## 3 Damage Detection and Localization Scheme
The \(k\)th closest point (k-nearest neighbor) and the associated distance \(d_{k}(o)\) are determined. This allows for assigning the reachability distance as:  
\[rd_{k}=\max(d_{k}(o),d(p,o)) \tag{10}\]  
The local reachability density (LRD) is calculated as the reciprocal of the average reachability distance using:  
\[LRD_{k}(p)=\frac{|N_{k}(p)|}{\sum rd_{k}(p,o)} \tag{11}\]  
where \(N_{k}(p)\) represents the \(k\)-nearest neighbors of observation \(p\) and is \(|N_{k}(p)|\) the number of observations in \(N_{k}(p)\). Finally, the LOF of observation \(p\) is computed as the average density ratio of the observation to its neighbors:  
\[LOF_{k}(p)=\frac{1}{|N_{k}(p)|}\frac{LRD_{k}(o)}{LRD_{k}(p)} \tag{12}\]  
For normal observations, the LOF values are less than or close to 1, indicating that the local reachability density of an observation is higher than or similar to that of its neighbors. A local outlier factor value greater than 1 indicates an anomaly. Similar to LOF, k-means relies on similarity of the data distribution. While designed specifically for partitioning the data into \(k\)-clusters, it can be adapted for OCC with a minor modification. By setting the number of clusters to one and using the 'normal' observations as belonging to this cluster, the centroid of this cluster is calculated. A distance measure (Euclidean) to this centroid can then be defined, allowing for the classification of new observations based on a user-specified threshold where those larger than the threshold are identified as outliers. By taking a boundary-based approach, OCSVM aims to create a decision boundary that encloses most of the training samples. This transformation facilitates finding a clear decision boundary. In this new space, the origin typically represents the outlier class, separate from the remaining data points, which represent the target class. The algorithm then identifies a maximum-margin hyperplane--a decision surface with the largest possible margin--that divides most of the data points (target class) from the origin (outliers). This hyperplane allows for classifying new data points as belonging to the target class or being potential outliers. The maximum-margin hyperplane with a normal vector \(w\), is determined through solving the following optimization problem [58]:  
\[\min_{w,\xi_{i},p}\frac{1}{2}\|w\|^{2}+\frac{1}{vm}{\sum}_{i=1}^{m}\xi_{i}-\rho \tag{13}\]  
subject to  
\[(w.\Phi(x_{i}))\geq\rho-\xi_{i}\hskip 28.452756 pt\xi_{i}\geq 0 \tag{14}\]  
where \(x_{i}\) are the training examples belonging to one-class, \(\xi_{i}\) are the slack variables introduced to relax the constraints and allow some error during training, \(v\) is the regularization coefficient between 0 and 1 providing an upper-bound for the ratio of outliers among all training data and \(\rho\) is the offset (or threshold). In contrast to detecting anomalies by effectively profiling normal observations, iForest explicitly uses isolation for finding anomalies [59]. It leverages the fact that the anomalies are fewer than the normal observations and have different attribute values. The algorithm employs a tree structure, where anomalies are isolated closer to the root, while normal instances reside deeper in the tree. Within the tree structure, each internal node represents a feature and a split value, and each leaf node indicates an isolated data point. Since anomalies are rare, they require fewer splits to be isolated from the majority of normal data points. The algorithm measures the average distance it takes to isolate each data point. The points with the shorter path distances, indicating faster isolation within the tree structure are identified as anomalies. Each of the one-class algorithms discussed above presents distinct advantages and limitations. LOF is effective for detecting local anomalies in non-uniform data distributions but requires careful tuning of neighborhood size. k-means clustering offers simplicity and efficiency for anomaly detection but relies on predefined cluster numbers and assumptions about data structure.