## 5 Comparison of the Proposed Model  
To check the prediction accuracy and resilience of the proposed models, three different statistical metrics, i.e., RMSE, MAE, and MAPE, are utilized. The mathematical expressions of these performance metrics are given as follows:  
\[RMSE=\sqrt{\frac{1}{N}\sum_{t=1}^{N}(A_{t}-P_{t})^{2}} \tag{16}\]  
\[MAE=\frac{1}{N}\sum_{t=1}^{N}\left|A_{t}-P_{t}\right| \tag{17}\]  
\[MAPE=\frac{1}{N}\sum_{t=1}^{N}\left|\frac{A_{t}-P_{t}}{A_{t}}\right| \tag{18}\]  
whereas \(A_{t}\), and \(P_{t}\) denote the actual and predicted values, and \(N\) is the total number of trading days used in the testing data set. The smaller values of these three statistical metrics signify that there is less variation between the actual and predicted values. Furthermore, among these three statistical measures, it is easy to interpret and understand the MAPE. Hence, it is used as a key evaluating metric to check the prediction performance of our  
Figure 10: Prediction results of (**a**) EMD-LSTM (**b**) EEMD-LSTM (**c**) SEMD-LSTM (**d**) Proposed Akima-EMD method. proposed model. In this research article, we used EMD, EEMD, and SEMD, proposed a new method to decompose the actual signal, and then built our model for the correlated subcomponents using the LSTM network. We calculated these three main statistical measures to compare all four hybrid models and select the best prediction model in terms of minimum values of RMSE, MAE, and MAPE. It can be seen from the investigational results presented in Table 11 that the best model to predict the KSE-100 index closing price is our proposed hybrid Akima-EMD-LSTM model having minimum values of RMSE, MAE, and MAPE as compared to the other hybrids models. After selecting the best model among the four proposed hybrid models, our next goal is to compare it with other models. The following subsection will describe the details of such a comparison. It is evident from the empirical results presented in Table 11 that the most efficient prediction model among the four hybrid models is our proposed Akima-EMD-LSTM. Now the performance of the proposed new model is compared with other models, including the single LSTM, EMD-SVM, EEMD-SVM, SEMD-SVM, Akima-EMD-SVM, EMD-Random Forest, EEMD-Random Forest, SEMD-Random Forest, Akima-EMD-Random Forest, EMD-Decision Tree, EEMD-decision tree, SEMD-decision tree, and Akima-EMD-decision tree. The deep learning LSTM network does not require any input features, and the actual stock price data can be used directly to build the single LSTM network, whereas the other ensemble models use the correlated subcomponents (IMFs), and the residue obtained by decomposing it with the help EMD, EEMD, SEMD and proposed new model. The prediction performance of the proposed model is similarly effective in all scenarios, i.e., for each IMF and residue obtained from the KSE-100 index actual closing price, which is a key strength of this research. The investigational results shown in Table 12 make it simple to confirm that the three statistical measures of the suggested model are minimal; for example, the value of RMSE is 299.5416, which is minimal among other models. Akima-EMD-SVM, which has an RMSE of 309.345, is the second-best model in this competition, and the same SVM model with the EEMD ensemble, which has an RMSE of 322.675, is in third place. Similarly, the MAE and MAPE values of our proposed hybrid model are 234.340 and 0.578, respectively, which are also minimum as compared to the other models. Interestingly, the ensemble Akima-EMD-SVM model stands second in comparison with the proposed model, but the performance of this model is much better than the other ensemble random forest and decision tree models. A snapshot will more accurately depict the scope of this research work rather than presenting the actual and forecasted values, which are not presented here. The actual and predicted values are schematically shown in Figures 6-10 to make the prediction simple and comprehensible for each subseries and the actual closing KSE-100 price.