درجہ بندی کلسٹرنگ

Hierarchical clustering represents relationships among observations at multiple levels rather than producing only one partition. In agglomerative clustering, each observation begins in its own cluster. The algorithm repeatedly merges the pair of clusters judged closest under a linkage rule until one cluster remains or a stopping condition is reached. Divisive methods start with one group and split it, though agglomerative approaches are more common. The linkage defines inter-cluster distance. Single linkage takes the minimum pairwise distance, which can connect elongated groups through a chain of nearby points. Complete linkage takes the maximum pairwise distance, favoring tighter groups but reacting to distant members. Average linkage averages pairwise distances. Ward linkage merges clusters to minimize the increase in within-cluster sum of squares and is tied to Euclidean geometry. These choices can lead to different trees from the same observations. A dendrogram visualizes the hierarchy. Leaves represent observations, and branch joins show which clusters merge and at what distance or linkage cost. Cutting the tree at a chosen height creates a flat clustering; the height is not automatically a statistical significance threshold. Large vertical gaps can suggest candidate cuts, but stability, group usefulness and domain context also matter. A dendrogram may become unreadable for large datasets, and implementations can differ in tie handling. Distances and scaling strongly influence the result. A feature with a much larger numeric range can dominate Euclidean distance unless scaling is appropriate. The method can be computationally expensive for large datasets, and many agglomerative procedures cannot naturally assign new observations to an existing tree without a separate extension. Choose a distance and linkage suited to the data, then assess sensitivity to those decisions. A hierarchy is descriptive structure under a specified geometry, not proof that nature contains a single correct set of groups.