## 6 Comprehensive Analysis Based on the FAEWM
### Determination of Weights Based on the Factor Analysis Method
If the test reveals that the variables are independent of each other, it implies that factor analysis cannot be performed. When the \(p\)-value is less than 0.05, it suggests that there is a correlation between the variables, and factor analysis can be performed. Step 3: Solve for Factor Loadings  
The principal ingredient analysis (PCA) is used to determine the factor loadings. In the PCA, the eigenvalue decomposition of the covariance matrix is performed to obtain the eigenvalues and eigenvectors of the correlation coefficient matrix R. The eigenvalues and eigenvectors are denoted as (12) and (13), respectively. \[\lambda=[\lambda_{1},\lambda_{2},\cdots,\lambda_{i}] \tag{12}\]  
\[U=[u_{1},u_{2},\cdots,u_{i}] \tag{13}\]  
Then, the loading matrix A of the principal ingredient analysis for the correlation coefficient matrix R is calculated using (14). \[A=\left(\sqrt{\lambda_{1}}u_{1},\sqrt{\lambda_{2}}u_{2},\cdots,\sqrt{\lambda_ {i}}u_{i}\right) \tag{14}\]  
Step 4: Factor Rotation  
The loading matrix obtained from the principal ingredient analysis solution in Step 3 is not unique. The original variables may have exhibited high loadings on the different common factors before factor rotation, leading to an inability to interpret the meaning of the common factors. The values in the loading matrix undergo appropriate factor transformation to polarize them, resulting in newly transformed factors that exhibit a more distinct difference. Step 5: Calculate Factor Scores  
To evaluate the different schemes, it is necessary to obtain the values of the principal ingredients. Therefore, it is necessary to calculate the scores of each original variable on the principal ingredients. The estimates of the principal ingredient scores can be calculated via the parameter estimation method. Using [PERSON]'s test of sphericity to examine the correlation among the original variables in the four schemes to determine whether FAM could be used, the test results are shown in Table 10.  
The higher the variance explained rate, the greater the importance of the principal ingredient and the higher the weight assigned to that ingredient. Typically, ingredients with a pre-rotation eigenroot greater than \(1\) are selected as principal ingredients while ensuring the cumulative variance explained rate is greater than 80%. Three ingredients are needed to express more than 95% of the information content of all variables in Tables 11-14, so three common factors are extracted. After identifying the three common factors, it is necessary to specify the importance of each original variable in each common factor. This is performed by calculating the factor loading coefficients, as shown in Tables 15-18. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Original Variable**} & \multicolumn{3}{c}{**Rotated Factor Loading Coefficients**} & \multirow{2}{*}{**Commonality**} \\ \cline{2-3} \cline{5-5}  & **Factor 1** & **Factor 2** & **Factor 3** & \\ \hline Delay & \(0.981\) & \(0.09\) & \(-0.157\) & \(0.995\) \\ Travel time & \(0.974\) & \(0.122\) & \(-0.167\) & \(0.992\) \\ CO emission & \(0.978\) & \(0.142\) & \(-0.139\) & \(0.997\) \\ Fuel consumption & \(0.978\) & \(0.142\) & \(-0.139\) & \(0.997\) \\ TTC & \(-0.04\) & \(0.96\) & \(0.177\) & \(0.955\) \\ PET & \(0.317\) & \(0.916\) & \(-0.092\) & \(0.949\) \\ CSI & \(-0.246\) & \(0.089\) & \(0.964\) & \(0.998\) \\ Number of conflicts & \(0.983\) & \(0.046\) & \(-0.155\) & \(0.992\) \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Factor loading coefficients after rotation (Scheme 2).