Establishing a direct connection between detrended fluctuation analysis and Fourier analysis

Area of Science:

• * Physics
• * Data Analysis
• * Signal Processing

Background:

• * Detrended fluctuation analysis (DFA) is a method for detecting long-range correlations in time series data.
• * Understanding the methodological limitations of DFA, particularly with higher-order polynomial fitting, is crucial for accurate analysis.
• * Fourier analysis provides a frequency-domain perspective on time series, offering complementary insights.

Purpose of the Study:

• * To establish a direct analytical connection between Detrended Fluctuation Analysis (DFA) and Fourier analysis.
• * To investigate the impact of higher-order polynomial fitting on DFA's ability to estimate scaling exponents.
• * To analyze and address scale distortion issues in DFA by relating time and frequency domains.

Main Methods:

• * Exact analytical calculation of the single-frequency response of DFA.
• * Establishing the relationship between DFA scaling exponent (α) and power spectral density (PSD) scaling exponent (β).
• * Analyzing the influence of polynomial order (m) on the detectable scaling exponent range.

Main Results:

• * Higher-order detrending in DFA does not adversely affect the estimation of the scaling exponent α for power-law PSDs, maintaining α=(β+1)/2.
• * The maximum detectable scaling exponent in DFA is analytically bounded by m+1, where m is the polynomial fit order.
• * Scale distortion between DFA's time scale and PSD's frequency scale is identified and can be corrected.

Conclusions:

• * DFA's accuracy in estimating scaling exponents is robust to higher-order detrending for specific stochastic processes.
• * The order of polynomial fitting in DFA directly influences the range of detectable scaling exponents.
• * Analytical insights enable characterization of DFA variants and correction of scale distortions for improved time series analysis.