# Traffic Demand Estimations Considering Route Trajectory Reconstruction in Congested Networks
## 4 Solving Algorithm
\[\frac{\partial L}{\partial c_{k}^{\omega 2}}=0,\forall k\in K^{\omega},\omega\in W \tag{16}\]0 In Figure 5, we compare three cases: a trajectory reconstruction based on the Gaussian mixture model, a k-means clustering trajectory, and only integrating data with known trajectories. The root mean square error is defined as: \(RMSE=\|\mathbf{d}^{tru}-\mathbf{d}^{est}\|/\sqrt{|\omega|}\), where \(\mathbf{d}^{tru}\), \(\mathbf{d}^{est}\), and \(|\omega|\) represent exact traffic demand, estimated traffic demand, and potential traffic demand (number of set elements), respectively. Compared with only integrating data with known trajectories, the root mean square error of traffic demand estimation is smaller, but its variance fluctuates significantly. Compared with the k-means clustering trajectory method, variance in the root mean square error of traffic demand estimations based on trajectory reconstruction using the Gaussian mixture model is larger, but the mean error stabilizes at approximately 1.3. Therefore, trajectory reconstruction using the Gaussian mixture model is more accurate for traffic demand estimation than using k-means cluster trajectory or only integrating data with known trajectories. ### Sioux Falls Network

The Sioux Falls network [24] was used to further test the applicability of the designed algorithm. This road network consists of 24 nodes, 76 links, and 550 OD pairs. Characteristics of this road link are described in the literature [24]. Observed variables, link maximum driving speed, link length, and congestion density are consistent with the Nguyen-Dupuis network. The sensitivity of traffic demand to link observed data is shown in \[\frac{\partial L}{\partial c_{k}^{\omega 2}}=0,\forall k\in K^{\omega},\omega\in W \tag{16}\]1 Observed traffic flow and speed on links 16 and 28 are disturbed, respectively. From these three figures, it can be seen that when the disturbance of observed variables is 0%, estimated traffic demand is close to its actual value (i.e., 120, 160, and 180). When disturbance increases, estimated traffic demand gradually diverges from actual demand. In this experiment, OD pairs 2-15 and 3-20 are more sensitive to observed data on link 28, whereas OD pair 6-11 is more susceptible to observed data on link 16. The evolution process of the Bayesian traffic demand estimation algorithm with route trajectory reconstruction in congested networks was plotted, and is shown in \[\frac{\partial L}{\partial c_{k}^{\omega 2}}=0,\forall k\in K^{\omega},\omega\in W \tag{16}\]2 The lower part of the figure illustrates a convergence situation of root mean square error of the entire iterative algorithm; the upper three figures represent the convergence process of the Bayesian update strategy (upper model) in the second, third, and fourth iterations of the algorithm, respectively. In the calculation process, the designed iterative algorithm and the Bayesian successive update method in the upper-level model can converge to accuracies of 0.06 and 0.003, respectively. The root mean square error of traffic demand decreases steadily during each iteration, whereas the root mean square error does not continue to decrease when solving the upper-level model. However, with an increase in observed information, the method ensures that the overall trend of root mean square error decreases. This indicates that the algorithm is robust. ## 6 Conclusions

In this paper, the relationship between traffic variables and travel characteristics in congested network environments is analyzed, and the Bayesian traffic demand estimation model with route trajectory reconstruction in congested networks is established. Data analysis of traffic environments reveals that there is a non-monotone relationship between link travel time and traffic flow in congested networks. Therefore, the three-parameter relationship of congested network flow is analyzed to transform observed variables to a uniform format. The Gaussian mixture model is used to reconstruct observed route trajectories using location data of observed route trajectories.