# Traffic Demand Estimations Considering Route Trajectory Reconstruction in Congested Networks
## 4 Solving Algorithm
The Bayesian estimation method is used to estimate OD, to modify prior OD demand distribution using observed data, and to obtain the posterior OD demand distribution in the current moment. The upper-level model is a maximized posterior probability distribution model, and the lower-level model is a stochastic network-user equilibrium model. The upper-level model transmits traffic demand to the lower-level model, and the lower-level model transmits the OD-link ratio and road flow properties to the upper-level model. The Nguyen-Dupuis and Sioux Falls networks were used to test the properties of the proposed model and algorithm. The results show that traffic demand estimations that consider traffic congestion are significantly better than those that ignore traffic congestion, with traffic estimation errors of almost 0 and within \(\pm 12\), respectively. Compared with

\[\frac{\partial L}{\partial c_{k}^{\omega 2}}=0,\forall k\in K^{\omega},\omega\in W \tag{16}\]3 Traffic demands between different OD pairs under fluctuations of observed data on links 16 and 28. \[\frac{\partial L}{\partial c_{k}^{\omega 2}}=0,\forall k\in K^{\omega},\omega\in W \tag{16}\]4 Evolution processes of root mean square errors of traffic demands. only integrating known trajectory data, the mean of root mean square error of traffic demand estimated by integrating unknown route trajectory data is smaller. Compared with the k-means clustering trajectory method, the mean of root mean square error of traffic demand estimated using the Gaussian mixture model reconstruction trajectory is smaller and stabilizes at approximately 1.3. The designed iterative algorithm and the Bayesian successive update method in the upper-level model can converge to 0.06 and 0.003, respectively. The Bayesian traffic demand estimation model with route trajectory reconstruction in congested networks can be applied to urban traffic system optimization projects; a variety of traffic demand distribution forms are adopted to establish the traffic demand estimation model. Conceptualization, W.T., J.C. and C.S. ; methodology, W.T., C.S. and G.L. ; writing--original draft preparation, W.T., J.C., H.W. and C.S. ; writing--review and editing, W.T. and C.S. All authors have read and agreed to the published version of the manuscript. This study was supported by the Natural Science Fund for Colleges and Universities in Jiangsu Province under Grant 20 KJB580016, the General Program of Philosophy and Social Science Research in Jiangsu Universities under Grant 2020 SJA0133, the MOE of PRC Industry-University Collaborative Education Program (Program No. 202102055014, Kingfar-CES \"Human Factors and Economics\" Program), and the Science and Technology Innovation Fund for Youth Scientists of Nanjing Forestry University under Grant CX2018011. Not applicable. Data used to support the findings of this study are available from the corresponding author upon request. The authors declare no conflict of interest. ## References

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