## 3 Experiments
### ADF, Granger Test, and Adjacency Matrix Construction  
For example, the EEMD and reconstruction process (outlined in Section 2.2) were applied to the training set data of sensor #0 within the context of this study. Subsequently, the results of this process were visualized, as shown in Figure 4, which illustrates the decomposition and reconstruction outcomes for sensor #0.  
The three components after treatment showed significant differences, as shown in Figure 4. The short-term component reflects non-linear fluctuations daily, indicating that the daily water demand recorded by this sensor was unstable. The medium-term component generally aligns with the ups and downs of the water demand record, suggesting that the weekly demand remains relatively stable when observed weekly. Furthermore, the long-term component exhibited a decreasing trend, possibly related to population decline resulting from neighboring factories relocating.  
The ADF test results for each sensor reconstruction decomposition are presented in Table 1. The statistical significance of both the short-term time-series components and the medium-term time-series components led to the rejection of the original hypothesis, suggesting that these reconstruction series exhibit smooth characteristics. In contrast, the results of the long-term time-series component, sensor #0 and sensor #2, aligned with the original hypothesis, and therefore, they were non-stationary. They could not be involved in subsequent Granger causality tests.  
Figure 4: Time series EEMD decomposition results for sensor #0 (ultrasonic water flow meter) and Merged results. Raw data is in orange, and decomposition results are in blue.  
The \(p\)-values obtained from the Granger causality test are detailed in Table 2. The process of the neighbor matrix was constructed according to the results of the Granger causality test, as shown in Figure 5. Following the experiment, the threshold for the \(p\)-value was predominantly set at 0.01; values above this threshold indicate an insignificant causal relationship, whereas values below it suggest a significant causal relationship between two variables. When constructing Granger causality, only relationships that demonstrate substantial causality, as determined by this threshold, are taken into account.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**Short-Term Component**} \\ \hline Sensor ID & \#0 & \#1 & \#2 & \#3 & \#4 \\ \#0 & N/A & 0.0063 & 0.0000 & 0.0005 & 0.7162 \\ \#1 & 0.0001 & N/A & 0.5182 & 0.0306 & 0.7539 \\ \#2 & 0.0000 & 0.2087 & N/A & 0.0000 & 0.6710 \\ \#3 & 0.0000 & 0.0009 & 0.0000 & N/A & 0.8560 \\ \#4 & 0.2399 & 0.1041 & 0.7045 & 0.7180 & N/A \\ \multicolumn{5}{c}{**Middle-Term component**} \\ Sensor ID & \#0 & \#1 & \#2 & \#3 & \#4 \\ \#0 & N/A & 0.0000 & 0.1025 & 0.0000 & 0.0000 \\ \#1 & 0.0000 & N/A & 0.0000 & 0.0003 & 0.0014 \\ \#2 & 0.0000 & 0.0000 & N/A & 0.0006 & 0.0055 \\ \#3 & 0.0000 & 0.0000 & 0.0023 & N/A & 0.0037 \\ \#4 & 0.0000 & 0.2030 & 0.0003 & 0.0000 & N/A \\ \multicolumn{5}{c}{**Long-Term component**} \\ Sensor ID & \#0 & \#1 & \#2 & \#3 & \#4 \\ \#0 & N/A & N/A & N/A & N/A & N/A \\ \#1 & N/A & N/A & N/A & N/A & N/A \\ \#2 & N/A & N/A & N/A & 0.0000 & 0.1523 \\ \#3 & N/A & 0.0033 & N/A & N/A & 0.0064 \\ \#4 & N/A & 0.0021 & N/A & 0.0002 & N/A \\ \hline \hline \end{tabular}
\end{table}
Table 2: Granger causality test results.  
Figure 5: Partial Granger-directed graph construction process.