# Bridge Health Monitoring Using Proper Orthogonal Decomposition and Transfer Learning
## 2 TL-Based Damage Identification
### TL Using JDA-kernel  
To investigate TL efficiency in transforming knowledge from the source bridge to the target bridge, the JDA-kernel method was implemented. This effective TL method was developed for image processing and pattern recognition applications [26] and it was selected for this study because of its demonstrated robustness in transforming knowledge between domains with different marginal and conditional distributions [26] and its classification performance effectiveness [23]. The main goal of JDA is to minimize distances between the joint and conditional distributions for the source and target to find the optimum mapping function [26]. The linear kernel is incorporated into the JDA algorithm to transfer feature components for the entire source domain into a kernel space for mapping purposes. The KNN classifier is then used to predict labels for part of the features in the target domain reserved for testing. Two important aspects of the JDA-kernel are its domain and task [24]. A domain is denoted by \(\mathcal{D}=\{\mathcal{X},\,P(X)\}\) and contains a feature space \(\mathcal{X}\) and a marginal probability distribution \(P(X)\) of a feature \(X\), where \(X\) represents the data matrix and is a sample set from feature space \(\mathcal{X}\). \(X\) is defined as \(X=\{\mathbf{x}_{i}\}_{i=1}^{n}\in\ R^{m\times n}\), where \(n\) denotes the total number of samples in the domain and \(m\) represents the number of observations. A task is represented by \(\mathcal{T}=\{\mathcal{Y},f(\cdot)\}\) and consists of the label space \(\mathcal{Y}\) and the classifier \(f(\cdot)\). The classifier \(f(\cdot)\) is also defined as the conditional distribution function denoted by \(P(y|X)\), where \(y\in\ \mathcal{Y}\). Consider a source domain (\(D_{s}\)) and task (\(T_{s}\)) and a target domain (\(D_{t}\)) and task (\(T_{t}\)). For homogeneous TL, it is assumed that the features and the label spaces between the domains are the same: \(\mathcal{X}_{s}=\mathcal{X}_{t}\) and \(\mathcal{Y}_{s}=\mathcal{Y}_{t}\). DA is a subset of transductive TL and is used to improve the target predictive function utilizing knowledge obtained from \(D_{s}\) and \(T_{s}\)[3]. DA also assumes \(\mathcal{X}_{s}=\mathcal{X}_{t}\) and \(\mathcal{Y}_{s}=\mathcal{Y}_{t}\) while \(P(X_{s})\
eq\ P(X_{t})\) and \(P(y_{s}|X_{s})\
eq\ P(y_{t}|X_{t})\). For a case with single source and target domains, assuming that no label is available in the target domain, the domains are defined as [26]:  
\[\mathcal{D}_{s}=\{(\mathbf{x}_{1},y_{1}),\ \ldots,\ (\mathbf{x}_{n_{t}},y_{n_{s}})\ \} \tag{2}\]  
\[\mathcal{D}_{t}=\left\{\mathbf{x}_{n_{s+1}},\ \ldots,\ \mathbf{x}_{n_{s}+n_{t}}\ \right\} \tag{3}\]  
where: \(n=n_{s}+n_{t}\), and \(n_{s}\) and \(n_{t}\) are the size of the samples available from the source and target domains, respectively. For nonlinear mapping, the problem is kernelized using a Reproducing Kernel Hilbert Space (RKHS) and source and target joint and conditional distributions are computed in a new space. The kernel is defined to map \(X\) to a corresponding \(\phi(X)\) in the kernel space: \(K(\mathbf{x}_{i},\ \mathbf{x}_{j})=\left\langle\phi(\mathbf{x}_{i}),\ \phi(\mathbf{x}_{j})\right\rangle\), where \(K(X,X)\ \in\ R^{(n_{s}+n_{t})\times(n_{s}+n_{t})}\). A k-dimensional orthogonal transformation matrix, \(\mathbf{A}\), in the kernel space for features \(X\) is determined using Kernel Principal Component Analysis (PCA). Using PCA, the variance of matrix \(A^{T}X\) is maximized by solving the following eigenvalue problem:  
\[\mathbf{K}\mathbf{H}\mathbf{K}^{T}\mathbf{A}=\mathbf{A}\mathbf{\Phi}, \tag{4}\]  
where \(\mathbf{H}\) denotes the centering matrix \(\mathbf{H}=\mathbf{I}-\frac{\mathbf{1}}{n}\mathbf{1}\), calculated using \(\mathbf{I}\) the Identity matrix, and \(\mathbf{1}\) is an \(n\times n\) matrix of ones; \(\mathbf{\Phi}\) represents the \(k\) largest eigenvalues; and \(\mathbf{A}\) contains corresponding eigenvectors of the decomposed matrix, \(\mathbf{K}\mathbf{H}\mathbf{K}^{T}\).