## 3 Methods
### Risk Factor Coupling Analysis
#### 3.4.1 N-K Model  
The N-K model was initially developed to address problems in biological adaptive evolution [55]. In the field of risk assessment, the N-K model has been applied to analyze  
Figure 3: Risk coupling types. the coupling relationships and quantify the extent of dependency among multiple risk factors [3; 56; 57]. In recent years, it has evolved into a theoretical framework for analyzing the interdependencies among elements in complex systems. In the N-K model, \(N\) represents the number of elements involved in the system, and \(K\) represents the number of coupling relationships among these \(N\) elements. Generally, \(K\in[0,N-1]\), and when \(K\in(0,N-1]\), the state of the system is affected by two or more elements. In our study, the system elements were the risk factor types, namely human (H), vehicle (V), road environment (R), weather (W), and management (M). The coupling relationship was determined by the co-occurrence relationship among risk types that led to an accident or further evolution. To this end, \(h,v,r,w\), and \(m\) with binary numbers 0 and 1, i.e., \(h,v,r,w,m\in\{0,1\}\), were used to represent the state of these five risk factor types in an accident. For example, if an accident involves human-related risk factors, then \(h=1\); otherwise, \(h=0\). A common approach to quantifying the coupling values between risk factors is to calculate the mutual information between them [3; 57]. Given that mutual information \(T_{n,m}(X_{1},X_{2},\ldots,X_{n})\) represents the coupling value of the combination of the \(n-\)th coupling scenario among \(N\) risk coupling types, \(X_{i}\) is the \(i-\)th risk type, \(i\in\{1,2\ldots,n\}\). In our study, \(N=5,1\leq i\leq 5\). For example, if \(n=2\), and there were 10 coupling scenarios (i.e., combinations of the five risk factor types) of the two-factor coupling type, \(m\in\{1,2,\ldots 10\}\). Specifically, \(m=1\), \(X_{1}=H\), and \(X_{2}=V\), \(T_{2,1}(H,V)\) represents the risk coupling value in the human-vehicle coupling scenario of the two-factor coupling type. Since single-factor coupling involves the exchange of information and energy within a single system, it is generally considered a special case of multifactor risk coupling and cannot be calculated using mutual information to determine the coupling values [58]. Therefore, there were four risk coupling types: two-factor coupling, three-factor coupling, four-factor coupling, and five-factor coupling. These four coupling types corresponded to 10, 10, 5, and 1 coupling scenarios, respectively. The risk coupling value for the two-factor coupling type was calculated as follows:  
\[\left\{ \begin{array}{l}T_{2,1}(H,V)=\sum_{h}\sum_{v}P(h,v)\cdot log_{2}\frac{ P(h,v)}{P(h)P(v)}\\ T_{2,2}(H,R)=\sum_{h}\sum_{v}P(h,r)\cdot log_{2}\frac{P(h,v)}{P(h)P(r)}\\ T_{2,3}(H,W)=\sum_{h}\sum_{w}P(h,w)\cdot log_{2}\frac{P(h,w)}{P(h)P(w)}\\ T_{2,4}(H,M)=\sum_{h}\sum_{m}P(h,m)\cdot log_{2}\frac{P(h,m)}{P(h)P(m)}\\ T_{2,5}(V,R)=\sum_{v}\sum_{r}P(v,r)\cdot log_{2}\frac{P(v,r)}{P(v)P(r)}\\ T_{2,6}(V,W)=\sum_{v}\sum_{w}P(v,w)\cdot log_{2}\frac{P(v,w)}{P(v)P(w)}\\ T_{2,7}(V,M)=\sum_{v}\sum_{m}P(v,m)\cdot log_{2}\frac{P(v,m)}{P(v)P(m)}\\ T_{2,8}(R,W)=\sum_{r}\sum_{w}P(r,w)\cdot log_{2}\frac{P(r,w)}{P(r)P(w)}\\ T_{2,9}(R,M)=\sum_{v}\sum_{m}P(r,m)\cdot log_{2}\frac{P(r,m)}{P(r)P(m)}\\ T_{2,10}(W,M)=\sum_{w}\sum_{m}P(w,m)\cdot log_{2}\frac{P(v,m)}{P(w)P(m)}\\ \end{array} \right. \tag{6}\]  
where \(P(h,v)\), \(P(h,r)\), \(P(h,w)\), \(P(h,m)\), \(P(v,r)\), \(P(v,w)\), \(P(v,m)\), \(P(r,w)\), \(P(r,m)\), and \(P(w,m)\) represent the joint probabilities (or coupling probabilities) of two-factor risk. Taking \(P(h,v)\) as an example, it is calculated as follows:  
\[P(h,v)=\sum_{r}\sum_{w}\sum_{m}P_{hvvww} \tag{7}\]where \(P_{n_{strvum}}\) represents the frequency of the combinations of the five risk factors. For example, in this study, there are no cases without risk coupling. This indicates that if \(h,v,r,w,m=0\), then \(P_{n_{strvum}}=P_{00000}=0\).