Abstract
The precise estimation of time-varying demand matrices using traffic data is an essential step for planning, scheduling, and evaluating advanced traffic management systems. This paper presents an innovative method, based on the least squares approach, to handle the inherent complexities of estimating the dynamic characteristics of changing demand flow over time while considering congestion conditions. The time-dependent origin–destination (OD) demand matrices of the network are estimated by exploiting the received partial paths data from an automated vehicle identification system and vehicle counts data from loop detectors on a subset of the links. A traffic assignment approach based on partial paths is embedded into the measurement equations of the least squares model. For all time intervals, the relation between the variable aspects of congestion (the temporal and spatial distribution of the OD traffic flows) is established by their variance–covariance matrices. The LSQR algorithm, an iterative algorithm that is logically equivalent to the conjugate gradient method, is employed for solving the proposed least squares problem. Numerical examples are performed on three different approaches: utilizing only link counts data, utilizing only partial path flows data, and utilizing both of them. The results demonstrate that using variance–covariance matrices provides more precise estimates for time-dependent OD matrices. The effectiveness of the solution algorithm and the main ideas of the model are examined using the Sioux-Falls and Sodermalm networks. This paper reports the features of the discussed model based on different data as a proof of concept that incorporating partial path flows significantly improves the results for solving time-dependent OD matrix estimation problems.
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References
Akçelik, R.: Travel time functions for transport planning purposes: davidson’s function, its time dependent form and an alternative travel time function. Aust. Road Res. 21(3), 49–59 (1991)
Antoniou, C., Ben-Akiva, M., Koutsopoulos, H.N.: Nonlinear Kalman filtering algorithms for on-line calibration of dynamic traffic assignment models. IEEE Trans. Intell. Transp. Syst. 8(4), 661–670 (2007)
Antoniou, C., Ben-Akiva, M., and Koutsopoulos, H. N.: Incorporating automated vehicle identification data into origin-destination estimation. Transp. Res. Rec. J. Transp. Res. Board, 1882(1), 37–44, (2004). https://doi.org/10.3141/1882-05
Antoniou, C., Balakrishna, R., Koutsopoulos, H. N.: A synthesis of emerging data collection technologies and their impact on traffic management applications. Eur. Transp. Res. Rev. / Trasporti Europei, 3(3), 139–148, (2011). https://doi.org/10.1007/s12544-011-0058-1
Asakura, Y., Hato, E., Kashiwadani, M.: Origin-destination matrices estimation model using automatic vehicle identification data and its application to the Han-Shin expressway network. Transp. 27(4), 419–438 (2000). https://doi.org/10.1023/A:1005239823771
Ashok, K.: Estimation and prediction of time-dependent origin-destination flows. Ph.D. Thesis, Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, Cambridge, USA, (1996). http://dspace.mit.edu/
Balakrishna, R., Ben-Akiva, M., Koutsopoulos, H.N.: Time-dependent origin-destination estimation without assignment matrices. Transp. Simul. Beyond Tradit. Approaches (2019). https://doi.org/10.1201/9780429093258-12
Barceló, J., Montero, L., Bullejos, M., Serch, O., Carmona, C.: A kalman filter approach for exploiting bluetooth traffic data when estimating time-dependent od matrices. J. Intell. Transp. Syst. 17(2), 123–141 (2013). https://doi.org/10.1080/15472450.2013.764793
Bell, M.G.: The estimation of origin-destination matrices by constrained generalised least squares. Transp. Res. Part B Methodol. 25(1), 13–22 (1991). https://doi.org/10.1016/0191-2615(91)90010-g
Bera, S., and Rao, K.: Estimation of origin-destination matrix from traffic counts: the state of the art. Eur. Transp. Res. Rev. / Trasporti Europei, 49(3–23, (2011).
