Detecting long-range correlations with detrended fluctuation analysis

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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.

Introduction

In recent years, the detrended fluctuation analysis (DFA) invented by Peng et al. [1] has been established as an important tool for the detection of long-range (auto-) correlations in time series with non-stationarities. It has successfully been applied to such diverse fields of interest as DNA [2], [3], [4], heart rate dynamics [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], neuron spiking [15], [16], human gait [9], [17], long-time weather records [18], [19], [20], cloud structure [21], [22], economical time series [23], [24], [25], [26], and even solid state physics [27], [28]. While the spectral analysis (Fourier transform) and the recently developed wavelet transform modulus maxima (WTMM) method [29], [30], [31] analyze the time series directly, the DFA is based on the random walk theory [32], [33], similar to the Hurst rescaled-range analysis [34] (see also [35]) and similar to another method based on the wavelet transform used, e.g. in [18], [19]. Since the time series are summed up in the methods based on the random walk theory (including the DFA), the noise level due to imperfect measurements in most records is reduced. The fano factor [36] and the allan factor [37], see also [8], have been employed in a similar context, but these methods do not remove trends in the data.

For the reliable detection of long-range correlations, it is essential to distinguish trends from the long-range fluctuations intrinsic in the data. Trends are caused by external effects—e.g. the greenhouse warming and seasonal variations for temperature records—and they are usually supposed to have a smooth and monotonous or slowly oscillating behavior. Strong trends in the data can lead to a false detection of long-range correlations if only one (non-detrending) method is used or if the results are not carefully interpreted. It is the advantage of the DFA that it can systematically eliminate trends of different order (like the method based on wavelet transform that has been applied, e.g. in [18], [19]). In this way, we can gain an insight into the scaling behavior of the natural variability as well as into the trends in the considered time series.

In this paper, we study systematically different orders of the DFA technique, that allow to eliminate different orders of trends. The paper is organized as follows: in Section 2, the method is described. In Section 3, we suggest a straightforward extension of the DFA that eliminates DFA specific deviations from scaling at small time scales. We describe how crossovers in the observed long-range correlation behavior can be detected and detail how the crossover time can be determined reliably. Finally, we show how several types of trends in the data affect the different orders of DFA. We summarize the results in the forth section of the paper.

Section snippets

Long-range correlations and the detrended fluctuation analysis

We consider a record (xi) of i=1,…,N equidistant measurements. In most applications, the index i will correspond to the time of the measurements. We are interested in the correlation of the values xi and xi+s for different time lags, i.e., correlations over different time scales s. In order to get rid of a constant offset in the data, the mean 〈x〉=1/Ni=1Nxi is usually subtracted, x̄i≡xi−〈x〉. Quantitatively, correlations between x-values separated by s steps are defined by the (auto-)

The correction function and a modified version of the DFA

Figs. 2a–c show that small deviations from the scaling law Eq. (7), i.e., deviations from a straight line in the log–log plot, occur for small scales s. These deviations are intrinsic to the usual DFA method, since the scaling behavior is only approached asymptotically. The deviations limit the capability of DFA to determine the correct correlation behavior in very short records and in the regime of small s. DFA6, e.g., is only defined for s⩾8, and significant deviations from the scaling law (7)

Note added in proof

After submission of the paper, a related preprint entitled ”Effect of trends on detrended fluctuation analysis” has been submitted [41].

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft and the project “KLIWA” of the Bayerische Landesamt für Wasserwirtschaft.

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    Present address: Dipartimento de Fisica, Universidade Federal do Rio Grande de Norte, 59072-970 Natal, Brazil.

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