Probabilistic properties of detrended fluctuation analysis for Gaussian processes

Area of Science:

• Time series analysis
• Statistical physics
• Signal processing

Background:

• Detrended fluctuation analysis (DFA) is a key method for detecting long-range dependence in time series data.
• Despite its wide application, the probabilistic underpinnings of DFA remain incompletely understood.
• Existing research highlights DFA's effectiveness as a detrending technique.

Purpose of the Study:

• To investigate the probabilistic properties of Detrended Fluctuation Analysis (DFA) specifically for Gaussian processes.
• To elucidate the distribution of the squared error sum in the detrended process.
• To establish a theoretical foundation for understanding DFA's behavior in statistical analysis.

Main Methods:

• Employing a probabilistic approach to analyze DFA.
• Deriving general formulas for the expected value and variance of the squared fluctuation function.
• Applying these methods to specific Gaussian processes including fractional Brownian motion.

Main Results:

• General formulas for the expected value and variance of DFA's squared fluctuation function for Gaussian processes were derived.
• Analytical results for the expected value were obtained for Gaussian white noise, fractional Gaussian noise, Brownian motion, and fractional Brownian motion.
• Numerical simulations confirmed the accuracy of the derived analytical formulas.

Conclusions:

• The derived formulas provide a theoretical basis for understanding DFA's statistical properties.
• These findings are crucial for analyzing DFA-based estimators of the fluctuation function and long-memory parameter.
• This work lays the groundwork for more rigorous statistical inference using DFA in time series analysis.