# Unequal Interval Dynamic Traffic Flow Prediction with Singular Point Detection
## 3 System Model  
This section presents an overview of a system model, including the framework and process of the proposed dynamic prediction for the system model. Notations in this paper are described in Table 1.  
Figure 1 shows the main framework for the unabridged dynamic unequal interval of a traffic flow prediction process. The entire process involves three categories of data, i.e., observed data, training data, and predicted results. Observed data are the real-time traffic flow data that are acquired via the VANETs based on our previous work, which depends on the vehicle-to-vehicle (V2V) communication [23], vehicle-to-roadside unit (V2R) communication [24], and roadside-unit-to-roadside unit (R2R) information sharing mechanism [25]. Real-time traffic information can be acquired and stored in the roadside unit (RSU). The obtained real-time traffic flow information is a continuous-time process and will be used to predict the traffic flow trend in a later specific duration. The continuous and incomplete real-time traffic flow information is also called observation value in the following discussion [26]. The complete historical traffic data of the same road segment on different days are used to analyze the singular point distribution of the real-time observed traffic flow signal. The system forecasts the next singular point position of the observation traffic flow sequence via the historical data analysis results. Therefore, multigroups of historical data are also called training data in the paper, which are used for predicting the next singular point of the observed traffic flow signal.  
The main idea of the proposed prediction framework is to analyze the singular points in a real-time traffic flow signal obtained by a wavelet transform. Singular points can be used to divide the observed traffic flow signal into several unequal slots. Meanwhile, the historical traffic information is used as the training data and processed via the same scale wavelet transform as observed data. Considering the distribution of singular points between the historical data and observed information, the position of the next singular point can be predicted via a similarity measurement with a weighted parameter and correction factor. Then the dynamic intervals of the observed traffic flow signal can be determined, including the analyzed singular points and the predicted next singular point. Finally, the system uses a Vasicek model to predict the traffic flow's trend from the current moment  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(a\) & scale parameter of wavelet \\ \(x(t)\) & original traffic flow signal \\ \(\varphi_{a}(t)\) & wavelet signal with scale \(a\) \\ \(T_{i}\) & \(i\)th singular point’s position in real-time traffic data \\ \(H(i)\) & \(i\)th singular point’s position in historical traffic data \\ \(w_{k}\) & weighed parameter of \(k\)th historical traffic data \\ \(\varepsilon_{k}\) & relative error of the \(k\)th historical traffic data \\ \(\lambda_{k}\) & antirandom interference factor of the \(k\)th historical traffic data \\ \(\eta_{k}\) & correction factor of the \(k\)th historical traffic data \\ \(X_{t}\) & current observation process \\ \(\overline{X_{i}}(i+1)\) & traffic flow prediction results from \(t\) to \(T_{i+1}\) \\ \(W_{t}\) & stochastic disturbance in the traffic flow process \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the main mathematical notations.  
to the predicted next singular point. According to a comparison between predicted and observed traffic flow results, the system corrects and adjusts the parameters of the model to improve the accuracy. The process is updated and looped.