# Higher-Order Multifractal Detrended Partial Cross-Correlation Analysis for the Correlation Estimator

Authors: Keqiang Dong; Xiaojie Gao
Journal: Complexity
 (2020)

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## Abstract

In this paper, we develop a new method to measure the nonlinear interactions between nonstationary time series based on the detrended cross-correlation coefficient analysis. We describe how a nonlinear interaction may be obtained by eliminating the influence of other variables on two simultaneous time series. By applying two artificially generated signals, we show that the new method is working reliably for determining the cross-correlation behavior of two signals. We also illustrate the application of this method in finance and aeroengine systems. These analyses suggest that the proposed measure, derived from the detrended cross-correlation coefficient analysis, may be used to remove the influence of other variables on the cross-correlation between two simultaneous time series.


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## Body

## 1. Introduction

There are numerous real-world systems where the output signals are nonstationary and exhibit complex self-correlation or cross-correlation over a broad range of time scales. The output signals can be characterized by power-law correlations. One method, which has proved to be quite useful to detect the degree of interrelation between two stationary variables, is Pearson’s correlation coefficient [1]:(1)r=X−X⋅Y−YσX⋅σY,where X is the arithmetic average of X and σX is its standard deviation and likewise for Y. Proposition of Pearson's correlation coefficient (PCC) has achieved great success in multivariate analysis, such as the principal component analysis [2], random matrix theory [3], and singular value decomposition [4].Nevertheless, in real-world systems, nonlinear and nonstationary characteristics are present. Therefore, PCC may not be suitable to describe the interrelation between two variables that are nonlinear and nonstationary. For dealing with the drawbacks of PCC, the detrended cross-correlation analysis (DCCA) method and the DCCA coefficient are proposed by Stanley and Podobnik [5, 6]. The advantage of the DCCA method is that it allows the detection of cross-correlations between noisy signals with embedded polynomial trends, which can mask the true cross-correlations in the fluctuations of signals. The DCCA method is widely applied to measure the cross-correlations in different fields, such as social sciences [7], biology [8], climatology [9], geophysics [10, 11], transportation [12, 13], seismic signals [11, 14], economics [15–20], and aeroengine dynamics [21–24].Recently, multifractal analysis is one of the major interests for researchers from interdisciplinary domains to uncover the scaling properties and understand the hidden information. Among these researchers, many of them applied the multifractal analysis to meteorology [25–27], electroencephalography [28], and economics [29–31]. Later, as some researchers thought of extending the research of multifractal analysis to the detrended cross-correlations between time series, the multifractal detrended cross-correlation analysis (MFDXA) was proposed [32–34].The cross-correlation between two variables may be influenced by other variables. Hence, we have to be alert to the possibilities of spurious correlation while investigating the cross-correlation. Then, the methods of partial correlation and partial correlation coefficient are therefore proposed to measure the degree of association between two random variables [35, 36]. The linear effect may be removed using the partial correlation coefficient (partial CC):(2)rXY,ξ=X′−X′⋅Y′−Y′σX′⋅σY′=rXY−rXξ⋅rYξ1−rXξ2⋅1−rYξ2,where X′=X−LXξ and LXξ=c0+c1ξ to minimize the mean EX−LXξ2 and likewise for Y′. If n additional variables are to be accounted for, say ξ1,ξ2,…,ξn, the nth-order partial CC can be computed by [36](3)rXY.12⋯n=rXY.12⋯n−1−rXn.12⋯n−1rYn.12⋯n−11−rXn.12⋯n−121−rYn.12⋯n−12.Lately, the detrended partial cross-correlation analysis and multifractal detrended partial cross-correlation analysis (MFDPXA) which can measure cross-correlations between nonlinear time series influenced by common external forces is proposed [37, 38].In order to remove the spurious correlation and improve the estimation performance for quantifying the intrinsic interactions between two nonstationary time series, this paper proposes the method ofnth-order multifractal detrended partial cross-correlation analysis by incorporating the partial correlation coefficient with the multifractal detrended cross-correlation analysis.The rest of the paper is organized as follows. In the next section, we introduce the multifractal DCCA coefficient method and propose the method ofnth-order multifractal detrended partial cross-correlation analysis. In Section 3, we show the data results for the randomly generated dataset and stock and engine dataset by the proposed methods. Finally, we draw some conclusions in Section 4.


