Visualizing Clusters

Each node T_i in a hierarchical cluster tree T represents a nested collection of enclosed data points or sub-clusters. The node contains a summary of all points and sub-clusters rooted from it. The following information may be directly obtained from T_i.

• n_i : the number of data points enclosed.
• m_i : the mean of the data points.
• B_i : the extents, i.e. the minimum and maximum bounds of the cluster for each dimension.
• v_i : a measure of the size of cluster T_i.
• l_i : the tree depth at node T_i

v_i is a computed measure of a cluster size and satisfies the following criteria: If T_i is an ancestor of T_j, then v_i > v_j

The value of v_i is directly dependent on the shape of the clusters produced by the clustering algorithm. For spherical clusters, v_i may be the radius of a cluster. For rectangular clusters, v_i may be the N-dimensional volume of the cluster.

In parallel coordinates, we display the information at a node by making use of variable-width opacity bands. Figure 2 shows a graduated band that visually encodes the information for a cluster. The mean stretches across the middle of the band and is encoded with the deepest opacity. The deepest opacity is a function of the density of a cluster, defined as the ratio $\frac{n_i}{v_i}$. The top and bottom edges of the band have full transparency. The opacity across the rest of the band is linearly interpolated. The thickness of the band across each axis section represents the extents of the cluster in that dimension.
 
 

Figure 2:A single multi-dimensional graduated band that visually encodes information at a cluster node.