## 3 Real Application Case
### Compared with Other Prediction Models  
The mean absolute error (MAE), the mean square error (MSE) and the R\({}^{2}\) are popular indicators for evaluating the prediction results [22; 23; 24; 25]. Thus, we use the three indicators to evaluate the  
Figure 5: Distribution of grey wolf population in the MGWO optimization process. (**a**) Initial population. (**b**) The fourth-generation population. (**c**) The eighth-generation population. Figure 6: Verification results of the MGWODBN model. (**a**) Fitting of training data. (**b**) 10-fold cross validation. proposed prediction model. Here, two evaluation tasks are to done. The first is to evaluate the optimization ability of the MGWO algorithm and the second is to evaluate the prediction accuracy of the MGWDBDN model. To verify the optimization ability of the MGWO algorithm, the particle swarm optimization (PSO) [26], differential evolution (DE) [27] and basic GWO algorithms are used for comparison. Considering the maximum optimization capability of each algorithm, each algorithm should achieve the convergence. Thus, the population size and maximum number of iterations of all algorithms are set to 20 and 50, respectively. In the PSO algorithm, the inertia weight is linearly reduced from 0.9 to 0.4 and the learning factor is set to 2.0. In the DE algorithm, the scaling factor and crossover probability factor are set to 0.5 and 0.2, respectively. Table 2 reports the optimization results of the PSO, DE, GWO, and MGWO algorithms. The structure parameters are obtained by the optimization algorithms in the 1028 training samples, while the MAE, MSE, and R\({}^{2}\) are obtained by the calculation of prediction results of DBN with optimized structure parameters in the 30 testing samples. The DBN optimized by the MGWO algorithm shows superiority in the structure complexity, fitting accuracy, and optimization time. More specifically, the MGWODBN model achieves 6 \(H_{1}\) nodes and 5 \(H_{2}\) nodes, which is more concise than the other three models. Regardless of the MAE or MSE, the fitting error of the MGWODBN model is lower than that of the other three models. For the R\({}^{2}\), the MGWODBN model also reaches a higher value. In particular, the optimization time of the MGWODBN model is 293.367 s, significantly lower than the 726.362 s of the PSODBN model and the 623.746 s of the DEDBN model. Compared with the GWODBN model, the optimization time of the MGWODBN model is slight higher. These results indicate that the MGWO algorithm has better optimization ability than the PSO, DE, and GWO algorithms. In addition, the lower optimization time implies that the MGWO algorithm has a lower complexity. The PSO algorithm performs parallel search by comparing the local extremum with the global extremum. When the population iterates to a certain value, the fitting error will no longer decrease. Thus, the PSO algorithm is easy to fall into the local optimum. The DE algorithm performs random search through the population differences, but the differences within the late population will decrease, resulting in slow convergence. Thus, the DE algorithm is also prone to falling into the local optimum. However, the GWO algorithm simulates the intelligent hunting activities of a grey wolf population. Through the surrounding, hunting, and attacking operations, it can search a good solution. In this paper, the basic GWO algorithm is modified by using the nonlinear convergence and position update strategies, so that the MGWO algorithm can jump out of the local optimum and get a better solution. To verify the superiority of the MGWDBDN model in prediction performance, the genetic algorithm optimized back propagation (GABP) neural network [28], differential evolution optimized support vector machine (DESVM) [29], and random forest [30] models are used for comparison. The prediction results of PM2.5 concentrations of 30 testing samples using the four models are shown in Figure 7. It can be seen that the predicted curve of the MGWODBN model are closer to the actual observed curve.