## 3 Materials and Method
### Detrended Cross-Correlation Analysis Coefficient  
The DCCAC was first introduced and implemented by [PERSON] [48]. It uses detrended fluctuation analysis (DFA) [49] and detrended cross-correlation analysis (DCCA) [50] to measure the degree of cross-correlation. These methods have been applied in mobility [51], biological processes [52], climate [53], econophysics [54; 55; 56; 57], big data [58], and epidemic data series [59; 60; 61]. It provides an index for quantifying the cross-correlation between two nonstationary time series [62]. Examining these fluctuations enables the identification of co-movements between two time series. Two studies may be highlighted regarding the application of DCCAC to evaluate different behaviors. [PERSON] et al. [51] proposed a method to assess the impact of rainfall on bus accessibility for passengers with disabilities and the elderly compared to other passengers. Using the DCCAC, they identified the bus lines and city areas with the worst accessibility for PWD and elderly passengers during the rainy season. [PERSON] and [PERSON] [63] analyzed a twin-engine aircraft and found that the fuel flow time series from the left and right engines exhibited significantly stronger cross-correlations than the exhaust gas temperature time series from the same engines. Compared to [PERSON]'s method, the DCCAC is more robust to contaminated noises and less sensitive to the amplitude ratio between slow and fast components [64]. This is an efficient correlation coefficient used to estimate correlations between non-stationary  
Figure 1: Filtered data collected for the 7 vehicles. The axis Y is the measured voltage (V) in each vehicle and the axis X is the time in milliseconds. time series. The method also follows the procedure of [65] to test the significance of the correlation and calculate the 95% confidence intervals for non-significant results. DCCAC is calculated using the following equations [48]:  
Consider two paired time series, \(\{x_{t}\}\) and \(\{y_{t}\}\), where \(t=1,2,\ldots,N\), with \(N\) representing the total length of the time series. To remove non-stationary trends and better analyze the cross-correlations between the two time series, the cumulative sum (integration) of each series is computed to obtain two new integrated time series, as shown in Equation (1). \[xx_{k}=\sum_{t=1}^{k}x_{t}\quad\text{and}\quad yy_{k}=\sum_{t=1}^{k}y_{t},\ k=1,2, \ldots,N \tag{1}\]  
Next, the integrated time series \(\{xx_{k}\}\) and \(\{yy_{k}\}\) are divided into overlapping boxes of equal length \(s\). The number of overlapping boxes is given by \((N-s)\), and the box length \(s\) is constrained within the range \(4\leq s\leq\frac{N}{4}\). Within each box, the local trend of the time series is estimated using a least-squares polynomial fit, denoted as \(xP_{i}(k)\) for \(\{xx_{k}\}\) and \(yP_{i}(k)\) for \(\{yy_{k}\}\). The residuals from these local trends represent detrended fluctuations, which are used to compute the covariance of residuals within each box, as given by Equation (2). \[f^{2}\ xy(s,i)=\frac{1}{s+1}\sum_{k=1}^{i+s}(xx_{k}-xP_{i}(k))(yy_{k}-yP_{i}(k)) \tag{2}\]  
To obtain a global measure of covariance across all overlapping boxes, the average of the covariance over all boxes is computed, yielding the detrended covariance function, as shown in Equation (3). \[F^{2}\ xy(s)=\frac{1}{N-s}\sum_{i=1}^{N-s}f^{2}\ xy(s,i) \tag{3}\]  
Finally, the Detrended Cross-Correlation Analysis Coefficient (DCCAC) is determined using Equation (4). The DCCAC normalizes the covariance function by the individual detrended fluctuation functions \(F_{xx}(s)\) and \(F_{yy}(s)\), which represent the auto-affinity of each time series. \[DCCAC(s)=\frac{F^{2}\ xy(s)}{F_{xx}(s)F_{yy}(s)} \tag{4}\]  
The DCCAC is a dimensionless coefficient. Its value ranges between \(-1\) and \(+1\), where:  
* DCCAC = \(+1\): the time series are perfectly cross-correlated;
* DCCAC = \(0\): indicates no cross-correlation between the two analyzed time series;
* DCCAC = \(-1\): the time series are perfectly anti-correlated.