## 6 Damage Detection of IASC-ASCE Structural Health Monitoring Benchmarks
### Third PhaseInfluence of Various Noise Levels on the Identification Results  
In the third phase, five sets of WWLS-SVM models are built to study the effect of different noise levels on the identification performance. The models are trained and tested with varying degrees of uncertainty. The rates of uncertainty for the acceleration response are (1) 0%, (2) 2% (SNR = 34 dB), (3) 5% (SNR = 26 dB), (4) 15% (SNR = 16.5 dB), and (5) 20% (SNR = 14 dB). Table 6 shows the PoDE and DMI values for different combinations of uncertainties in the training and testing data. Testing datasets are archived from damage pattern 2-damage case4. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **RMSE** & **Total Time (s)** \\ \hline FE model (PDE) & \(5.02\times 10^{-04}\) & 1202 \\ FE model (PoDE) & \(5.02\times 10^{-04}\) & 978 \\ Probabilistic WWLS-SVM surrogate model (PDE) & \(3.16\times 10^{-03}\) & 561 \\ Non-Probabilistic WWLS-SVM surrogate model (PoDE) & \(2.01\times 10^{-03}\) & 251 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison the results in terms of computational cost and accuracy. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Damage Case**} & \multirow{2}{*}{**Damage Patten**} & \multicolumn{4}{c}{**Story 1**} & \multicolumn{4}{c}{**Story 3**} \\ \cline{3-10}  & & \(\theta_{i-y}\) & \(\theta_{i+x}\) & \(\theta_{i+y}\) & \(\theta_{i-x}\) & \(\theta_{i-y}\) & \(\theta_{i+x}\) & \(\theta_{i+y}\) & \(\theta_{i-x}\) \\ \hline \multirow{4}{*}{Target Damage} & 1 & 0.45 & 0.71 & 0.45 & 0.71 & 0.00 & 0.00 & 0.00 & 0.00 \\  & 2 & 0.45 & 0.71 & 0.45 & 0.71 & 0.45 & 0.71 & 0.45 & 0.71 \\ Damage & 3 & 0.00 & 0.36 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ Cases & 4 & 0.00 & 0.36 & 0.00 & 0.00 & 0.23 & 0.00 & 0.00 & 0.00 \\
4\&5 & 5 & 0.00 & 0.36 & 0.00 & 0.00 & 0.23 & 0.00 & 0.00 & 0.00 \\  & 6 & 0.00 & 0.23 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \multirow{4}{*}{Prediction Damage} & 1 & 0.44 & 0.70 & 0.44 & 0.70 & 0.00 & 0.00 & 0.00 & 0.00 \\  & 2 & 0.43 & 0.70 & 0.44 & 0.69 & 0.44 & 0.74 & 0.42 & 0.68 \\ Damage & 3 & 0.05 & 0.35 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ Case 4 & 4 & 0.00 & 0.32 & 0.00 & 0.01 & 0.19 & 0.00 & 0.02 & 0.00 \\ \hline \multirow{4}{*}{Prediction Damage} & 1 & 0.42 & 0.69 & 0.40 & 0.67 & 0.00 & 0.00 & 0.00 & 0.00 \\  & 2 & 0.40 & 0.68 & 0.44 & 0.70 & 0.44 & 0.67 & 0.40 & 0.68 \\ Prediction & 3 & 0.00 & 0.30 & 0.00 & 0.04 & 0.00 & 0.00 & 0.00 & 0.00 \\ Damage & 4 & 0.06 & 0.38 & 0.00 & 0.07 & 0.19 & 0.01 & 0.03 & 0.00 \\ Case 5 & 5 & 0.00 & 0.29 & 0.00 & 0.00 & 0.20 & 0.00 & 0.04 & 0.00 \\  & 6 & 0.00 & 0.18 & 0.00 & 0.04 & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \hline \end{tabular} Maximum damage severity corresponding to structural members in story 2 and 4 (not listed in Table 5) is 0.09.  
\end{table} 
Table 5: Estimates of Damage Severities for Damage Cases 4–5 (DMI). Table 6 shows that at the damaged locations, higher PoDE and DMI values occurred, indicating that the proposed approach can detect damage with noisy data. The table also shows that as the degree of uncertainty increases, the accuracy of the proposed surrogate model decreases. For example, if a WWLS-SVM model trained with 0% uncertainty level in extracted features of acceleration response (deterministic model), the PoDE values will decrease when tested using high-noise data (e.g., 20%). A similar trend can be observed for the DMI index, i.e., it cannot show the exact severity of damage when the relative percentage of noise in the test data increases. The same trend is observed when the model is trained with training data that has a higher noise percentage. For instance, as shown on Table 1, when the WWLS-SVM model is trained with extracted features of the acceleration response, with 20% noise, and tested with 0% noise in the testing data, the accuracy of PoDE and DMI indices decreases.