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

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. .ese analyses suggest that the proposed measure, derived from the detrended crosscorrelation coefficient analysis, may be used to remove the influence of other variables on the cross-correlation between two simultaneous time series.


Introduction
ere 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. e 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]: where 〈X〉 is the arithmetic average of X and σ X 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. erefore, 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]. e advantage of the DCCA method is that it allows the detection of crosscorrelations between noisy signals with embedded polynomial trends, which can mask the true cross-correlations in the fluctuations of signals. e 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][16][17][18][19][20], and aeroengine dynamics [21][22][23][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][26][27], electroencephalography [28], and economics [29][30][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][33][34]. e 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. en, the methods of partial correlation and partial correlation coefficient are therefore proposed to measure the degree of association between two random variables [35,36]. e linear effect may be removed using the partial correlation coefficient (partial CC): where X ′ � X − L X (ξ) and L X (ξ) � c 0 + c 1 ξ to minimize the mean E(X − L X (ξ)) 2 and likewise for Y ′ . If n additional variables are to be accounted for, say ξ 1 , ξ 2 , . . . , ξ n , the nthorder partial CC can be computed by [36] 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 of nth-order multifractal detrended partial cross-correlation analysis by incorporating the partial correlation coefficient with the multifractal detrended cross-correlation analysis. e rest of the paper is organized as follows. In the next section, we introduce the multifractal DCCA coefficient method and propose the method of nth-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.

Multifractal Detrended Partial Cross-Correlation
Analysis. For the sake of clarity, we begin with a summary of the multifractal DCCA coefficient algorithm. For two series r i (t) and r j (t) 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: where 〈r i 〉 and 〈r j 〉 are the average values of r i (t) and r j (t) , respectively.
Step 2: divide the profiles R i (k) and R j (k) into N s � int(N/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. us, we obtain 2N s segments of equal length s. In this paper, we follow the previous literature practice and set 10 ≤ s ≤ N/4.
Step 3: for each segment v(v � 1, 2, . . . , N s , N s + 1, . . . , 2N s ), the local trends R v i (k) and R v j (k) are estimated on the basis of a least-squares fit of the sequences R i (k) and R j (k) , respectively. e corresponding detrended covariance for v � 1, 2, . . . , N s is where R v i (k) and R v j (k) are the fitting polynomials in the segment v.
Step 4: calculate the average of multifractal detrended covariance fluctuation function F q DCCA (s, v) over all segments: Generally, q can take any real value, except zero. For q � 0, the equation becomes For q � 2, F q DCCA (s, v) is equal to the detrended crosscorrelation fluctuation function F 2 DCCA (s).
Step 6: compute the multifractal detrended partial cross-correlation coefficient between X and Y by eliminating the influence of the controlling variable ξ 1 on X and Y analogous to the generalization of the correlation coefficient to partial correlation coefficient: named the first-order multifractal detrended partial cross-correlation coefficient (first-order MFDPCC coefficient), where X, Y are random variables, ξ 1 is the controlling variable, and ρ q XY , ρ q X1 , ρ q Y1 represent the mean of MFDCCA coefficients for X and Y, X and ξ 1 , and Y and ξ 1 , respectively.
For q � 2, the first-order detrended partial cross-correlation coefficient (first-order DPCC coefficient) is retrieved.

Data and Analysis
3.1. Two-Component ARFIMA Process. In order to test the robustness of the proposed n-controlling-variables MFDPCC coefficient method, power-law cross-correlated time series u i and v i are generated by using the twocomponent ARFIMA stochastic process in this section [18,39,40]. In this model, the series is defined by weight, W is a free parameter to control the coupling strength between u i and v i (0.5 ≤ W ≤ 1), and ε i and ε i are independent and identically distributed (i.i.d.) Gaussian variables with 〈ε i 〉 � 〈ε i 〉 � 0 and 〈ε i 2 〉 � 〈ε i 2 〉 � 1 [18,39]. For different values of W, the different coupling strength between the variables u i and v i is 1 − W. In this section, the two-component ARFIMA series u i and v i with parameter ρ 1 � ρ 2 � 0.3 and W � 0.5, denoted by X and Y, are employed to detect the interactions between two time series. en, the effect of white noise sequence ξ 1 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 I(n, q). e influence degree function is defined as We calculate the influence degree function I(n, q) of the synthetical signals using the proposed first-order MFDPCC coefficient and present the influence degree function I(1, q) vs. parameter q in Figure 1.
e 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.