## 2. Methodologies

### 2.1. Multifractal Detrended Partial Cross-Correlation Analysis

For the sake of clarity, we begin with a summary of the multifractal DCCA coefficient algorithm. For two seriesrit and rjt with equal length N, where t=1,2,…,N, the computational procedure of the multifractal DCCA coefficient is as follows:Step 1: construct the profile of each series by eliminating the mean value:(4)Rit=∑k=1trik−ri,Rjt=∑k=1trjk−rj,t=1,2,…,N,whereri and rj are the average values of rit and rjt, respectively.Step 2: divide the profilesRik and Rjk into Ns=intN/s nonoverlapping units of equal length s. Considering that N is usually not a multiple of the time scale s, we repeat the same procedure by starting from the opposite end of the sequence in order to take the whole series into account. Thus, we obtain 2Ns segments of equal length s. In this paper, we follow the previous literature practice and set 10≤s≤N/4.Step 3: for each segmentvv=1,2,…,Ns,Ns+1,…,2Ns, the local trends R˜ivk and R˜jvk are estimated on the basis of a least-squares fit of the sequences Rik and Rjk, respectively. The corresponding detrended covariance for v=1,2,…,Ns is(5)fDCCA2s,v=1s∑t=1sRiv−1s+tt−R˜ivtRjv−1s+tt−R˜jvt,and forv=Ns+1,Ns+2,…,2Ns is(6)fDCCA2s,v=1s∑t=1sRiN−v−Nss+tt−R˜ivtRjN−v−Nss+tt−R˜jvt,whereR˜ivk and R˜jvk are the fitting polynomials in the segment v.Step 4: calculate the average of multifractal detrended covariance fluctuation functionFDCCAqs,v over all segments:(7)FDCCAqs,v=12Ns∑v=12NsfDCCA2s,vq/21/q.Generally,q can take any real value, except zero. For q=0, the equation becomes(8)F0s=exp12Ns∑m=1NslnFs,m.Forq=2, FDCCAqs,v is equal to the detrended cross-correlation fluctuation function FDCCA2s.Step 5: estimate the multifractal DCCA coefficient:(9)ρijqs=FDCCAqsFDFAritqsFDFArjtqs.Forq=2, the standard DCCA coefficient ρijs is retrieved.Step 6: compute the multifractal detrended partial cross-correlation coefficient betweenX and Y by eliminating the influence of the controlling variable ξ1 on X and Y analogous to the generalization of the correlation coefficient to partial correlation coefficient:(10)ρXY,1q=ρXYq−ρX1qρY1q1−ρX1q21−ρY1q2,named the first-order multifractal detrended partial cross-correlation coefficient (first-order MFDPCC coefficient), whereX,Y are random variables, ξ1 is the controlling variable, and ρXYq,ρX1q,ρY1q represent the mean of MFDCCA coefficients for X and Y, X and ξ1, and Y and ξ1, respectively.Forq=2, the first-order detrended partial cross-correlation coefficient (first-order DPCC coefficient) is retrieved.


### 2.2. Thenth-Order Multifractal Detrended Partial Cross-Correlation Analysis and n-Controlling-Variables Detrended Partial Cross-Correlation Coefficient

Considering the cross-correlation betweenX and Y affected by more than one variable in complex systems, we define the second-order multifractal detrended partial cross-correlation coefficient (second-order MFDPCC coefficient) by using the partial correlation method [36]:(11)ρXY,12q=ρXY,1q−ρX2,1qρY2,1q1−ρX1q21−ρY2,1q2,where X,Y are random variables, controlling variables ξ1,ξ2 are not related to each other, and ρXY.1,ρX2.1, and ρY2.1 are first-order MFDPCC coefficients.Generally, thenth-order multifractal detrended partial cross-correlation coefficient (nth-order MFDPCC coefficient) is as follows:(12)ρXY,12⋯nq=ρXY,12⋯n−1q−ρXn,12⋯n−1qρYn,12⋯n−1q1−ρXn,12⋯n−1q21−ρYn,12⋯n−1q2,where ρXY.12⋯n−1q, ρXn.12⋯n−1q, and ρYn.12⋯n−1q are (n − 1) th-order MFDPCC coefficients and controlling variables ξ1,ξ2,…,ξn are not related to each other. For q=2, the nth-order detrended partial cross-correlation coefficient (nth-order DPCC coefficient) is retrieved.In general, thenth-order partial cross-correlation is necessary when these controlling variables ξ1,ξ2,…,ξn are not related to each other. Nevertheless, in real-world systems, the variables ξ1,ξ2,…,ξn generated by large number of interacting units are cross-correlated. Therefore, we define the n-controlling-variables multifractal detrended partial cross-correlation coefficient (n-variables MFDPCC) by equation (12) for related controlling variables ξ1,ξ2,…,ξn. Note that when the controlling variables ξ1,ξ2,…,ξn are not related to each other, the n-variables MFDPCC is equivalent to the nth-order MFDPCC.


