## 3 The Proposed Generalized SOMs with Splitting-Merging Structures  
Finally, we would like to briefly present our two original approaches, i.e., (i) Generalized SOMs with 1-Dimensional Neighborhood (GeSOMs with 1 DN [11] also referred to as Dynamic SOMs (DSOMs) in [12; 13]) which operate on splitting-merging neuron chains (see Figure 15) and (ii) Generalized SOMs with splitting-merging Tree-Like Structures (GeSOMs with T-LSs [14; 15; 16; 17; 18; 19]; see Figure 16). Main features, main objectives, general concept, and implementation of both approaches are the following:  
_Main features_: Our approaches work in a fully unsupervised way, i.e., (i) they do not need to predefine the number of clusters and (ii) they use unlabeled data. _Main objectives_: (i) an automatic determination of the number of clusters in a given data set and (ii) an automatic generation of multi-point prototypes for particular clusters. _General concept_: In comparison with conventional SOMs, the proposed approaches are also equipped with three additional mechanisms (allowing for data segmentation) such as: (i) automatic adjustment of the number of neurons in the network (removing low-active neurons and adding new neurons in the areas of the existing high-active neurons), (ii) automatic disconnection of the tree-like structure into subnetworks, and (iii) automatic reconnection of some of the subnetworks preserving--in the case of GeSOMs with T-LSs--the no-loop spanning-tree properties. Such generalized SOMs are able to detect data clusters of virtually any shape and density (including both volumetric and thin, shell-type clusters) by locating a single disconnected subnetwork in the area of the data space occupied by a given cluster. Hence, the number of automatically generated subnetworks is equal to the number of clusters. Moreover, the set of neurons in a given subnetwork is a multi-prototype of the corresponding cluster. Such prototypes can be directly used in clustering/classification tasks using the well-known nearest multi-prototype algorithms [41]. _Implementation:_ Figure 2 (Initialization- and WTM-algorithm-blocks as well as blocks concerning GeSOM with 1 DN and GeSOM with T-LSs) presents generalizations of the conventional-SOM learning algorithm for both structures considered. Both approaches start with a few neurons (usually two--see the Initialization block of Figure 2). In turn, after each learning epoch, the mechanism (i), (ii), and (iii) listed in the _General-concept_ part of this section are implemented by conditional activation of five successive operations (see also Figure 2; for details, see the afore-listed references):  
1. The removal of single, low-active neurons preserving the network continuity: a given neuron is removed if its activity--measured by the number of its wins--is below an assumed level. 2. The disconnection of the network (subnetwork) into two subnetworks: the disconnection of two neighboring neurons takes place if the Euclidean distance between them exceeds an assumed level. 3. The removal of very small subnetworks with two or three neurons (usually, they represent noise). 4. The insertion of additional neurons into the neighborhood of high-active neurons in order to take over some of their activities (it results in distributing more evenly the system's activity across the network). 5. The reconnection of two selected subnetworks: 5.1. The GeSOM with 1 DN case: the nearest end-neurons from two neighboring sub-chains are connected if the Euclidean distance between them is below an assumed level. Figure 16: GeSOM with T-LSs. Figure 15: Generalized SOMs (GeSOM) with 1 DN (DSOM). The GeSOM with T-LSs case: the nearest single neurons from two neighboring subnetworks are connected if the Euclidean distance between them is below an assumed level (this mechanism supports growing tree-like structure of the network). Figure 17a-i present the selected stages of the evolution of GeSOM with 1 DN (DSOM) for the more complex data set of Figure 6a.