## 3 Results and Discussion
### Examples of Generated Prototypes and Extracted Strong Elements  
In order to explain the classification process using MCLNN in a practical way, we use in this section the Iris data set, and we show by numerical data and graphical diagrams the positions of the learned prototypes and the extracted strong elements.  
The Iris dataset contains 150 examples of Iris flowers originated from three different classes, each represented by a vector of four measurements. To visually present this dataset, we choose to use 2D projections (Figure 3).  
Regardless of the used projection and the selected features, it is easy to see that one of the three Iris classes is well separated from the other two, which are very overlapping. We note that the main difficulty in classifying the Iris data consists in separating the two overlapping classes.  
Among 192,000 simulations carried out by classification of the Iris data, according to the proposed learning scheme, we select two different simulations giving an error rate equal to 0 (Table 3), and we illustrate the learned prototypes and the extracted strong elements. The meaning of the parameters given in Table 3 are: \(m\) is the initial value of the fuzziness factor; \(\zeta\) is the acceptance threshold for the processed object to be classified by  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **Nb. of Elements** & **Nb. of Features** & **Nb. of Clusters** & **Sources** \\ \hline Breast Cancer (1) & 250/27 & 9 & 2 & [23] \\ Breast Cancer (2) & 616/67 & 9 & 2 & [24,25] \\ Diabetes & 692/76 & 8 & 2 & [26] \\ German Credit & 900/100 & 20 & 2 & [27] \\ Ionosphere & 317/34 & 34 & 2 & [28] \\ Iris & 135/15 & 4 & 3 & [29] \\ Switzerland & 97/8 & 10 & 5 & [30] \\ Unbalanced & 771/85 & 32 & 2 & [31] \\ Va & 121/9 & 10 & 5 & [30] \\ Vote & 210/22 & 16 & 2 & [32] \\ Wdbc & 513/56 & 30 & 2 & [33] \\ Wine & 162/16 & 13 & 3 & [34,35] \\ Wpbc & 176/18 & 33 & 2 & [36] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Characteristics of the different training and test datasets used in this work.  
the first competitive layer; \(\delta\) is the acceptance step used to adjust the acceptance threshold through iterations; \(C^{tot}\) is the total number of neurons in MCLNN; \(C^{\prime}\) is the number of neurons of the \(r\)th layer; \(S^{tot}\) is the total number of strong elements extracted after training MCLNN; and \(S^{r}\) is the number of strong elements extracted by the \(r\)th layer.  
The first simulation gives five prototypes and 15 strong elements (Table 4), and simulation 2, two more prototypes and only three strong elements (Table 5). The positions of these prototypes and strong elements are graphically represented in Figure 4.  
We note that the well separated class is represented by a single prototype in the two simulations, while the two overlapping classes are represented by more than one prototype. Moreover, all strong elements belong exclusively to the two overlapping classes. In fact, strong elements can be seen as objects that cannot be correctly classified based on the prototypes directly learned by the layers of MCLNN. These elements are considered as noisy objects and do not contribute to the adjustment of the prototypes.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \multirow{2}{*}{**Simul.**} & \multicolumn{4}{c}{**Parameters**} & \multicolumn{4}{c}{**Results**} \\ \cline{2-13}  & \(m\) & \(\xi\) & \(\delta\) & \(N\) & \(C^{tot}\) & \(C^{1}\) & \(C^{2}\) & \(C^{3}\) & \(\xi^{tot}\) & \(S^{1}\) & \(S^{2}\) & \(S^{3}\) \\ \hline
1 & 1.10 & 0.990 & 0.715 & 2 & 5 & 3 & 2 & - & 15 & 12 & 3 & - \\ \hline
2 & 1.60 & 0.885 & 0.260 & 3 & 7 & 3 & 2 & 2 & 3 & 1 & 2 & 0 \\ \end{tabular}
\end{table}
Table 3: Two different simulations (with error rate equal to 0) chosen as examples for giving the positions of prototypes and strong elements.  
Figure 3: 2D projections of Iris data using a set of 3 features in different order each time (**a**) Features 2, 1 and 4 (**b**) Features 3, 2 and 4 (**c**) Features 3, 4 and 1 (**d**) Features 1, 3 and 2.