Bayesian Inference on Dynamic Linear Models of Day-to-Day Origin-Destination Flows in Transportation Networks
Abstract
:1. Introduction
2. Modeling Framework
2.1. A Dynamic Linear Model for Day-to-Day OD Flows
2.1.1. Model Definition
2.1.2. Estimation of Mean OD Flows
- Starting from , set ;
- For to T, do:
- (a)
- (b)
- Compute the assignment matrix by means of Equation (7).
- (c)
- (d)
- (e)
2.2. Bayesian Inference on Route Choice Parameters
2.2.1. Utility Model and Route Choice Probabilities
2.2.2. An MCMC Algorithm
- (backward sampling) Starting from , for , sample each backwards from the conditionals , in which:
- Initialize vectors and .
- From iteration onwards, repeat until convergence:
3. Results and Discussion
3.1. Generation of Simulated Data
- At , set values for , , , , , , for to T and past route costs , where r is the size of users’ memory and denotes route costs computed assuming free flow in the network.
- For to T do:
3.2. Application of the MCMC Algorithm
3.3. Unknown Evolution Matrix
3.4. Observation of Traffic Volumes on Partial Links
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Vector of mean OD flows at time t; | |
Vector of realized origin–destination (OD) flows at time t; | |
Vector of route flows at time t; | |
Vector of traffic volumes on links at time t; | |
Vector of route choice probabilities at time t; | |
Vector of route costs at time t; | |
Set of routes for OD pair i; | |
Vector of parameters in the utility equation of a route given past route costs; | |
Probability of not using a route in a given route choice set; | |
Performance (cost) function of a link; | |
Vector of random errors in the observational model at time t; | |
Vector of evolution errors in the as the dynamic model at time t; | |
State evolution matrix (system matrix); | |
Assignment matrix at time t; | |
Evolution covariance matrix of mean OD flows at time t; | |
Covariance matrix of traffic volumes on links at time t; | |
A covariance matrix at time t; | |
Route choice matrix at time t; | |
A link-path incidence matrix; | |
v | Location vector of the prior distribution of mean OD flows at time t; |
Covariance matrix of the prior distribution of mean OD flows at time t; | |
Location vector of the posterior distribution of mean OD flows at time t; | |
Covariance matrix of the posterior distribution of mean OD flows at time t; | |
The identity matrix; | |
A matrix whose main diagonal is the vector and entries off the main diagonal are all zero; | |
A block-diagonal matrix composed of submatrices ; | |
Multivariate normal distribution with mean vector and covariance matrix ; | |
Multinomial distribution with parameters x and . |
References
- De Dios Ortúzar, J.; Willumsen, L. Modelling Transport, 4th ed.; Wiley: Chichester, UK, 2011. [Google Scholar]
- Cascetta, E. Transportation Systems Analysis: Models and Applications, 2nd ed.; Springer: New York, NY, USA, 2009. [Google Scholar]
- Robillard, P. Estimating the OD matrix from observed link volumes. Transp. Res. 1975, 9, 123–128. [Google Scholar] [CrossRef]
- Nguyen, S. Estimating an OD Matrix from Network Data: A Network Equilibrium Approach; Technical Report; Centre de Recherche sur les Transports, Université de Montreal: Montreal, QC, Canada, 1977. [Google Scholar]
- Beckmann, M.; McGuire, C.; Winsten, C.B. Studies in Economics of Transportation; Yale University Press: Ann Arbor, MI, USA, 1956. [Google Scholar]
- Van Zuylen, H.J.; Willumsen, L.G. The most likely trip matrix estimated from traffic counts. Transp. Res. Part B 1980, 14, 281–293. [Google Scholar] [CrossRef] [Green Version]
- Cascetta, E. Estimation of Trip Matrices from Traffic Counts and Survey Data: A Generalized Least Squares Estimator. Transp. Res. Part B 1984, 16, 289–299. [Google Scholar] [CrossRef]
- Cascetta, E.; Nguyen, S. A unified framework for estimating or updating origin/destination matrices from traffic counts. Transp. Res. Part B 1988, 22B, 437–455. [Google Scholar] [CrossRef]
- Brenninger-Göthe, M.; Jörnsten, K.O. Estimation of origin–destination matrices from traffic counts using multiobjective programming formulations. Transp. Res. Part B 1989, 23B, 257–269. [Google Scholar] [CrossRef]
