# Causality Analysis and Risk Assessment of Haze Disaster in Beijing
## 2 Materials and Methods
### Nonlinear Dynamical Models  
The urban haze system is a special complex system with multi-elements, multi-levels and multi-feedback with human participation. Various variables constitute multiple causal feedback loops and are self-generating within the system, which need a nonlinear model to analyze and study the interaction between the internal structure, function and behavior of the system and the feedback mechanism [41]. System dynamics is one of the important simulation methods used to study the dynamic complexity and regards the research object  
Figure 8: Causality diagram of sulfur dioxide emission system in Hebei and Tianjin. Figure 9: Overall causality diagram of Beijing urban haze system. as a dynamic open system, which is suitable to be applied to studying the problem of macroscopic haze management [39; 40]. The system dynamics model can observe the changing trend of the research system with the change in the control variable by setting and adjusting the variables, and can then obtain the sensitivity of the system to different control variables [41; 42]. When conducting dynamic simulation experiments on urban haze systems, the different states and trends of the system can be obtained by changing the input parameters, and the influence mechanism of different control factors on the operation of the main system can be obtained by adjusting the control variables. Flow level and flow rate are the main parameters in the system dynamics model, in which, flow level is an accumulation variable and flow rate is the speed that causes the change in flow level. Flow level \(y(t)\) gradually accumulates on the basis of the initial value with the change in time, and the accumulation speed is affected by flow rate \(x(t)\). In a real complex system, the flow level \(y(t)\) as an observable measurement will contain some observation environment errors. Generally, this observation error can be modeled as Gaussian random noise \(n(t)\) with mean value of 0 and variance of \(\sigma_{n}^{2}\), and its amplitude probability distribution function is as follows:  
\[P\{n(t)\}=\frac{1}{\sqrt{2\pi}\sigma_{n}}\exp\left[-n^{2}(t)/\sigma_{n}^{2}\right]\]  
Generally, the observation noises at different times are independent of each other, that is, \(E\{n(t)\times n(t+\tau)\}=0\). In the process of complex nonlinear dynamics evolution, the flow level also affects the flow rate, which is a nonlinear saturation effect. When the flow level \(y(t)\) is small, the flow rate \(x(t)\) may change more quickly, and when the flow level \(y(t)\) exceeds a certain threshold \(\gamma\), the change in flow rate \(x(t)\) will slow down significantly. In order to make the simulation more consistent with the actual haze pollution system, nonlinear dynamic models were applied. The input was the flow rate \(x(t)\), which represents the evolution rate of pollutant emission and haze pollution. The output was the flow level \(y(t)\), which represents the level of pollutant emission and haze pollution. The model performance was validated with the actual values by simulating the results of past years. The above nonlinear effects can be described by Sigmoid function. For the convenience of expression, the nonlinear saturation effect function was recorded as \(F(x)=1/(1+\exp(x-x_{0})\), where \(x_{0}\) represents a specific nonlinear threshold. To sum up, the following stochastic nonlinear differential equations can be used to describe the nonlinear dynamics system composed of flow level and flow rate:  
\[ \begin{array}{l}\frac{dy(t)}{dt}=x(t)+n(t),\\ x(t)=F(g_{x}(t)),\\ g_{x}(t)=-\sum\limits_{j=1}^{J}\alpha_{j}c_{j}(t)+\alpha_{y}y(t),\end{array} \]  
where \(J\) is the number of factors that affect the flow rate change, \(c_{j}(t)\) is the value of the \(j-th\) factor, \(\alpha_{j}\) is the correlation strength of the \(j-th\) factor and the flow rate, \(\alpha_{y}\) is the correlation strength of the flow level and the flow rate change and \(g_{x}(t)\) is the flow rate without nonlinear saturation effect.