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Delay differential analysis of time series.

Claudia Lainscsek1, Terrence J Sejnowski

  • 1Howard Hughes Medical Institute, Computational Neurobiology Laboratory, Salk Institute for Biological Studies, La Jolla, CA 92037, U.S.A. and Institute for Neural Computation, University of California San Diego, La Jolla, CA 92093, U.S.A. claudia@salk.edu.

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This study introduces a novel time-domain spectral analysis for time series, utilizing delay differential equations (DDEs) for enhanced signal detection and classification. The method offers improved temporal resolution and robustness to noise for nonlinear dynamical systems.

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Area of Science:

  • Dynamical Systems and Time Series Analysis
  • Nonlinear Signal Processing
  • Computational Neuroscience

Background:

  • Nonlinear dynamical system analysis, traditionally using delay or derivative embeddings, is crucial for time series modeling, prediction, detection, and classification.
  • Existing embedding methods provide insights into system dynamics but can be extended to incorporate multiple timescales and functional relationships.
  • Delay differential analysis (DDA) offers a framework combining derivative and nonuniform delay embeddings for more comprehensive system representation.

Purpose of the Study:

  • To develop and validate a novel time-domain spectral analysis toolbox based on delay differential equations (DDEs) for signal detection and classification.
  • To demonstrate the capability of DDEs to support spectral analysis, including frequency detection, phase coupling, and bispectra computation.
  • To provide a robust, high-temporal-resolution alternative to traditional frequency-domain methods for analyzing short, sparse, or noisy time series data.

Main Methods:

  • Utilized functional embeddings, combining derivative and nonuniform delay embeddings, to construct small delay differential equation (DDE) models from time series data.
  • Employed nonlinear correlation functions in the time domain for spectral analysis, including frequency detection, frequency and phase couplings, and bispectra.
  • Developed a multivariate extension of the discrete Fourier transform (DFT) for frequency analysis and a linear, multivariate alternative to multidimensional fast Fourier transform for higher-order spectra.

Main Results:

  • Showcased that DDE properties enable efficient and noise-robust spectral analysis in the time domain using short time windows.
  • Demonstrated that the DDE-based framework extends to cross-trial and cross-channel spectra when multiple short data segments are available.
  • Achieved higher temporal resolution and increased information on frequency and phase couplings compared to traditional frequency-based methods like DFT.

Conclusions:

  • The proposed time-domain toolbox based on delay differential equations offers a powerful and flexible approach for analyzing nonlinear dynamical systems.
  • This method provides a straightforward implementation for higher-order spectra across time, surpassing limitations of frequency-domain techniques.
  • The DDE-based spectral analysis is suitable for diverse applications involving short, sparse, or noisy time series data, enhancing signal detection and classification capabilities.