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

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.