# Unequal Interval Dynamic Traffic Flow Prediction with Singular Point Detection
## 5 Multiscale Traffic Flow Prediction via Vasicek Model
### Unequal Interval Dynamic Traffic Flow Prediction Design Based on Vasicek Model  
The Vasicek model is a mathematical model used to describe the evolution of interest rates in the financial field [33]. It is a single-factor short-term interest rate model that describes the interest rate change in the case of only one source of market risk. The general model denotes  
\[dr_{t}=l(b-r_{t})dt+\sigma dW_{t}. \tag{15}\]  
Here, \(W_{t}\) denotes the Wiener process under the risk-neutral framework. \(\sigma\) denotes the parameter of a standard deviation, which influences the fluctuation of the interest. \(b\) denotes the long-term mean value, which produces a series of orbital values. \(l\) presents the return rate, and \(r_{t}\) is the momentary fluctuation that measures the amplitude of random factors in the system at each time point. Because the short-term prediction of a traffic flow variation trend also has a similar slope fluctuation, [PERSON] et al., predicted the short-term traffic flow variation trend by the Vasicek model via a stochastic differential equation [26]. Considering the singular point detection and prediction in Section 4, the model's elements denote unequal size. The first term in Equation (15) represents the average trend of the predicted traffic flow, and the second term denotes the random disturbance to the predicted traffic flow. It can be assumed that the trend of the traffic flow in the current interval is relatively stable except for random disturbance. Additionally, the parameters of the differential equation of traffic flow in this interval remain unchanged until the next singular point of the traffic flow occurs. The parameters of the prediction model should be recalculated once the next interval generation is divided via a singular point. Our proposed model can not only consider the accuracy of the traffic flow prediction but also reduce the times of reprediction that saves the computing resource in the system. Further, the prediction is a dynamic unequal interval according to the position of traffic flow singular points. The relationship between time and the traffic flow can be described via the Vasicek model as  
\[dX_{t}=(\alpha_{0}(t)+\alpha_{1}(t)X_{t})dt+\beta_{0}dW_{t}. \tag{16}\]  
Here, \(X_{t}\) denotes the observed process of the real-time traffic flow. Both \(\alpha_{0}(t)\) and \(\alpha_{1}(t)\) are the parameters that relate to the trend of the traffic flow. The parameters depend on the time and present the traffic flow's slope and amplitude. \(W_{t}\) denotes a random process that disturbs the traffic flow and \(\beta_{0}\) denotes its system factor. Figure 3 shows the definition and division of the observation process, in which slots are unequal and nonuniform intervals. The dynamic division for the interval depends on the position of the detected and predicted singular points. The observation process is divided into \(i+1\) intervals, which are \(i\) intervals by detected singular points and the last predicted slot. The traffic flow in each interval is sampled without loss of authenticity. For the interval \(TS_{k}\), \(k\in\{1,2,\ldots,i\}\), the length of the sampled traffic flow is \(\triangle t_{k}\). Additionally, the increment of the traffic flow in \(\triangle t_{k}\) denotes \(\triangle_{k}X\), which is described as  
\[\triangle_{k}X=(\alpha_{0}(k)+\alpha_{1}(k)X_{k})\triangle_{k}t+\beta_{0}(k) \triangle_{k}W. \tag{17}\]  
Figure 3: Illustration of dynamic interval division. The last term in Equation (17) is regarded as the random disturbance, and Equation (17) can be described as  
\[\frac{\triangle_{k}X}{\triangle_{k}t}=(\alpha_{0}(k)+\alpha_{1}(k)X_{k})+\zeta_{ k}. \tag{18}\]  
The increment of the traffic flow in the interval \(i+1\) can be attained when the parameters \(\alpha_{0}(k)\) and \(\alpha_{1}(k)\) are determined. Then the system predicts the traffic flow from the current time to the next predicted singular point.