embedding.

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Area of Science:

  • Nonlinear dynamics
  • Time series analysis
  • Chaos theory

Background:

  • Conventional delay coordinates have limitations in reconstructing high-dimensional dynamics.
  • Infinite-dimensional delay coordinates (InDDeCs) offer potential advantages in computational speed and prediction accuracy.
  • The applicability of InDDeCs beyond short-term prediction in nonlinear time series analysis remains underexplored.

Purpose of the Study:

  • To provide theoretical justification for using InDDeCs in various nonlinear time series analysis applications.
  • To demonstrate the utility of InDDeCs for reconstructing underlying dynamics from scalar time series.
  • To showcase the performance of InDDeCs using real-world weather data.

Main Methods:

  • Theoretical analysis supporting the use of InDDeCs.
  • Numerical simulations for applications including recurrence plots, correlation dimensions, and maximal Lyapunov exponents.
  • Testing directional couplings and extracting slow-driving forces from time series data.
  • Validation using empirical weather data.

Main Results:

  • InDDeCs provide theoretical support for their application in nonlinear time series analysis.
  • Demonstrated successful application of InDDeCs for calculating key dynamical invariants (recurrence plots, correlation dimensions, Lyapunov exponents).
  • Validated the effectiveness of InDDeCs in identifying directional couplings and extracting driving forces.
  • Showcased robust performance of InDDeCs on complex, real-world weather data.

Conclusions:

  • InDDeCs are theoretically justified and practically effective for a wide range of nonlinear time series analysis tasks.
  • InDDeCs enable faster and more reliable computations compared to conventional methods.
  • The findings support the potential of InDDeCs for achieving 'dimensionless embedding' in dynamical system analysis.