## 4 Application and Results
### Performance of Hybrid Models
#### 4.3.2 VMD-Based Soft Computing Models  
To decompose the input dataset using the VMD algorithm, the number of IMFs (\(K\)) and the quadratic penalty factor (\(\alpha\)) have to be implemented in advance. In this study, different sets of parameters were investigated, and three sets of parameters with higher correlations between the original and predicted data (i.e., the aggregation of decomposed series) were selected. The three sets were (\(K\), \(\alpha\)) = {(\(3,5\)), (\(4,5\)), (\(4,10\))}. Among them, the optimal parameters (\(K\) = 4 and \(\alpha\) = 10) yielding the best performance of the VMD-based soft computing models were chosen finally. Figure 11 shows the original WT series and the IMFs decomposed using the VMD algorithm (\(K\) = 4 and \(\alpha\) = 10).  
Figure 11: _Cont._  
Figure 10: Scatter plots of observed and predicted DO values using DWT-based soft computing models for the testing phase.  
Table 5 shows the performance statistics using the VMD-MLP-III and VMD-CCNN-III models during the training and testing phases. Tables 3 and 5 indicate that all of the VMD-MLP-III models enhanced the performance of the MLP-III model enormously, while all of the VMD-CCNN-III models did not enhance the performance of the CCNN-III model during the testing phase. In addition, the VMD-MLP-III (\(K=4\) and \(\alpha=10\)) model (e.g., RMSE = 0.107 (mg/L), NSE = 0.992, WI = 0.998, R\({}^{2}\) = 0.993, and MAE = 0.034 (mg/L)) provided the best results among all of the VMD-MLP-III models during the testing phase. The conjugation of VMD into the MLP-III model ensured the model accuracy for predicting the DO concentration. However, the CCNN-III model provided better results compared to all of the VMD-CCNN-III models. A comparison explained that all of the VMD-MLP-III produced better results compared with all of the VMD-CCNN-III models.  
Tables 4 and 5 explain that the statistical results of the DWT- and VMD-MLP-III models showed similar statistical patterns. A comparison of the best models revealed, however, that the VMD-MLP-III (\(K=4\) and \(\alpha=10\)) model yielded slightly better results than the DWT-MLP-III (D6 and S6) model. Figure 11(a),b shows scatter plots of the testing dataset between the observed and predicted values using the VMD-MLP-III (\(K=4\) and \(\alpha=10\)) and VMD-CCNN-III (\(K=4\) and \(\alpha=10\)) models. From visual interpretation of Figure 11(a),b, it can be explained that the dots in the VMD-CCNN-III (\(K=4\) and \(\alpha=10\)) plot were extremely sporadic compared to those in the VMD-MLP-III (\(K=4\) and \(\alpha=10\)) plot. Based on the coefficient of determination and the slope of trend lines to the unity, it can be concluded that the VMD-MLP-III (\(K=4\) and \(\alpha=10\)) model acted better in predicting the DO concentration.  
In general, various studies have reported that the combination of soft computing models and decomposition approaches improved the accuracy and reliability of model performance for predicting DO concentration [12; 47; 48; 49; 50; 51; 52]. Even if the CCNN model showed the outstanding performance for standalone models, the combination of the CCNN model and decomposition approaches cannot improve the model performance. The special model structure (e.g., adding hidden nodes) can prevent the model performance of complex nonlinear signals. To confirm the model performance, continued studies are required using different data, soft computing models, and decomposition approaches for predicting diverse environmental parameters (e.g., BOD, COD, TP, and TN etc.).  
Figure 11: Original water temperature and intrinsic mode functions (IMF1, IMF2, IMF3, and IMF4) decomposed using the VMD algorithm (\(K=4\) and \(\alpha=10\)).