Detrended fluctuation analysis

Detrended fluctuation analysis (DFA) is a method for quantifying the long-range correlations and scaling properties of a time series. Introduced by C.-K. Peng and colleagues in 1994, DFA was designed to address a key weakness of classical rescaled-range analysis: the sensitivity of the Hurst exponent estimator to polynomial trends and non-stationarities in the data. The method works by integrating the time series, dividing it into windows of size n, detrending each window (typically with a linear or polynomial fit), and measuring the root-mean-square fluctuation F(n). For a process with power-law correlations, F(n) scales as n^α, where the scaling exponent α is related to the Hurst exponent by α = H for stationary processes.

DFA has become a standard tool in the analysis of complex systems precisely because it separates genuine long-range dependence from artifacts of trend. In cardiac physiology, DFA reveals scaling exponents in heart rate variability that predict mortality risk. In neuroscience, it characterizes the temporal structure of neural spike trains and EEG signals. The method is not without limitations: the choice of polynomial order, the minimum and maximum window sizes, and the presence of crossovers between scaling regimes all affect the estimated exponent. DFA measures correlation structure, but it does not reveal mechanism.

DFA is a filter, not a theory. It tells you that a signal has memory, but it cannot tell you whether that memory arises from feedback loops, network topology, or external forcing. The same scaling exponent can emerge from a neuron, a market, or a climate model — and conflating these mechanisms is the most common error in applied DFA.