Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series

By J. Kantelhardt, S. Zschiegner, Eva Koscielny-Bunde, Shlomo Havlin, Armin Bunde, H. Eugene Stanley. 2002

3,234 citations recorded by the source corpus.

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 with 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 method, and show that the results are equivalent.

Subfields: Economics and Econometrics, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3702630f1bb236789437eda94b2fb532897345aa

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Citations in the corpus

p-exponent and p-leaders, Part II: Multifractal Analysis. Relations to Detrended Fluctuation Analysis (2015, methodology): “[23] for the founding article, and [24, 25] for related developments).”
Multifractality in combustion noise: predicting an impending combustion instability (2014, methodology): “Multifractality in a time series can come about in two ways: (i) due to a broad probability distribution of the data points, e.g. a Levy distribution, and (ii) due to different long-term correlations of the small- and large-scale fluctuations (Kantelhardt et al. 2002).”
Detrending moving-average cross-correlation coefficient: Measuring cross-correlations between non-stationary series (2013, methodology): “Apart from the analysis of the power laws in the cross-correlation function itself, Zebende [17] proposed the DCCA cross-correlation coefficient as a combination of DCCA and the detrended fluctuation analysis (DFA) [10, 11, 7].”
Persistence in eye movement during visual search (2016, methodology): “The MF-DFA provides the multifractal spectrum identifying the q-order fluctuation function, Fq, in the fractal structure within time periods with large and small fluctuations.”
Statistical properties and multifractality of Bitcoin (2017, methodology): “The relationship between the generalized Hurst exponent and the multifractal scaling exponent τ(q), defined from the standard multifractal formalism, is given by[15] τ(q) = qh(q)− 1.”
Multifractal methodology (2016, methodology): “In details, MDFA consists of the following steps.”
Multifractal detrended cross-correlations between the Chinese exchange market and stock market (2012, methodology): “Since then, DFA and its multifractal generalization, MF-DFA [16], have been widely used in the detection of long-range auto-correlations in financial markets, including the stock market [17,18] and foreign exchange market [19–21].”
A multifractal approach for stock market inefficiency (2008, methodology): “[32].”
A new approach to quantify power-law cross-correlation and its application to commodity markets (2011, background): “Empirical evidence supports the existence of fractals or multifractals in commodity or financial markets [1-6].”
An analysis of stock market efficiency: Developed vs Islamic stock markets using MF-DFA (2014, background): “[19], to take into account a global detection of multifractal behavior in measuring long-range dependence.”
A multifractal description of wind speed records (2004, methodology): “(7), we get, [22] αh = h(q) + q dh(q) dq , f(αh) = q[αh − h(q)] + 1 (9) The generalized exponents h(q) in Eqn.”