## 2.1. Multifractal Detrended Partial Cross-Correlation Analysis

For the sake of clarity, we begin with a summary of the multifractal DCCA coefficient algorithm. For two seriesrit and rjt with equal length N, where t=1,2,…,N, the computational procedure of the multifractal DCCA coefficient is as follows:Step 1: construct the profile of each series by eliminating the mean value:(4)Rit=∑k=1trik−ri,Rjt=∑k=1trjk−rj,t=1,2,…,N,whereri and rj are the average values of rit and rjt, respectively.Step 2: divide the profilesRik and Rjk into Ns=intN/s nonoverlapping units of equal length s. Considering that N is usually not a multiple of the time scale s, we repeat the same procedure by starting from the opposite end of the sequence in order to take the whole series into account. Thus, we obtain 2Ns segments of equal length s. In this paper, we follow the previous literature practice and set 10≤s≤N/4.Step 3: for each segmentvv=1,2,…,Ns,Ns+1,…,2Ns, the local trends R˜ivk and R˜jvk are estimated on the basis of a least-squares fit of the sequences Rik and Rjk, respectively. The corresponding detrended covariance for v=1,2,…,Ns is(5)fDCCA2s,v=1s∑t=1sRiv−1s+tt−R˜ivtRjv−1s+tt−R˜jvt,and forv=Ns+1,Ns+2,…,2Ns is(6)fDCCA2s,v=1s∑t=1sRiN−v−Nss+tt−R˜ivtRjN−v−Nss+tt−R˜jvt,whereR˜ivk and R˜jvk are the fitting polynomials in the segment v.Step 4: calculate the average of multifractal detrended covariance fluctuation functionFDCCAqs,v over all segments:(7)FDCCAqs,v=12Ns∑v=12NsfDCCA2s,vq/21/q.Generally,q can take any real value, except zero. For q=0, the equation becomes(8)F0s=exp12Ns∑m=1NslnFs,m.Forq=2, FDCCAqs,v is equal to the detrended cross-correlation fluctuation function FDCCA2s.Step 5: estimate the multifractal DCCA coefficient:(9)ρijqs=FDCCAqsFDFAritqsFDFArjtqs.Forq=2, the standard DCCA coefficient ρijs is retrieved.Step 6: compute the multifractal detrended partial cross-correlation coefficient betweenX and Y by eliminating the influence of the controlling variable ξ1 on X and Y analogous to the generalization of the correlation coefficient to partial correlation coefficient:(10)ρXY,1q=ρXYq−ρX1qρY1q1−ρX1q21−ρY1q2,named the first-order multifractal detrended partial cross-correlation coefficient (first-order MFDPCC coefficient), whereX,Y are random variables, ξ1 is the controlling variable, and ρXYq,ρX1q,ρY1q represent the mean of MFDCCA coefficients for X and Y, X and ξ1, and Y and ξ1, respectively.Forq=2, the first-order detrended partial cross-correlation coefficient (first-order DPCC coefficient) is retrieved.


## 2.2. Thenth-Order Multifractal Detrended Partial Cross-Correlation Analysis and n-Controlling-Variables Detrended Partial Cross-Correlation Coefficient

Considering the cross-correlation betweenX and Y affected by more than one variable in complex systems, we define the second-order multifractal detrended partial cross-correlation coefficient (second-order MFDPCC coefficient) by using the partial correlation method [36]:(11)ρXY,12q=ρXY,1q−ρX2,1qρY2,1q1−ρX1q21−ρY2,1q2,where X,Y are random variables, controlling variables ξ1,ξ2 are not related to each other, and ρXY.1,ρX2.1, and ρY2.1 are first-order MFDPCC coefficients.Generally, thenth-order multifractal detrended partial cross-correlation coefficient (nth-order MFDPCC coefficient) is as follows:(12)ρXY,12⋯nq=ρXY,12⋯n−1q−ρXn,12⋯n−1qρYn,12⋯n−1q1−ρXn,12⋯n−1q21−ρYn,12⋯n−1q2,where ρXY.12⋯n−1q, ρXn.12⋯n−1q, and ρYn.12⋯n−1q are (n − 1) th-order MFDPCC coefficients and controlling variables ξ1,ξ2,…,ξn are not related to each other. For q=2, the nth-order detrended partial cross-correlation coefficient (nth-order DPCC coefficient) is retrieved.In general, thenth-order partial cross-correlation is necessary when these controlling variables ξ1,ξ2,…,ξn are not related to each other. Nevertheless, in real-world systems, the variables ξ1,ξ2,…,ξn generated by large number of interacting units are cross-correlated. Therefore, we define the n-controlling-variables multifractal detrended partial cross-correlation coefficient (n-variables MFDPCC) by equation (12) for related controlling variables ξ1,ξ2,…,ξn. Note that when the controlling variables ξ1,ξ2,…,ξn are not related to each other, the n-variables MFDPCC is equivalent to the nth-order MFDPCC.


## 3. Data and Analysis

### 3.1. Two-Component ARFIMA Process

In order to test the robustness of the proposedn-controlling-variables MFDPCC coefficient method, power-law cross-correlated time series ui and vi are generated by using the two-component ARFIMA stochastic process in this section [18, 39, 40]. In this model, the series is defined by(13)ui=WUi+1−WVi+εi,vi=WVi+1−WUi+ε˜i,Ui=∑j=1∞ajρ1ui−j,Vi=∑j=1∞ajρ2vi−j,where ajρ=Γj−ρ/Γ−ρΓi+j0<ρ<0.5 is weight, W is a free parameter to control the coupling strength between ui and vi (0.5 ≤ W ≤ 1), and εi and ε˜i are independent and identically distributed (i.i.d.) Gaussian variables with εi=ε˜i=0 and εi2=ε˜i2=1 [18, 39]. For different values of W, the different coupling strength between the variables ui and vi is 1−W. In this section, the two-component ARFIMA series ui and vi with parameter ρ1=ρ2=0.3 and W = 0.5, denoted by X and Y, are employed to detect the interactions between two time series. Then, the effect of white noise sequence ξ1 on the cross-correlation of the two series X and Y is tested to investigate the validity of the n-controlling-variables MFDPCC coefficient analysis mentioned in this paper. For this purpose, we study the difference between the mean of the MFDCCA coefficient and the n-controlling-variables MFDPCC coefficient for any parameter q by using the influence degree function In,q. The influence degree function is defined as(14)In,q=ρXY,12⋯nq−ρXYq.We calculate the influence degree functionIn,q of the synthetical signals using the proposed first-order MFDPCC coefficient and present the influence degree function I1,q vs. parameter q in Figure 1. The results of the influence degree values of different q are just about nil, which indicates that there is hardly any effect of white noise sequence on cross-correlation of the two series X and Y.Figure 1
The effect of white noise sequenceξ1 on cross-correlation of the two series X and Y.


