## 3 Collaborative Monitoring Algorithms for Situational Assessment
### Collaborative Traffic Trend Prediction Approach
#### 3.2.2 Monitoring Indicator Weight Calculation
\[CI=\frac{\lambda_{\text{max}}-M}{M-1} \tag{21}\]  
\[CR=\frac{CI}{RI} \tag{22}\]  
where \(CI\) is the consistency indicator, \(CR\) is the consistency ratio and \(RI\) is the average randomness indicator. In Table 3, 1000 simulations of Satty obtained random consistency indicator \(RI\) values are shown, and the \(RI\) values can be known by checking this table. The final indicator weight based on the AHP method is obtained as \(W_{1}=(w_{11},w_{12},\ldots,w_{1M})^{\text{T}}\). Step 2: Calculation of weights using the EM method. The \(d_{ij}^{*}\) of the normalized dataset \(D^{*}\) obtained by data pre-processing in Section 3.2.1 is used to calculate the entropy value \(e_{j}\) of each indicator. It is shown in Equation (23). \[e_{j}=-k\sum_{i=1}^{N}d_{ij}^{*}\ln\bigl{(}p_{ij}\bigr{)} \tag{23}\]  
where \(d_{ij}^{*}\) is the \(i\)th road and the \(j\)th indicator value \((i=1,2,\ldots,N;j=1,2,\ldots,M)\), \(N\) is the number of road and \(M\) is the number of indicators. \(k\) is a constant related to \(N,k=\frac{1}{\ln N}\). \(d_{ij}^{*}\) satisfies \(\sum_{j=1}^{N}d_{ij}^{*}=1\), and when \(d_{ij}^{*}=0\), let \(p_{ij}\ln\Bigl{(}d_{ij}^{*}\Bigr{)}=0\). Calculate the coefficient of variation \(g_{j}\) and the objective weight \(w_{2j}\) of the \(j\)th indicator, which are shown in Equations (24) and (25). \[g_{j}=1-e_{j} \tag{24}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Orders n** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** \\ \hline \(RI\) & 0 & 0 & 0.58 & 0.90 & 1.12 & 1.24 & 1.32 & 1.41 & 1.45 & 1.49 & 1.51 & 1.54 & 1.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Table of Random Consistency Indicator RI Values. \[w_{2j}=\frac{g_{j}}{\sum_{i=1}^{N}g_{i}},(i=1,2,\ldots,N) \tag{25}\]  
The final indicator weight based on the EM method is obtained as \(W_{2}=(w_{21},w_{22},\ldots,w_{2M})^{T}\). Step 3: Calculate the combination coefficient of the multiple assignment methods. The weights \(W_{1}=(w_{11},w_{12},\ldots,w_{1M})^{T}\) calculated by the AHP method and the weights \(W_{2}=(w_{21},w_{22},\ldots,w_{2M})^{T}\) calculated by the EM method form the basic weight set \(W=\{W_{1},W_{2}\}\); the two indicator weights \(W_{1}\) and \(W_{2}\) can be regarded as the two sides of the game, and the optimal combination of weights can be considered as the two sides of the game reaching the equilibrium state. The method is shown in Equation (26). \[W=\sum_{k=1}^{2}\alpha_{k}W_{k}^{T},\alpha_{k}>0 \tag{26}\]  
where \(\alpha_{k}\) is the combination coefficient of the assignment method, and \(W_{k}^{T}\) is a set of assignment vectors of the weight set. \(\alpha=\{\alpha_{1},\alpha_{2}\}\) is the linear combination coefficient, so Equation (18) is equivalent to Equation (27). \[W=\alpha_{1}W_{1}^{T}+\alpha_{2}W_{2}^{T} \tag{27}\]  
According to the theory of the game assembly model, the objective is to optimize the two linear combination coefficients \(\alpha_{1}\) and \(\alpha_{2}\) in Equation (18) by minimizing the deviation of \(W\) and \(W_{k}\) to obtain the optimal weight \(W^{*}\). The Game Model is shown in Equation (28). \[\min\left\|\sum_{k=1}^{2}\alpha_{k}W_{k}^{T}-W_{k}\right\|_{2} \tag{28}\] \[\text{s.t. }\alpha_{1}+\alpha_{2}=1,\quad\alpha_{1},\alpha_{2}\geq 0\]  
According to the matrix differentiation property, the expression of Equation (27) is converted into a system of linear differential equations under the optimal first-order derivative condition, and the result is shown in Equation (29). \[\left[ \begin{array}{cc}W_{1}{W_{1}}^{T}&W_{1}W_{2}^{T}\\ W_{2}W_{1}^{T}&W_{2}W_{2}^{T}\end{array} \right]\left[ \begin{array}{c}\alpha _{1}\\ \alpha_{2}\end{array} \right]=\left[ \begin{array}{c}W_{1}{W_{1}}^{T}\\ W_{2}W_{2}^{T}\end{array} \right] \tag{29}\]  
The optimized combination coefficients \(\alpha_{1}\) and \(\alpha_{2}\) calculated by Equation (20) are normalized as shown in Equation (30).