# Spherical Tree-Structured SOM and Its Application to Hierarchical Clustering
## 3 Results
### Quantitative Evaluation of Clustering  
Here, we quantitatively compare the clustering performance of the S-TS-SOM and S-SOM. We used the purity and normalized mutual information (NMI) clustering indicators to evaluate the performance [18]. These two indicators take a value from 0 to 1, and values close to 1 indicate a high performance. Purity is calculated by counting the most dominant labels in each cluster and dividing by the number of data. Purity is simple to calculate. However, purity can easily be increased to 1 by increasing the number of clusters. By contrast, the calculation of NMI is complicated. However, it is calculated considering the number of clusters. NMI is close to 1 when the labels for each cluster are not mixed, and the number of clusters is close to the number of original categories. To calculate the purity and NMI clustering indicators, the true category labels are required. We used a total of 70,000 MNIST images as the experimental data. Therefore, in this experiment, the purity and NMI will be high if the data are correctly classified.  
We set the number of outermost nodes to 80, 320, 1280, and 5120 for both the S-TS-SOM and S-SOM. We calculated the purity and NMI by treating the data placed on the same winner node as belonging to one cluster. We set the number of S-TS-SOM updates to 100 for each layer and the number of S-SOM updates to 100. We set \(\sigma_{l+1}\) and \(\sigma(t)\) in the same  
Figure 8: Result of the S-SOM.  
Figure 7: Result of the S-ST-SOM.  
way as in Section 3.1. The 0 th layer of the S-TS-SOM and S-SOM were initialized with random values. Therefore, we present the average of the results of the five trials in Table 1. This table reveals that the S-TS-SOM and S-SOM have almost the same performance with respect to purity and NMI.  
In addition to the calculation of purity and NMI, we compared the learning time of the S-TS-SOM and S-SOM to examine whether the S-TS-SOM could be trained more quickly than the S-SOM. We used Python for model creation and calculation and used a PC with an Intel(R) Core (TM) i7-10870 2.20 GHz/5.0 GHz and 64 GB DDR4 memory. The \"Time\" column in Table 1 lists the learning time of each model. Because the number of updates for each layer of the S-TS-SOM was set to 100, the number of nodes searched during learning for one input vector is (20 + 4L) \(\times\) 101 from Formula (10). By contrast, in the S-SOM, the number of nodes searched for one input vector is 20 \(\times\) 4L \(\times\) 101.  
In addition to the difference in the number of searched nodes, the S-TS-SOM initializes and updates the reference vectors in multiple layers. Moreover, because of the differences in the algorithms, the number of searched nodes does not directly reflect the difference in the learning times of the S-TS-SOM and S-SOM. However, the learning time of the S-TS-SOM was shorter than the learning time of the S-SOM. Moreover, as the number of nodes increased, the difference between the S-TS-SOM and S-SOM learning times increased.  
From the experiments, we confirmed that an S-TS-SOM has almost the same clustering performance as a S-SOM but a shorter learning time.