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. 2012:2:315.
doi: 10.1038/srep00315. Epub 2012 Mar 14.

Revisiting detrended fluctuation analysis

R M Bryce  1 , K B Sprague
Affiliations

Revisiting detrended fluctuation analysis

R M Bryce et al. Sci Rep. 2012.

Abstract

Half a century ago Hurst introduced Rescaled Range (R/S) Analysis to study fluctuations in time series. Thousands of works have investigated or applied the original methodology and similar techniques, with Detrended Fluctuation Analysis becoming preferred due to its purported ability to mitigate nonstationaries. We show Detrended Fluctuation Analysis introduces artifacts for nonlinear trends, in contrast to common expectation, and demonstrate that the empirically observed curvature induced is a serious finite-size effect which will always be present. Explicit detrending followed by measurement of the diffusional spread of a signals' associated random walk is preferable, a surprising conclusion given that Detrended Fluctuation Analysis was crafted specifically to replace this approach. The implications are simple yet sweeping: there is no compelling reason to apply Detrended Fluctuation Analysis as it 1) introduces uncontrolled bias; 2) is computationally more expensive than the unbiased estimator; and 3) cannot provide generic or useful protection against nonstationaries.

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Figures

Figure 1
Figure 1. Fluctuation plots.
DFA (left column), and FA (right column) characterize a signal by integrating a fluctuating signal and analyzing the resulting integrated walk. The top row shows the fluctuation plots (diffused distance) for fBn with H = 0.3; significant curvature for small time windows can be seen for DFA. Unfortunately, this regime provides the most statistics. The local slope (middle row) corresponds to the estimated Hurst exponent; the black line indicates the specified value and the dashed line 2X this value. In the bottom row, the fluctuation plot is scaled by the nominal power law fit resulting in a normalized fluctuation plot. The superimposed grey lines are values averaged over 50 trials.
Figure 2
Figure 2. Source of fluctuation plot curvature.
The range (top panel) and standard deviation (bottom panel) for uncorrelated (H = 0.5) unit Gaussian noise display rapid decay with significant underestimation of the dispersion as the sample size becomes small and falls towards zero. This sample size dependent bias induces curvature in fluctuation plots. Note that the curvature is more pronounced in the range and does not saturate; this is reflected in the empirically noted improvement of DFA (which uses the standard deviation) over R/S Analysis (range).
Figure 3
Figure 3. Fluctuation plots for λ-DNA.
Following Ref. the FA (dots) and DFA (squares) fluctuation plots are shown, normalized here by (Δn)0.5 (top panel). Note the concave up curvature of the FA results, and the approximately flat DFA results. Karlin & Brendel model predictions (here recast into linear form, formula image) for λ-DNA (middle, lower panels) are close to the observed FA measured fluctuations (dots). A linear fit (grey line) indicates the Karlin & Brendal model is suitable; for two patch types an analytic prediction of fluctuations (grey squares) captures the observed λ-DNA fluctuations (black dots) surprisingly well, where the patch biases required for prediction are empirically estimated from the sequence. Some short-range structure, visible when inspecting the residual between the model and the data (bottom panel), is not accounted for by the two-type patch model, and is attributable to a notable short-range correlation in the DNA sequence, where base-pairs which are separated by two others are (positively) correlated.
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