## 3 Results and Discussion
### A Protection Effect Prediction Model Based on Projection Pursuit (PPR) Analysis Method  
#### 3.5.1 Introduction to the Principles of the PPR Model  
The principle of projection pursuit regression (PPR) involves using computer algorithms to project high-dimensional experimental data into a low-dimensional subspaces (typically 1-3 dimensions), thereby achieving effective data dimensionality reduction. By systematically analyzing the reduced-dimension data, the method identifies characteristic projection directions that preserve the essential structural features of the original high-dimensional data. These optimal projections then serve as the basis for constructing regression models, ultimately enabling effective research and analysis of high-dimensional datasets [32].  
The mathematical expression of the projection pursuit regression (PPR) model is given as follows [33]:  
\[\mathrm{f}(\mathrm{x})\ \sim\ \sum\limits_{\mathrm{m}}^{M}\beta_{\mathrm{m}}G_{ \mathrm{m}}(Z_{\mathrm{m}})=\sum\limits_{\mathrm{m}}^{M}\beta_{\mathrm{m}}G_{ \mathrm{m}}\left(\sum\limits_{\mathrm{m}}^{P}\alpha_{\mathrm{mj}}^{T}X\right) \tag{2}\]  
\(G_{\mathrm{m}}(Z_{\mathrm{m}})\) denotes the m-th ridge function; \(\beta_{\mathrm{m}}\) represents the weighting coefficient, indicating the contribution of the m-th ridge function to the output; \(Z_{\mathrm{m}}=\sum\limits_{\mathrm{m}}^{P}\alpha_{\mathrm{mj}}^{T}X\) is the argument of the ridge function, representing the projection of the P-dimensional input vector \(X\) onto the direction \(\alpha_{\mathrm{m}}\); \(\alpha_{\mathrm{mj}}\) is the j-th component of the m-th projection direction vector \(\alpha_{\mathrm{m}}\); and \(p\) denotes the dimensionality of the input space. Requirement \(\sum\limits_{\mathrm{j}}^{P}\alpha_{\mathrm{j}}^{2}=1\).  
Projection pursuit regression (PPR) analysis hinges on identifying optimal projection directions, with genetic algorithms currently being the predominant method for optimizing both projection directions and ridge function parameters. The solution procedure [34] is as follows:  
(1) Given an initial projection direction \(\alpha\), compute \(\mathrm{w_{j}}=\sum\limits_{\mathrm{j}}^{P}\alpha_{1}\), \(x_{\mathrm{j}}\), \(j=1\), \(L\), \(n\). by projecting the independent variables onto direction \(\alpha\), yielding \(n\) one-dimensional projection values.  
(2) The ridge function \(f_{1}\) is estimated via least squares minimization of the residual sum of squares \(Q=\sum\limits_{\mathrm{j}}^{n}\left(\mathrm{y_{j}}-\hat{y_{\mathrm{j}}}\right) ^{2}\).  
(3) Iteratively repeat steps (1)-(2) until convergence is achieved, obtaining the optimal projection direction \(\alpha_{1}\) and ridge function \(f_{1}\) to globally minimize \(Q\).  
(4) Verify whether the model meets convergence criteria. If satisfied, terminate computation; otherwise, proceed with subsequent calculations.  
(5) Sequentially identify subsequent ridge functions by replacing the dependent variable with residuals and re-establishing relationships with independent variables. Repeat steps (1)-(3) iteratively until obtaining the m-th optimal projection direction \(\alpha_{1}\) and corresponding optimal ridge function \(f_{1}\).  
(6) Iteratively repeat steps (4)-(5) until the model satisfies all convergence criteria and performance requirements.