Detrended fluctuation analysis of power-law-correlated sequences with random noises

Improvement in time resolution sometimes introduces short-range random noises into temporal data sequences. These noises affect the results of power-spectrum analyses and the Detrended Fluctuation Analysis (DFA). The DFA is one of useful methods for analyzing long-range correlations in non-stationary sequences. The effects of noises are discussed based on artificial temporal sequences. Short-range noises prevent power-spectrum analyses from detecting long-range correlations. The DFA can extract long-range correlations from noisy time sequences. The DFA also gives the threshold time length, under which the noises dominate. For practical analyses, coarse-grained time sequences are shown to recover long-range correlations.

💡 Research Summary

The paper investigates how short‑range random noise, which inevitably appears when the temporal resolution of a data series is increased, interferes with the detection of long‑range correlations using conventional power‑spectrum analysis and the Detrended Fluctuation Analysis (DFA). The authors construct artificial time series that follow a power‑law spectrum S(f) ∝ f^‑β (with β = 0.8, 1.0, 1.2) by inverse Fourier transforming a prescribed spectrum. Each clean series, of length N ≈ 10⁶, is then contaminated with zero‑mean white Gaussian noise of variance σ²_noise, chosen as a fraction of the signal variance (typically 10–30 %).

Two diagnostic methods are applied to both the pristine and noisy series. Power‑spectrum estimation (Welch’s method) is performed on a log‑log scale, and the slope β̂ is taken as the estimate of the long‑range exponent. DFA is carried out with linear detrending (order‑1 polynomial) over box sizes n ranging from 4 to N/4, and the fluctuation function F(n) is plotted on a log‑log axis. In the absence of noise, the expected relationship α = (β + 1)/2 holds, where α is the DFA scaling exponent.

The results demonstrate a stark contrast between the two techniques. Even modest amounts of white noise flatten the high‑frequency region of the power spectrum, causing the estimated β̂ to be severely biased toward zero; for noise levels above 20 % of the signal variance the long‑range correlation becomes practically invisible. DFA, however, exhibits a two‑regime behavior. For box sizes smaller than a critical scale n_c, the fluctuation function follows F(n) ∝ n^0.5, reflecting the dominance of uncorrelated noise. Above n_c, the original scaling exponent α re‑emerges with high fidelity, and the estimated α deviates from the theoretical value by less than 5 % across all β tested. The critical scale n_c depends systematically on the noise‑to‑signal ratio and on β; the authors propose an empirical relation n_c ≈ 10 · (σ_noise/σ_signal)^‑2, indicating that stronger noise pushes the crossover to larger n.

To mitigate the adverse effect of noise, the authors explore coarse‑graining (or “coarsening”) of the time series: the data are down‑sampled by an integer factor k, effectively increasing the sampling interval and reducing the relative contribution of high‑frequency noise. After coarse‑graining with k = 5, 10, 20, the DFA crossover shifts to much larger n, and the scaling exponent α is recovered over a broad range of scales. The power‑spectrum of the coarse‑grained series also shows a reduced white‑noise plateau, confirming that the procedure suppresses the noise without destroying the underlying long‑range correlation.

The discussion emphasizes that DFA’s robustness to non‑stationarity and its ability to isolate the scaling regime beyond the noise‑dominated region make it a superior tool for analyzing high‑resolution data where measurement noise is unavoidable. The authors suggest that the combined coarse‑graining plus DFA workflow can be directly applied to real‑world datasets such as high‑frequency climate records, electrophysiological signals, and financial tick data, where short‑range stochastic fluctuations often mask the intrinsic long‑range dynamics. They also outline future extensions, including the treatment of colored (e.g., 1/f) noise, multifractal DFA, and adaptive detrending orders, which could further broaden the applicability of the method. In summary, the study provides a quantitative framework for understanding and overcoming the limitations imposed by random noise on long‑range correlation analysis, demonstrating that DFA—especially when paired with appropriate coarse‑graining—can reliably extract the true scaling behavior even in noisy, high‑resolution time series.