# Meta-Extreme Learning Machine for Short-Term Traffic Flow Forecasting
## 3 Experiments
### Performance Evaluation
In addition to the numerical advantages reflected in the evaluation criteria, the ANN model and the ABC-ELM model also include:  
1. Since the connection weights of the input and implicit layers and the threshold of the implicit layer of ELM can be set randomly and do not need to be adjusted after setting, the connection weights \(\beta\) do not need to be adjusted iteratively. The above are determined once by solving a system of equations, which makes the learning speed of ELM significantly higher than that of ANN. 2. ANN, as a traditional gradient-based learning algorithm, also faces more complex problems, such as falling into local optimum, overfitting, etc. Overcoming these problems often requires some cost, while ELM does not need to overcome these problems, which leads to a much simpler network structure for ELM than ANN. 3. ELM can use non-differentiable functions as activation functions, but ANN is only applicable to differentiable functions. We also compare traditional ELM and the improved optimization models based on the traditional ELM, including the GA-ELM model and GSA-ELM [15]. As shown in Figure 5, \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Models** & **Data A1** & **Data A2** & **Data A4** & **Data A8** \\ \hline SVR & 329.09 & 259.74 & 253.66 & 190.30 \\ AR & 301.44 & 214.22 & 226.12 & 166.71 \\ ANN & 299.64 & 212.95 & 225.86 & 166.50 \\ GM & 347.94 & 261.36 & 275.35 & 189.57 \\ KF & 322.03 & 239.87 & 250.51 & 187.48 \\ DT & 316.57 & 224.79 & 243.19 & 238.35 \\ ELM & 300.67 & 208.84 & 224.54 & 172.69 \\ GA-ELM & 291.42 & 211.43 & 228.57 & 169.25 \\ GSA-ELM & 287.89 & 203.04 & 221.39 & 163.24 \\ ABC-ELM & **286.25** & **200.42** & **220.07** & **163.67** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The RMSE (vehs/h) of different forecasting models under four datasets. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Models** & **Data A1** & **Data A2** & **Data A4** & **Data A8** \\ \hline SVR & 14.34 & 12.22 & 12.23 & 12.47 \\ AR & 13.57 & 11.59 & 12.70 & 12.71 \\ ANN & 12.61 & 10.89 & 12.49 & 12.53 \\ GM & 12.49 & 10.90 & 13.22 & 12.89 \\ KF & 12.46 & 10.72 & 12.62 & 12.63 \\ DT & 12.08 & 10.86 & 12.34 & 13.62 \\ ELM & 11.92 & 10.32 & 12.09 & 12.58 \\ GA-ELM & 11.86 & 10.30 & 11.87 & 12.26 \\ GSA-ELM & 11.69 & 10.25 & 11.72 & 12.05 \\ ABC-ELM & **11.40** & **9.95** & **11.26** & **11.90** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The MAPE(%) of different forecasting models under four datasets. it is clear that the ABC-ELM model outperforms the GSA-ELM in all four MAPE evaluation criteria with the same dataset, and ABC-ELM still matches the performance of GSA-ELM in the RMSE evaluation criteria. In addition, we also compared the Akaike information criterion (AIC) values of traditional ELM, GA-ELM, GSA-ELM, and ABC-ELM for the four datasets, which is a measure of the goodness of fit of statistical models. As shown in Table 5, the ABC-ELM model has lower AIC values compared to the GA-ELM model and GSA-ELM model for the same dataset. Therefore, we believe that the ABC-ELM has better forecasting accuracy. Figure 6a-h represent the forecasting-measurement value comparison plots and the forecasting-measurement value relative error of the ABC-ELM model under four training sets A1, A2, A4, and A8, respectively. Where Figure 6a,b,e,f represent the forecasting-measurement value comparison plots of ABC-ELM, the blue line represents the forecasting, and the red line represents the measurement. As can be seen from the figure, the ABC-ELM model performs well in the traffic flow forecasting model, and the predicted results are almost consistent with the actual results. However, there are still shortcomings, especially when the traffic flow changes dramatically in a short period. Figure 6c,d,g,h represent the forecasting-measurement relative error of ABC-ELM. Its calculation formula can be expressed as:  
\[\epsilon_{i}=m_{i}-p_{i}, \tag{11}\]  
where \(e\) stands for relative error, and \(m\) and \(p\) represent the measurement and forecasting, respectively.