### 3.2. Stock Market

To further exemplify the potential utility of then-controlling-variables MFDPCC coefficient method for analyzing real-world data, we study daily closing prices of fifteen stock markets including the São Paulo Index (IBOV), the Dow Jones Index (DJI), the NASDAQ Index (IXIC), the Standard & Poor 500 Composite Stock Price Index (SPX), the FTSE Global Equity Index Series (FISE), the French CAC 40 (FCHI), German DAX Index (GDAXI), Nikkei 255 Index (N255), Korea Composite Index (KS11), Hang Seng Index (HSI), Australian Standard & Poor’s 200 (AS51), Mumbai Index (SENSEX), Russian Index (RTS), Shanghai Composite Index (SSEC), and Shenzhen Composite Index (SZI). Datasets are from January 04, 1993, to January 03, 2019.Figure2 shows the mean of DCCA coefficients for the stock series. The mean of DCCA coefficient between DJI and SPX is 0.97, which performs relevantly different from other DCCA coefficients. It indicates the close cross-correlation between the American stock markets. The next largest DCCA coefficient ρ=0.92 is obtained by SSEC and SZI, which indicates the close cross-correlation in Chinese mainland stock markets.Figure 2
The mean of DCCA coefficients between the stock series.The mean of DCCA coefficients between SZI and stock markets in developed countries (GDAXI, N225, KS11, and AS51) is less than 0.3. It shows that SZI has a weak relationship with stock markets in developed countries. The mean of DCCA coefficients between SZI and HSI is in an intermediate state, which indicates the existence of cross-correlation in Chinese stock markets.Next, we analyze the effect of the other thirteen stock markets on cross-correlation characteristics between SSEC and SZI, by applying the influence degree of first-order DPCC coefficient. For the effect on cross-correlation characteristics between the SSEC and SZI, the largest influence degreeI = 0.05 is obtained by HSI, which shows the information exchange between the Chinese stock market, as seen in Figure 3. The next largest I = 0.04 is acquired by SENSEX, which indicates the association between the stock markets in developing countries (Indian and Chinese stock markets). The I values of other stock time series are less than 0.1, which indicates little information exchange between the Chinese mainland stock market and other stock markets. The influence degree values of 13 stock markets for first-order MFDPCC coefficient with q=1,2,…,10 are also demonstrated in the upper left of Figure 3.Figure 3
The influence degree of the first-order detrended partial cross-correlation in stock market and the influence degree of the first-order multifractal detrended partial cross-correlation coefficient in stock market (inset).During the analysis, we observe the effect of HSI on cross-correlation characteristics between SSEC and SZI from influence degree functionI1,q that decreases as the scale q increases. And this infers the change of multifractal cross-correlation.In order to capture the change of multifractal cross-correlation between two nonstationary time series influenced by common external forces, multifractal detrended partial cross-correlation analysis (MFDPXA) is employed [38]. We also investigate the multifractal behavior between the bivariate time series through MFDPXA method for comparison. The result shows that both the corresponding spectra fxyα and fxy,zα are wide, but the latter is narrower than the former, which is presented in Figure 4.Figure 4
The multifractal spectrafxyα of bivariate time series obtained through MFDXA method and fxy,zα of bivariate time series obtained through MFDPXA, where x, y, and z denote the SSEC, SZI, and HSI.Here, we perform cross-correlation analysis using MFDPXA method and give the multifractal spectrum for SZI and SSEC time series in which HSI shows significant influence on multifractal spectrum, as seen in Figure4. We compare the obtained influence degrees with the aforementioned method and infer that the HSI has significant influence on SZI and SSEC time series. These similar results imply that the partial cross-correlation method is quite efficient in eliminating external common influence factor.Applied to scalar variables, the first-order MFDPCC will detect the intrinsic interactions by removing the correlations of controlling variables. When variables are time series, this application is equivalent to removal of zero delay correlations, whereas delayed correlations are not considered [36, 37, 40, 41]. Therefore, we investigate the delayed effect of variable ξ1 on the correlation between variables X and Y. Because the two variables X and Y in question may themselves be correlated at nonzero delays, we write the multifractal detrended partial cross-correlation between X and Y, given ξ1, as a function of two time delays:(15)ρXYτ1,1τ2q=ρXYτ1q−ρX1τ2qρYτ11τ2q1−ρX1τ2q21−ρYτ11τ2q2,where τ1 is the delay between variables X and Y and τ2 is the delay between variables X and ξ1.In this section, we estimate the delayed effect of HSI on the correlation between SSEC and SZI by using the time delay influence degreeIq,τ1,τ2=ρXYτ1,1τ2q−ρXYq. Figure 5 shows the time delay influence degree for q=2. The effect of τ1 on influence degree is weaker than that of τ2 on influence degree.Figure 5
The time delay influence degree for delayτ1 between SSEC and SZI and delay τ2 between SSEC and HSI.We now analyze the 2-controlling-variables effect of the other thirteen stock markets on cross-correlation characteristics between SSEC and SZI, by giving a set of two controlling variables. In Figure6, we illustrate the comparative relation of the influence degree of 2-controlling-variables DPCC coefficients for 13×13 elements by the matrix diagram.Figure 6
The influence degree of the 2-controlling-variables detrended partial cross-correlation.We note that the structure of the matrix is symmetrical and that element at the intersection of rowi and column j represents the influence of controlling variables ξi,ξj on the cross-correlation of SSEC and SZI, where the 2-controlling-variables ξi,ξji,j=1,2,…,13 are the stock time series from IBOV, DJI, IXIC, SPX, FISE, FCHI, GDAXI, N255, KS11, HSI, AS51, SENSEX, and RTS. Therefore, we analyze the top left corner of the matrix. It can be seen that the largest element is the intersection of row 2 and column 10, i.e., SENSEX and HSI, which indicates the association between the Indian and Chinese stock markets. This is consistent with our result of first-order MFDPCC coefficient method.Concerning the influence degreeI2,q of the 2-controlling-variables MFDPCC, we demonstrate 5 cases (HSI and SENSEX, HSI and RTS, SENSEX and KS11, FCHI and N255, and IXIC and FISE) for q=1,2,…,10 in Figure 7. The largest influence degree is the case SENSEX and HSI, which is consistent with the 2-controlling-variables DPCC method.Figure 7
The influence degree of the 2-controlling-variables multifractal detrended partial cross-correlation.


