# Frost Resistance Prediction of Concrete Based on Dynamic Multi-Stage Optimisation Algorithm
## 5 Model Evaluation
### Model Evaluation Indicators  
This study uses coefficient of determination (R\({}^{2}\)), mean square error (MSE), root mean square error (RMSE), mean absolute error (MAE), correlation coefficient (CC), and standard deviation ratio (SDR) as the main evaluation indicators. These evaluation metrics are widely used in relevant studies and can comprehensively reflect the goodness of fit, error level, correlation, and stability of the prediction model from multiple dimensions. In terms of indicator selection, this study also considers other evaluation methods such as symmetric mean absolute percentage error and the Theil index. However, since the dataset of this study is not in the form of percentage or normalised structure, and the main goals are regression accuracy and stability, the above six indicators are finally selected as the comprehensive evaluation criteria. R\({}^{2}\) represents the model's ability to explain data variability. The closer the R\({}^{2}\) value is to 1, the better the model fits; MSE is the mean of the square of the difference between the predicted and actual values, providing a square measure of error and helping amplify the impact of larger errors. RMSE is the square root of the mean of the square of the difference between the predicted and actual values, providing a standard measure of error. MAE represents the average of the absolute value of the difference between the predicted and actual values, which is less sensitive to the response to outliers and can provide the average level of error. The lower the values of MSE, RMSE, and MAE, the more significant the model's advantages in error control and prediction stability. CC measures the degree of linear correlation between the two variables and reflects the strength of the linear relationship between the predicted and actual values. SDR is used to compare the standard deviation of the predicted and actual values to measure the volatility of the prediction accuracy [33]. These indicators jointly evaluate the prediction accuracy and explanatory power of the model from different perspectives. The formula is defined as follows:  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}{{{{(y_{i}-\hat{y_{i})}}^{2}}}}}{\sum \limits_{i=1}^{n}{{{{(y_{i}-\overline{y})}^{2}}}}} \tag{9}\]  
\[MSE=\frac{1}{n}{\sum\limits_{i=1}^{n}{{{{(y_{i}-\hat{y_{i})}}^{2}}}}} \tag{10}\]  
\[RMSE=\sqrt{\frac{1}{n}{\sum\limits_{i=1}^{n}{{{{(y_{i}-\hat{y_{i})}}^{2}}}}}} \tag{11}\]  
\[MAE=\frac{1}{n}{\sum\limits_{i=1}^{n}{{{{|y_{i}-\hat{y_{i}|}}}}}} \tag{12}\]\[CC=\frac{\sum{(y_{i}-\overline{y})(\hat{y}_{i}-\overline{y})}}{\sqrt{\sum{(y_{i}- \overline{y})^{2}}\sum{(\hat{y}_{i}-\overline{y})^{2}}}} \tag{13}\]  
\[SDR=\frac{s_{g}}{s_{y}} \tag{14}\]  
where \(y_{i}\) is the actual value, \(\hat{y}_{i}\) is the predicted value, \(n\) is the total number of samples, \(|y_{i}-\hat{y}_{i}|\) is the absolute value of the difference between the actual and predicted values, \(\overline{y}\) is the average value of the actual value, the numerator is the sum of the square difference between the predicted and actual values, the denominator is the sum of the square difference between the actual and actual mean values, \(s_{g}\) is the standard deviation of the predicted value, and \(s_{y}\) is the standard deviation of the actual value.