Eva Koscielny-Bunde

Bio: Eva Koscielny-Bunde is an academic researcher from University of Giessen. The author has contributed to research in topics: Multifractal system & Detrended fluctuation analysis. The author has an hindex of 10, co-authored 16 publications receiving 6720 citations. Previous affiliations of Eva Koscielny-Bunde include Bar-Ilan University & Potsdam Institute for Climate Impact Research.

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 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.

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.

Detecting long-range correlations with detrended fluctuation analysis

Abstract: We examine the detrended fluctuation analysis (DFA), which is a well-established method for the detection of long-range correlations in time series. We show that deviations from scaling which appear at small time scales become stronger in higher orders of DFA, and suggest a modified DFA method to remove them. The improvement is necessary especially for short records that are affected by non-stationarities. Furthermore, we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of DFA.

Long-term persistence and multifractality of precipitation and river runoff records

Abstract: [1] We discuss and compare the multifractal temporal scaling properties of precipitation and river discharge records on large timescales. To detect long-term correlations and multifractal behavior in the presence of trends, we apply recently developed methods (detrended fluctuation analysis (DFA) and multifractal DFA) that can systematically detect nonstationarities and overcome trends in the data at all timescales. We find that above some crossover time that usually is several weeks, the daily runoffs are characterized by an asymptotic scaling exponent that indicates a slow power law decay of the runoff autocorrelation function and varies from river to river in a wide range. Below the crossovers, pronounced short-term correlations occur. In contrast, most of the precipitation series show scaling behavior corresponding to a rapid decay of the autocorrelation function. For the multifractal characterization of the data we determine the generalized Hurst exponents and fit them by three operational models. While the fits based on the universal multifractal model describe well the scaling behavior of the positive moments in nearly all runoff and precipitation records, positive as well as negative moments are consistent with two-parameter fits from a modified version of the multiplicative cascade model for all runoff records and most of the precipitation records. For some precipitation records with weak multifractality, however, a simple bifractal characterization gives the best fit of the data.

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 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.

Effect of trends on detrended fluctuation analysis.

Abstract: scaling behavior. We find that crossovers result from the competition between the scaling of the noise and the ‘‘apparent’’ scaling of the trend. We study how the characteristics of these crossovers depend on ~i! the slope of the linear trend; ~ii! the amplitude and period of the periodic trend; ~iii! the amplitude and power of the power-law trend, and ~iv! the length as well as the correlation properties of the noise. Surprisingly, we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws—i.e., long-range power-law dependence of the position of the crossover on the parameters of the trends. We show that the DFA result of noise with a trend can be exactly determined by the superposition of the separate results of the DFA on the noise and on the trend, assuming that the noise and the trend are not correlated. If this superposition rule is not followed, this is an indication that the noise and the superposed trend are not independent, so that removing the trend could lead to changes in the correlation properties of the noise. In addition, we show how to use DFA appropriately to minimize the effects of trends, how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation, or if the crossover is due to a trend without any transition in the dynamical properties of the noise.

Effect of nonstationarities on detrended fluctuation analysis.

Abstract: Detrended fluctuation analysis ~DFA! is a scaling analysis method used to quantify long-range power-law correlations in signals. Many physical and biological signals are ‘‘noisy,’’ heterogeneous, and exhibit different types of nonstationarities, which can affect the correlation properties of these signals. We systematically study the effects of three types of nonstationarities often encountered in real data. Specifically, we consider nonstationary sequences formed in three ways: ~i! stitching together segments of data obtained from discontinuous experimental recordings, or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts—a ‘‘cutting’’ procedure commonly used in preparing data prior to signal analysis; ~ii! adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes; and ~iii! generating a signal comprised of segments with different properties—e.g., different standard deviations or different correlation exponents. We compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities. We find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on ~a! the fraction and size of the parts cut out from the signal, ~b! the concentration of spikes and their amplitudes ~c! the proportion between segments with different standard deviations or different correlations and ~d! the correlation properties of the stationary signal. We show how to develop strategies for preprocessing ‘‘raw’’ data prior to analysis, which will minimize the effects of nonstationarities on the scaling properties of the data, and how to interpret the results of DFA for complex signals with different local characteristics.

Effect of Nonstationarities on Detrended Fluctuation Analysis

Abstract: Detrended fluctuation analysis ~DFA! is a scaling analysis method used to quantify long-range power-law correlations in signals. Many physical and biological signals are ‘‘noisy,’’ heterogeneous, and exhibit different types of nonstationarities, which can affect the correlation properties of these signals. We systematically study the effects of three types of nonstationarities often encountered in real data. Specifically, we consider nonstationary sequences formed in three ways: ~i! stitching together segments of data obtained from discontinuous experimental recordings, or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts—a ‘‘cutting’’ procedure commonly used in preparing data prior to signal analysis; ~ii! adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes; and ~iii! generating a signal comprised of segments with different properties—e.g., different standard deviations or different correlation exponents. We compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities. We find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on ~a! the fraction and size of the parts cut out from the signal, ~b! the concentration of spikes and their amplitudes ~c! the proportion between segments with different standard deviations or different correlations and ~d! the correlation properties of the stationary signal. We show how to develop strategies for preprocessing ‘‘raw’’ data prior to analysis, which will minimize the effects of nonstationarities on the scaling properties of the data, and how to interpret the results of DFA for complex signals with different local characteristics.

Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

Abstract: Physiological and behavioural phenomena are often complex, characterized by variations in time series. Variations in time series reflect how these phenomena organize into coherent structures by interactions that span multiple scales in both time and space. The present tutorial is an introduction to multifractal analyses that can identify these scale invariant interactions within time series by its multifractal spectrum. The multifractal spectrum can be estimated directly from scale-dependent measurements or from its q-order statistics. The tutorial emphasizes the most common scale-dependent measurements defined by the wavelet transforms and the detrended fluctuation analyses. The tutorial also emphasizes common features of all multifractal analyses, like the choice of linear regression method, scaling range and elimination of spurious singularities, which are important for a robust estimation of the multifractal spectrum. The tutorial ends with two brief examples where multifractal analyses are employed to time series from multifractal models and the complex phenomena of cognitive performance. References to available software for multifractal analyses are included at the end of the tutorial. The main aim of the tutorial is to give the reader an introduction to multifractal analyses without the extensive technicalities typically provided in mathematical journals.