Bert, E.: Dynamic Urban Origin-Destination Matrix Estimation Methodology. Ph.D. Thesis, Faculté Environnment Naturel, Architectural et Construit, École Polytechnique Fédérale de Lausanne, Laussane, Switzerland, (2010). https://infoscience.epfl.ch
Bierlaire, M., Crittin, F.: An efficient algorithm for real-time estimation and prediction of dynamic OD tables. Oper. Res. 52(1), 116–127 (2004). https://doi.org/10.1287/opre.1030.0071
Bierlaire, M., Toint, P.L.: Meuse: an origin-destination matrix estimator that exploits structure. Transp. Res. Part B Methodol. 29(1), 47–60 (1995). https://doi.org/10.1016/0191-2615(94)00025-u
Bierlaire, M., Toint, P. L., and Tuyttens, D.: On iterative algorithms for linear least squares problems with bound constraints. Linear Algebra Appl., 143(111–143, (1991). https://doi.org/10.1016/0024-3795(91)90009-l
BPR. United States. Bureau of Public Roads: Traffic Assignment Manual. Department of Commerce, Urban Planning Division, Washington, D.C. USA, (1964).
Burghout, W., Koutsopoulos, H.N., Andreasson, I.: Incident management and traffic information: tools and methods for simulation-based traffic prediction. Transp. Res. Rec. 2161(1), 20–28 (2010). https://doi.org/10.1109/vnis.1991.205776
Burghout, W., Andreasson, I., Koutsopoulos, H. N.: A discrete-event mesoscopic traffic simulation model for hybrid traffic simulation. In: 9th IEEE Intelligent Transportation Systems Conference (ITSC 2006), Toronto, Canada, pp. 1102–1107 (2006). https://doi.org/10.1109/itsc.2006.1707369
Burghout, W.: Hybrid microscopic-mesoscopic traffic simulation. doctoral dissertation, KTH Royal Institute of Technology, Dept. Infrastructure, Div. Transport planning, Stockholm, Sweden (2004). https://www.kth.se
Caggiani, L., Ottomanelli, M., Sassanelli, D.: A fixed point approach to origin–destination matrices estimation using uncertain data and fuzzy programming on congested networks. Transp. Res. Part C Emerging Technol. 28, 130–141 (2013). https://doi.org/10.1016/j.trc.2010.12.005
Cantelmo, G., Cipriani, E., Gemma, A., Nigro, M.: An adaptive bi-level gradient procedure for the estimation of dynamic traffic demand. IEEE Trans. Intell. Transp. Syst. 15(3), 1348–1361 (2014). https://doi.org/10.1109/tits.2014.2299734
Cao, Y., Tang, K., Sun, J., and Ji, Y.: Day-to-day dynamic origin–destination flow estimation using connected vehicle trajectories and automatic vehicle identification data. Transp. Res. Part C Emerging Technol., 129(2), 103241, (2021). https://doi.org/10.1016/j.trc.2021.103241
Carrese, S., Cipriani, E., Mannini, L., and Nigro, M.: Dynamic demand estimation and prediction for traffic urban networks adopting new data sources. Transp. Res. Part C Emerging Technol., 81(83–98, (2017). https://doi.org/10.1016/j.trc.2017.05.013
Cascetta, E.: Estimation of trip matrices from traffic counts and survey data: A generalized least squares estimator. Transp. Res. Part B Methodol. 18(4–5), 289–299 (1984). https://doi.org/10.1016/0191-2615(84)90012-2
Cascetta, E., Nguyen, S.: A unified framework for estimating or updating origin/destination matrices from traffic counts. Transp. Res. Part B Methodol. 22(6), 437–455 (1988). https://doi.org/10.1016/0191-2615(88)90024-0
Cascetta, E., Inaudi, D., Marquis, G.: Dynamic estimators of origin-destination matrices using traffic counts. Transp. Sci. 27(4), 363–373 (1993). https://doi.org/10.1287/trsc.27.4.363
