## 4 Simulation Results and Discussions
### Numerical Optimization of the SS-ELM Hyper-Parameters  
The proposed scheme has several parameters that need to be optimized, and some of these parameters have a correlation between them, so they need to be optimized together. Throughout the paper, we adopt the sigmoid mapping function, since it has reported an excellent generalization ability [18; 22]. We initially optimized the regularization parameter \(C\) and the number of hidden neurons \(n_{h}\), by considering \(\lambda=0\) and \(\mu=0\). This means that the first optimization is basically the same for a supervised ELM. To select the best parameters, we use contour plots, which are obtained for four \(E_{b}/N_{0}\) values: 5, 10, 15, and 20 dB for the two channel configurations presented in Table 2. In Figures 8 and 9, each point of the contour plot corresponds to the representative BER value obtained by following the Monte Carlo simulation explained at the beginning of this Section. Then, to select the sub-optimal values of the regularization parameter and number of hidden neurons that minimize the system performance, we use visual inspection. Namely, the areas with the best performance (blue zones) and overlap among most of the figures are chosen. All sub-figures in Figures 8 and 9 are considered in order to obtain a BER invariant to the movement angles, acceleration angles, and velocities of the transmitter and receiver. As expected, these plots also verify that the BER decreases as the relationship \(E_{b}/N_{0}\) increases. observed that the BER metric decreases from configuration 1 to 2 and that the area of the optimal parameters increases from configuration 1 to 2. Based on the previous observations, we selected \(C=10^{1}\) and \(n_{h}=64\), namely, a pair of hyper-parameters that are inside the optimal area of both configurations. This adoption also considers that the ELM architecture is kept simple, namely, a minimum number of hidden neurons is selected. It is important to notice that the apparent optimal area increases from configuration 2 to 1, which indicates that, for bigger velocity values, the optimal area may be bigger and will probably contain the ones shown above. Figure 8: BER countour plot in terms of the regularization parameter \(C\), number of hidden neurons, and different \(E_{b}/N_{0}\) values for system configuration 1. Figure 9: _Cont._  
Once the values of \(C\) and \(n_{h}\) were optimized, we proceed to optimize the parameters \(\lambda\) and \(\mu\) simultaneously for the SS-ELM training/testing, in the same way as the previous ones for the standard ELM. Remember that \(\lambda\) represents the trade-off parameter between the supervised and no-supervised learning and \(\mu\) proportionally represents the number of unlabeled samples; refer to the end of Section 3. Note that, for these results, we use a linear representation of the BER instead of a logarithmic one because the results are closer to each other and the resulting areas are smaller. Looking at the results of Figures 10 and 11, it can bee seen that the optimal areas are relatively smaller when compared with the ones shown in Figures 8 and 9. This means that there is a smaller range of values that can be used without compromising the BER performance. Furthermore, the values obtained are more sensitive to different channel configurations. It can also be seen in Figures 10 and 11 that, when the value of the parameter \(\mu\) is higher than 7, the system performs poorly. This result indicates that the proposed SS-ELM algorithm suffers over-fitting, or that there is too much noise added to the training process with the unlabeled data inclusion. When considering the parameter \(\lambda\), it can be seen that the better values tend to be bigger than \(10^{0}\) for the first configuration, and bigger than \(10^{1}\) for the second configuration. This behavior means that SS-ELM benefits from bigger values of \(\lambda\), meaning that the non-labeled data have a big impact on a one-to-one comparison with the labeled data.