Detrended cross-correlation analysis (DCCA) is a scaling method commonly used to estimate long-range power-law cross-correlation in nonstationary signals. Recent studies have reported signals superimposed with trends, which often lead to the complexity of the signals and the susceptibility of DCCA. This paper artificially generates long-range cross-correlated signals and systematically investigates the effect of seasonal trends. Specifically, for the crossovers raised by trends, we propose a smoothing algorithm based on empirical mode decomposition (EMD) method which decomposes underlying signals into several intrinsic mode functions (IMFs) and a residual trend. After the removal of slowly oscillating components and residual term, seasonal trends are eliminated.
Recently, the existence of cross-correlations in complex systems has been extensively studied. Many studies provide strong empirical evidence for the existence of cross-correlations between time series. For example, in seismology, Campillo and Paul discussed the degree of cross-correlation among noise signals taken at different antennas of detector arrays [
For determining the scaling exponent of cross-correlational time series in the presence of nonstationarities, detrended cross-correlation analysis method was developed by Podobnik et al. [
However, many noisy signals in real systems exhibit trends so that the scaling phenomenon obtained by DCCA method becomes intricate to reveal the intrinsic dynamics of the real systems due to many trends changing the scaling exponent for different range of scales. Specifically, seasonal trends usually lead to a change of cross-correlations in signals. A technique, developed by Hajian and Movahed, based on singular-value decomposition (SVD) of the trajectory matrix was applied on the method specifically to eliminate possible crossovers in cross-correlation exponents [
In this work, the DCCA procedure is pieced together with empirical mode decomposition (EMD), which is a well-established and promising method to analyze nonlinear and nonstationary signals [
Here we first investigate the scaling characteristic of noisy signals with sinusoidal trends. And then, we obtain the scaling properties of the reconstructed data by applying the EMD-based method, and we compare the difference in the scaling results. The results indicate that sinusoidal trends embedded in original data are extracted and the intrinsic fluctuations are retrieved.
The organization of this paper is as follows. First, the DCCA method is given in Section
Many physical and biological time series are often nonstationary or have means, variances
DCCA is a new method to quantify the cross-correlations between two nonstationary time series. This method is an extension of detrended fluctuation analysis (DFA) method. The DCCA procedure consists of four steps. Both methods are based on random walk theory. For two nonstationary time series
Compute the profiles of underlying time series using
Cut the profiles
Calculate the covariance of the residuals in each segment as follows:
Determine the power-law relationship between the detrended cross-correlation function
The empirical mode decomposition (EMD) first proposed by Huang et al. is designed for the time-frequency analysis of real-world signals [
Specifically, for EMD method, if
The procedure used for the extraction of IMFs from a signal
Identify all the extrema (maxima and minima) of the series
Obtain the signal envelope passing through the minima
Compute the local mean series by point averaging of the two envelopes
Subtract the mean from the signal to obtain an IMF candidate
Check the properties of
Repeat the procedure from Step
As previously demonstrated [
As we expect to eliminate the trend from original signal
The effectiveness of the proposed technique in minimizing the sinusoidal trends will be discussed in this section. Specifically, we consider two types of time series, artificial and aeroengine time series, in this paper.
In this section, we employ two-component ARFIMA method [
First, we use ARFIMA method to produce correlated signals
DFA and DCCA on signals generated by ARFIMA when
For reliable detection of the cross-correlations, it is essential to distinguish trends from the intrinsic fluctuations in data. Generally, trends embedded in measurements are of two types: polynomial and sinusoidal trends. Although the DCCA method eliminates the polynomial trends, the sinusoidal trends remain [
Inspired from [
The generated data sets with sinusoidal trends,
DFA on signals with sinusoidal trends (circles), reconstructed data using the smoothing filter (triangles), and original signals generated by ARFIMA method (plus signs).
We, respectively, use EMD to decompose original signals with sinusoidal trends into several IMFs. The IMFs are achieved by sifting out rapidly oscillating components from the data, by subtracting slowly oscillating components as shown in Figures
The IMFs for
The IMFs for
Next, we apply DCCA on the original signals superimposed with sinusoidal trend. As shown in Figure
DCCA on signals with sinusoidal trends (circles), reconstructed data using the smoothing filter (triangles), and original signals generated by ARFIMA method (plus signs).
We note that crossovers caused by any sinusoidal trend in artificial time series can be solved by EMD. We also test the effect of real time series on the method and find the expected results in the next section.
In the following discussion, we study the effectiveness of the proposed method on correlated signals with seasonal trend. Given a seasonal trend,
DCCA on signals with seasonal trends (circles), reconstructed data using the smoothing filter (triangles), and original signals generated by ARFIMA method (plus signs).
In this section, the effectiveness of the proposed method is analyzed by applying the aeroengine series. The sinusoidal trends superimposed on the exhaust gas temperature (EGT) and engine fan speeds (N1), respectively, denoted by
The DCCA of the original aeroengine data, original data with the sinusoidal trend, and reconstructed data is shown in Figure
DCCA on aeroengine time series with sinusoidal trends (circles), reconstructed data using the smoothing filter (triangles), and original signals generated by ARFIMA method (plus signs).
In the paper, we consider EMD-based method to eliminate sinusoidal trends which introduce crossovers in cross-correlation exponent in DFA and DCCA. The effectiveness of the proposed EMD-based method is applied to long-range power-law cross-correlated signals which are generated by two-component ARFIMA artificially. After the removal of slowly oscillating components, sinusoidal trend is successfully eliminated. EMD shows advantage in eliminating sinusoidal trend in artificial time series.
EMD together with DCCA method can be applied to real dynamics to reliably detect long-range cross-correlation where sinusoidal trend exists. In aeroengine dynamics, we apply EMD method by decomposing underlying signals into several intrinsic mode functions and present similar results with artificial time series. We note that EMD is of great advantage when dealing with sinusoidal trend. Therefore, we do believe that our smoothing filter algorithm may provide some help to minimize the effect of the trends introduced by the sinusoidal trends and facilitate a reliable extraction of the scaling exponent.
The financial support from the funds of the Fundamental Research Funds for the Central Universities under Grant no. ZXH 2012C002, the National Natural Science Foundation of China under Grant no. U1233201, and the Natural Science Foundation of Hebei Province (E2012402013) are gratefully acknowledged.