# Quantum Neural Networks Approach for Water Discharge Forecast
## 3 Results and Discussion
### Results Comparisons  
Table 1 contains comparisons of the modeling results. The models' evaluation was based on the MAE, MSE, and R2. With respect to MAE, the best results on the training set were obtained for S and So. The model built for So (S1o, and S2o, respectively) is better than that for S (S1, and S2, respectively). The models performed better on the test series S, S1, and S2o compared to the training sets, whereas for the rest of the models, the best results were those on the training sets. Figure 8: The QNN model for S2o (**a**) on the training set and (**b**) test set. (**c**) The evolution of the objective function during the training (8 iterations). The MAE and MSE are significantly higher when modeling the raw data series than those without aberrant values. Opposite, R2 is lower in the same case. In terms of R2, all the models have a very high accuracy, the lowest being that of the S2 model. These observations indicate a worse fit of the initial series, presenting a higher variability than those without aberrations. Overall, the best models were those for So and S1o, followed by those for S and S2o, even though we expected the best accuracy for S2o given that the training and test sets belong to the period after building the dam and removing the aberrant values. The run time and the corresponding number of epochs are presented in the last two columns of Table 1. The smallest were those associated with S2o, S2, and S1o (the shortest series), and the highest were those for So (not S, as expected given the length). The following compares the results of the previous studies [60; 65] performed for the same series and the same training and test sets. On the S series,  
On the training set, the lowest MAE, 5.7250, was obtained using a hybrid Sparrow Search Algorithm-Backpropagation Neural Network (SSA-BP), and on the test set, 4.2351, by a Convolutional Neural Network-Long Short-Term Memory (CNN-LSTM). On the training set, the lowest MSE, 80.5765, resulted from an Echo State Network (ESN) algorithm, and on the test set, 32.4993, from SSA-BP. On the training set, the highest R2, 0.9976, resulted after fitting a Sparrow Search Algorithm-Echo State Network (SSA-ESN) model, and on the test set, 0.9983, after using a Long Short-Term Memory (LSTM) model. On the S1 series,  
On the training set, the lowest MAE, 6.5177, was obtained using CNN-LSTM, and on the test set, 4.4784, by the same algorithm. On the training set, the lowest MSE, 102.9393, resulted from an ESN algorithm, and on the test set, 39.7982, from CNN-LSTM. On the training and test sets, the highest R2, 0.9899 and 0.9917, respectively, resulted from fitting an LSTM model. On the S2 series,  
On the training and test sets, the lowest MAEs, 4.7433 and 3.5245, respectively, were obtained using CNN-LSTM. On the training set, the lowest MSE, 57.3421, resulted from an SSA-ESN algorithm, and on the test set, 29.8323, from CNN-LSTM. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Series** & **Set** & **MAE** & **MSE** & **R2** & **Time (s)** & **Epochs** \\ \hline S & Training & 1.6851 & 4.4144 & 0.9863 & 10.8472 & 9 \\  & Test & 1.1937 & 2.3815 & 0.9858 & 10.8472 & 9 \\ S1 & Training & 3.8257 & 18.1368 & 0.9479 & 7.0415 & 10 \\  & Test & 3.7392 & 16.9383 & 0.8991 & 7.0415 & 10 \\ S2 & Training & 4.5597 & 22.9951 & 0.9179 & 4.6218 & 8 \\  & Test & 4.8917 & 26.2494 & 0.8436 & 4.6218 & 8 \\ So & Training & 0.9616 & 1.4660 & 0.9917 & 12.1807 & 11 \\  & Test & 1.0406 & 1.6954 & 0.9869 & 12.1807 & 11 \\ S1o & Training & 1.0460 & 1.7586 & 0.9910 & 5.5552 & 8 \\  & Test & 1.0791 & 1.6857 & 0.9870 & 5.5552 & 8 \\ S2o & Training & 3.5237 & 13.4814 & 0.9188 & 4.3140 & 8 \\  & Test & 3.2577 & 11.6611 & 0.9098 & 4.3140 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the models’ performance. * On both training and test sets, the highest R\({}^{2}\), 0.9992 and 0997, respectively, resulted from an LSTM model fitting.