## 4 Applications on Electroencephalogram (EEG) Prediction
### Organization of the ANN-ALO  
As was indicated before, an optimum ANN model was first determined by evaluating the accuracy with the EEG data used in [10], and then five different ALO adjustments were used for the ANN model structure for the prediction applications in the study. After performing the prediction applications with each hybrid system, the system with the best results was used for the comparative evaluations done in further stages. Here, a remarkable consideration was also given to obtain an ANN-ALO \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Application** & & & \\ (EEG Time Series) & **Lyapunov Exponent 1** & **Lyapunov Exponent 2** & **Lyapunov Exponent 3** \\ \hline A1 & \(\mathbf{0.6515\times 10^{-3}}\) & \(\mathbf{0.5078\times 10^{-4}}\) & \(-0.7469\times 10^{-3}\) \\ A2 & \(\mathbf{0.2002\times 10^{-3}}\) & \(-0.2354\times 10^{-4}\) & \(-0.4054\times 10^{-3}\) \\ A3 & \(-0.0480\times 10^{-3}\) & \(-0.9309\times 10^{-4}\) & \(\mathbf{0.8753\times 10^{-3}}\) \\ A4 & \(\mathbf{0.2552\times 10^{-3}}\) & \(-0.2734\times 10^{-4}\) & \(\mathbf{0.5395\times 10^{-3}}\) \\ A5 & \(\mathbf{0.4205\times 10^{-3}}\) & \(\mathbf{0.5442\times 10^{-4}}\) & \(-0.0385\times 10^{-3}\) \\ A6 & \(-0.8610\times 10^{-3}\) & \(\mathbf{0.7984\times 10^{-4}}\) & \(-0.1260\times 10^{-3}\) \\ A7 & \(\mathbf{0.1245\times 10^{-3}}\) & \(-0.0166\times 10^{-4}\) & \(\mathbf{0.2455\times 10^{-3}}\) \\ A8 & \(-0.3045\times 10^{-3}\) & \(\mathbf{0.7614\times 10^{-4}}\) & \(\mathbf{0.3279\times 10^{-3}}\) \\ A9 & \(\mathbf{0.6972\times 10^{-3}}\) & \(-0.1718\times 10^{-4}\) & \(\mathbf{0.9491\times 10^{-3}}\) \\ A10 & \(\mathbf{0.2330\times 10^{-3}}\) & \(-0.4401\times 10^{-4}\) & \(-0.8605\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lyapunov exponents for each EEG time series application (Figure 5) in the study. Figure 6: Phase-space state portraits of the EEG time series (Figure 5) included in the study. system which does not have an overfitting problem. This was controlled by prediction applications done over different ANN-ALO adjustments by evaluating the relation between training and testing errors. At the end of the process, more consideration was also given to employing the best adjustment of ANN-ALO with no overfitting problem. Table 2 shows a report on the mean accuracy of the prediction shown by different BPA-trained ANN models (the best model is in bold style). The models were run 10 times over four different EEG data (explained in [10]) and in order to make the evaluation simpler, the mean accuracy of prediction for the test data only (which was 50% of the whole EEG data) was used. Furthermore, in order to obtain one single value of prediction accuracy, the mean of 10 runs was calculated over a single mean accuracy value calculated for each ANN model by considering the entire EEG time series. Two of different lag alternatives are the ones reported by [10; 65]. As a widely used activation function, sigmoid was used in all ANN models in this study. It can be seen from Table 2 that a greater the number of inputs (previous data points) does not always lead to a better prediction performance for the ANN model. On the other hand, it is also important to determine the optimum number of total hidden layers and artificial neurons included within each of them. Since it is not ideal to use different numbers of neurons in different hidden layers for the training and test performances of an ANN model, the same number of neurons was included in each hidden layer in the context of related works by the author. Here, it is considered that the alternative ANN model formation No. 3 provided better prediction accuracy than the other alternative formations. Additionally, the lag structure chosen in model no. 3 led to better findings than the lag structure preferred in [10; 65]. Moving on from the obtained findings, Table 3 provides information about the essential adjustments of the optimum ANN model structure that are used in the prediction applications of chaotic EEG data.