## 4 Experimental Results and Discussion
### Evaluation Indicators  
In order to substantiate the precision of the predictive methodologies, an array of rigorous scientific evaluative indicators were meticulously selected to appraise the forecast outcomes' efficiency. These indicators include the mean absolute error (MAE), the root mean squared error (RMSE), and the symmetric mean absolute percentage error (SMAPE).  
MAE represents the average magnitude of the errors between the estimated values and the original data, without considering their direction. MAE is defined as follows:  
\[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}\lvert x_{i}(t)-y_{i}(t)\rvert \tag{49}\]  
where \(x_{i}(t)\) represents the original values, \(y_{i}(t)\) represents the estimated values, and \(n\) represents the number of observations.  
RMSE is another commonly used metric that represents the square root of the average squared differences between the estimated values and the original data. It is defined as follows:  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}(t)-y_{i}(t))^{2}} \tag{50}\]  
where \(x_{i}(t)\) represents the original values, \(y_{i}(t)\) represents the estimated values, and \(n\) represents the number of observations.  
RMSE is sensitive to larger errors and, thus, provides a measure that emphasizes larger discrepancies between the predicted and observed values.  
Additionally, the symmetric mean absolute percentage error (SMAPE) is used to further evaluate the performance of the prediction methods. SMAPE is described as follows:  
\[\text{SMAPE}=\frac{100\%}{n}\sum_{i=1}^{n}\frac{\lvert x_{i}(t)-y_{i}(t) \rvert}{(\lvert x_{i}(t)\rvert+\lvert y_{i}(t)\rvert)/2} \tag{51}\]  
where \(x_{i}(t)\) represents the original values, \(y_{i}(t)\) represents the estimated values, and \(n\) represents the number of observations.  
Generally, lower MAE, RMSE, and SMAPE values indicate a better performance on the part of the prediction method.  
To further assess the predictive accuracy, Theil's inequality coefficient (TIC) and the Index of Agreement (IA) were also employed.  
TIC is a measure of the relative accuracy of the predictions, where a lower TIC indicates better predictive performance. It is defined as follows:  
\[\text{TIC}=\frac{\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}(t)-y_{i}(t))^{2}}}{ \sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}(t)^{2}}+\sqrt{\frac{1}{n}\sum_{i=1}^{n}y _{i}(t)^{2}}} \tag{52}\]  
where \(x_{i}(t)\) represents the original values, \(y_{i}(t)\) represents the predicted values, and \(n\) represents the number of observations. TIC ranges between 0 and 1, with values closer to 0 indicating better model accuracy.  
The Index of Agreement (IA) is another indicator of model performance, providing a normalized measure of the match between observed and predicted values. The IA is defined as follows:  
\[\text{IA}=1-\frac{\sum_{i=1}^{n}(x_{i}(t)-y_{i}(t))^{2}}{\sum_{i=1}^{n}(\lvert x _{i}(t)-\bar{x}(t)\rvert+\lvert y_{i}(t)-\bar{x}(t)\rvert)^{2}} \tag{53}\]where \(\vec{x}(t)\) denotes the mean of the observed values. IA values range from 0 to 1, with values closer to 1 indicating better agreement between the predicted and observed data.  
By combining the indicators MAE, RMSE, SMAPE, TIC, and IA, a comprehensive evaluation of the prediction method's performance can be achieved.