# Hybrid Machine Learning Model for Predicting Shear Strength of Rock Joints
## 4 Results and Analysis
### Discussion on Data Processing Method  
The data normalization method has a significant impact on both the predictive accuracy and training stability of machine learning models. In general, normalization not only eliminates the magnitude discrepancies inherent in raw data, but also markedly improves the computational efficiency of learning algorithms. Therefore, this study investigates the influence of different normalization strategies on the prediction performance of the SMA-MLP model. Three commonly used normalization techniques were selected for comparison:  
1. Z-score normalization (Equation (7)), which standardizes the data to have zero mean and unit variance;
2. Min-Max normalization (Equation (13)), which linearly scales features into the [0, 1] interval;
3. Arctangent normalization (Equation (14)), which applies a nonlinear transformation to compress input values into the (\(-1\), 1) interval to suppress the influence of outliers. \[\overline{X}=\frac{X-X_{min}}{X_{max}-X_{\min}}\] (13) \[\overline{X}=2\frac{arctanX}{\pi}\] (14)  
where \(X\) and \(\overline{X}\) denote the original and normalized values, respectively, while X\({}_{\max}\) and X\({}_{\min}\) denote the maximum and minimum values of the original dataset, respectively. Figure 11 presents the distributional characteristics of the input and output variables under three normalization strategies. As shown, the Z-score normalization defined by Equation (7) transforms each variable to have zero mean and unit variance, while preserving the original relative variability and outlier patterns. This method is particularly suitable for models that are sensitive to the covariance structure of the input data. The Min-Max normalization described in Equation (13) linearly scales all features into the range [0, 1], enhancing the comparability between variables, and is especially appropriate for models--such as neural networks--that are sensitive to feature magnitudes. However, this method is highly sensitive to extreme values, which may compress the main data distribution and cause most samples to cluster around the center, potentially masking meaningful distributional information. In contrast, the Arctangent normalization method (Equation (14)) demonstrates stronger compressive capacity through nonlinear transformation, mapping data into the (0, 1) interval while significantly suppressing high-magnitude values. This makes it more robust against outliers and better suited to datasets with long-tailed distributions. In summary, each normalization technique has its own strengths in terms of scale unification and outlier resistance. The choice of method should be based on the characteristics of the dataset and the specific requirements of the predictive model. Each normalization method was independently applied to the SMA-MLP model. After preprocessing with each normalization method, the same modeling procedure was applied. Each model was trained using an identical training set and evaluated on the same testing set. The predictive results under the three normalization schemes on the test set are illustrated in Figure 12. It is worth mentioning that the prediction result of Equation (7) is the model prediction result shown in Section 4.1. It can be observed from Figure 12 that the Z-score normalization yielded the highest overall prediction accuracy, as indicated by the highest R\({}^{2}\) value and the lowest RMSE. This superior performance may be attributed to its ability to maintain a balanced dynamic range of input features while preserving the statistical distribution of the data, which facilitates stable gradient propagation and effective parameter learning in MLP networks. Figure 11: Boxplot comparison of three normalization methods. Figure 12: Performance evaluation indicators results for different data processing method. The Min-Max normalization also shows higher prediction accuracy, but the prediction accuracy is obviously lower than Z-score normalization.