## 2 Methods
### Detrended Cross-Correlation Analysis (DCCA)  
In order to quantify long-range cross correlations between two non-stationary time series, a modification of the covariance analysis, called DCCA, was proposed by [PERSON] and [PERSON] [23]. The authors firstly considered two long-range cross-correlated time series \(x(i)\) and \(y(i)\) of equal length \(N_{\max}\), from which the authors computed two integrated signals as follows:  
\[\begin{array}{l}X(k)=\sum\limits_{i=1}^{k}\big{[}x(i)-\langle x\rangle\big{]},\\ Y(k)=\sum\limits_{i=1}^{k}\big{[}y(i)-\langle y\rangle\big{]}.\end{array} \tag{8}\]  
The authors divided the integrated time series into \(N_{n}\) (\(=N_{\max}-n+1\)) overlapping boxes of equal size \(n\) and defined the local trends \(\overline{X}_{\
u}(k)\) and \(\overline{Y}_{\
u}(k)\) of a polynomial function for the \(\
u\)-th box by using a linear least-squares fit. Next, by defining the detrended walk as the difference between the integrated series and the local trend, the authors calculated the covariance of the residuals in the \(\
u\)-th box as follows:  
\[F_{n,DCCA}^{2}(\
u)=\frac{1}{n}\sum\limits_{i=1}^{n}\Big{\{}X[(\
u-1)n+i]- \overline{X}_{\
u}(i)\Big{\}}\Big{\{}Y[(\
u-1)n+i]-\overline{Y}_{\
u}(i)\Big{\}}, \tag{9}\]  
and the detrended covariance function was obtained by averaging the above covariance functions as follows:  
\[F_{DCCA}(n)=\Bigg{\{}\frac{1}{N_{n}}\sum\limits_{\
u=1}^{N_{n}}F_{n,DCCA}^{2} (\
u)\Bigg{\}}^{1/2}. \tag{10}\]  
In order to examine the behavior of the covariance function versus \(n\), the authors considered two cases: one with no power-law cross-correlations in both time series, and the other with power-law cross-correlations in both series. For the former, the exponent of \(F_{DCCA}(n)\) is not defined because it has complex values; thus, the authors defined a new measure to quantify the degree of cross-correlation in both time series with non-stationaries. The DCCA coefficient \(\rho_{DCCA}(\alpha,\ \alpha^{\prime},\ N_{\max},\ n)\), defined below, was introduced by [PERSON] [27].  
\[\rho_{DCCA}(\alpha,\ \alpha^{\prime},\ N_{\max},\ n)=\frac{F_{DCCA}^{2}(n)}{F_{ DFA}(n)F_{DFA}^{\prime}(n)}, \tag{11}\]  
which is dependent on the length of two time series, their respective DFA exponents \((\alpha,\ \alpha^{\prime})\), and the box size \(n\). [PERSON] et al. presented the framework to determine the statistical significance of \(\rho_{DCCA}(\alpha,\ \alpha^{\prime},\ N_{\max},\ n)\) via the computed critical point \(\rho_{c}(N_{\max},\ n)\) for the 95% confidence level by varying the data length \(N_{\max}\) and the box size \(n\)[28]. For the latter approach, the authors could find a power-law scaling between both time series, as shown below.  
\[F_{DCCA}(n)\sim n^{\lambda}. \tag{12}\]  
In particular, for two persistent time series, the DCCA exponent \(\lambda\) was approximately equal to the average of both DFA exponents, i.e., \(\lambda\approx(\alpha+\alpha^{\prime})/2\)[24]. However, for a pair of persistent and anti-persistent series, the authors had no clear support for the previous relationship. According to the numerical analysis, the DCCA exponent \(\lambda\) tended to approximately 0.5. Lastly, the authorsaddressed the fact that there should be a common error term in both time series if the authors observed a power-law cross-correlation between them.