## 3 Results and Discussion
### A Protection Effect Prediction Model Based on Projection Pursuit (PPR) Analysis Method
#### 3.5.2 Establishment and Analysis of the PPR Model  
This study employed a dataset comprising nine groups for model construction, while reserving eight independent groups for validation. To evaluate model performance, we first applied the projection pursuit regression (PPR) algorithm to analyze the modeling samples. The relative errors between measured and fitted values were then calculated using the \"reverse fitting\" method, with an acceptance criterion of \(1\delta 1<10\%\)[35]. Upon confirming that the modeling samples met the required qualification rate, we subsequently tested the PPR model using the reserved eight validation datasets. When the validationsamples similarly satisfied the predetermined criteria, the PPR model was confirmed to demonstrate robust reliability and stability.  
During model operation, five critical projection parameters must be specified: \(P=3\): number of influencing factors (independent variables); \(Q=1\): number of response variables (dependent variables); \(S=0.1\): smoothing coefficient (range: 0-1), where smaller values indicate higher model sensitivity to data variations; \(M=5\): maximum allowable number of ridge functions; \(M_{u}=3\): optimal number of ridge functions. The combination of \(M\) and \(M_{u}\) determines the model's precision in capturing the underlying data structure. The derived projection directions obtained through computational modeling are given in Equation (3), the corresponding weight coefficients (\(\beta\)) are listed in Equation (4), and the ridge functions are visualized in Figure 11.  
\[\left(\begin{array}{c}\overset{\alpha}{\rightarrow}_{1}\\ \overset{\alpha}{\rightarrow}_{2}\\ \overset{\alpha}{\alpha}_{3}\end{array}\right)=\left[\begin{array}{cc}0.9225 0833&0.11691377&0.64317800\\ 0.08619222&0.99618779&0.01344602\\ 0.26885471&0.29706425&0.91622594\end{array}\right] \tag{3}\]  
\[\beta=(0.99965939\,,\,0.74574278,\,0.82382684) \tag{4}\]  
Figure 11: Ridge function diagram of PPR model: (**a**) ridge function T-f\({}_{1}\); (**b**) ridge function T-f\({}_{2}\); (**c**) Ridge function T-f\({}_{3}\).  
Additionally, the PPR program was used to calculate the thermal insulation performance of the slurry-coated geotextiles. The contribution weight coefficients of each influencing factor to the thermal insulation temperature were determined as follows: geotextile thickness = 1.208, clay content = 1.105, and slurry thickness = 1.408. This indicates that the order of contribution weights of the three factors to thermal insulation temperature is as follows: slurry thickness > geotextile thickness > clay content.  
Based on the PPR calculation results, the predictive model formula for the thermal insulation temperature of slurry-coated geotextiles is as follows:  
\[S=17.11-0.0033\left[\text{f}_{1}(\sum_{j=1}^{3}\alpha_{1}\text{x}_{j})+0.746 \text{f}_{2}(\sum_{j=1}^{3}\alpha_{1}\text{x}_{j})+0.824\text{f}_{3}(\sum_{j=1 }^{3}\alpha_{1}\text{x}_{j})\right] \tag{5}\]  
In the formula, \(S\) is the predicted value of insulation temperature; \(f\) represents the ridge function; \(\alpha\) is the projection direction; and \(x\) is the horizontal factor.  
As illustrated in Figure 12, based on the aforementioned PPR modeling process, the predicted thermal insulation temperatures for both the modeling and validation samples demonstrate that the relative error (\(\delta\)) between the predicted and measured values remains within 10%. This outcome robustly validates the reliability of the PPR model. Consequently, this model can be utilized to accurately predict the thermal insulation temperature of slurry-coated geotextile bags in untested conditions and to analyze the optimal parameter configurations for thermal insulation performance across different scenarios.