### 3.3. Aeroengine Time Series

Previous research studies show that the aeroengine gas path parameters such as low-pressure rotor speed (N1), high-pressure rotor speed (N2), and fuel flow (WF) play an important role in understanding the aeroengine system [21, 42]. The mean of DCCA coefficients for the aeroengine time series is shown in Figure 8, where the average DCCA coefficient between N1 and N2 is 0.85, which shows the close cross-correlation between N1 and N2.Figure 8
The mean of standard DCCA coefficients for the time series of aeroengine system gas path parameters.We here investigate the partial correlation betweenN1 and N2 given a set of eight controlling variables, including WF, exhaust gas temperature (EGT), N2 tracked vibration channel B (N2TB), inlet air pressure (P2), outlet temperature of high-pressure compressor (T3), outlet temperature of low-pressure compressor (T2.5), and other temperatures (T2 and T2.95).In Figure9, we plot the influence degree of first-order DPCC coefficient, investigating the effect of the other eight controlling variables on cross-correlation characteristics between N1 and N2. The largest influence degree I = 0.51, obtained by T3, shows the information exchange between the outlet temperature of high-pressure compressor and the rotor speed system. The next largest I = 0.22 is acquired by WF, which indicates the association between the fuel flow system and rotor speed system.Figure 9
The influence degree of the first-order detrended partial cross-correlation in aeroengine system and the influence degree of the first-order multifractal detrended partial cross-correlation coefficient in aeroengine system (the upper left figure).The result of the influence degreeI1,q for eight aeroengine parameters applying by first-order MFDPCC coefficient with q=1,2,…,10 is also demonstrated in the upper left of Figure 9. The effect of T3 on cross-correlation characteristics, observed from influence degree function I1,q, decreases as the scale q increases. It indicates that the multifractal cross-correlation differs across values of q.Further, we apply the MFDPXA method on aforementionedN1 and N2 time series considering the T3 as common influencing factor. It is observed from Figure 10 that the corresponding spectra fxyα and fxy,zα are wide which shows the strength of multifractal behavior in analyzed time series. We observe that the width of singularity spectrum fxy,zα is narrower, and this implies the strength of multifractal nature is weak in analyzed bivariate time series.Figure 10
The multifractal spectrumfxyα of bivariate time series obtained through MFDXA method and fxy,zα of bivariate time series obtained through MFDPXA, where x, y, and z denote N1, N2, and T3.Here, we estimate the delayed effect of T3 on the correlation between N1 and N2 by using the time delay influence degreeIq,τ1,τ2. Figure 11 shows the time delay influence degree for q=2. It is obvious that the time delay influence degree gradually increases and then declines as a single-peak curve when τ2 remains constant. As τ2 increases, the peak value of time delay influence degree shifts rightward.Figure 11
The time delay influence degree for delayτ1 between N1 and N2 and delay τ2 between N1 and T3.The next observation concerns the influence degree of 2-controlling-variables DPCC coefficient in the aeroengine system. We now analyze the influence of two controlling parameters on the cross-correlation betweenN1 and N2. In Figure 12, we illustrate the comparative relation of the influence degree of 2-controlling-variables DPCC coefficient for aeroengine system. It can be seen that the larger elements in the symmetrical matrix are located at row 3 or column 6, which denote T3 has a greater impact on the correlation between N1 and N2.Figure 12
The influence degree of the 2-controlling-variables detrended partial cross-correlation for aeroengine system.Concerning the influence degreeI2,q of the 2-controlling-variables MFDPCC, we demonstrate 7 cases (T3 and WF, T3 and N2TB, T3 and T2, T3 and T2.5, T2.95 and N2TB, T2.95 and P2, and T2.95 and T2) for q=1,2,…,10 in Figure 13. Larger influence degrees exist in the cases with the presence of T3 (T3 and WF, T3 and N2TB, T3 and T2, and T3 and T2.5), which is consistent with the 2-controlling-variables DPCC method, as seen in Figure 12.Figure 13
The influence degree of the 2-controlling-variables multifractal detrended partial cross-correlation for the aeroengine system.For the aeroengine, the parametersN1 and N2 are chosen to indicate the engine thrust which depends on the throttle lever angle. Hence, the cross-correlation between them is strong. The temperature and pressure parameters are linked with many factors, including the compressor power, combustion efficiency, throttle lever angle, etc. Therefore, the dynamic interaction of these three groups makes the aeroengine function. These results estimate the influence of temperature and pressure parameters on the cross-correlation between N1 and N2.


