## 5 Forecasting Results  
The time series data was used in short term prediction simulation to evaluate the performance of the proposed DRBM-FFA approach. To objectively benchmark the performance accuracy of the proposed approach it was compared with two other well know predictive modelling approaches namely a conventional Multi-Layer Perceptron (MLP) neural network and a linear ARIMA model [45]. To also have a fair evaluation of the hybrid FFA optimization strategy used, hybrid variants of MLP optimized using FFA (MLP-FFA) and ARIMA optimized using particle swarm optimization (ARIMA-PSO) were also compared with the proposed DRBM-FFA method. Figure 5 shows the designing an optimized predictor of DBN via the best firefly. Table 2 lists the parameters and values for each of the algorithms evaluated in our prediction experiments. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Model Elements/Parameters** & **Quantity** \\ \hline Population of PSO & P & 10 \\ The number of RBM & RBM1,RBM2 & 2 \\ The number of input layer & N (\(1\ \leq\ N\ \leq\ 20\)) & Given by FFA \\ Absorption coefficient & \(\gamma\) & 0.1 \\ Velocity coefficient & \(c_{1},c_{2}\) & 1.0 \\ The number of hidden layer & M (\(1\ \leq\ M\ \leq\ 20\)) & Given by FFA \\ The number of output & - & 1 \\ Interval of input data & \(\tau\) & 1 \\ Learning rate of RBM & \(\varepsilon\) (_Step 2_) & Given by FFA \\ Learning rate of BP & - & Given by FFA \\ Population of FFA & P & 10 \\ Iteration times of BP & L & 100 \(<\) L \(<\) 5000 \\ Biases of units & \(b_{i},b_{j}\) & 0.0 \\ Convergence parameter of BP & \(\alpha\) & 0.05 \\ Convergence parameter of RBM & \(\beta\) & 0.0005 \\ Convergence period of RBM & k & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The used parameters in the prediction experiments. Short-term prediction accuracy of the DRBM-FFA model compared against the ARIMA, MLP-FFA and ARIMA-PSO are shown in Figure 6. Each algorithm is used to predict traffic flows in all five links in the traffic network where traffic data is utilized to predict traffic flow for the whole transportation network. The short-term prediction precision of the DRBM-FFA is compared against each of the other models and the results are shown in Figures 7-10. The flowchart of the processing steps is shown in Figure 5.  
Different statistical estimators are applied to assess the performance of the proposed DRBM-FFA method. These estimators are as follows: the (MSE) shown in Equation (6), the coefficient of determination (\(R^{2}\)) presented in Equation (7), the root mean square (RMSE) presented in Equation (8), correlation coefficient (\(r\)) presented in Equation (9), mean absolute percentage error (MAPE) shown in Equation (10), root-mean square percentage error (RMSPE) displayed in Equation (11). Figure 5: Designing an optimized predictor of DBN via the best firefly. Figure 6: Prediction results by DRBM-FFA, ARIMA-PSO, ARIMA and MLP-FFA methods. \[\text{MSE}=\frac{1}{r}\sum_{i=1}^{r}(D_{pi}-D_{ai})^{2} \tag{6}\]  
\[R^{2}=1-\frac{\sum_{i=1}^{r}\bigl{(}D_{pi}-D_{ai}\bigr{)}^{2}}{\sum_{i=1}^{r} \bigl{(}D_{pi}-D_{av}\bigr{)}^{2}} \tag{7}\]  
\[\text{RMSE}=\sqrt{\frac{1}{r}\sum_{i=1}^{n}\bigl{(}D_{pi}-D_{ai}\bigr{)}^{2}} \tag{8}\]  
\[r=\frac{\sum_{i=1}^{n}\bigl{(}D_{pi}-\overline{D_{pi}}\bigr{)}.\bigl{(}D_{ai}- \overline{D_{ai}}\bigr{)}}{\sqrt{\sum_{i=1}^{n}\bigl{(}D_{pi}-\overline{D_{pi}} \bigr{)}.\sum_{i=1}^{n}\bigl{(}D_{pi}-\overline{D_{pi}}\bigr{)}}} \tag{9}\]  
\[\text{MAPE}=\frac{1}{r}\sum_{i=1}^{n}\Bigl{|}\frac{D_{pi}-D_{ai}}{D_{ai}} \Bigr{|}\times 100 \tag{10}\]  
Figure 6.0  
where \(n\) is the quantity of data, \(D_{pi}\) is the predicted value; \(D_{av}\) is the average of the actual values; \(D_{ai}\) is the actual value; \(\overline{D}_{pl}\) is the predicted traffic flow; \(D_{l}\) shows the measured traffic flow for link \(l\); \(\overline{D_{pi}}\) and \(\overline{D_{ai}}\) are the mean value of \(D_{pi}\) and \(D_{ai}\), respectively.