Cascetta, E.: Transportation Systems Analysis: Models and Applications (2nd Edition). Springer, Berlin (2009). https://doi.org/10.1007/978-0-387-75857-2
Cascetta, E.: Transportation Systems Engineering: Theory and Methods. Springer, Berlin (2013). https://doi.org/10.1007/978-1-4757-6873-2
Castillo, E., Menéndez, J.M., Jiménez, P.: Trip matrix and path flow reconstruction and estimation based on plate scanning and link observations. Transp. Res. Part B Methodol. 42(5), 455–481 (2008). https://doi.org/10.1016/j.trb.2007.09.004
Castillo, E., Jiménez, P., Menéndez, J.M., Nogal, M.: A Bayesian method for estimating traffic flows based on plate scanning. Transp. 40(1), 173–201 (2013). https://doi.org/10.1007/s11116-012-9443-4
Cipriani, E., Florian, M., Mahut, M., Nigro, M.: A gradient approximation approach for adjusting temporal origin–destination matrices. Transp. Res. Part C Emerging Technol. 19(2), 270–282 (2011). https://doi.org/10.1016/j.trc.2010.05.013
Dantsuji, T., Hoang, N.H., Zheng, N., Vu, H.L.: A novel metamodel-based framework for large-scale dynamic origin–destination demand calibration. Transp. Res. Part C Emerging Technol. 136, 103545 (2022). https://doi.org/10.1016/j.trc.2021.103545
Del Castillo, J., Benitez, F.: On the functional form of the speed-density relationship-I: general theory. Transp. Res. Part B Methodol. 29(5), 373–389 (1995). https://doi.org/10.1016/0191-2615(95)00008-2
Dey, S., Winter, S., Tomko, M.: Origin-destination flow estimation from link count data only. Sensors 20(18), 5226 (2020). https://doi.org/10.3390/s20185226
Dios Ortúzar, J., Willumsen, L. G.: Modelling Transport (4th Edn.). Wiley, Chichester, (2011). https://doi.org/10.1002/9781119993308
Doblas, J., Benitez, F.G.: An approach to estimating and updating origin–destination matrices based upon traffic counts preserving the prior structure of a survey matrix. Transp. Res. Part B Methodol. 39(7), 565–591 (2005). https://doi.org/10.1016/j.trb.2004.06.006
El-Assi, W., Morency, C., Miller, E.J., Habib, K.N.: Investigating the capacity of continuous household travel surveys in capturing the temporal rhythms of travel demand. Transportation 47(4), 1787–1808 (2020). https://doi.org/10.1007/s11116-019-09981-x
Florian, M., Chen, Y.: A Coordinate descent method for the bi-level OD matrix adjustment problem. Int. Trans. Oper. Res. 2(2), 165–179 (1995). https://doi.org/10.1016/0969-6016(95)00001-n
Flötteröd, G., Liu, R.: Disaggregate path flow estimation in an iterated dynamic traffic assignment microsimulation. J. Intell. Transp. Syst. 18(2), 204–214 (2014). https://doi.org/10.1080/15472450.2013.806854
Frederix, R., Viti, F., Tampère, C.M.: Dynamic origin–destination estimation in congested networks: theoretical findings and implications in practice. Transportmetrica Transp. Sci/ 9(6), 494–513 (2013). https://doi.org/10.1080/18128602.2011.619587
Frederix, R., Viti, F., Corthout, R., Tampère, C. M.: New gradient approximation method for dynamic origin–destination matrix estimation on congested networks. Transp. Res. Rec. J. Transp. Res. Board, 2263(1), 19–25 (2011). https://doi.org/10.3141/2263-03
Gentile, G., Meschini, L., Papola, N.: Spillback congestion in dynamic traffic assignment: a macroscopic flow model with time-varying bottlenecks. Transp. Res. Part B Methodol. 41(10), 1114–1138 (2007). https://doi.org/10.1016/j.trb.2007.04.011
