# Frost Resistance Prediction of Concrete Based on Dynamic Multi-Stage Optimisation Algorithm
## 6 Results and Discussion  
This study constructs four traditional prediction models and four optimised prediction models to evaluate the performance of the models through the following six indicators: R\({}^{2}\), MSE, RMSE, MAE, CC, and SDR ratio. Tables 5 and 6 show the average performance results of the test set after ten-fold cross-validation. From the two tables, the optimisation model using the DMSOA shows better performance than the traditional model, with lower MSE, RMSE, and MAE values, and higher R\({}^{2}\), CC, and SDR values, which indicate that the model with DMSOA has a significant improvement in accuracy and prediction error. Among all models, the DMSOA-CatBoost model shows the best performance, and R\({}^{2}\) reached 0.864 and 0.885 on the test set of the two datasets, respectively. Moreover, the prediction results were the most accurate among all models. more directly, the paper uses a Taylor diagram to compare the prediction effects of different prediction models in Figures 20 and 21. In the figures, the ratio of the standard deviation of the predicted value to the measured value is the radial distance from the origin. The correlation coefficient between the predicted value and the measured value is the angle in the polar diagram, and the root mean square error is the radial distance from the reference point. The red dashed line in the figure is the standard deviation of the reference data point. The model prediction close to this line has a standard deviation similar to the observed value, which means that the model can capture the variability of the data well. The angles of model points show the degree of correlation between their predicted and actual values, and models with angles close to 1 show high correlations. The dashed grey line represents the contours of the RMSE. Similarly, from the two Taylor diagrams, DMSOA-CatBoost tends to the reference point more and has the most outstanding performance. Figures 22 and 23 are the comparative line charts of the predicted and true values of the DMSOA-CatBoost model on the relative dynamic elastic modulus and mass loss rate  
Figure 21: Taylor diagram (MLR). Figure 20: Taylor diagram (RDEM). test set at a certain fold in the ten-fold cross-validation. From the figure, the predicted and true values are in good agreement, and the expected effect is achieved [35]. To compare the performance of DMSOA with other well-known optimisation algorithms, the CatBoost model was optimised using DMSOOA, the sine cosine algorithm (SCA), and the Archimedean optimisation algorithm (AOA), respectively. After 10-fold cross-validation, the average performance results of different optimisation algorithms on the test set were obtained, as shown in Tables 7 and 8. The experimental results indicate that DMSOOA-CatBoost outperforms the SCA- and AOA-optimised models on both the RDEM and MLR datasets, exhibiting better performance across all average metrics. Among them, the DMSOOA-CatBoost model consistently maintains the highest SDR, indicating that it has smaller fluctuations across different folds and demonstrates greater stability and robustness under different data partitions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & \(\mathbf{R^{2}}\) & **MSE** & **RMSE** & **MAE** & **CC** & **SDR** \\ \hline SCA-CatBoost & 0.842 & 21.056 & 4.590 & 2.562 & 0.921 & 0.912 \\ AOA-CatBoost & 0.854 & 20.110 & 4.484 & 2.493 & 0.925 & 0.918 \\ DMSOOA-CatBoost & 0.864 & 19.739 & 4.369 & 2.424 & 0.930 & 0.924 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Evaluation of different optimised models (RDEM). Figure 23: Comparison of predicted and true values of DMSOOA-CatBoost partial data (MLR). Figure 22: Comparison of predicted and true values of DMSOOA-CatBoost partial data (RDEM). In addition, to further verify whether the differences in prediction performance among the optimised algorithm models are statistically significant, the R\({}^{2}\) values from the 10-fold cross-validation in the experiment are collected.