# Unequal Interval Dynamic Traffic Flow Prediction with Singular Point Detection
## 6 Performance Evaluations
### Traffic Flow Prediction Performance Comparison  
Figure 9 shows the traffic flow prediction performances of our proposed algorithm and the other three comparing methods. Compared with the realistic real-time observation data, the ranking in terms of prediction performance is our proposed algorithm, kernel regression, Fourier method, and linear regression. The prediction accuracy of our proposed algorithm increases by 21.3%, 29.1%, and 83.4% respectively. The reason for the prediction performance improvement is that the unequal interval division via singular points can reflect the hidden traffic condition variations of the flow signal. Moreover, the Wiener process in the Vasicek model expresses the heuristic fluctuation of the traffic flow signal. The details of the traffic flow prediction performances of our proposed algorithm and the comparing methods are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Our Proposed** & **Linear Regression** & **Kernel Regression** & **Fourier** \\ \hline unequal interval? & **Y** & N & N & N \\ reprediction times & 46 & 90 & **10** & **10** \\ multiscale? & **Y** & N & N & N \\ degree & **7** & - & - & 8 \\ MRE & **0.841** & 1.543 & 1.021 & 1.087 \\ SSE & **137.009** & 432.321 & 216.668 & 283.275 \\ RMSE & **1.064** & 1.890 & 1.338 & 1.530 \\ normalized computing time & **0.256** & 0.500 & 1.000 & 0.500 \\ \hline \end{tabular}
\end{table} 
Table 3: Details of prediction performance comparisons in four methods. Figure 8: The next singular point prediction MRE results in different types of wavelet. As shown in Table 3, only our proposed algorithm predicts the traffic flow within dynamic unequal intervals. Linear regression adjusts the prediction interval to guarantee the confidence interval. However, it can predict the main trend of the traffic flow but cannot be a detailed expression, in which the SSE of the linear regression in limited to 3.15 times of our proposed algorithm's performance. Kernel regression uses KNN (k-nearest neighbor) to train the parameter, which presents the best MRE in three comparing algorithms. It might outperform our proposed algorithm when it increases the number of iterations. However, the increment of training costs more computing time. Our proposed algorithm's computing time is reduced by 74.4%. To obtain a better performance, we used an 8-degree Fourier model. The gap between our proposed algorithm and Fourier remains 22.4% because Fourier cannot detect the abnormal points of a traffic flow signal. Furthermore, as LSTM needs parts of an observation as the input training data, the prediction performance comparison between our proposed algorithm and LSTM is separated as Figure 10. Train degree relates to the input training ratio of the observation, which means that the higher the train degree is, the more observation is used as training data. To be consistent with LSTM, our proposed algorithm selects eight different wavelet scales. Figure 10a shows an MRE comparison between the two algorithms. The MRE of both algorithms is limited to decrement and stability when it increases the training degree and wavelet frequency. The average MREs of both algorithms are similar, which are 0.093 and 0.096, respectively. However, the trend of MRE decrement of our proposed algorithm outperforms LSTM. LSTM has a tiny increment in the 5 th and 6 th degrees because of the limitation of iterations. Our proposed algorithm can maintain a stable trend via the determination of singular points. Figure 10b shows a normalized computing time comparison between the two algorithms. Because of the omission of repeat and iteration, our proposed algorithm saves 59.2% computing time compared with LSTM. Moreover, our proposed algorithm's increase in computing time is also better than LSTM. The growth of computing time becomes sharp when the wavelet scale frequency rises relatively. Figure 10c shows the traffic prediction performance between two algorithms.