# Resilience Assessment for the Northern Sea Route Based on a Fuzzy Bayesian Network
## 3 Model Established for NSR Resilience Measurement
### Uncertainty Analysis Based on Information Entropy Theory  
Based on the discussion in Section 3.3, theoretically, the NSR security system is conditionally dependent on the absorptive capacity, adaptive capacity, and restorative capacity, which can be illustrated as shown in Figure 11. Actually, there is no direct and obvious causal relationship between system resilience and the decomposed capacities based on the analysis of the fuzzy Bayesian network. The system resilience and its supporting capacities are connected by the dotted line in Figure 8.  
As shown in Figure 11, the system resilience is set as the target variable \(X\), and the decomposed absorptive capacity, adaptive capacity, and restorative capacity are regarded as \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\), respectively. In this section, \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\) are set as the predictive variables (predictor) associated with system resilience \(X\). The predictive importance of each predictor can be determined by calculating the entropy and mutual information \(I(X,Y_{i})\), and the most important predictor will provide maximum information gain for system resilience. The prior probability of system resilience can be obtained by simulating the developed Bayesian network as:  
\[P(X\ =\ true)\ =\ 0.785,\ P(X\ =\ false)\ =\ 0.205 \tag{37}\]  
Then, the entropy of system resilience can be calculated by:  
\[H(X) =-\sum\limits_{i\ =\ 1}^{2}P_{Y_{i}}(X)\cdot\log_{2}P_{X_{i}}(X)\ =\ 0.785\times\log_{2}(0.785)+0.205\times\log_{2}(0.205)\ =\ 0.743 \tag{38}\] \[H(X|Y_{1})\ =\ \sum\limits_{i\ =\ 1}^{2}P_{Y_{i}}(X)H_{i}(X|Y_{1})\ =\] (39) \[P(X\ =\ true)\cdot H(X\ =\ true|Y_{1}\ =\ true)+P(X\ =\ false)\cdot H(X\ =\ false|Y_{1}\ =\ false)\]  
Figure 11: Mutual information links between resilience and the corresponding decomposed capacities. where the conditional entropies \(H(\,X\,=\,\,true|Y_{1}\,=\,\,true)\) and \(H(\,X\,=\,\,false|Y_{1}\,=\,\,false)\) are unknown terms to be solved according to the developed Bayesian network. Notably, the state of the absorptive capacity in the network is set as \"true\" and \"false\", and the BN is simulated with GeNle 2.3 software; the conditional entropy can be calculated as:  
\[H(\,X\,=\,\,true|Y_{1}\,=\,\,true)\,=\,\,-[0.805\times\log_{2}(0.805)+0.185 \times\log_{2}(0.185)]\,\,=\,\,0.304 \tag{40}\]  
\[H(\,X\,=\,\,false|Y_{1}\,=\,\,false)\,=\,\,-[0.298\times\log_{2}(0.298)+0.692 \times\log_{2}(0.692)]\,\,=\,\,0.89 \tag{41}\]  
By substituting the results of Equations (37), (40) and (41) into Equation (39), we are able to obtain the conditional entropy of \(H(X|Y_{1}\,)\) by:  
\[H(X|Y_{1}\,)\,=\,\,0.785\times 0.304+0.205\times 0.888\,\,=\,\,0.421 \tag{42}\]  
Finally, the mutual information between \(X\) and \(Y_{1}\) can be evaluated based on the results of Equations (38) and (42) by:  
Figure 8.0  
In the framework of information entropy, the result obtained from Equation (40) indicates that the absorptive capacity can reduce the uncertainty of system resilience by approximately 32.2%; that is, if we have good knowledge associated with the absorptive capacity, the extent of uncertainty of the system resilience can be reduced by 32.2%. Similarly, the conditional entropy and mutual information of the adaptive capacity and restorative capacity can also be calculated. Finally, the results for all three capacities obtained from information entropy theory are listed in Figure 8.1  
Based on the information presented in Table 15, by comparing the values of mutual information between system resilience and its three supporting capacities, it can be observed that the absorptive capacity of system resilience is able to reduce the extent of uncertainty regarding the resilience level of the NSR, and the restorative capacity is the least valuable in reducing the uncertainty of the resilience level. Generally, the results of the present study agree with the general belief that the absorptive capacity is complex and expandable and that the restorative capacity is not as difficult to identify.