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

We introduce Kernel Detrended Fluctuation Analysis (kDFA), a multivariate, nonlinear generalization of detrended fluctuation analysis for quantifying long-range persistence in complex systems. We show that kDFA generalizes the traditional variance-based fluctuation functional by replacing it with a kernel cross-covariance-based measure. This formulation connects the estimator to the Hilbert-Schmidt norm of the covariance operator in the reproducing kernel Hilbert space. This allows persistence to be inferred from linear to strongly nonlinear regimes via kernel learning. We showcase kDFA in synthetic and real experiments. On synthetic data, kDFA accurately retrieves Hurst exponents and generalizes the standard DFA to nonlinear cases. Comparisons of the Lorenz systems (both L63 and L96) against iterated amplitude-adjusted Fourier transform surrogates reveal genuine nonlinear persistence beyond linear autocorrelation. Applied to ecosystems, kDFA uncovers robust, long-term coupling between vegetation activity and its drivers across European vegetation sites, and detects patterns in this coupling relative to the long-term vegetation trend. kDFA, thus, provides a scalable, theory-grounded tool to uncover hidden multivariate memory in natural and engineered systems.