Multifractal detrended fluctuation analysis of nonstationary time series

Abstract

We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series to those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima (WTMM) method, and show that the results are equivalent.Comment: 14 pages (RevTex) with 10 figures (eps

Statistical properties of the seismic noise field: influence of soil heterogeneities

“…We define the de-trended time-series for segment duration τ , denoted by F i (τ ), as the difference between the original time-series and the fit. Since different degrees of the fitting polynomial only differ in their capability of eliminating trends in the data without changing the scaling behaviour of the de-trended profile, we only refer to which is equivalent to Hurst's analysis (Kantelhardt et al 2002). Finally, we average over all segments to obtain the qth order fluctuation function…”

“…Thus, we usually exclude scales τ > N/6 when calculating α. On the other hand, systematic deviations from the scaling behaviour will occur for very small timescales τ ≤ 10/(sampling rate) since the scaling behaviour is only approached asymptotically (Kantelhardt et al 2002). These deviations, being intrinsic to the standard detrended fluctuation analysis, would lead to an overestimation of the fluctuation exponent for very short timescales and, therefore, would limit our capability of determining the correct correlation behaviour.…”

“…These deviations, being intrinsic to the standard detrended fluctuation analysis, would lead to an overestimation of the fluctuation exponent for very short timescales and, therefore, would limit our capability of determining the correct correlation behaviour. To improve the reliability for small timescales (0.04 s ≤ τ ≤ 0.1 s), Kantelhardt et al (2002) suggested using a correction function consisting of uncorrelated data. This means that the fluctuation function F q (τ ) has to be divided by the shuffled fluctuation function F q shuff (τ ) in which the data are put into a random order, thus leading to the destruction of all correlations.…”

“…The dependence on q can have two major origins, the distribution of the data itself and their correlation structure (Kantelhardt et al 2002). Since it is quite difficult to elucidate any information onto the correlation structure of the data just by the analysis of F q (τ ), we will limit our study to the standard detrended fluctuation analysis using q = 2, meaning that the qth order moments of the time-series are considered stationary (Kantelhardt et al 2002).…”

“…Since it is quite difficult to elucidate any information onto the correlation structure of the data just by the analysis of F q (τ ), we will limit our study to the standard detrended fluctuation analysis using q = 2, meaning that the qth order moments of the time-series are considered stationary (Kantelhardt et al 2002). In this case, the exponent α can be directly expressed by means of α ≡ 2H with H being the Hurst exponent (Hurst 1951), synonymously characterizing the type of self-affinity, that is, scaleinvariance of the wavefield.…”