## 2 PM2.5 Concentration Prediction Approach
### Modified Grey Wolf Optimization Algorithm  
Grey wolf optimization (GWO) algorithm [PERSON] et al. (2016) simulates intelligent hunting behavior of grey wolves. According to hierarchical mechanism, grey wolf population can be divided into chief wolf \(\alpha\), deputy chief wolf \(\beta\), ordinary wolf \(\delta\), and bottom wolf \(\omega\) from high to low. When the wolves capture prey, the other individuals are organized to besiege the prey under the leadership of wolf \(\alpha\). In the GWO algorithm, the \(\alpha\), \(\beta\), and \(\delta\) denote the individuals with best fitness, the second best fitness and the third best fitness respectively, while the remaining individuals are noted as \(\omega\). The position of the prey is defined as the global optimal solution of optimization problem. Then, the GWO algorithm can be briefly described as follows. In a \(D\)-dimensional search space, suppose that the position of the \(i\)th grey wolf is \(X_{i}=(X_{i}^{1},X_{i}^{2},\cdots,X_{i}^{d},\cdots,X_{i}^{D})\), where \(X_{i}^{d}\) denotes the position of the \(i\)th grey wolf in the \(d\)th dimension. First of all, the grey wolf population surrounds the prey. The mathematical description of the process is  
\[X_{i}^{d}(t+1)=X_{p}^{d}(t)-A_{i}^{d}|C_{i}^{d}X_{p}^{d}(t)-X_{i}^{d}(t)| \tag{9}\]  
where \(t\) is the current number of iterations; \(X_{p}=(X_{p}^{1},X_{p}^{2},\cdots,X_{p}^{D})\  
Next, the grey wolf population hunts. The process, which is guided by \(\alpha\), \(\beta\), and \(\delta\), aims to update the all individual positions. The mathematical description is  
\[ \begin{array}{l}X_{i,\alpha}^{d}(t+1)=X_{\alpha}^{d}(t)-A_{i,1}^{d}|C_{i,1}^{ d}X_{\alpha}^{d}(t)-X_{i}^{d}(t)|\\ X_{i,\beta}^{d}(t+1)=X_{\beta}^{d}(t)-A_{i,2}^{d}|C_{i,2}^{d}X_{\beta}^{d}(t)-X_ {i}^{d}(t)|\\ X_{i,\delta}^{d}(t+1)=X_{\delta}^{d}(t)-A_{i,3}^{d}|C_{i,3}^{d}X_{\delta}^{d}(t )-X_{i}^{d}(t)|\end{array}  \tag{13}\]  
\[X_{i}^{d}(t+1)=\frac{X_{i,\alpha}^{d}(t+1)+X_{i,\beta}^{d}(t+1)+X_{i,\delta}^{d }(t+1)}{3} \tag{14}\]  
Finally, the grey wolves attack and capture the prey. The attack behavior is mainly achieved by linearly decreasing the convergence factor \(a\) from 2 to 0. When \(|A|\leq 1\), the grey wolves concentrate on attacking the prey, corresponding to the local search of the algorithm, while when \(|A|>1\), the grey wolves disperse for the global search. How to balance the local search and the global search is a very important problem often encountered in swarm intelligence algorithm [19]. The local search ability can speed up the convergence of the algorithm; while the global search can maintain the diversity of the population. For the GWO algorithm, we find that the convergence process in practice is not linear. Instead, a nonlinear convergence factor may be more appropriate for balancing the local search and the global search. Thus, a new nonlinear convergence factor is proposed to replace the original linear convergence factor (Equation (12)) and is given by  
\[a=2-2\Big{(}\text{tan}\Big{(}\frac{\pi t}{4T}\Big{)}\Big{)}^{k} \tag{15}\]  
where \(k\) denotes the attenuation order of the nonlinear convergent factor and takes the integer between 0 and 5. The nonlinear convergence factor with larger \(k\) decreases more sharply. Figure 3 shows the convergence factors with different values of \(k\). At the beginning of the iterations, the attenuation degree of \(a\) is reduced for constructing the global search. In the later stage of the iterations, the attenuation degree of \(a\) is improved for constructing the local accurate search. In this paper, value of \(k\) is taken as 3. The original GWO algorithm updates the grey wolf positions through calculating the average of the three best grey wolf positions. However, the update strategy ignores the characteristics of the different solutions, which may achieve the final solution with lower accuracy. In this paper, we design a weighting factor considering the contribution of each solution.