## 4 Application and Results
### Performance of Hybrid Models  
#### 4.3.1 DWT-Based Soft Computing Models  
To decompose the input dataset using the DWT algorithm, the optimal level of decomposition (\(L\)) should be selected. In this study, Equation (8) was used to calculate the optimal level of decomposition [68; 70; 72]. Although the optimal level of decomposition can be implemented using a trial-and-error method, it is time-consuming and a waste of energy:  
\[L=\text{int}[\log(N)], \tag{8}\]  
where \(N\) is the length of the time series, \(\text{int}[k]\) returns the integer portion of \(k\), and \(k\) is a real number.  
In this study, \(L=2\) was determined using Equation (8). In addition, mother wavelets have to be set before DWT-based soft computing models are employed. Using different mother wavelets, the dataset was decomposed with a details (D\({}_{1}\) and D\({}_{2}\)) and an approximation (A\({}_{2}\)) components for individual input data [72; 75]. For the DWT algorithm, Daubechies, Symmlets, and Coiflets have been frequently used as mother wavelets in previous studies [72; 75; 89; 90]. Therefore, the underlying mother wavelets, including Coiflet-6 (C6), Coiflet-12 (C12), Coiflet-18 (C18), Daubechies-6 (D6), Daubechies-12 (D12), Daubechies-18 (D18), Symmlet-6 (S6), Symmlet-12 (S12), and Symmlet-18 (S18), were implemented. For each DWT-based soft computing model, the optimal mother wavelet yielding the best model performance was recommended. Figure 9 shows an approximation and details decomposed using the Symmlet-6 (S6) mother wavelet for the original water temperature (WT).  
Figure 8: Error residual values for the standalone models for the testing phase.  
Figure 9: \(Cont\).  
Table 4 gives the performance statistics using the DWT-MLP-III and DWT-CCNN-III models during the training and testing phases. Tables 3 and 4 suggested that all of the DWT-MLP-III models improved the performance of the MLP-III model significantly, while all of the DWT-CCNN-III models did not improve the performance of the CCNN-III model during the testing phase. In addition, the DWT-MLP-III (D6) and DWT-MLP-III (S6) (e.g., RMSE = 0.161 (mg/L), NSE = 0.983, WI = 0.996, R\({}^{2}\) = 0.983, and MAE = 0.061 (mg/L) for D6 and S6) models produced the best results among all of the DWT-MLP-III models during the testing phase. The combination of the DWT into the MLP-III model could confirm the model accuracy for the prediction of the DO concentration. However, the CCNN-III model provided better results than any of the DWT-CCNN-III models. A comparison explained that all the DWT-MLP-III models yielded better results compared with all the DWT-CCNN-III models.  
Figure 9(a),b shows the scatter plots of the testing dataset between the observed and predicted values using the DWT-MLP-III (S6) and DWT-CCNN-III (S6) models. From visual interpretation of Figure 9(a),b, it can be explained that the dots in the DWT-CCNN-III (S6) plot were extremely sporadic compared to those in the DWT-MLP-III (S6) plot. In addition, based on the coefficient of determination and the slope of trend lines to the unity, it can be concluded that the DWT-MLP-III (S6) model acted better in predicting the DO concentration.  
Figure 10: Cont.  
Figure 9: Original water temperature and sub series (details (D\({}_{1}\) and D\({}_{2}\)) and approximation (A\({}_{2}\)) components) decomposed using Symmlet-6 (S6) mother wavelet.