## 2 PM2.5 Concentration Prediction Approach
### DBN Structure Parameters Determined by MGWO Algorithm  
Due to lack of the effective training algorithms for the DBN structure parameters containing the number of hidden layers, the number of hidden nodes, the learning rate and the momentum coefficient, the selection of structure parameters mainly relies on the manual experience or the multiple experiments. To address this issue, the modified grey wolf optimization (MGWO) algorithm is proposed to optimize the number of hidden layers, the number of hidden nodes, the learning rate, and the momentum coefficient. Figure 4 shows the combination process of MGWO algorithm and DBN. Wherein N grey wolves are selected and the DBN structure parameters are searched in parallel.  
The parameter optimization process of DBN based on MGWO algorithm is as follows.  
Step 1: Initialize the grey wolf population. Each individual position consists of the number of hidden layers \(l\), the number of hidden nodes \(n\), the learning rate \(\sigma\), and the momentum coefficient \(\lambda\).  
Step 2: Learn the training samples and take the mean square error of prediction results using DBN as the individual fitness function of MGWO algorithm.  
Step 3: Calculate \(a\) according to Equation (15) and update \(A\) and \(C\).  
Step 4: Calculate \(w\) according to Equation (17) and update the individual position according to Equations (13) and (16).  
Step 5: Return the optimal individual position if the termination condition is reached; otherwise, repeat the Step 3-Step 5.  
The key to finding the global optimal solution in the MGWO algorithm is to determine the termination condition and the fitness function [20,21]. In this paper, the training error of the DBN is used to calculate the fitness function of the MGWO algorithm, and the training error threshold of  
Figure 4: Modified grey wolf optimization (MGWO) algorithm-based optimization flow chart.  
the DBN is taken as the termination condition of the MGWO algorithm. The calculation steps of the training error are as follows.  
Step 1: If the fitness of the grey wolf of the \(t\)th generation is \(f(l_{t},n_{t},\sigma_{t},\lambda_{t})\), then the number of hidden layers, the number of hidden nodes, the learning rate, and the momentum coefficient can be noted as \(l_{t}\), \(n_{t}\), \(\sigma_{t}\), and \(\lambda_{t}\). Then, initialize the DBN parameter set \(\theta\) consisting of weights and biases.  
Step 2: Let \(v_{0}\) be the input sample vector, \(q\) be the iteration number of the DBN and \(e\) be the training error of the DBN.  
Step 3: Calculate the feature vectors \(h_{0},v_{0},h_{1},v_{1},\cdots,h_{l_{t}}\) of the visible and hidden layers of the RBM according to Equations (4) and (5).  
Step 4: Get the joint probability distribution of the initial state and the update state of the RBM according to Equation (7), and then substitute it into Equation (8) to modify the parameter set \(\theta\).  
Step 5: Iterate the training set by \(q\) times with random batches, and repeat the Step 3-Step 4.  
Step 6: Fine-tune the \(\theta\) using the BP algorithm.  
Step 7: Calculate the \(h_{l_{t}}\) using the \(\theta\) and get the training error \(e=\|h_{l_{t}}-v_{0}\|\).  
Thus, the MGWO algorithm is associated with the DBN through the fitness function. The fitness value can reflect the quality of DBN structure parameters, so as to generate a suitable predictor.