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Kernel detrended fluctuation analysis: A nonlinear, multivariate method for detecting long-range persistence.

Tristan K E Williams1, Homer Durand1, Tobias Braun2

  • 1Image Processing Laboratory, Universitat de València, València, Spain.

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Summary
This summary is machine-generated.

We developed Kernel Detrended Fluctuation Analysis (kDFA), a new method to detect long-range persistence in complex systems. This nonlinear approach reveals hidden memory in ecological and engineered systems.

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Area of Science:

  • Complex systems analysis
  • Nonlinear dynamics
  • Statistical physics

Background:

  • Traditional Detrended Fluctuation Analysis (DFA) quantifies long-range persistence but is limited to linear systems.
  • Complex systems often exhibit nonlinear dynamics and multivariate interactions that are not captured by linear methods.
  • Quantifying nonlinear long-range persistence is crucial for understanding memory in various natural and engineered systems.

Purpose of the Study:

  • Introduce Kernel Detrended Fluctuation Analysis (kDFA), a novel multivariate, nonlinear generalization of DFA.
  • Extend the capability of fluctuation analysis to detect persistence in strongly nonlinear regimes.
  • Provide a scalable and theory-grounded tool for uncovering hidden multivariate memory.

Main Methods:

  • Generalize traditional DFA by replacing variance-based fluctuation functions with kernel cross-covariance measures.
  • Utilize kernel learning to infer persistence across linear to nonlinear regimes.
  • Connect the kDFA estimator to the Hilbert-Schmidt norm of the covariance operator in reproducing kernel Hilbert spaces.

Main Results:

  • kDFA accurately retrieves Hurst exponents on synthetic data, generalizing standard DFA to nonlinear cases.
  • Analysis of Lorenz systems (L63, L96) using kDFA reveals genuine nonlinear persistence beyond linear autocorrelation.
  • Application to European vegetation sites uncovers robust, long-term coupling between vegetation activity and its drivers, revealing patterns relative to trends.

Conclusions:

  • kDFA is a powerful, scalable, and theory-grounded method for quantifying multivariate, nonlinear long-range persistence.
  • The method successfully identifies hidden memory in complex systems, including ecological and engineered domains.
  • kDFA offers a significant advancement over traditional linear methods for analyzing complex system dynamics.