# Bridge Health Monitoring Using Proper Orthogonal Decomposition and Transfer Learning
## 4 TL Using Coupled POD and JDA-kernel  
Before implementing the JDA-kernel method, two models were selected from the experiment set to map knowledge from one model, identified as the source domain, to another, target domain, model to predict labels for the target domain. The base model, \(M_{0}\), was selected as the source domain, and each model with MU from Table 1 as the target domain. Each model had eleven damage labels, one for the healthy state (0% damage) and ten imposed damage levels with different intensities \(DI=\{10,\,20,\,\ldots,\,100\%\}\) at different locations \(DL=\{1,\,2,\,\ldots,\,20\}\), identified in Figure 1b. Feature spaces were the first POMs. The base model (\(M_{0}\)) was simulated for all 4824 scenarios using SAP2000 OAPI and validated using field measured data from the bridge. The JDA-kernel method and KNN classifier were implemented for damage identification in the target domain using trained knowledge from the source domain. Each TL implementation assumed that estimated POMs for source and target domains from [22] were classified and labels for the source domain known. Predicted target domain labels from this supervised TL implementation were compared with those from previously known labels to determine their accuracy. Since the JDA-kernel generates pseudo labels, the algorithm was implemented over several iterations until it converged. Three scenarios were investigated. Scenario 1 was where DI was known, and the algorithm determined DLs. The feature space in both the source and target domains contain matrices of the first POMs where each column represented the associated first POM for a particular DL, where \(\mathbf{x}_{n_{s}}\ \in\ R^{m\times n_{s}}\) and \(n_{s}=480\), and the label space included \(\mathcal{Y}_{s}=\mathcal{Y}_{t}\ \in\ \{1,\,2,\ldots,20\}\). Scenario 2 identified DIs in the target domain with DL being known. Here, the feature space in both the source and target domains contain matrices of the first POMs where each column represented the associated first POM for a particular DI, where \(\mathbf{x}_{n_{s}}\ \in\ R^{m\times n_{s}}\) and \(n_{s}=264\), and the label space included \(\mathcal{Y}_{s}=\mathcal{Y}_{t}\ \in\ \{0,\,1,\,2,\ldots,10\}\). For Scenario 3, both DIs and DLs were unknown and the algorithm was applied to the entire dataset in the source and target domains to determine target domain labels. Categorization was performed for various DIs and the label space was defined as \(\mathcal{Y}_{s}=\mathcal{Y}_{t}\ \in\ \{0,\,1,\,2,\ldots,10\}\). The feature space was defined as \(\mathbf{x}_{n_{s}}\ \in\ R^{m\times n_{s}}\) and \(n_{s}=4824\) (24 features for healthy case and 480 features for each DI). In this scenario, redundant healthy POMs were added to the feature space so that the same number of features from each damage state were included to prevent the training process from underestimating healthy states. This provided \(n_{s}=5280\). Figure 2: Healthy average first POMs for each model. Cross-validation of the JDA-kernel algorithm was implemented using the k-fold method [39]. Four partitions were used, with one partition being reserved for cross-validation and training being carried out using the remaining three partitions. Hyperparameter tuning was also implemented using the grid search method [40] to find optimum values for the number of transferred components (\(k\)) and the regularization parameter (\(\lambda\)) used in each simulation. Figure 3 details the TL process. As illustrated in the figure, the process consisted of four steps. Preparing features and defining source and target models were addressed in Step 1. JDA-kernel implementation, cross-validation, classifier training, and hyperparameter tuning were implemented in Step 2. In Steps 3 and 4, the JDA-kernel and classifier were reimplemented using partial data and entire data sets, respectively, from optimized parameters obtained in Step 2. The score in Step 3 and 4 represents the calculated mean accuracy of the JDA-kernel and classifier by applying the optimized parameters in the algorithm.