# Spherical Tree-Structured SOM and Its Application to Hierarchical Clustering
## 2 Materials and Methods
### S-Ts-Som  
In this section, we describe the S-TS-SOM. The basic structure is the same as that of a TS-SOM. However, in an S-TS-SOM, nodes are evenly arranged on a spherical surface to eliminate the edges of the map. In this paper, we use polyhedrons inscribed in a sphere to arrange the nodes on the spherical surface. The nodes are arranged as follows. First, we divide the icosahedron and stack it, as shown in Figure 5. Next, nodes are placed on each surface of the polyhedrons, and links are added from the nodes of the inner layer to the nodes of the outer layer. Using this process, the tree structure is constructed for each face of the polyhedron, as shown in Figure 6. In the method proposed in this paper, except for the outermost layer's nodes, the tree structure is created such that each node has four links to the outer layer's nodes. The direction of layering is not important because it does not change the underlying concept. However, for the sake of explanation, we create the structure such that the outer layer has more nodes. An S-TS-SOM has multiple layers, each layer has a node \(m_{i_{l}}(\,i_{l}=1,\ldots I_{l})\), and node \(m_{i_{l}}\) has a reference vector \(m_{i_{l}}\). Here, \(l\) indicates the level of the layer, and \(l=0,1,2,\ldots L\) in order from the inner layer to the outer layer. In addition, \(i_{l}\) indicates the node number of the \(l\)th layer. We used an icosahedron for the innermost polyhedron, and because we divide the icosahedron to increase the number of layers, the number of nodes in each layer is expressed by the following formula. \[I_{l}=20\times 4^{l} \tag{5}\]  
Figure 4: Map shapes the extended SOMs. The learning algorithm of the S-TS-SOM is as follows. First, we initialize the reference vectors of the 0 th layer nodes with random values. The 0 th layer is trained using the SOM algorithm. In a SOM, the node with the reference vector that is closest to the input vector that was selected from the input set is determined to be the winner node. After determining the winner node, the reference vectors of the winner node and its neighborhood nodes are updated so that they are closer to the value of the input vector. By repeatedly determining the winner node and updating the reference vectors, the SOM is trained. In this paper, Euclidean distance is used to measure the similarity between the input and reference vectors. For layers other than the 0 th layer, the winner node is determined using the winner node of the inner layer. If the winner node in the \(l\)th layer of \(x_{j}(j=1,\dots J)\) selected from the input vector set is \(m_{c_{l},\ j}\), the winner node in the \(l+1\)th layer is determined from the child node set \(c_{l},\ j\), which has links from node \(m_{c_{l},\ j}\). The update method of each layer is the same as that used in a SOM. Update methods include online learning and batch learning methods. We adopt batch learning because it is not affected by the order of the input data. After determining the winner nodes of all the input vectors in the \(l\)th layer, the reference vectors of the \(l\)th layer are updated by the following formula. \[m_{i_{l}}(t+1)=\frac{\sum_{j=1}^{J}h_{c_{l}j_{l}}\ x_{j}}{\sum_{j=1}^{J}h_{c_{ l}j_{l}}} \tag{6}\]  
Figure 5: Layers of the S-TS-SOM. Figure 6: Structure of the S-TS-SOM. \[h_{c_{ij}j_{i_{j}}}\left(t\right)=\exp\left(-\frac{\theta_{c_{ij}j_{i_{j}}}^{2}}{2 \sigma^{2}(t)}\right), \tag{7}\]  
Here, \(h_{c_{ij}j_{i_{j}}}\) is a neighborhood function obtained from the positions of winner node \(m_{c_{i},\ j}\) and node \(m_{i_{j}}\). In addition, \(\theta_{c_{ij}j_{i_{j}}}\) is the angle between winner node \(m_{c_{i_{j}},\ j}\) and node \(m_{i_{j}}\), as seen from the center of the sphere; \(\sigma(t)\) is a function that determines the size of the update area; \(\sigma(t)\) is set to decrease as the number of updates increases.