# Spherical Tree-Structured SOM and Its Application to Hierarchical Clustering
## 2 Materials and Methods
### Related Work
The structure of a TS-SOM is shown on the left-hand side of Figure 2. A TS-SOM is a SOM composed of multiple layers and has a structure in which the number of nodes increases from the upper layers to the lower layers. The nodes of the upper layer have links to the nodes of lower layer, and the winner nodes determined in the upper layer assist the search for the lower layer's winner nodes  
Figure 1: The structure of the SOM. to speed up the search for the winner nodes. The right-hand side of Figure 2 shows how to determine the winner node in a TS-SOM. The lower winner node is determined from the child node group that has a link from the winner node determined in the upper layer. As in a SOM, the nodes of each layer in a TS-SOM are arranged in a two-dimensional grid pattern. Therefore, a TS-SOM also has map edges. Next, we describe the shape of the SOM. Normally, the SOM arranges nodes on a two-dimensional grid pattern for learning. Therefore, a SOM is a map with edges. The presence of the edges of the map causes a difference in the size of the update area of a winner node at the center of the map and a winner node at the edge of the map. Figure 3 shows the update area according to the position of the winner node. In addition, because of the characteristics of the SOM, if the map has an edge, the SOM tends to collect data at the edges of the map. Therefore, to eliminate differences in the size of the update area, the torus SOM, which has a map without edges, and the spherical SOM (S-SOM), which arranges nodes on a spherical surface, were proposed [11; 12]. Figure 4 shows the map shapes of the torus SOM and the S-SOM. In the torus SOM, by connecting the edges of the map as shown in the Figure 4, the edges of the map are eliminated. The torus SOM can learn uniformly regardless of the position of the winner nodes. However, when the map is visualized, it is necessary to consider the connections between the edges of the map, which make it difficult for humans to intuitively understand the result. In this study, we chose to arrange nodes on a sphere for intuitive visibility. Intuitive visibility is achieved by drawing the spherical map in three dimensions and rotating the spherical map for viewing. Figure 3: Difference in the update area depending on the positions of the winner nodes. Figure 2: Structure of the TS-SOM and how to determine the winner node. When clustering data, how finely the data should be clustered strongly depends on the target task. For example, if the data are uniformly distributed, it is difficult to automatically determine the optimal number of clusters. Furthermore, for data with small categories within large categories, the hyperparameters need to be adjusted depending on the size of the target category. Generally, for such data, a dendrogram is created by performing hierarchical clustering, and the number of clusters is arbitrarily determined according to the task. In addition to hierarchical clustering, a SOM with multiple layers can be used to arbitrarily determine the number of clusters. The TS-SOM, which is the basis of the method proposed in this paper, is composed of multiple layers, and has a structure in which the number of nodes increases from the upper layers to the lower layers. Therefore, the data are classified more finely toward the lower layers, and clustering results with different granularities can be obtained in each layer. In addition to the TS-SOM, the hierarchal feature map (HFM) has also been proposed [13; 14]. Moreover, the GH-SOM, which enables the HFM to learn adaptively, and a method that makes the TS-SOM learn adaptively, have been proposed [15; 16; 17].