## 5 Forecasting Results
The coefficient of determination, \(R^{2}\) represents the linear regression line among the predicted values of the neural network model. The essential output, is applied as a measure of performance. Expressed differently, \(R^{2}\) is the square of the correlation between the response values and the predicted response values. The closer \(R^{2}\) is to 1, the better the model can fit the actual data [46]. This measurement controls the degree of success the fit has in stating the change of the data. It can be indicated as the square of the multiple correlation coefficients, and the coefficient of multiple determinations. The smaller amount of MAPE has a superior performance model, and conversely, in the case of \(r\). The detail prediction errors (MSEs) for the original data are shown in Figure 6.1  
Table 3 shows that the DRBM was able to outperform in comparison to the other approaches based on obtaining the lowest learning MSE and short-term prediction MSE based on the time series results shown in Figures 7-10. The MLP with FFA obtained the next lowest learning MSE and short-term prediction MSE followed by the ARIMA. The Monte Carlo method was used to acquire a more objective evaluation of the performance of each approach that is based on sampling testing data based on sub-blocks to evaluate the forecasting efficiency of the algorithm. Experiments to determine traffic flow prediction performance over five time horizons were carried out to evaluate the performance of the MLP-FFA, ARIMA, ARIMA-PSO and DRBM-FFA methods. Let \(P(t+1)\) represent the estimated flow for the \((i+1)th\) time interval in the future. For the first forecasting interval \((i+0)\), the flow is represented by \(P(t)\). Table 4 shows the forecasting outcomes from the test data for 5 links. In Table 4, each \"\(t\)\", \"\(t+1\)\", \"\(t+2\)\"_and_ \"\(t+3\)\" is a 15-min interval into the future. The results show that the performance of all four prediction models improves when forecasting further into the future. Values in bold style show the minimum quantities for RMSE, \(r\) and MAPE. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Structure and Evaluation** & **MLP-FFA** & **ARIMA** & **ARIMA-PSO** & **DRBM-FFA** \\ \hline Learning rates & 0.85 & 0.64 & 0.73 & 0.98 \\ Iterations & 336 & 350 & 298 & 200 \\ Learning MSE & 109.21 & 122.4 & 108.9 & 98.70 \\ Short-term prediction MSE & 234.38 & 280.50 & 126.11 & 109.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The detail prediction errors (MSEs). Table 4 shows that all error measurement for DRBM-FFA are less than those for the other algorithms for all 15-min prediction intervals. As shown in Table 4, DRBM-FFA outperformed MLP-FFA, ARIMA, and ARIMA-PSO forecasters for all three time intervals. As anticipated, the PSO improved prediction accuracy of the ARIMA model. Figure 7 further illustrates the prediction results of selected links every 5 min for DRBM-FFA for the next 30 min which was determined using root-mean square percentage Error (RMSPE). Figure 7 demonstrates the RMSPE for the selected links. In addition to the given experiments, Monte Carlo [47] method is applied to assess the sensitivity and accuracy of each predictive algorithm due to the stochastic variation of traffic data. Firstly, in each experiment, the ratio of traffic flow for links is calculated. Secondly, 50% of the ratio of traffic flow for links are designated randomly. Thirdly, selected data is increased by a Gaussian random variable \(r\). Fourthly, the new ratio of traffic flow for each link are served to the predictive method, and the results are recorded. The final stage is where, the four previous stages should be repeated 1000 times per data sample. Hence, the standard deviation of the Monte Carlo results are calculated and the coefficient \(r\) is supposed to be a Gaussian random variable \(r\sim N(1,0.1)\). Figure 8 shows the error for a particular data sample for our prediction model.