## 3.1. Two-Component ARFIMA Process

In order to test the robustness of the proposedn-controlling-variables MFDPCC coefficient method, power-law cross-correlated time series ui and vi are generated by using the two-component ARFIMA stochastic process in this section [18, 39, 40]. In this model, the series is defined by(13)ui=WUi+1−WVi+εi,vi=WVi+1−WUi+ε˜i,Ui=∑j=1∞ajρ1ui−j,Vi=∑j=1∞ajρ2vi−j,where ajρ=Γj−ρ/Γ−ρΓi+j0<ρ<0.5 is weight, W is a free parameter to control the coupling strength between ui and vi (0.5 ≤ W ≤ 1), and εi and ε˜i are independent and identically distributed (i.i.d.) Gaussian variables with εi=ε˜i=0 and εi2=ε˜i2=1 [18, 39]. For different values of W, the different coupling strength between the variables ui and vi is 1−W. In this section, the two-component ARFIMA series ui and vi with parameter ρ1=ρ2=0.3 and W = 0.5, denoted by X and Y, are employed to detect the interactions between two time series. Then, the effect of white noise sequence ξ1 on the cross-correlation of the two series X and Y is tested to investigate the validity of the n-controlling-variables MFDPCC coefficient analysis mentioned in this paper. For this purpose, we study the difference between the mean of the MFDCCA coefficient and the n-controlling-variables MFDPCC coefficient for any parameter q by using the influence degree function In,q. The influence degree function is defined as(14)In,q=ρXY,12⋯nq−ρXYq.We calculate the influence degree functionIn,q of the synthetical signals using the proposed first-order MFDPCC coefficient and present the influence degree function I1,q vs. parameter q in Figure 1. The results of the influence degree values of different q are just about nil, which indicates that there is hardly any effect of white noise sequence on cross-correlation of the two series X and Y.Figure 1
The effect of white noise sequenceξ1 on cross-correlation of the two series X and Y.


