## 4 Model Training and Evaluation Methods
### Evaluation Metrics  
Determining the appropriate metrics is an important step in evaluating the performance of machine learning models, measuring the accuracy of the models, and comparing different algorithms. In this study, the coefficient of determination (R\({}^{2}\)), mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), root mean squared error (RMSE), mean log squared error (MSLE), maximum error (MaxEr), and explained variance (ExVar) evaluation metrics, in addition to ten additional metrics, were used to  
Figure 2: Scatter plots of GPA versus the top four most-correlated features: absences, grade class, parental support, and weekly study time. compare predicted student achievement with actual values and analyze the effectiveness of the models. These metrics will be described in Table 2.  
In this table, \(y_{i}\) is the actual value of the \(i\)-th sample, \(\hat{y}_{i}\) is the \(i\)-th value predicted by the model, \(\overline{y_{i}}\) is the mean of the actual values, \(\hat{y}_{i}\) is the arithmetic mean value, and \(n\) is the total number of observations. The R\({}^{2}\) value is a measure of how much of the total variance in the dependent variable (GPA) the model can explain. A high R\({}^{2}\) value indicates that the model can provide a better explanation of the information associated with the independent variables, while a low R\({}^{2}\) value indicates that the predictive power of the model is weak [26]. The MAE value is the average of the absolute MAE values of the differences between the model's predicted values and the actual values. A low value indicates that the model makes more accurate predictions. The absence of negative values makes the evaluation more understandable [27]. The MSE value is calculated by averaging the squares of the prediction errors. Due to the squaring process, it gives higher weight to large errors and therefore reveals large deviations more clearly [28]. \begin{table}
\begin{tabular}{l l} \hline \hline
**Metrics** & **Formula** \\ \hline Determination coefficient (R\({}^{2}\)) & \(R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}-\hat{ y})^{2}}\) \\ \hline Mean absolute error (MAE) & MAE \(=\frac{1}{n}\sum_{i=1}^{n}|y_{i}-\hat{y}_{i}|\) \\ \hline Mean squared error (MSE) & \(MSE=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}\) \\ \hline Root mean squared error (RMSE) & \(RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}\) \\ \hline Mean squared logarithmic error (MSLE) & \(MSLE=\frac{1}{n}\sum_{i=0}^{n-1}(\ln(1+y_{i})-\ln(1+\hat{y}_{i}))^{2}\) \\ \hline Maximum error (MaxEr) & \(MaxEr=max\{y_{i}-\hat{y}_{i}\}\) \\ \hline Explained variance (ExVar) & \(ExVar=1-\frac{Var(y_{i}-\hat{y}_{i})}{Var(y_{i})}\) \\ \hline Median absolute error (MedAE) & \(MedAE=median(y_{i}-\hat{y}_{i})|\) \\ \hline Mean square relative error (MARE) & \(MARE=\frac{1}{n}\sum\limits_{i=1}^{n}\left|\frac{y_{i}-\hat{y}_{i}}{y_{i}}\right|\) \\ \hline Mean square relative error (MSE) & \(MRSE=\frac{1}{n}\sum\limits_{i=1}^{n}\left|\frac{y_{i}-\hat{y}_{i}}{y_{i}} \right|^{2}\) \\ \hline Root mean square relative error (RMSRE) & \(RMRSE=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}\left|\frac{y_{i}-\hat{y}_{i}}{y_{i }}\right|^{2}}\) \\ \hline Relative root mean square error (RRMSE) & \(RRMSE=100\times\frac{\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}(y_{i}-\hat{y}_{i})^{ 2}}}{\sum\limits_{i=1}^{n}y_{i}}\) \\ \hline Mean bias error (MBE) & \(MBE=\frac{1}{n}\sum\limits_{i=1}^{n}(y_{i}-\hat{y}_{i})\) \\ \hline Maximum absolute relative error (eMAX) & \(erMAX=\max\left|\frac{y_{i}-\hat{y}_{i}}{y_{i}}\right|\) \\ \hline Standard deviation (SD) & \(SD=\sqrt{\frac{\sum\limits_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{n}}\) \\ \hline t-statistic (t-stat) & \(t-stat=\sqrt{\frac{(n-1)MBE^{2}}{RMSE^{2}-MBE^{2}}}\) \\ \hline Uncertainty at 95\% (U\textsubscript{{}_{95}}) & \(U\textsubscript{{}_{95}}=\sqrt{1.96(SD^{2}+RMSE^{2})}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical metrics [10].