## 4 Results and Discussion  
Regression analysis was performed for training data, testing data, validation data, and all data for fifty times, and the performances were averaged. MSEs and state transition performances were determined, and error histograms and regression plots were obtained. For the sake of brevity, only the results of the LM for Monday are given in the following figures. Figure 12 shows the MSE performance for training, validation, and test data as a function of epoch for one randomly selected run of the LM. As seen from the figure, the best validation performance was achieved at four epochs, with an MSE value of 3.47 \(\times\) 10\({}^{-3}\). Figure 11: The change in MSE versus number of neurons. Figure 12: MSE performance of the LM for Monday (for one randomly selected run). Figure 13 illustrates the training state values for one randomly selected run, in which the value of the gradient after ten epochs was 2.22 \(\times\) 10\({}^{-3}\) and mu was 1.00 \(\times\) 10\({}^{-5}\). The training process has been ended after six validation failures. The error histogram of one run with a reference zero error line is shown in Figure 14. Though there are some outliers; these data points are not largely different, and the error histogram roughly follows a normal distribution. The regression plots of the average of fifty runs for training, validation, testing, and all data are given in Figure 15. It can be seen from the figures that the average regression coefficient, \(R\), is 0.981 for training data, and the average \(R\) for validation, testing, and all data are 0.982, 0.977, and 0.980, respectively, which indicates a strong correlation and high performance. Figure 14: Error histogram of the LM for Monday (for one randomly selected run). Figure 13: Training state of the LM for Monday (for one randomly selected run). In order to evaluate the overall performance of LM and BR, the average MSE values, epochs, and performances are tabulated in Tables 3 and 4. Additionally, R values for LM and BR are listed in Tables 5 and 6, respectively. As seen from Tables 3 and 4, the performances of both methods are approximately 3 \(\times\) 10\({}^{-3}\). The MSE values for training, validation, and testing change between 2.58 \(\times\) 10\({}^{-3}\) to 4.08 \(\times\) 10\({}^{-3}\) for the LM method, while the values for training and testing for the BR method varies between 2.28 \(\times\) 10\({}^{-3}\) and 5.73 \(\times\) 10\({}^{-3}\). Although there is no noticeable difference between the MSE values of both methods, the LM-based method reaches a similar validation and training error with a smaller average number of epochs. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Days**} & \multicolumn{3}{c}{**MSE ***} & \multirow{2}{*}{**Epoch ***} & \multirow{2}{*}{**Performance ***} \\ \cline{2-2} \cline{5-6}  & **Training** & **Validation** & **Testing** & & \\ \hline Mon. & 2.71 \(\times\) 10\({}^{-3}\) & 2.89 \(\times\) 10\({}^{-3}\) & 2.84 \(\times\) 10\({}^{-3}\) & 6.30 & 2.76 \(\times\) 10\({}^{-3}\) \\ Tue. & 3.91 \(\times\) 10\({}^{-3}\) & 4.00 \(\times\) 10\({}^{-3}\) & 4.06 \(\times\) 10\({}^{-3}\) & 6.42 & 3.95 \(\times\) 10\({}^{-3}\) \\ Wed. & 3.55 \(\times\) 10\({}^{-3}\) & 3.66 \(\times\) 10\({}^{-3}\) & 3.85 \(\times\) 10\({}^{-3}\) & 6.38 & 3.61 \(\times\) 10\({}^{-3}\) \\ Thu. & 3.41 \(\times\) 10\({}^{-3}\) & 3.78 \(\times\) 10\({}^{-3}\) & 3.59 \(\times\) 10\({}^{-3}\) & 6.18 & 3.49 \(\times\) 10\({}^{-3}\) \\ Fri. & 2.91 \(\times\) 10\({}^{-3}\) & 2.97 \(\times\) 10\({}^{-3}\) & 3.21 \(\times\) 10\({}^{-3}\) & 6.46 & 2.96 \(\times\) 10\({}^{-3}\) \\ Sat. & 3.77 \(\times\) 10\({}^{-3}\) & 3.85 \(\times\) 10\({}^{-3}\) & 4.08 \(\times\) 10\({}^{-3}\) & 6.24 & 3.83 \(\times\) 10\({}^{-3}\) \\ Sun. & 2.58 \(\times\) 10\({}^{-3}\) & 2.74 \(\times\) 10\({}^{-3}\) & 2.78 \(\times\) 10\({}^{-3}\) & 6.86 & 2.64 \(\times\) 10\({}^{-3}\) \\ \hline \end{tabular}  
* Average values.  
\end{table} 
Table 3: Performance evaluation of the LM.