## 3.2. Stock Market

To further exemplify the potential utility of then-controlling-variables MFDPCC coefficient method for analyzing real-world data, we study daily closing prices of fifteen stock markets including the São Paulo Index (IBOV), the Dow Jones Index (DJI), the NASDAQ Index (IXIC), the Standard & Poor 500 Composite Stock Price Index (SPX), the FTSE Global Equity Index Series (FISE), the French CAC 40 (FCHI), German DAX Index (GDAXI), Nikkei 255 Index (N255), Korea Composite Index (KS11), Hang Seng Index (HSI), Australian Standard & Poor’s 200 (AS51), Mumbai Index (SENSEX), Russian Index (RTS), Shanghai Composite Index (SSEC), and Shenzhen Composite Index (SZI). Datasets are from January 04, 1993, to January 03, 2019.Figure2 shows the mean of DCCA coefficients for the stock series. The mean of DCCA coefficient between DJI and SPX is 0.97, which performs relevantly different from other DCCA coefficients. It indicates the close cross-correlation between the American stock markets. The next largest DCCA coefficient ρ=0.92 is obtained by SSEC and SZI, which indicates the close cross-correlation in Chinese mainland stock markets.Figure 2
The mean of DCCA coefficients between the stock series.The mean of DCCA coefficients between SZI and stock markets in developed countries (GDAXI, N225, KS11, and AS51) is less than 0.3. It shows that SZI has a weak relationship with stock markets in developed countries. The mean of DCCA coefficients between SZI and HSI is in an intermediate state, which indicates the existence of cross-correlation in Chinese stock markets.Next, we analyze the effect of the other thirteen stock markets on cross-correlation characteristics between SSEC and SZI, by applying the influence degree of first-order DPCC coefficient. For the effect on cross-correlation characteristics between the SSEC and SZI, the largest influence degreeI = 0.05 is obtained by HSI, which shows the information exchange between the Chinese stock market, as seen in Figure 3. The next largest I = 0.04 is acquired by SENSEX, which indicates the association between the stock markets in developing countries (Indian and Chinese stock markets). The I values of other stock time series are less than 0.1, which indicates little information exchange between the Chinese mainland stock market and other stock markets. The influence degree values of 13 stock markets for first-order MFDPCC coefficient with q=1,2,…,10 are also demonstrated in the upper left of Figure 3.Figure 3
The influence degree of the first-order detrended partial cross-correlation in stock market and the influence degree of the first-order multifractal detrended partial cross-correlation coefficient in stock market (inset).During the analysis, we observe the effect of HSI on cross-correlation characteristics between SSEC and SZI from influence degree functionI1,q that decreases as the scale q increases. And this infers the change of multifractal cross-correlation.In order to capture the change of multifractal cross-correlation between two nonstationary time series influenced by common external forces, multifractal detrended partial cross-correlation analysis (MFDPXA) is employed [38]. We also investigate the multifractal behavior between the bivariate time series through MFDPXA method for comparison. The result shows that both the corresponding spectra fxyα and fxy,zα are wide, but the latter is narrower than the former, which is presented in Figure 4.Figure 4
The multifractal spectrafxyα of bivariate time series obtained through MFDXA method and fxy,zα of bivariate time series obtained through MFDPXA, where x, y, and z denote the SSEC, SZI, and HSI.Here, we perform cross-correlation analysis using MFDPXA method and give the multifractal spectrum for SZI and SSEC time series in which HSI shows significant influence on multifractal spectrum, as seen in Figure4. We compare the obtained influence degrees with the aforementioned method and infer that the HSI has significant influence on SZI and SSEC time series. These similar results imply that the partial cross-correlation method is quite efficient in eliminating external common influence factor.Applied to scalar variables, the first-order MFDPCC will detect the intrinsic interactions by removing the correlations of controlling variables. When variables are time series, this application is equivalent to removal of zero delay correlations, whereas delayed correlations are not considered [36, 37, 40, 41]. Therefore, we investigate the delayed effect of variable ξ1 on the correlation between variables X and Y. Because the two variables X and Y in question may themselves be correlated at nonzero delays, we write the multifractal detrended partial cross-correlation between X and Y, given ξ1, as a function of two time delays:(15)ρXYτ1,1τ2q=ρXYτ1q−ρX1τ2qρYτ11τ2q1−ρX1τ2q21−ρYτ11τ2q2,where τ1 is the delay between variables X and Y and τ2 is the delay between variables X and ξ1.In this section, we estimate the delayed effect of HSI on the correlation between SSEC and SZI by using the time delay influence degreeIq,τ1,τ2=ρXYτ1,1τ2q−ρXYq. Figure 5 shows the time delay influence degree for q=2. The effect of τ1 on influence degree is weaker than that of τ2 on influence degree.Figure 5
The time delay influence degree for delayτ1 between SSEC and SZI and delay τ2 between SSEC and HSI.We now analyze the 2-controlling-variables effect of the other thirteen stock markets on cross-correlation characteristics between SSEC and SZI, by giving a set of two controlling variables. In Figure6, we illustrate the comparative relation of the influence degree of 2-controlling-variables DPCC coefficients for 13×13 elements by the matrix diagram.Figure 6
The influence degree of the 2-controlling-variables detrended partial cross-correlation.We note that the structure of the matrix is symmetrical and that element at the intersection of rowi and column j represents the influence of controlling variables ξi,ξj on the cross-correlation of SSEC and SZI, where the 2-controlling-variables ξi,ξji,j=1,2,…,13 are the stock time series from IBOV, DJI, IXIC, SPX, FISE, FCHI, GDAXI, N255, KS11, HSI, AS51, SENSEX, and RTS. Therefore, we analyze the top left corner of the matrix. It can be seen that the largest element is the intersection of row 2 and column 10, i.e., SENSEX and HSI, which indicates the association between the Indian and Chinese stock markets. This is consistent with our result of first-order MFDPCC coefficient method.Concerning the influence degreeI2,q of the 2-controlling-variables MFDPCC, we demonstrate 5 cases (HSI and SENSEX, HSI and RTS, SENSEX and KS11, FCHI and N255, and IXIC and FISE) for q=1,2,…,10 in Figure 7. The largest influence degree is the case SENSEX and HSI, which is consistent with the 2-controlling-variables DPCC method.Figure 7
The influence degree of the 2-controlling-variables multifractal detrended partial cross-correlation.


