## 4 Results and Discussion
### Optimization of Modeling Results
#### 4.1.2 Characteristics of Variables Based on Optimization Modeling Results  
In the dust-spraying experiment, regression coefficients for the factors were estimated using the least squares method as shown in Equation (4) to ascertain the influence of four factors (A: spray pressure; B: speed; C: spray distance; D: number of sprays) on the response variable (point cloud). \[minL=\sum_{i=1}^{n}\epsilon_{i}^{2}=(y-X\beta)^{\mathsf{T}}(y-X\beta), \tag{4}\]  
where  
\[\frac{\partial L}{\partial\beta}=-2X^{\mathsf{T}}y+2X^{\mathsf{T}}X\beta\ =\ 0,\]  
\[\beta\cong\left(X^{\mathsf{T}}X\right)^{-1}X^{\mathsf{T}}y.\]  
The resulting second-order polynomial equation is as follows:  
\[ \begin{array}{l}\text{Reduced Point Cloud}=1594.8+572.4\text{A}+61.2\text{B }-85\text{C}+610.6\text{D}-235\text{A}^{2}+181\text{B}^{2}\\ -357\text{C}^{2}-227\text{D}^{2}+352.9\text{AD}-115.8\text{CD}\end{array}  \tag{5}\]  
As presented in Table 5, the ANOVA results for dust dispersion factors demonstrate that Spray pressure (A) and number of sprays (D) had statistically highly significant positive \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**RMSE** & **R-Squared** & **Adj. R-Squared** & **F-Value** & **\(p\)-Value** \\ \hline
166.08 & 0.9712 & 0.9560 & 64.05 & \textless{}0.0001 *** \\ \hline \hline \end{tabular}  
* \({}^{\mathsf{avg}}p<0.01\).  
\end{table} 
Table 4: ANOVA results for the initial dust dispersion model. Figure 5: Results of modeling: (**a**) normal probability plot, and (**b**) histogram. coefficients, indicating that increasing these factors leads to a pronounced increase in the point cloud value. Conversely, spray distance (C) exerted a negative influence, suggesting that increasing the distance reduces particle density and may impair spray consistency. Spray speed (B) was not significant as a main effect (\(p\) = 0.134); however, its quadratic term (B\({}^{2}\)) approached significance at the 10% level (\(p\) = 0.096), implying possible saturation or adverse effects under certain conditions. The quadratic terms for spray pressure (A\({}^{2}\)), distance (C\({}^{2}\)), and number of sprays (D\({}^{2}\)) were all statistically significant. Notably, the curvature effect for spray distance (C\({}^{2}\)) was the most pronounced, indicating that both excessively short and excessively long distances reduce the response. As illustrated in Figure 6b,\(t\), the optimal spray distance likely resides within the 270-300 mm range. Interaction analysis showed that spray pressure-number of sprays (AD) had a strong positive interaction, meaning simultaneous increases in both variables substantially augment the point cloud. By contrast, spray distance-number of sprays (CD) showed a negative interaction, suggesting the response may decrease under such combined conditions. In conclusion, setting spray pressure (A) and number of sprays (D) to their highest feasible levels while maintaining spray distance (C) at an intermediate level was identified as the most effective condition for optimizing the point cloud. The 3D surface plots in Figure 6 illustrate the interaction effects between pairs of spray parameters on the resulting point cloud value. (a) Spray Pressure-Speed Interaction  
Figure 6a illustrates the interaction between spray pressure (A) and spray speed (B). An increase in spray pressure consistently results in higher point cloud values, which indicate improved cleaning effectiveness. Conversely, the effect of spray speed is less pronounced and may exhibit a slight nonlinear behavior, thereby suggesting that excessively high speeds do not necessarily enhance performance under certain conditions. (b) Spray Pressure-Distance Interaction  
Figure 6b presents the combined effect of spray pressure (A) and spray distance (C) on LiDAR point cloud values. While increased spray pressure enhances cleaning performance, spray distance exhibits a nonlinear trend with a clear curvature.