## 6 More Comparisons with Alternative Approaches  
In addition to the evaluation of predictions done by ANN-ALO, a comparison-based evaluation was performed to get more idea about the effectiveness of the ANN-ALO model. At this point, the chosen (best) ANN-ALO system (ANN-ALO A\({}_{4}\)) from the former prediction results was compared with alternative hybrid systems with the same ANN model structure (Table 3), but with different optimization algorithms as trainers. The evaluation was done according to the same criteria under Equations (8) and (9). However, this time, a total of 30 different runs were performed on each system to eliminate the findings obtained by chance. For the evaluation, the means of MAE and MSE were used for each EEG time series via different ANN-based systems. As this was done while determining the appropriate ANN-ALO adjustment, the chaotic flow was checked again and again for all alternative approaches, considering the points in the prediction phase of the related processes. Additionally, the whole approaches were checked for any possible overfitting issue that could have affected objective evaluation, and further analyses were done when it was certain that they did not have an overfitting problem. As for the trainers, the particle swarm optimization (PSO) [98; 109; 110], cuckoo search (CS) [111; 112], firefly algorithm (FA) [113; 114], bat algorithm (BA) [115; 116] and, also, the traditional back-propagation algorithm (BPA) [117] were used for the ANN, and each different system was employed over the same time series considered in this study (the ANN-ALO A\({}_{4}\) system was run, except for the previously obtained findings, and a mean of 30 different runs was considered for this evaluation in order determine the level of stability within all runs). The mean MAE and mean MSE values (with standard deviations) obtained over the time series via different ANN-based systems are reported in Tables 7 and 8 respectively (the best values are in bold style). It can be seen from the findings provided in Tables 7 and 8 that ANN-ALO A\({}_{4}\) outperformed the other systems in six out of the 10 applications. In detail, the findings obtained for the four remaining applications were also in the top three findings obtained by all different ANN-based hybrid systems. Here, CS seems to be a competitive algorithm for ALO (ALO A\({}_{4}\)) in the context of chaotic EEG time series. However, the other algorithms do not seem to be strong enough to even get near findings with the same ANN, compared with the ANN-ALO A\({}_{4}\) system. As the traditional trainer, BPA also seems to be generally different from the other systems in terms of the obtained findings. After the comparison with different ANN-based hybrid systems, the ANN-ALO A\({}_{4}\) was compared with some alternative chaotic time series prediction approaches. In detail, the alternative time series prediction approaches were the dynamic Boltzmann machine (DyBM) [118], the support vector machine (SVM) [119], the hidden Markov model (HMM) [120], the Bayesian learning on Gaussian process model (BG) [121], the autoregressive integrated moving average (ARIMA) [122], the autoregressive model (ARM) [123], and the K-nearest neighbor algorithm (K-NN) [124]. The same approach of using a total of 30 different runs was performed over the objective 10 chaotic EEG time series, and the mean values of MAE obtained for each approach were assessed to determine the 'outperforming approaches'. In order to use the general ranking after that, the mean MAE values for ANN-ALO A\({}_{4}\) (as reported in Table 7) were directly used, which means only seven alternative approaches were run and compared with ANN-ALO A\({}_{4}\)  
Tables 9 and 10 report the mean values of MAE that were obtained for the time series via ANN-ALO A\({}_{4}\) and the other alternative prediction approaches (best values are in bold style).