# Traffic Demand Estimations Considering Route Trajectory Reconstruction in Congested Networks
## 4 Solving Algorithm
## 5 Numerical Experiment

### Nguyen-Dupuis Network

Topology structures, road properties and traffic demand of the Nguyen-Dupuis network [23] are shown in Figure 2. Traffic demand in the figure is \(d^{\omega}=k^{\omega}U\), and traffic operation status of 60 time windows within two hours is observed using Vissim simulation technology, where traffic demand in the first half hour and the last half hour is set as \(1/2d^{\omega}\), and traffic demand in the middle hour is set as \(d^{\omega}\). Each time window captured 50 groups of link flow, 30 groups of link travel time, and 20 groups of route travel time with unknown trajectories. The congested links are set as 5-6, 6-7, 10-11, and 11-2, and the maximum driving speed \(S_{f,\mu}\) of each link is 40. According to the Green Shields model, the link length and value of congestion density in Equation (1) are: (free flow travel time \(\times\) maximum driving speed) and (4 \(\times\) road capacity/maximum driving speed), respectively. Traffic demand variance is \(\sigma_{U}^{2}=10\). The initial demand weight matrix is \(\mathbf{k}^{(0)}=[6,10,8,6]\). Random error parameters are not considered. The discrete parameter of travel perception error is set as \(\theta=1\). The learning process of the Bayesian traffic demand estimation model solution in a congested environment was tested. Figure 3 depicts the distribution status of traffic demand in the 10 th, 20 th, 30 th, 40 th, 50 th, and 60 th time windows, respectively, where the abscissa is the number of the OD demand and the ordinate is the probability that the OD demand is the corresponding value. Different colors and lines represent testing different

Figure 2: Topology, link characteristics, and OD demands of the Nguyen–Dupuis network. time windows (see Figure 3b). Note that the predicted traffic demand of the first and last half hours (the 10 th, 50 th, and 60 th time windows) is lower, and the traffic demand of the middle hour (the 20 th, 30 th, and 40 th time windows) is higher. In the 20 th time window, the variance in traffic demand is relatively large, because the traffic demand has just changed from \(1/2d^{\omega}\) to \(d^{\omega}\), and observed data makes a significant difference. After traffic demand stabilizes, its variance is low (the 60 th period, for example). Figure 3 also shows that the proposed model can reflect the change state of real-time traffic demand. Bayesian traffic demand estimation models for estimated errors under cases when traffic congestion is and is not considered are compared in Figure 4. Figure 4a depicts errors in traffic demand estimations in the 30 th time window, and Figure 4b is the probability distribution of errors in traffic demand estimations in all 60 time windows. The ordinates in the graph are the differences between estimated traffic demand and actual traffic demand. Figure 4a illustrates that traffic demand estimations considering congestion are significantly superior to estimations ignoring congestion. Figure 4b illustrates that errors in traffic demand estimation without considering network congestion are within \(\pm 12\), and that the average error in traffic demand estimations considering congestion is near zero. This is because traffic demand estimations ignoring congestion are based on monotonous road travel time increases with link flow, which violates actual traffic flow characteristics. To illustrate the role of unknown route trajectory data in traffic demand estimations, root mean square errors of traffic demand with different observed data are shown in

Figure 4: Comparison of traffic demand estimation errors when traffic congestion is and is not considered. (**a**) Traffic demand errors in the 30 th time window. (**b**) Probability distribution of traffic demand errors in all time periods. Figure 3: Probability distribution of Bayesian learning-based OD demands under different time intervals. (**a**) OD pair 1-2. (**b**) OD pair 1-3.