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

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.

Detecting long-range correlations with detrended fluctuation analysis

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.

https://arxiv.org/pdf/physics/0103018.pdf

Effect of trends on detrended fluctuation analysis.

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.

Effect of Nonstationarities on Detrended Fluctuation Analysis

We study the temporal correlations in the atmospheric variability by 14 meteorological stations around the globe, the variations of the daily maximum temperatures from their average values. We apply several methods that can systematically overcome possible nonstationarities in the data. We find that the persistence, characterized by the correlation C(s) of temperature variations separated by s days, approximately decays $C(s)\ensuremath{\sim}{s}^{\ensuremath{-}\ensuremath{\gamma}}$, with roughly the same exponent $\ensuremath{\gamma}\ensuremath{\cong}0.7$ for all stations considered. The range of this universal persistence law seems to exceed one decade, and is possibly even larger than the range of the temperature series considered.

Indication of a Universal Persistence Law Governing Atmospheric Variability

We have studied long-time daily temperature records (between 56 and 218 years) obtained from 12 meteorological stations in Europe and North America from various climatological zones. To analyse the fluctuations of the daily temperatures around their average values, we have calculated directly the autocorrelation function C(l) between two days separated by l days, and applied also random-walk fluctuation analysis and wavelets methods, which can systematically overcome non-stationarities in the data. In addition, we have also analysed the distribution of the daily temperature fluctuations, which in most cases is well approximated by a Gaussian. Our analysis suggests that the persistence, characterized by the autocorrelation function C(l), is long ranged and approximately decays with a power law C(l) ≃ l -γ, with roughly the same exponent γ ≈ 2/3 for all stations considered. This universal persistence law seems to be valid at least for one decade of years, but we cannot exclude the possibility that ...

Long-range power-law correlations in local daily temperature fluctuations

The persistence of short term weather states is a well known phenomenon: A warm day is more likely to be followed by a warm day than by a cold one and vice versa. Using advanced methods from statistical physics that are able to distinguish between trends and persistence we have shown recently that this rule may well extend to months, years and decades, and on these scales the decay of the persistence seems to follow a universal power law. Here we review these studies and discuss, how the law can be used as an (uncomfortable) test bed for the state-of-the-art climate models. It turns out that the models considered display wide performance differences and actually fail to reproduce the universal power law behavior of the persistence. It seems that the models tend to underestimate persistence while overestimating trends, and this fact may imply that the models exaggerate the expected global warming of the atmosphere.

Long term persistence in the atmosphere: global laws and tests of climate models

A Comment on the Letter by Klaus Fraedrich and Richard Blender, Phys. Rev. Lett. 90, 108501 (2003). The authors of the Letter offer a Reply.

Comment on "Scaling of atmosphere and ocean temperature correlations in observations and climate models".

In a recent letter [K. Fraedrich and R. Blender, Phys. Rev. Lett. 90, 108501 (2003)], Fraedrich and Blender studied the scaling of atmosphere and ocean temperature. They analyzed the fluctuation functions F(s) ~ s^alpha of monthly temperature records (mostly from grid data) by using the detrended fluctuation analysis (DFA2) and claim that the scaling exponent alpha over the inner continents is equal to 0.5, being characteristic of uncorrelated random sequences. Here we show that this statement is (i) not supported by their own analysis and (ii) disagrees with the analysis of the daily observational data from which the grid monthly data have been derived. We conclude that also for the inner continents, the exponent is between 0.6 and 0.7, similar as for the coastline-stations.

E. Koscielny-Bunde

Papers

Indication of a Universal Persistence Law Governing Atmospheric Variability

Long-range power-law correlations in local daily temperature fluctuations

Long term persistence in the atmosphere: global laws and tests of climate models

Comment on "Scaling of atmosphere and ocean temperature correlations in observations and climate models".

Comment on: Scaling of atmosphere and ocean temperature correlations in observations and climate models. Authors' reply