# Review of Vibration-Based Structural Health Monitoring Using Deep Learning
## 3 Application of Machine Learning to System Identification by Vibration Responses  
Machine learning algorithms were used to verify applicability of vibration-based health monitoring, including k-means, PCA, agglomerative clustering and density-based spatial clustering of application with noise (DBSCAN). The method that was applied to identify the bolt clamping force and their accuracies are compared for illustration in this section [72]. The k-means algorithm requires setup of k cluster centroids. The distance from each centroid is calculated and allocated to the nearest cluster. This operation is called clustering. After allocation,the centroid is reassigned in each cluster. The basic concept of the k-means algorithm is to converge the cluster by repeating the above process using the newly-designated centroids. With PCA, the principal component of correlated data is calculated. Instead of analyzing each component of data, it analyzes a principal component through data distribution. By the PCA, the direction vector with the largest variance of the data is calculated. Therefore, the result obtained by PCA returns the same number of vertical vectors as the system dimension. PCA replaces \(p\) measured variables \(x_{1}\), \(x_{2}\), \(\cdots\), \(x_{p}\), by \(m\) derived variables, \(z_{1}\), \(z_{2}\), \(\cdots\), \(z_{m}\). All possible variance of linear functions, \(z_{k}\), is given as [73]:  
\[z_{k}=\ \alpha_{k1}x_{1}+\alpha_{k2}x_{2}+\cdots+\alpha_{kp}x_{p} \tag{1}\]  
where \(\alpha_{k1},\alpha_{k2},\cdots,\ \alpha_{kp}\) is the eigenvector of \(x\). \(z_{1}\) shows maximum possible variance among all possible linear functions. \(z_{2}\) is the second largest possible variance orthogonal to \(z_{1}\). The constraint to normalize the variance is defined as:  
\[\sum_{i=1}^{k}\alpha_{ki}^{2}=1 \tag{2}\]  
or equivalently, \(\alpha_{k}^{\prime}\alpha_{k}=1\), using standard vector notation. In case of the agglomerative clustering algorithms, all data are initially set to clusters. When the distance between the data is closer than the specified value, it is merged into the same cluster. Hierarchical and nonhierarchical methods exist for calculating the distance of data in a cluster during a merge process [74]. The hierarchical method, such as agglomerative clustering, proceeds with the memory of the previous cluster when clustering again after the merger. The relationship between clusters is used in the clustering process. The nonhierarchical method such as k-means clustering and DBSCAN considers the clustered data in exactly the same group. There are methods to calculate the minimum, maximum and mean distance between all points in different clusters [75]. The algorithm ends when the number of clusters falls below the initially set value. DBSCAN algorithm performs clustering based on the density of data [76]. When a certain number of data exist within a radius of data points, the core point is designated with border points in the surrounding points. When the calculation is applied to all data, and if the border point of the core point (c\({}_{1}\)) is the same as the core point (c\({}_{2}\)) of another cluster, it falls into the same group and shares all the border points. Data not included in any cluster are considered as a noise. In the case of DBSCAN, there is no configuration process for the number of clusters. It is possible to find clusters with geometric shapes and outliers for detection by noisy measurements [77]. The use of clustering for status identification using vibration responses is an effective methodology, as was illustrated for the clamping force estimation during bolt tightening process [72]. Figure 1 shows the vibration responses induced at the nut-runner when the bolt was tightened using different torques. During the bolt tightening, the magnitude of the vibration response increases and lasts for a short period of time. With these short vibration responses, the clamping force in the bolt was identified accurately compared to those using the tightening torque.