Gentili, M., Mirchandani, P.B.: Locating sensors on traffic networks: models, challenges and research opportunities. Transp. Res. Part C Emerging Technol. 24, 227–255 (2012). https://doi.org/10.1016/j.trc.2012.01.004
Hadavi, M., Shafahi, Y.: Vehicle identification sensor models for origin–destination estimation. Transp. Res. Part B Methodol. 89, 82–106 (2016). https://doi.org/10.1016/j.trb.2016.03.011
Hanke, M.: Conjugate Gradient Type Methods for Ill-posed Problems. Chapman and Hall/CRC, New York (2017)
Hazelton, M.L.: Inference for origin–destination matrices: estimation, prediction and reconstruction. Transp. Res. Part B Methodol. 35(7), 667–676 (2001). https://doi.org/10.1016/s0191-2615(00)00009-6
Huang, S., Sadek, A.W., Guo, L.: Computational-based approach to estimating travel demand in large-scale microscopic traffic simulation models. J. Comput. Civ. Eng. 27(1), 78–86 (2013). https://doi.org/10.1061/(asce)cp.1943-5487.0000202
Kattan, L., and Abdulhai, B.: Noniterative approach to dynamic traffic origin–destination estimation with parallel evolutionary algorithms. Transp. Res. Rec. J. Transp. Res. Board, 1964(1), 201–210, (2006). https://doi.org/10.1177/0361198106196400122
Krishnakumari, P., Van Lint, H., Djukic, T., Cats, O.: A data driven method for OD matrix estimation. Transp. Res. Part C Emerging Technol. 113, 38–56 (2020). https://doi.org/10.1016/j.trc.2019.05.014
Lee, J.-B., and Ozbay, K.: New calibration methodology for microscopic traffic simulation using enhanced simultaneous perturbation stochastic approximation approach. Transp. Res. Rec. J. Transp. Res. Board, 2124(1), 233–240, (2009). https://doi.org/10.3141/2124-23
Lu, C.-C., Zhou, X., Zhang, K.: Dynamic origin–destination demand flow estimation under congested traffic conditions. Transp. Res. Part C Emerging Technol. 34, 16–37 (2013). https://doi.org/10.1016/j.trc.2013.05.006
Lu, L., Xu, Y., Antoniou, C., Ben-Akiva, M.: An enhanced SPSA algorithm for the calibration of dynamic traffic assignment models. Transp. Res. Part C Emerging Technol. 51, 149–166 (2015). https://doi.org/10.1016/j.trc.2014.11.006
Ma, W., and Qian, Z.: A Generalized single-level formulation for dynamic origin–destination estimation under stochastic user equilibrium. Transp. Res. Rec. J. Transp. Res. Board, 2672(48), 58–68, (2018). https://doi.org/10.3141/1882-05
Maher, M.J., Zhang, X., Van Vliet, D.: A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows. Transp. Res. Part B Methodol. 35(1), 23–40 (2001). https://doi.org/10.1016/s0191-2615(00)00017-5
Mo, B., Li, R., and Dai, J.: Estimating dynamic origin–destination demand: a hybrid framework using license plate recognition data. Comput.-Aided Civ. Infrastruct. Eng., 35(7), 734–752, (2020). https://doi.org/10.1111/mice.12526
Nigro, M., Cipriani, E., Abdelfatah, A., Colombaroni, C., Fusco, G., Gemma, A.: Dynamic O-D demand estimation: application of SPSA AD-PI method in conjunction with different assignment strategies. J. Adv. Transp. 2018, 1–18 (2018). https://doi.org/10.1155/2018/2085625
Osorio, C.: Dynamic origin-destination matrix calibration for large-scale network simulators. Transp. Res. Part C Emerging Technol. 98, 186–206 (2019a). https://doi.org/10.1016/j.trc.2018.09.023
Osorio, C.: High-dimensional offline origin-destination (OD) demand calibration for stochastic traffic simulators of large-scale road networks. Transp. Res. Part B Methodol. 124, 18–43 (2019b). https://doi.org/10.1016/j.trb.2019.01.005
