2.3.5. Dendrogram

A dendrogram is a visualization in the form of a tree that shows the hierarchical relationship like the order and distance (dissimilarity) between samples [62]. The individual samples are located along the bottom of the dendrogram and referred to leaf nodes. The hierarchical clusters are formed by merging individual samples or existing lower-level clusters. In a dendrogram, the vertical axis is labeled distance and refers to a dissimilarity measure between individual samples or clusters. Generally, in a dendrogram, horizontal lines can be regarded as places where clusters merge, while vertical lines show the distance at which lower-level clusters were merged, forming a new higher-level cluster. The dissimilarity measure between two groups is calculated as Equation (12):

$$\text{Dis} = 1 - \mathbb{C} \tag{12}$$

where Dis means the dissimilarity or distance among objects and C means the correlation degree between clusters.

If clusters are highly correlated to each other, they will have a correlation value close to 1. To that, Dis = 1 − C will be given a value close to zero. Therefore, highly related clusters are nearer to the bottom of the dendrogram. Those clusters that are not correlated have a correlation value close to zero. Clusters that are negatively correlated will give a distance value larger than 1 in the dendrogram. The dendrogram can be used to visually allocate correlated objects to clusters or to detect outliers and anomaly in a diagram [47]. In the dendrogram, each sample is treated as a single cluster and then successively combines pairs of clusters until all clusters have been merged into a single cluster. In this process, the dendrogram shows how the aggregations are performed from bottom to top of the dendrogram statically. This procedure allows the cut-off points to flexibly and efficiently represent the number of clusters. Therefore, this study used the number of cut-off points in the dendrogram to validate the cluster number of the agglomerative clustering.
