## 3 Methodology  
### Detrended cross Correlation Analysis (DCCA)  
The detrended cross correlation analysis (DCCA) is a method for finding long range cross correlation properties in the non-stationary time series that were proposed by [PERSON] [3]. In fact, Pearson correlation coefficient is a sum of product that removes 0 th order global trend (i.e., average) for the whole box size. On the other hand, DCCA is a sum of product that removes each local trend (usually 1 st order) for given sized boxes. Therefore, DCCA has advantages that it can analyze the noise effect by removing various types of trend for various box sizes when compared to Pearson method [12; 26]. Here, we used _MATLAB_ as a simulation tool to apply the DCCA method, and \(R\) for network analysis and visualization.  
The first step in the DCCA is to accumulate two different time series \(\{x_{i}\}\) and \(\{y_{i}\}\) of equal lengths \(N\).  
\[X_{k}=\sum\limits_{i=1}^{k}x_{i},\ Y_{k}=\sum\limits_{i=1}^{k}y_{i}\qquad(k=1,\ 2,\ \ldots,\ N) \tag{2}\]  
Next, the local fluctuations are calculated, as follows.  
\[\begin{array}{l}f_{DFA}^{2}(n,i)=\frac{1}{n-1}\sum\limits_{i=1}^{i+n-1}\left( X_{k}-\widetilde{X}_{k,i}\right)^{2}\ \text{and}\\ f_{DCCA}^{2}(n,i)=\frac{1}{n-1}\sum\limits_{k=i}^{i+n-1}\left(X_{k}-\widetilde{ X}_{k,i}\right)\left(Y_{k}-\widetilde{Y}_{k,i}\right)\end{array} \tag{3}\]  
Equation (3) refers to the following process. We divide total time series by a specific box-length \(n\) to create \(N-n+1\) overlapping boxes and the limit size of the box-lengths is set to \(1/5\) of the total time series [27]. Here, \(\widetilde{X}_{k,i}\) and \(\widetilde{Y}_{k,i}\) are linear least squares in the \(i\)-th box and they are called 'local trends'. Subtracting 'local trends' \(\widetilde{X}_{k,i}\) from the accumulated data \(X_{k}\) and averaging the sum of squares make the local fluctuation \(f_{DFA}^{2}(n,i)\) of \(i\)-th box with box-length \(n+1\). If we want to see the cross fluctuation of the two time series, we can calculate \(f_{DCCA}^{2}(n,i)\) by replacing it with the mutual product for \(X_{k}\) and \(Y_{k}\) instead of the square for \(X_{k}\).  
Averaging all of the local fluctuations for \(N-n\) overlapping boxes, the fluctuations \(F_{DFA}^{2}(n)\) and \(F_{DCCA}^{2}(n)\) are induced as in Equation (4).  
\[\begin{array}{l}F_{DFA}^{2}(n)=\frac{1}{N-n+1}\sum\limits_{i=1}^{N-n+1}f_{ DFA}^{2}(n,i)\ \text{and}\\ F_{DCCA}^{2}(n)=\frac{1}{N-n+1}\sum\limits_{i=1}^{N-n+1}f_{DCCA}^{2}(n,i)\end{array} \tag{4}\]  
Equation (4) is, in fact, an extended version of variance and co-variance. 'local trend' can be any functions as well as a linear function. If the 'local trend' is a 0-th order monomial of length \(n=N\), the trend becomes the average value of time series, and \(F_{DFA}^{2}(n)\) and \(F_{DCCA}^{2}(n)\) are exactly same as variance and co-variance of time series. Therefore, the correlation coefficient \(\rho_{DCCA}(n)\) of box-length \(n\) can be made as Equation (5). Unlike the Pearson correlation coefficient, \(\rho_{DCCA}(n)\) can be obtained several values with varying box-length \(n\).  
\[\rho_{DCCA}(n)=\frac{F_{DCCA}^{2}(n)}{\sqrt{F_{DFA}^{2}(n)}\sqrt{F_{DFA}^{2}( n)}} \tag{5}\]The correlation coefficient has a value of \(-1\leq\rho_{DCCA}(n)\leq 1\). If \(\rho_{DCCA}(n)\) is close to 1, two time series are strongly correlated. Conversely, if the value is close to -1, they are strongly anti-correlated. In both cases, we can easily infer information regarding another time series from one. If \(\rho_{DCCA}(n)\) is close to 0, there is no correlation between the two time series.