## 3 Multi-Dimensional Safety Risk Theory
### Calculation of Multi-Dimensional Safety Risk Model  
Traditional risk assessment often adopts qualitative/semi-quantitative methods. The basic rule is to classify risk factors into different levels qualitatively based on experience, and then refer to the risk assessment model for semi-quantitative risk assessment. The core reason for using the qualitative/semi-quantitative risk assessment method is that the risk factors have different dimensional units, and the resulting risk values can be considered as the normalized result after empirical classification. In GJB 900A [64], the probability and severity are classified into five levels and four levels, respectively, and based on expert scoring method [65; 66], different probability levels and severity levels corresponding to different risk values are shown in Table 1.  
Therefore, risk value in Table 1 mapping to the two-dimensional space, then Euclidean distance between the risk assessment point \(R(P,S)\) as shown in Figure 2 and the space origin is introduced to calculate risk evaluation values as shown in Equation (12). a, b is the preference correction factors.  
\[R=\sqrt[2]{\left(a*P\right)^{2}+\left(b*S\right)^{2}} \tag{12}\]  
According to Table 1, \(R=f(P,S)\), \(1=f(1,\,1)\), \(2=f(2,\,1)\), \(3=f(4,\,1)\)..., and based on Equation (12),'regress' function in MATLAB is applied to implement multiple linear regression fitting, obtaining \(a=2.2\), \(b=3.3\).  
Similarly, based on the multi-dimensional safety risk theory, this paper uses a five-level risk factor level method based on expert experience [67]. In other words, the degree from light to heavy is level 1 to level 5. Similarly, based on GJB 900A and expert scoring method, different risk values corresponding to different propagation probability, severity and propagation time are obtained as shown in Table 2. Therefore, the actual parameter values of the safety risk factor propagation probability \(P\), potential severity \(S\), and propagation time \(T\) can be quantified into risk factor level.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Probability Level**} & \multicolumn{4}{c}{**Severity Level**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline
**1** & 1 & 3 & 7 & 13 \\
**2** & 2 & 5 & 9 & 16 \\
**3** & 4 & 6 & 11 & 18 \\
**4** & 8 & 10 & 14 & 19 \\
**5** & 12 & 15 & 17 & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Risk index matrix based on GJB 900A.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Risk**} & \multirow{2}{*}{**Probability Level**} & \multirow{2}{*}{**Severity Level**} & \multirow{2}{*}{**Propagation Time**} \\ \cline{2-2} \cline{4-5}  & & & 1 & 1 \\
1 & 1 & 2 & 1 \\
1 & 3 & 1 & 1 \\
2 & 2 & 1 & 2 \\... &... &... &... \\
13 & 3 & 3 & 3 \\... &... &... &... \\
25 & 5 & 5 & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Risk index of multi-dimensional safety risk model.  
Figure 2: Traditional risk model space of risk factors.  
Based on multi-dimensional safety risk model, the risk factors \(P\), \(S\), \(T\) are mapped to the three-dimensional space shown as Figure 3. The improved Euclidean distance between the risk assessment point \(R(p,s,t)\) and the space origin is introduced to calculate risk evaluation values as shown in Equation (13).  
\[R=\sqrt[2]{[a*f_{1}(p)]^{2}+[b*f_{2}(s)]^{2}+[c*f_{3}(t)]^{2}} \tag{13}\]  
where \(f_{1}(p)\), \(f_{2}(s)\), \(f_{3}(t)\) are risk factor levels, which actual parameter values of risk factors \(P\), \(S\), \(T\) are classified into, respectively; \(a\), \(b\), \(c\) is the preference correction factors. Based on Table 2 and Equation (13),'regress' function in MATLAB is applied to implement multiple linear regression fitting, obtaining \(a=2.2\), \(b=3.3\), \(c=2.7\).  
In addition, total safety risk value \(R_{N}\) of the system and safety risk ratio \(\eta_{i}\) of element \(i\) is calculated as shown in Equations (14) and (15).  
\[R_{N}=\sum_{i=1}^{n}R_{i} \tag{14}\]  
\[\eta_{i}=\frac{R_{i}}{R_{N}}\times 100\% \tag{15}\]