## 2 Conventional SOM, Its Selected Generalizations, and Related Work
### Conventional SOM
The number of classes (equal to 7) and the class assignments are known here; it allows us to directly verify the obtained results (obviously, the SOM--trained in an unsupervised way--does not utilize the knowledge of class assignments during the learning process). Figure 4 presents the U-matrix (with \(m_{1}=m_{2}=30\)) after performing the map calibration (see [1] for comments) for the considered data set. It is a two-dimensional display preserving, in general, the distance- and density structure of the 16-dimensional data set and showing, roughly, 7 data clusters (the darker areas of the U-matrix represent borders between them). The expressions \(a(b/c)\) in Figure 4 should be read as follows: \(a\) is the class label, \(b\) is the number of \(a\)'s records assigned to \(a\) in a given cluster, and \(c\) is the overall number of \(a\)'s records (hence, the overall classification accuracy is equal to 95%). In turn, we consider two two-dimensional synthetic data sets characterized by different complexities and types of their data clusters. Obviously, two-dimensional data do not have to be visualized. However, they allow us to directly evaluate the performance of data visualization methods by comparing the visualization effects with original data. The first data set (see Figure 5a) contains five well-separated volume-type clusters whereas the second set (see Figure 6a) includes both thin, shell-like as well as volume-type clusters; their distribution and mutual relations are much more complex than in the first set. Figure 5b-i present the evolution (in selected learning epochs) of the conventional, two-dimensional SOM for the synthetic data set of Figure 5a. Figure 7 presents the U-matrix (with \(m_{1}=m_{2}=30\)) for that data set. Analogous plots--for the synthetic data set of Figure 6a--are shown in Figure 6b-i and Figure 8.  
Figure 4: Gray-level U-matrix map for the visualization of the _Zoo_ data set. For simple data set of Figure 5a, the conventional SOM generates clear and correct image of distance- and density structures in those data, detecting five distinct clusters as shown in Figure 7. However, for more complex data of Figure 6a, their image shown in Figure 8 is much less clear; the borders between some clusters are much less distinct (see, e.g., some regions between clusters 3 and 5 as well as 1 and 3). Moreover, Figure 6c,d show that SOM has some problems in \"unfolding\" its structure at the initial stage of learning (or, equivalently, exhibits some tendency to \"twist up\" its structure) which may end up in wrong mapping of the learning data. It is caused by the processing of the whole map of \(m_{1}\times m_{2}\) neurons in each learning epoch (particularly, in the first phase of the map formation in which it should correctly \"unfold\" itself over the learning data). The processing of the whole map is also a very time-consuming operation. The introduction of Growing SOMs (GSOMs) [7] and Growing Grid Networks (GGNs) [8] is an attempt to address those problems. \[\mathbf{w}_{j}(k+1)=\mathbf{w}_{j}(k)+\eta_{j}(k)N(j,j_{\mathbf{x}},k)[\mathbf{x}(k)-\mathbf{w}_{j }(k)], \tag{2}\]0 Synthetic data set (**a**) and the evolution of the conventional SOM in it in learning epochs: (**b**) No. 0 (start of learning), (**c**) No. 1, (**d**) No. 2, (**e**) No. 3, (**f**) No. 5, (**g**) No. 500, (**h**) No. 800, and (**i**) No. 1000 (end of learning). \[\mathbf{w}_{j}(k+1)=\mathbf{w}_{j}(k)+\eta_{j}(k)N(j,j_{\mathbf{x}},k)[\mathbf{x}(k)-\mathbf{w}_{j }(k)], \tag{2}\]1 Synthetic data set (**a**) and the evolution of the conventional SOM in it in learning epochs: (**b**) No. 0 (start of learning), (**c**) No. 1, (**d**) No. 2, (**e**) No. 3, (**f**) No. 5, (**g**) No. 500, (**h**) No. 800, and (**i**) No. 1000 (end of learning). \[\mathbf{w}_{j}(k+1)=\mathbf{w}_{j}(k)+\eta_{j}(k)N(j,j_{\mathbf{x}},k)[\mathbf{x}(k)-\mathbf{w}_{j }(k)], \tag{2}\]2 Gray-level U-matrix map for the visualization of the synthetic data set of Figure 5a (U-matrix is calculated for the neuron map of Figure 5i).