## 2 Research Methodology
### Data Clustering
#### 2.1.1 ILI Data Clustering Using DENC  
DENC employs proximity thresholds, like in epsilon neighborhood graphs or DBSCAN, but instead of relying on an isotropic proximity threshold (\(\varepsilon\)), DENC proximity thresholds are the directional components of \(\varepsilon\): \(\varepsilon_{x}\) along the \(x\)-axis and \(\varepsilon_{y}\) along the \(y\)-axis. The method constructs a directed graph of the ILI data by labeling features as anchors and nodes. Adjacency boundaries are determined based on the directional thresholds applied to the anchors, enabling the identification of adjacent nodes that fall within these boundaries. This adjacency criterion determines the structure of the graph and the boundaries of the clusters, represented in a binary adjacency matrix (\(\mathbb{B}\)). The first step in DENC involves dividing the feature tallies into anchors and nodes. Features in the reference set \(Q\) are designated as anchors, while features in the moving set \(P\) are treated as nodes. Adjacency is established exclusively between anchors and nodes, with no adjacency permitted between anchors themselves or between nodes. This controlled adjacency forms the basis for constructing adjacency relationships. It is worth noting that the moving set can be designated as anchors and the reference set as nodes without affecting the outcomes. However, the proposed approach in the following steps must be consistently realigned to account for such a swap. The adjacency relationships between anchors and nodes are established using a directional proximity criterion along the axial (x) and circumferential (y) directions. As illustrated in Figure 9, this criterion implies that for a node \(\mathrm{p}_{i}\) to be considered adjacent to anchor \(\mathrm{q}_{i}\) it must fall within the rectangular boundary surrounding \(\mathrm{q}_{j}\), defined by the thresholds \(\varepsilon_{x}\) and \(\varepsilon_{y}\). Here, the directional distance matrix \(\mathbb{D}=\left(\left[\left(\mathrm{d}_{\mathrm{\tilde{p}}}^{x},\mathrm{d}_{ \mathrm{\tilde{p}}}^{y}\right)\right],\forall\mathrm{j}=1,2,\ldots,m,\forall \mathrm{i}=1,2,\ldots n\right)\) is established. Each element \(\left(\mathrm{d}_{\mathrm{\tilde{p}}}^{x},\mathrm{d}_{\mathrm{\tilde{p}}}^{y}\right)\) in matrix \(\mathbb{D}\) represents the displacements between anchors \(\mathrm{Q}\) and nodes \(\mathrm{P}\) along the x and y directions, respectively. The displacements \(\mathrm{d}_{\mathrm{\tilde{p}}}^{x}\) and \(\mathrm{d}_{\mathrm{\tilde{p}}}^{y}\) are computed as follows: \(\mathrm{d}_{\mathrm{\tilde{p}}}^{x}=\left|\mathrm{x}_{\mathrm{q}}-\mathrm{x}_ {\mathrm{P}_{\mathrm{\tilde{p}}}}\right|\) and \(\mathrm{d}_{\mathrm{\tilde{p}}}^{y}=\left|\mathrm{y}_{\mathrm{q}}-\mathrm{y}_ {\mathrm{P}_{\mathrm{\tilde{p}}}}\right|\). The binary adjacency matrix \(\mathbb{B}=\left(\left[\mathrm{B}_{\mathrm{\tilde{p}}}\right],\forall\mathrm{ j}=1,2,\ldots,m,\forall\mathrm{i}=1,2,\ldots n\right)\) is constructed such that each element \(\mathrm{B}_{\mathrm{\tilde{p}}}\) represents a binary value when the directional distance \(\left(\mathrm{d}_{\mathrm{\tilde{p}}}^{x},\mathrm{d}_{\mathrm{\tilde{p}}}^{y}\right)\) is within the directional thresholds \(\varepsilon_{x}\) and \(\varepsilon_{y}\), and it is given by the following:  
\[\mathrm{B}_{\mathrm{\tilde{p}}}= \begin{cases}1;&\mathrm{d}_{\mathrm{\tilde{p}}}^ {x}\leq\varepsilon_{x}\text{ and }\mathrm{d}_{\mathrm{\tilde{p}}}^{y}\leq \varepsilon_{y}\\ 0;&\mathrm{otherwise}\end{cases}  \tag{4}\]  
Figure 9: Establishing adjacency relationships in DENC based on boundaries defined by the directional proximity thresholds \(\varepsilon_{x}=0.300\) m and \(\varepsilon_{y}=0.150\) m—example problem. In the context of ILI data, \(\mathrm{d}_{\mathrm{ji}}^{\mathrm{y}}\) can be slightly modified to account for the pipeline's circumferential continuity, as denoted in Equation (5).