## 3 Numerical Example
### Damage Severity Prediction of Suspension Bridge Hangers  
The absolute identification error (AIE) of the damage severity of the undamaged and damaged hangers under different damage scenarios is calculated in Table 5.  
It can be seen from Table 5 that the AIE of damage severity prediction of undamaged hangers based on the CNN-LSTM model is in the range of 0.001-0.229%, where the average AIE of damage scenario 1 is the largest, and those of scenarios 12 and 13 are the smallest. Comparing the AIE of different damage types, the three damaged hangers are the lowest, the two damaged hangers are the second, and the single damaged hanger is relatively high. The AIE of the CNN-LSTM model for predicting the damage severity of damaged hangers is between 0.563 and 2.470%, where the AIE of each scenario of the single damage type is higher than that of the other damage types. According to the above analysis, the proposed model has high accuracy in predicting the severity of the undamaged hangers. The proposed model predicts that the damage severity of the damaged hangers generally performs well, but the prediction result for a single damaged hanger is relatively poor.  
To evaluate the performance of the proposed damage identification method, the average relative identification error (ARIE) is used to evaluate the recognition effect of the proposed model on the prediction of the damage severity of hangers, which is expressed as:  
\[ARIE=\sum_{i}^{n}\left|\frac{y_{degrad}-y_{degrad}}{y_{degrad}}\right|\times \frac{100\%}{n} \tag{15}\]  
where \(y_{degrad}\) is the prediction vector of damage severity, \(y_{degrad}\) is the target vector, and \(n\) is the number of hangers under the corresponding scenarios.  
When calculating the ARIE, this study only focuses on the damaged hangers. The value of the evaluation index ARIE is calculated according to Equation (15) and presented in Figure 10.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Scenarios** & **Average AIE of Undamaged Hangers (\%)** & **Average AIE of Damaged Hangers (\%)** \\ \hline
1 & 5\% & 0.229 & 1.726 \\
2 & 15\% & 0.068 & 2.047 \\
3 & Single damaged hanger & 25\% & 0.026 & 1.711 \\
4 & 35\% & 0.017 & 2.070 \\
5 & 45\% & 0.024 & 2.470 \\ \hline
6 & 5\% & 0.061 & 0.648 \\
7 & 15\% & 0.005 & 0.913 \\
8 & Two damaged hangers & 25\% & 0.005 & 1.102 \\
9 & 35\% & 0.002 & 1.159 \\
10 & 45\% & 0.010 & 0.852 \\ \hline
11 & 5\% & 0.018 & 0.563 \\
12 & Three damaged & 15\% & 0.001 & 1.006 \\
13 & hangers & 25\% & 0.001 & 0.804 \\
14 & 35\% & 0.003 & 1.012 \\
15 & 45\% & 0.016 & 0.405 \\ \hline \hline \end{tabular}
\end{table}
Table 5: AIE of the damage severity of hangers under different damage scenarios.  
The following conclusions can be drawn from Figure 10.  
1. The ARIE of the CNN-LSTM model for the damage severity prediction of hangers is 8.0% in the test set. The ARIE under different damage scenarios is between 0.90 and 34.52%, and the ARIE under most damage scenarios is below 7.0%.
2. For the same damage type, the ARIE decreases as the damage severity increases. The single damage scenarios decrease the fastest, with a decrease of approximately 29.03%, and the two and three damaged scenarios decrease by 11.06 and 10.37%, respectively. It can also be found from the ARIE results of the different damage severities that the CNN-LSTM model has relatively small prediction errors for larger damage severities (\(\geq\)15%), while the accuracy of damage prediction for smaller damage severities (such as 5%) is poor.
3. For the same damage severity, as the number of damaged hangers increases, ARIE shows an overall downward trend.