# Unequal Interval Dynamic Traffic Flow Prediction with Singular Point Detection
## 6 Performance Evaluations
### Performance Evaluation of Singular Point Detection and Prediction by Multiscale Wavelet Transform  
Figure 5 shows the traffic flow in 2 h of the observation that is preprocessed via average calculation per minute. Figure 6 shows the wavelet transform results of the observation of Figure 5 based on the Haar wavelet, which contains three different scales \(a\) of the selected wavelet. Figure 6 provides the different singular point detection results in different wavelet scales.  
As shown in Figure 6, the positions of the max extreme points of the results are the traffic flow singular points because the system uses the first deviation of the low-pass function as the wavelet. The bottom subfigure detects the most singular points with the smallest scale. Additionally, the top of the subfigure detects the fewest singular points as its wavelet scale is the largest. Parts of the singular points are regarded as tiny noises and ignored. The numbers of singular points in three different wavelet scales are 18, 32, and 65. Moreover, all the singular points detected via a larger wavelet scale are included in the detected singular point set of a smaller wavelet scale. Therefore, it validated that our proposed algorithm can detect the traffic flow signal's singular points adaptively by the scale \(a\), and the intervals can be divided into different amounts and lengths.  
Since the detected singular points divide the process into unequal intervals, Figure 7 shows the results of the the next singular point prediction MRE in different types of wavelets. Haar-1 and Haar-6 share the same envelope of a continuous signal but the different sampled intervals of the discrete signal. The sampled interval of Haar-1 is larger than Haar-6's. It can be observed that both the Haar and Daubechies wavelets can predict the next singular point accurately and efficiently when the number of input-detected singular points is enough. The average MRE of the next singular point prediction is less than 20.0%. Additionally, in all wavelets, the MRE approaches stability when the number of input-detected singular points increases. This is because the smaller sampled interval leads to a better low-pass property of the wavelet. In the same type, the wavelet that has a smaller sampled interval outperforms the one with larger intervals. The next singular point prediction MRE of Daubechies-6 reduces by 83.6% compared with the MRE of Daubechies-1. Further, comparing Haar-1 and Haar-6 increases the accuracy of the next singular point prediction by 22.6%. The performance for the next singular point prediction by using a Haar  
Figure 5: Traffic flow of the observation process.  
Figure 6: Observation wavelet transform results and singular point detection with different scale parameters.  
wavelet is better than a Daubechies's. Specifically for the 10 th singular point prediction, the MRE of Haar is decreased, respectively, by 98.8% and 76.5% compared with Daubechies in the same sampled interval. It is because the low-pass performance of Haar's first derivative is better than Daubechies's. Especially, parts of the predicted singular points MRE increase, for instance, the 6 th and 10 th singular point predictions. It is caused by a traffic accident that never occurs in the training data. Therefore, the MRE is less than 15.0%, and increasing the number of training data can improve the performance.  
Overall, the evaluations verify that the proposed multiscale wavelet transform can detect and predict the singular points of the traffic flow signal accurately and efficiently. In the next subsection, the comprehensive traffic flow prediction of our proposed algorithm will be discussed.