Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes

Stochastic fractal signals can be characterized by the Hurst coefficient $H,$ which is related to the exponents of various power-law statistics characteristic of these processes. Two techniques widely used to estimate $H$ are spectral analysis and detrended fluctuation analysis (DFA). This paper examines the analytical link between these two measures and shows that they are related through an integral transform. Numerical simulations confirm this relationship for ideal synthesized fractal signals. Their performance as estimators of $H$ is compared based on a mean square error criterion and found to be similar. DFA measures are derived for physiological signals of heartbeat $R-R$ intervals through the integral transform of a spectral density estimate. These agree with directly calculated DFA estimates, indicating that the relationship holds for signals with nonideal fractal properties. It is concluded that DFA and spectral measures provide equivalent characterizations of stochastic signals with long-term correlation.