Abstract
Detrended fluctuation analysis (DFA) is a popular tool in physiological and medical studies for estimating the self-similarity coefficient, α, of time series. Recent researches extended its use for evaluating multifractality (where α is a function of the multifractal parameter q) at different scales n. In this way, the multifractal-multiscale DFA provides a bidimensional surface α(q,n) to quantify the level of multifractality at each scale separately. We recently showed that scale resolution and estimation variability of α(q,n) can be improved at each scale n by splitting the series into maximally overlapped blocks. This, however, increases the computational load making DFA estimations unfeasible in most applications. Our aim is to provide a DFA algorithm sufficiently fast to evaluate the multifractal DFA with maximally overlapped blocks even on long time series, as usually recorded in physiological or clinical settings, therefore improving the quality of the α(q,n) estimate. For this aim, we revise the analytic formulas for multifractal DFA with first- and second-order detrending polynomials (i.e., DFA1 and DFA2) and propose a faster algorithm than the currently available codes. Applying it on synthesized fractal/multifractal series we demonstrate its numerical stability and a computational time about 1% that required by traditional codes. Analyzing long physiological signals (heart-rate tachograms from a 24-h Holter recording and electroencephalographic traces from a sleep study), we illustrate its capability to provide high-resolution α(q,n) surfaces that better describe the multifractal/multiscale properties of time series in physiology. The proposed fast algorithm might, therefore, make it easier deriving richer information on the complex dynamics of clinical signals, possibly improving risk stratification or the assessment of medical interventions and rehabilitation protocols.
Highlights
In the mid of the 90s, the algorithm of detrended fluctuation analysis (DFA) gave a great boost to the study of fractal physiology providing an easy-to-calculate method for evaluating the Hurst’s exponent of physiological times series (Castiglioni et al, 2017)
These findings suggest that if the time series is composed by fractional Gaussian noises and/or fractional Brownian motions, DFA1 is preferable to DFA2 at the shorter scales, while at the larger scales the choice may depend on the sign of q, and if q = 0 averaging the DFA1 and DFA2 coefficients might improve the estimate
Since the DFA introduction, hundredths of works in physiological or clinical settings quantified the fractal dynamics by α
Summary
In the mid of the 90s, the algorithm of detrended fluctuation analysis (DFA) gave a great boost to the study of fractal physiology providing an easy-to-calculate method for evaluating the Hurst’s exponent of physiological times series (Castiglioni et al, 2017). Multifractal series are composed by interwoven fractal processes and the multifractal DFA approach extends the calculation of the secondorder fluctuations function to a range of positive and negative moments that includes q = 2 (the second-order moment): positive q moments amplify the contribution of fractal components with larger amplitude and negative q moments the contribution of fractal components with smaller amplitude (Kantelhardt et al, 2002) This led to the calculation of the local slope of the qth-order fluctuation function, α(q,n), as the slope of the regression lines fitting Fq(n) over a running window at each q separately (Gieraltowski et al, 2012). It should be considered that the scale resolution achievable with the regression line method is limited because the window width cannot be too short to make α insensitive to the estimation variability
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