Ou, J., Lu, J., **a, J., An, C., Lu, Z.: Learn, assign, and search: real-time estimation of dynamic origin-destination flows using machine learning algorithms. IEEE Access 7, 26967–26983 (2019). https://doi.org/10.1109/access.2019.2901289
Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software 8(1), 43–71 (1982). https://doi.org/10.1145/355984.355989
Parry, K., Hazelton, M.L.: Estimation of origin–destination matrices from link counts and sporadic routing data. Transp. Res. Part B Methodol. 46(1), 175–188 (2012). https://doi.org/10.1016/j.trb.2011.09.009
Perrakis, K., Karlis, D., Cools, M., Janssens, D., Vanhoof, K., Wets, G.: A Bayesian approach for modeling origin–destination matrices. Transp. Res. Part A Policy Pract. 46(1), 200–212 (2012). https://doi.org/10.1016/j.tra.2011.06.005
Pitombeira Neto, A. R., Oliveira Neto, F. M. d., and Loureiro, C. F. G.: Statistical models for the estimatio of the origin-destination matrix from traffic counts. Transportes, 25(4), 1–12, (2017). https://doi.org/10.14295/transportes.v25i4.1344
Prakash, A. A., Seshadri, R., Antoniou, C., Pereira, F. C., Ben-Akiva, M.: Improving scalability of generic online calibration for real-time dynamic traffic assignment systems. Transp. Res. Rec. J. Transp. Res. Board; 2672(48), 79–92 (2018). https://doi.org/10.1177/0361198118791360
Rao, W., Wu, Y.-J., **a, J., Ou, J., Kluger, R.: Origin-destination pattern estimation based on trajectory reconstruction using automatic license plate recognition data. Transp. Res. Part C Emerging Technol. 95, 29–46 (2018). https://doi.org/10.1016/j.trc.2018.07.002
Ros-Roca, X., Montero, L., Barceló, J.: Investigating the quality of Spiess-like and SPSA approaches for dynamic OD matrix estimation. Transportmetrica Transp. Sci. 17(3), 235–257 (2021). https://doi.org/10.1080/23249935.2020.1722282
Ros-Roca, X., Montero, L., Barceló, J., Nökel, K., Gentile, G.: A practical approach to assignment-free dynamic origin-destination matrix estimation problem. Transp. Res. Part C Emerging Technol. 134, 103477 (2022). https://doi.org/10.1016/j.trc.2021.103477
Rostami Nasab, M., Shafahi, Y.: Estimation of origin–destination matrices using link counts and partial path data. Transp. 47(6), 2923–2950 (2020). https://doi.org/10.1007/s11116-019-09999-1
Sadegh, P., Spall, J.C.: Optimal random perturbations for stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control 43(10), 1480–1484 (1998). https://doi.org/10.1109/acc.1997.609490
Scheffer, A., Cantelmo, G., Viti, F.: Generating macroscopic, purpose-dependent trips through Monte Carlo sampling techniques. Transp. Res. Procedia 27, 585–592 (2017). https://doi.org/10.1016/j.trpro.2017.12.111
Shafiei, S., Gu, Z., Saberi, M.: Calibration and validation of a simulation-based dynamic traffic assignment model for a large-scale congested network. Simul. Model. Pract. Theory 86, 169–186 (2018). https://doi.org/10.1016/j.simpat.2018.04.006
Spiess, H.: A maximum likelihood model for estimating origin-destination matrices. Transp. Res. Part B Methodol. 21(5), 395–412 (1987). https://doi.org/10.1016/0191-2615(87)90037-3
Spiess, H.: Conical volume-delay functions. Transp. Sci. 24(2), 153–158 (1990a). https://doi.org/10.1287/trsc.24.2.153
Spiess, H.: A Gradient Approach for the OD Matrix Adjustment Problem. Publication No. 693, 1–11, Centre for Research on Transportation, Université de Montréal, Canada (1990b).