## 3.3. Aeroengine Time Series

Previous research studies show that the aeroengine gas path parameters such as low-pressure rotor speed (N1), high-pressure rotor speed (N2), and fuel flow (WF) play an important role in understanding the aeroengine system [21, 42]. The mean of DCCA coefficients for the aeroengine time series is shown in Figure 8, where the average DCCA coefficient between N1 and N2 is 0.85, which shows the close cross-correlation between N1 and N2.Figure 8
The mean of standard DCCA coefficients for the time series of aeroengine system gas path parameters.We here investigate the partial correlation betweenN1 and N2 given a set of eight controlling variables, including WF, exhaust gas temperature (EGT), N2 tracked vibration channel B (N2TB), inlet air pressure (P2), outlet temperature of high-pressure compressor (T3), outlet temperature of low-pressure compressor (T2.5), and other temperatures (T2 and T2.95).In Figure9, we plot the influence degree of first-order DPCC coefficient, investigating the effect of the other eight controlling variables on cross-correlation characteristics between N1 and N2. The largest influence degree I = 0.51, obtained by T3, shows the information exchange between the outlet temperature of high-pressure compressor and the rotor speed system. The next largest I = 0.22 is acquired by WF, which indicates the association between the fuel flow system and rotor speed system.Figure 9
The influence degree of the first-order detrended partial cross-correlation in aeroengine system and the influence degree of the first-order multifractal detrended partial cross-correlation coefficient in aeroengine system (the upper left figure).The result of the influence degreeI1,q for eight aeroengine parameters applying by first-order MFDPCC coefficient with q=1,2,…,10 is also demonstrated in the upper left of Figure 9. The effect of T3 on cross-correlation characteristics, observed from influence degree function I1,q, decreases as the scale q increases. It indicates that the multifractal cross-correlation differs across values of q.Further, we apply the MFDPXA method on aforementionedN1 and N2 time series considering the T3 as common influencing factor. It is observed from Figure 10 that the corresponding spectra fxyα and fxy,zα are wide which shows the strength of multifractal behavior in analyzed time series. We observe that the width of singularity spectrum fxy,zα is narrower, and this implies the strength of multifractal nature is weak in analyzed bivariate time series.Figure 10
The multifractal spectrumfxyα of bivariate time series obtained through MFDXA method and fxy,zα of bivariate time series obtained through MFDPXA, where x, y, and z denote N1, N2, and T3.Here, we estimate the delayed effect of T3 on the correlation between N1 and N2 by using the time delay influence degreeIq,τ1,τ2. Figure 11 shows the time delay influence degree for q=2. It is obvious that the time delay influence degree gradually increases and then declines as a single-peak curve when τ2 remains constant. As τ2 increases, the peak value of time delay influence degree shifts rightward.Figure 11
The time delay influence degree for delayτ1 between N1 and N2 and delay τ2 between N1 and T3.The next observation concerns the influence degree of 2-controlling-variables DPCC coefficient in the aeroengine system. We now analyze the influence of two controlling parameters on the cross-correlation betweenN1 and N2. In Figure 12, we illustrate the comparative relation of the influence degree of 2-controlling-variables DPCC coefficient for aeroengine system. It can be seen that the larger elements in the symmetrical matrix are located at row 3 or column 6, which denote T3 has a greater impact on the correlation between N1 and N2.Figure 12
The influence degree of the 2-controlling-variables detrended partial cross-correlation for aeroengine system.Concerning the influence degreeI2,q of the 2-controlling-variables MFDPCC, we demonstrate 7 cases (T3 and WF, T3 and N2TB, T3 and T2, T3 and T2.5, T2.95 and N2TB, T2.95 and P2, and T2.95 and T2) for q=1,2,…,10 in Figure 13. Larger influence degrees exist in the cases with the presence of T3 (T3 and WF, T3 and N2TB, T3 and T2, and T3 and T2.5), which is consistent with the 2-controlling-variables DPCC method, as seen in Figure 12.Figure 13
The influence degree of the 2-controlling-variables multifractal detrended partial cross-correlation for the aeroengine system.For the aeroengine, the parametersN1 and N2 are chosen to indicate the engine thrust which depends on the throttle lever angle. Hence, the cross-correlation between them is strong. The temperature and pressure parameters are linked with many factors, including the compressor power, combustion efficiency, throttle lever angle, etc. Therefore, the dynamic interaction of these three groups makes the aeroengine function. These results estimate the influence of temperature and pressure parameters on the cross-correlation between N1 and N2.


## 4. Conclusion

In this paper, we propose thenth-order multifractal detrended partial cross-correlation analysis method and the n-controlling-variables multifractal detrended partial cross-correlation analysis method for understanding the interactions between two nonstationary time series. For comparing these new methods with classical measures, we introduce the influence degree function. We then apply the n-controlling-variables multifractal detrended partial cross-correlation analysis of stock markets and aeroengine performance parameters and measure the influence degree function of the partial cross-correlation in a dynamic system.To understand the numerous real-world systems where the output signals exhibit complex cross-correlation, both cross-correlation and partial correlation are subjects of investigation. The information ofn-variables MFDPCC helps people to research information exchange in complex systems. This paper gives two examples, stock markets and aeroengine systems. For stock time series, our results indicate that, concerning closing index values, there is little information exchange between the Chinese stock markets and the American-European stock markets, whereas the SSEC, SZI, and HSI, by first-order MFDPCC method and 2-controlling-variables MFDPCC, show frequent and abundant information exchange in Chinese stock markets. For aeroengine performance parameters, our results show that there is some information exchange between the engine rotor system and the aeroengine parameters, such as the outlet temperature of the high-pressure compressor and the fuel flow.We believe that the MFDPCC method can be used to detect the intrinsic interactions among multiple dynamical systems, and therefore it can be widely applied to many research fields such as the aeroengine health monitoring systems and the investment portfolio where the covariance is employed to explore the interaction of assets income.The multifractal detrended partial cross-correlation analysis is used to delete the possible indirect correlation, but it may also delete valuable information. This problem required further investigation, both experimental and theoretical. Hence, the results of this paper should be considered as preliminary results on the multifractal detrended partial cross-correlation analysis. Therefore, we hope that this study will be extended to analyze the filtered information.



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