# Quantum Neural Networks Approach for Water Discharge Forecast
## 3 Results and Discussion  
### Results for the Raw Series  
Figure 3 presents the real and fitted model for S series, on the training (Figure 3a) and test (Figure 3b) sets, together with the variation of the objective function (Figure 3c).  
We notice the quick convergence of the objective function to a value under 0.4, which, together with the chart from Figure 3a, indicates a good algorithm performance on the training set. Figure 3b shows that the simulated series has the same shape as the recorded one, except for a few periods when the actual series is increasing while the simulated one is decreasing (e.g., 2007-06, 2009-01, 2009-02, etc.). In most cases, the bias between the forecast series and the recorded one is positive, indicating a slight overestimation of the raw data series. We also remark that there is a very good fit of the highest values in the test series. Comparisons of the training and test results indicate a better fit in the first case. In the second one, the highest bias is noticed in 2009-October and 2010-April, August, November, and December.  
Figure 4 presents the results of QNN on S1.  
Figure 3: The QNN model for S (**a**) on the training and (**b**) test sets. (**c**) The evolution of the objective function during the training (8 iterations).  
The maximum value of the objective function was higher in this situation than for S, but it dropped under 0.5 after two iterations, remaining almost constant. The model did not very well fit the river's discharge extreme values of the training set (Figure 4a), mainly underestimating them. The shapes of the raw and simulated series are similar on the test set (Figure 4b). The highest underestimated values of the recorded series are in March-April, August-October 2006, January-February, November 2008, etc.  
Comparisons of Figures 3b and 4b show a better performance of the model on S compared to that on S1, in the first case the extremes being better fitted. These results were somehow expected because, in the case of S, the training set contained values recorded before 1984 and a part after January 1984, whereas in the case of S1, the training set was  
Figure 4: The QNN model for S1 (**a**) on the training set and (**b**) test set. (**c**) The evolution of the objective function during the training (10 iterations).  
in the period before the river flow regularization, and the test set was recorded after that moment. Therefore, the inhomogeneity of the data series was higher when modeling S1.  
Figure 5 illustrates the behavior of the simulated series against the recorded S2.  
Despite the homogeneity of the training and test sets (both belonging to the same period, after building the Siriu Dam), we notice that there are higher deviations of the simulated series from the registered one on the test set (Figure 5b) compared to the case of the entire series S (that contains values from both periods, before and after the change in the water discharge regimen). The objective function stabilizes at about 0.1, a higher value compared to the cases of S and S1, indicating a worse fit of the test series of S2. Moreover,  
Figure 5: The QNN model for S2 (**a**) on the training set and (**b**) test set. (**c**) The evolution of the objective function during the training (8 iterations).  
the training and test datasets are both underestimated. This behavior was unexpected because both sets (training and test) belonged to the same period (after 1984). The result is opposite to those obtained using other algorithms on the same data series [60; 63; 65].