## 4 BP Neural Network and Pearson Correlation Analysis
### Computation Analysis
#### 4.2.3 Correlation Analysis of Individual Influencing Factors  
The Pearson correlation calculation method quantitatively reflects the positive or negative correlations between individual factors and experimental data, serving as an effective counterpart to extreme variance analysis. Pearson's linear correlation coefficient (R) was computed as follows:  
\[\mathrm{R=}\mathrm{\sum_{i=1}^{S}\left(X_{i}-\overline{X_{i}}\right)\cdot\left( Y_{i}-\overline{Y_{i}}\right)/\sqrt{\mathrm{\sum_{i=1}^{S}\left(X_{i}- \overline{X_{i}}\right)^{2}\cdot\mathrm{\sum_{i=1}^{S}\left(Y_{i}-\overline{Y_ {i}}\right)^{2}}}} \tag{20}\]  
In Equation (20), [PERSON]'s linear correlation coefficient R showed a value ranging from \(-1\) to \(1\), where the interval of 0-1 could be subdivided as follows: 0.0-0.2 was considered very weak or no correlation, 0.2-0.4 indicated a weak correlation, 0.4-0.6 represented a moderate correlation, 0.6-0.8 signified a strong correlation, and 0.8-1.0 denoted a very strong correlation. This classification was similarly applied to the \(-1\)-0 interval. The correlation between the individual factors and the ballast tank drainage percentage was verified for the 18 orthogonal experimental conditions [30]. The parameters selected for generating [PERSON]'s correlation analysis heat map included blowing duration (s), the cylinder group with gas pressure (MPa) and volume (L), the ballast tank liquid level (cm), gas supply pipeline with length (m) and inner diameter (mm), the sea valve inner diameter (mm), the ballast tank with gas pressure (MPa) and temperature (MPa), the sea tank back pressure (MPa), and the ballast tank drainage percentage (dimensionless), as illustrated in Figure 40.  
Figure 40: Pearson analysis heat map. Figure 39: Prediction precision percentages of working condition 18. The logical relationship between each factor and the blowing process provided supplementary insights to the conclusions drawn from the orthogonal experiments in Sections 3.1 and 3.2:  
* First, the blowing duration exhibited a correlation coefficient of 0.6535, indicating a strong positive correlation. Theoretically, a longer gas supply duration resulted in a greater amount of water being expelled [31];
* Second, the sea tank back pressure had a value of 0.8105, reflecting an extremely strong positive correlation. This suggests that the ballast tank drainage was closely related to the outboard back pressure through a complex relationship. Specifically, the variations in back pressure influenced the pressure changes during blowing within the ballast tank [12; 32]. Additionally, back pressure affected the dynamic balance between gas and water, as well as the corresponding blowing efficiency [2; 31]. Finally, it impacted energy consumption and system efficiency during blowing operations [12; 32];
* Third, the volume of the gas cylinder group was associated with a correlation coefficient of 0.3587, indicating a weak positive correlation. According to the aerodynamic computation theories, there exists a direct relationship between gas consumption and ballast tank drainage [32];
* Fourth, the gas pressure within the cylinder group registered at 0.5569, indicating a moderately strong positive correlation and suggesting that higher gas pressure significantly affected blowing effectiveness [13];
* Fifth, the inner diameter of the gas supply pipeline showed a value of 0.02499, and this represents a very weak positive correlation. Theoretically speaking, a larger inner diameter would have increased the gas supply efficiency over time, which would facilitate the quicker establishment of gas cushions at the top of the ballast tank, thereby improving the overall blowing performance [12]. However, due to size constraints on the flow dynamics caused by only having pipe diameters available at 6 (mm), 8 (mm), and 10 (mm), with lengths limited to just 0.3 (m);
* Sixth, the length of the gas supply pipeline was \(-\)0.3741, indicating a medium-strength negative correlation.