## 5 Findings from Prediction Applications  
To be sure about which ANN-ALO solution is more successful at predicting the used time series, the prediction errors obtained with five different applications were compared. In order to make comparisons, errors in the prediction performances were calculated for the whole dataset (including both training and test data) with the mean absolute error (MAE), while the mean squared error (MSE) was used for ANN training processes. MAE and MSE can be represented as follows [107,108]:  
Let \(o_{i}\) be the observation (\(i\)) and \(p_{i}\) be the prediction regarding \(o_{i}\). \[MAE=MeanValue(|e_{i}|) \tag{8}\]  
where \(e_{i}=o_{i}-p_{i}\). \[MSE=MeanValue(\sum_{1}^{n}e_{i}) \tag{9}\]  
where \(e_{i}=o_{i}-p_{i}\), and \(n\) is the total data (row) considered. The MAE and MSE values obtained with five different ANN-ALO systems are presented in Tables 5 and 6 respectively (best values are in bold style). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & **Adjust. 1** & **Adjust. 2** & **Adjust. 3** & **Adjust. 4** & **Adjust. 5** \\  & **(ALO-A\({}_{1}\))** & **(ALO-A\({}_{2}\))** & **(ALO-A\({}_{3}\))** & **(ALO-A\({}_{4}\))** & **(ALO-A\({}_{5}\))** \\ \hline \hline Number of ant-lions (particles) & 50 & 75 & 100 & 100 & 125 \\ \hline Total iterations (stopping criteria) & 2000 & 3000 & 4000 & 5000 & 6000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Five different ALO adjustments considered for training ANN models along the prediction applications for the EEG time series (Figure 5) in the study. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Total Neurons** & **Values of** & **Total Neurons** & **Value of ANN** & **Total** & **Total Neurons in** & **Activation** \\
**in the Input** & **ANN Input** & **in the Output** & **Output Neuron** & **Hidden** & **Each Hidden** & **Function** \\
**Layer** & **Neurons** & **Layer** & & **Layers** & **Layer** & **Neurons** \\ \hline \hline \multirow{3}{*}{4} & \(x(time)\); & & & & & \\ \(x(time-3)\); & & & & & & \\ \(x(time-6)\); & & & & & & \\ \(x(time-9)\) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Adjustments of ANN models employed in prediction applications for the EEG time series (Figure 5) in the study. Tables 5 and 6 show that the ANN-ALO system with Adjustment 4 (ALO-A\({}_{4}\)/Table 3) had the best performance in seven out of 10 prediction applications. Figure 7 presents the prediction flow for the related EEG time series (Figure 5) in order to give visual idea about the success of the employed hybrid system. In Figures 7 and 8, the predictions done for both the training and test data of the considered chaotic EEG time series are shown (predictions of training and test data are separated with a red line). The original data is shown with a white line, and predictions are represented with red dots (line). Additionally, some visible and remarkable periods including errors in prediction are represented by blue squares. Visualization of the prediction performances (Figures 7 and 8) showed that the ANN-ALO system is successful enough to predict the future states of the considered chaotic EEG time series data flow. In order to be sure, the chaotic flow was also checked by considering the points in the prediction phase of the processes. Additionally, there no overfitting problem observed when the error values were evaluated again. On the other hand, it is also still important to evaluate the system in detail by comparing it with some alternative approaches, and even validating it statistically.