## 2 Methods
### Data Preprocessing
#### 2.2.1 EEMD and ADFuller Stationarity Test  
EMD is an adaptive and efficient technique used to handle time series. Its main contribution is that any complex series can be decomposed into a finite and typically small number of IMF \(imf\) and residual \(r(t)\), which are nearly non-overlapping orthogonal components in the frequency domain. The process of EMD is shown in Equation (2):  
\[x(t)=\sum_{i}^{k}imf_{i}+r(t) \tag{2}\]  
To enhance the reductions in mode mixing and improve the noise robustness of the decomposition process, EEMD performs multiple iterations of EMD on a given signal \(x(t)\)  
Figure 1: Data preprocessing process. This process constructs a new distribution of sensor relationships without an a priori sensor distribution.  
introducing different random noises into the signal each time. The \(imfs\) obtained from each iteration are then averaged to reduce the randomness introduced in a single run, resulting in the final \(imfs\).  
After EEMD, the original time series is decomposed into IMFs from high to low frequency. This implies that these \(imfs\) contain information ranging from short-time scale to long-timescale [16]. Subsequently, the decomposed \(imfs\) are reconstructed into three relatively independent components, each carrying different time-scale information. The reconstruction process is illustrated in Equation (3):  
\[\left\{\begin{aligned} I_{1}(t)=\sum im_{i}\frac{l_{i}}{n_{i}}\leq 2 88\\ I_{2}(t)=\sum im_{i},288<\frac{l_{i}}{n_{i}}\leq 2016\\ I_{3}(t)=\sum im_{i},\frac{l_{i}}{n_{i}}>2016\end{aligned}\right. \tag{3}\]  
where \(I_{1}(t)\) represents the short-term component, \(I_{2}(t)\) represents the middle-term component, and \(I_{3}(t)\) represents the long-term component. \(n_{i}\) is the total number of local extrema detected for each \(imf_{i}\), and \(l_{i}\) is the length of \(imf_{i}\).  
Since the sensor recordings are taken every 5 min, each day consists of 288 time granularities. If the ratio of \(l_{i}\) to \(n_{i}\) is below 288, it can be deemed a daily periodicity, suggesting a short-term component. This principle applies to other term components as well.  
Since the Granger causality test is only applicable to stationary series, the reconstructed time series components need to undergo the Augmented Dickey-Fuller test for stationarity testing. The process is illustrated in Equation (4):  
\[\Delta y_{t}=\rho y_{t-1}+\gamma+\beta_{1}\Delta y_{t-1}+\beta_{2}\Delta y_{t -2}+\dots+\beta_{p}\Delta y_{t-p}+\epsilon_{t} \tag{4}\]  
where \(\Delta\) represents the first-order difference, \(y_{t}\) is the value of the time series, \(\rho\) is the test statistic, \(\gamma\) is the intercept, \(\beta_{1},\beta_{2},\ \dots,\beta_{p}\) are the coefficients of the difference terms, and \(\epsilon_{t}\) is the introduced white noise. The null hypothesis of the ADF test is that the time series has a unit root, indicating non-stationarity, while the alternative hypothesis is that the series is stationary. During the ADF test, attention is focused on whether the test statistic \(\rho\) is is significant. If the null hypothesis is significantly rejected, it can be concluded that the time series is stationary.