Stathopoulos, A., Tsekeris, T.: Hybrid meta‐heuristic algorithm for the simultaneous optimization of the O–D trip matrix estimation. Comput.-Aided Civ. Infrastruct. Eng., 19(6), 421–435, (2004). https://doi.org/10.1111/j.1467-8667.2004.00367.x
Tang, K., Cao, Y., Chen, C., Yao, J., Tan, C., Sun, J.: Dynamic origin‐destination flow estimation using automatic vehicle identification data: A 3D convolutional neural network approach. Comput.-Aided Civ. Infrastruct. Eng., 36(1): 30–46, (2021). https://doi.org/10.1111/mice.12559
Tavana, H.: Internally consistent estimation of dynamic network origin-destination flows from intelligent transportation systems data using bi-level optimization. Ph.D. Dissertation, The University of Texas at Austin, (2001). https://repositories.lib.utexas.edu
Tebaldi, C., West, M.: Bayesian inference on network traffic using link count data. J. Am. Stat. Assoc. 93(442), 557–573 (1998). https://doi.org/10.1080/01621459.1998.10473707
Toledo, T., Kolechkina, T.: Estimation of dynamic origin–destination matrices using linear assignment matrix approximations. IEEE Trans. Intell. Transp. Syst. 14(2), 618–626 (2013). https://doi.org/10.1109/tits.2012.2226211
Tympakianaki, A., Koutsopoulos, H.N., Jenelius, E.: c-SPSA: cluster-wise simultaneous perturbation stochastic approximation algorithm and its application to dynamic origin–destination matrix estimation. Transp. Res. Part C Emerging Technol. 55, 231–245 (2015). https://doi.org/10.1016/j.trc.2015.01.016
Van Zuylen, H.J., Willumsen, L.G.: The most likely trip matrix estimated from traffic counts. Transp. Res. Part B Methodol. 14(3), 281–293 (1980). https://doi.org/10.1016/0191-2615(80)90008-9
Vasko, F.J., Lu, Y., Zyma, K.: What is the best greedy-like heuristic for the weighted set covering problem? Oper. Res. Lett. 44(3), 366–369 (2016). https://doi.org/10.1016/j.orl.2016.03.007
Wei, C., Asakura, Y.: A Bayesian approach to traffic estimation in stochastic user equilibrium networks. Transp. Res. Part C Emerging Technol. 36, 446–459 (2013). https://doi.org/10.1016/j.trc.2013.06.013
**e, C., Kockelman, K.M., Waller, S.T.: A maximum entropy-least squares estimator for elastic origin-destination trip matrix estimation. Transp. Res. Part B Methodol. 45(9), 1465–1482 (2011). https://doi.org/10.1016/j.trb.2011.05.018
Yang, H., Sasaki, T., Iida, Y., Asakura, Y.: Estimation of origin-destination matrices from link traffic counts on congested networks. Transp. Res. Part B Methodol. 26(6), 417–434 (1992). https://doi.org/10.1016/0191-2615(92)90008-k
Yang, X., Lu, Y., Hao, W.: Origin-destination estimation using probe vehicle trajectory and link counts. J. Adv. Transp. (2017). https://doi.org/10.1155/2017/4341532
Yazdi, P. T., Shafahi, Y.: Solving location problem for vehicle identification sensors to observe and estimate path flows in large-scale networks. In: European Conference on Modelling and Simulation, pp. 323–328 (2018). https://doi.org/10.7148/2018-0323
Zhang, C., Osorio, C.: Efficient Offline Calibration of Origin-Destination (Demand) for Large-Scale Stochastic Traffic Models. Massachusetts Institute of Technology (2017). https://web.mit.edu
Zhang, H., Seshadri, R., Prakash, A.A., Antoniou, C., Pereira, F.C., Ben-Akiva, M.: Improving the accuracy and efficiency of online calibration for simulation-based dynamic traffic assignment. Transp. Res. Part C Emerging Technol. 128, 103195 (2021). https://doi.org/10.1016/j.trc.2021.103195
Zhou, X., Mahmassani, H.S.: Dynamic origin-destination demand estimation using automatic vehicle identification data. IEEE Trans. Intell. Transp. Syst. 7(1), 105–114 (2006). https://doi.org/10.1109/tits.2006.869629
Zhou, X., Mahmassani, H.S.: A structural state space model for real-time traffic origin–destination demand estimation and prediction in a day-to-day learning framework. Transp. Res. Part B Methodol. 41(8), 823–840 (2007). https://doi.org/10.1016/j.trb.2007.02.004
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Vahidi, M., Shafahi, Y. Time-dependent estimation of origin–destination matrices using partial path data and link counts. Transportation (2023). https://doi.org/10.1007/s11116-023-10412-1
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DOI: https://doi.org/10.1007/s11116-023-10412-1