Stock Market.
To further exemplify the potential utility of the n-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),  Figure 2 shows the mean of DCCA coefficients for the stock series. e 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. e next largest DCCA coefficient ρ � 0.92 is obtained by SSEC and SZI, which indicates the close cross-correlation in Chinese mainland stock markets. e 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. e mean of DCCA coefficients between SZI and HSI is in an intermediate state, which indicates the existence of crosscorrelation 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 degree I � 0.05 is obtained by HSI, which shows the information exchange between the Chinese stock market, as seen in Figure 3. e 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). e 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. e influence degree values of 13 stock markets for firstorder MFDPCC coefficient with q � 1, 2, . . . , 10 are also demonstrated in the upper left of Figure 3.
During the analysis, we observe the effect of HSI on cross-correlation characteristics between SSEC and SZI from influence degree function I(1, q) that decreases as the scale q increases. And this infers the change of multifractal crosscorrelation.
In order to capture the change of multifractal crosscorrelation 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. e result shows that both the corresponding spectra f xy (α) and f xy,z (α) are wide, but the latter is narrower than the former, which is presented in Figure 4.
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   Figure 3: e 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).  Complexity influence on multifractal spectrum, as seen in Figure 4. We compare the obtained influence degrees with the aforementioned method and infer that the HSI has significant influence on SZI and SSEC time series. ese 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]. erefore, we investigate the delayed effect of variable ξ 1 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 ξ 1 , as a function of two time delays: where τ 1 is the delay between variables X and Y and τ 2 is the delay between variables X and ξ 1 .
In this section, we estimate the delayed effect of HSI on the correlation between SSEC and SZI by using the time delay influence degree I(q, τ 1 , τ 2 ) � |ρ q XY(τ 1 ),1(τ 2 ) − ρ q XY |. Figure 5 shows the time delay influence degree for q � 2. e effect of τ 1 on influence degree is weaker than that of τ 2 on influence degree.
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 Figure 6, we illustrate the comparative relation of the influence degree of 2-controllingvariables DPCC coefficients for 13 × 13 elements by the matrix diagram.
We note that the structure of the matrix is symmetrical and that element at the intersection of row i and column j represents the influence of controlling variables ξ i , ξ j on the cross-correlation of SSEC and SZI, where the 2-controllingvariables ξ i , ξ j (i, j � 1, 2, . . . , 13) are the stock time series from IBOV, DJI, IXIC, SPX, FISE, FCHI, GDAXI, N255, KS11, HSI, AS51, SENSEX, and RTS. erefore, 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. is is consistent with our result of first-order MFDPCC coefficient method.
Concerning the influence degree I(2, 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. e largest influence degree is the case SENSEX and HSI, which is consistent with the 2-controlling-variables DPCC method.

Aeroengine Time Series.
Previous research studies show that the aeroengine gas path parameters such as lowpressure rotor speed (N1), high-pressure rotor speed (N2), and fuel flow (WF) play an important role in understanding the aeroengine system [21,42]. e 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.
In Figure 9, 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. e 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. e next largest I � 0.22 is acquired by WF, which indicates the association between the fuel flow system and rotor speed system. e result of the influence degree I(1, 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. e effect of T3 on cross-correlation characteristics, observed from influence degree function I (1, 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 aforementioned N1 and N2 time series considering the T3 as common influencing factor. It is observed from Figure 10 that the corresponding spectra f xy (α) and f xy,z (α) are wide which shows the strength of multifractal behavior in analyzed time series. We observe that the width of singularity spectrum f xy,z (α) is narrower, and this implies the strength  e influence degree of the 2-controlling-variables detrended partial cross-correlation. 6 Complexity of multifractal nature is weak in analyzed bivariate time series.
Here, we estimate the delayed effect of T3 on the correlation between N1 and N2 by using the time delay influence degree I(q, τ 1 , τ 2 ). 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 τ 2 remains constant. As τ 2 increases, the peak value of time delay influence degree shifts rightward. e 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 between N1 and N2.
In Figure 12, we illustrate the comparative relation of the influence degree of 2-controlling-variables DPCC   Complexity 7 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.
For the aeroengine, the parameters N1 and N2 are chosen to indicate the engine thrust which depends on the throttle lever angle. Hence, the cross-correlation between them is strong. e temperature and pressure parameters are linked with many factors, including the compressor power, combustion efficiency, throttle lever angle, etc. erefore, the dynamic interaction of these three groups makes the aeroengine function. ese results estimate the influence of temperature and pressure parameters on the cross-correlation between N1 and N2.

Conclusion
In this paper, we propose the nth-order multifractal detrended partial cross-correlation analysis method and the n-controlling-variables multifractal detrended partial crosscorrelation 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 crosscorrelation 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. e information of n-variables MFDPCC helps people to research information exchange in complex systems.
is 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 2controlling-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. e multifractal detrended partial cross-correlation analysis is used to delete the possible indirect correlation, but it may also delete valuable information.
is problem   e influence degree of the 2-controlling-variables detrended partial cross-correlation for aeroengine system. 8 Complexity 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. erefore, we hope that this study will be extended to analyze the filtered information.
Data Availability e stock market data used to support the findings of this study are available from the corresponding author upon request. e aeroengine data used to support the findings of this study have not been made available because of commercial secrets.

Conflicts of Interest
e authors declare that they have no conflicts of interest.