## 2 Theoretical Background
### Line Classification Using Clustering Analysis  
Following the detection and classification of the impedance discontinuity points, the clustering algorithm was used for differentiating faulty lines in the multi-core cable. The input features for clustering were the TFCC results and the phase synchrony index of each reflected data, which estimates the phase difference between the incident signal and the reflected signal to distinguish the characteristics of the fault. The phase synchrony between the incident and the reflected signal was estimated using the reduced interference Rihaczek time-frequency distribution, which provides a better localization for the phase-modulated signals and time-varying phases [25]. The phase difference between the incident signal and the reflected signal is computed as follows:  
\[\phi_{sr}(t,\omega)=arg\left[\frac{R_{s}(t,\omega)}{|R_{s}(t,\omega)|}\frac{R _{r}^{*}(t,\omega)}{|R_{r}(t,\omega)|}\right]. \tag{9}\]  
The TFCC and the phase synchrony of the real impedance discontinuity point differ with the crosstalk signals. While, the TFCC values are related with the degree of the impedance discontinuity, phase difference is relevant to the type of impedance discontinuity. When at least one generation-victim line pair is defective, a reflected signal with a faulty line pair at the real impedance discontinuity point shows a high TFCC value. Estimated phase values are either the same (open) or the opposite (short) with those of the cable endpoints. Conversely, reflected signals caused by crosstalk have relatively lower TFCC values and show a different phase from actual defective lines. Therefore, a clustering analysis using the energy and phase properties of the reflected signal can be utilized to differentiate the faulty line. We used the hierarchy agglomerative clustering algorithm which allows each data to fit into multiple clusters with varying degrees of similarities. Hierarchy clustering is used for maximizing both the similarity within the cluster and the dissimilarities among clusters. First, the starting point for each data is in its own cluster, and the cluster pair merges as it rises above the hierarchy which continues until only a final cluster remains. Let \(X=\{x_{1},x_{2},...,x_{N}\}\) be the data set. The hierarchy clustering process was done as follows:  
1. Create the similarity matrix \(D=[d_{ij}]\), where \(d_{ij}\) is the distance between data samples using the following formula: \[d_{ij}=\left(\sum_{K}(x_{i}-x_{j})^{2}\right)^{1/2}\] (10) where \(K\) is dimension of the data;
2. Find the closest pair of clusters and merge them into a single cluster;
3. Compute similarities between the new cluster and each of the old clusters;
4. Repeat steps (2) and (3) until all data are clustered into a single cluster of size \(N\). In the clustering process, the similarities among clusters are defined as the shortest distance between two points in each cluster which is expressed as follows:  
\[L(X,Y)=\min_{x\in X,y\in Y}d(x,y), \tag{11}\]  
where \(X\) and \(Y\) indicate separate clusters. In addition, the silhouette index was calculated for each data cluster in order to validate its accuracy and internal consistency. The silhouette of a cluster measures the similarity of its own data and compares it with other clusters. The silhouette index for data \(l\), \(s(l)\) can be expressed as follows: [26]  
\[s(l)=\frac{b(l)-a(l)}{max\left(a(l),b(l)\right)} \tag{12}\]  
where \(a(l)\) is the average dissimilarity of l with all the other data within the same cluster, \(b(l)\) is the lowest average dissimilarity of data \(l\) with the data in the other clusters, (of which data \(l\) is not a member). \(a(l)\) indicates how well data \(l\) is assigned to its cluster and \(b(l)\) indicates how similar data \(l\) is to the data in neighboring clusters. The silhouette index ranges from \(-1\) to \(1\); a higher silhouette value indicates clustering results of higher quality. An average silhouette index greater than 0.5 indicates reasonable partitioning of the data [27].