# Resilience Assessment of Beijing Subway Lines under Extreme Precipitation Weather
## 3 Resilience Assessment with Multiple Performance Indicators
### Bayesian Network Modeling for Resilience Assessment  
In the previous subsection, the historical operational data of each line of the Beijing subway system required for the subway resilience assessment were obtained and pre-processed, and the time series of different key performance indicators of each line for the time periods before and after each of the severe weather events were generated. The corresponding resilience assessment indicators were also defined. Here, the Bayesian network model required for the resilience assessment is developed based on these data. Above all, a multi-dimensional method for resilience assessment at the subway line level can be completed with the subway resilience indicator system and FMEA proposed in this work.  
First, we create the variables of parent and child nodes for the Bayesian network model. The structure of this Bayesian network is schematically shown in Figure 5. The child node in the Bayesian network (the red node in Figure 5) represents the resilience values for each key performance indicator of each line estimated during severe weather. In order to estimate a resilience value, the main influencing factors associated with it are selected as each parent node (each blue node in Figure 5). This work focuses on the effects of different types and severity levels of severe weather events on the resilience values of subway system lines. Therefore, several influencing factors such as the type of severe weather, the time when the severe weather occurs, and the spatial distribution of the severe weather are considered here as the parent nodes in the constructed Bayesian network.  
After that, we define random variables for all parent and child nodes in the Bayesian network. Some of them can be discrete variables, while others are continuous variables. Moreover, the continuous variables among them are transformed into discrete variables by discretization to facilitate calculation. In addition, it is noted that the spatial distribution of severe weather may be multi-dimensional in distribution. Therefore, it is also necessary to  
Figure 4: Schematic of the variation of line key indicators in the Beijing subway system over time.  
be decomposed into multiple one-dimensional variables. At this point, the structure of the Bayesian network has been determined, and it can be assumed that the listed individual parent nodes have casual relationships with the child node.  
Finally, it is necessary to obtain the historical time series for each parent and child variable based on the historical data obtained in the previous section. For the child node (line resilience) of the Bayesian network, it is possible to directly start from the data obtained in the previous section on the change in each key performance indicator for different lines before and after each severe weather event, and calculate the area of the resilience triangle for that indicator for a period after each event as the line's resilience value for that key performance indicator during that severe weather event. Thus, the resilience value of each key performance indicator for each line can be obtained after \(L\) severe weather events in history, thus forming a time series of length \(L\) for the child node. For each parent variable, a value for that variable can be obtained similarly after each severe weather event in history, thus also forming a historical time series of length \(L\) for each parent. We denote the time series of parent and child nodes by \(x_{e,j}(t)\) and \(x_{f}(t)\), \(i=1,\ldots,i_{\max}\), \(t=1,\ldots,L\), where \(i_{\max}\) represents the number of parent nodes.