- Fisk, C. On combining maximum entropy trip matrix estimation with user optimal assignment. Transp. Res. Part B 1988, 22B, 69–79. [Google Scholar] [CrossRef]
- Fisk, C. Trip matrix estimation from link counts: the congested network case. Transp. Res. Part B 1989, 23B, 331–336. [Google Scholar] [CrossRef]
- Wardrop, J.G. Some theoretical aspects of traffic research. In Proceedings of the Institution of Civil Engineers Part II; Institution of Civil Engineers: London, UK, 1952; Volume 1, pp. 325–378. [Google Scholar]
- Smith, M. The existence, uniqueness and stability of traffic equilibria. Transp. Res. Part B 1979, 13B, 295–304. [Google Scholar] [CrossRef]
- Yang, H.; Sasaki, T.; Iida, Y.; Asakura, Y. Estimation of origin–destination matrices from link counts on congested networks. Transp. Res. Part B 1992, 26B, 417–434. [Google Scholar] [CrossRef]
- Yang, H. Heuristic algorithms for the bilevel origin–destination matrix estimation problem. Transp. Res. Part B 1995, 29B, 231–242. [Google Scholar] [CrossRef]
- Cascetta, E.; Postorino, N. Fixed point approaches to the estimation of O/D matrices using traffic counts on congested networks. Transp. Sci. 2001, 35, 134–147. [Google Scholar] [CrossRef]
- Sheffi, Y. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods; Prentice-Hall: Englewood Cliffs, NJ, USA, 1985. [Google Scholar]
- Vardi, Y. Network tomography: Estimating source-destination traffic intensities from link data. J. Am. Stat. Assoc. 1996, 91, 365–377. [Google Scholar] [CrossRef]
- Tebaldi, C.; West, M. Bayesian inference on network traffic using link count data. J. Am. Stat. Assoc. 1998, 93, 557–573. [Google Scholar] [CrossRef]
- Hazelton, M.L. Estimation of origin–destination matrices from link flows on uncongested networks. Transp. Res. Part B 2000, 34, 549–566. [Google Scholar] [CrossRef]
- Hazelton, M.L. Some comments on origin–destination matrix estimation. Transp. Res. Part A 2003, 37, 811–822. [Google Scholar] [CrossRef]
- Hazelton, M.L. Inference for origin–destination matrices: Estimation, prediction and reconstruction. Transp. Res. Part B 2001, 35, 667–676. [Google Scholar] [CrossRef]
- Cremer, M.; Keller, H. A new class of dynamic methods for the identification of origin–destination flows. Transp. Res. Part B 1987, 21, 117–132. [Google Scholar] [CrossRef]
- Cascetta, E.; Inaudi, D.; Marquis, G. Dynamic estimators of origin–destination matrices using traffic counts. Transp. Sci. 1993, 27, 363–373. [Google Scholar] [CrossRef]
- Ashok, K.; Ben-Akiva, M.E. Estimation and prediction of time-dependent origin–destination flows with a stochastic mapping to path flows and link flows. Transp. Sci. 2002, 36, 184–198. [Google Scholar] [CrossRef]
- Hazelton, M.L. Statistical inference for time varying origin–destination matrices. Transp. Res. Part B 2008, 42, 542–552. [Google Scholar] [CrossRef]
- Parry, K.; Hazelton, M. Bayesian inference for day-to-day dynamic traffic models. Transp. Res. Part B 2013, 50, 104–115. [Google Scholar] [CrossRef]
- Hazelton, M.L.; Parry, K. Statistical methods for comparison of day-to-day traffic models. Special issue: Day-to-Day Dynamics in Transportation Networks. Transp. Res. Part B: Methodol. 2016, 92, 22–34. [Google Scholar] [CrossRef]
- Watling, D.P.; Cantarella, G.E. Modelling sources of variation in transportation systems: Theoretical foundations of day-to-day dynamic models. Transp. B Transp. Dyn. 2013, 1, 3–32. [Google Scholar] [CrossRef]
- Cantarella, G.E.; Watling, D.P. Modelling road traffic assignment as a day-to-day dynamic, deterministic process: A unified approach to discrete- and continuous-time models. EURO J. Transp. Logist. 2015, 5, 69–98. [Google Scholar] [CrossRef]
- West, M.; Harrison, J. Bayesian Forecasting and Dynamic Models, 2nd ed.; Springer: New York, NY, USA, 1997. [Google Scholar]
- Särkkä, S. Bayesian Filtering and Smoothing; Cambridge University Press: New York, NY, USA, 2013. [Google Scholar]
- Ben-Akiva, M.; Lerman, S.R. Discrete Choice Analysis: Theory and Application to Travel Demand; MIT Press: Cambridge, MA, USA, 1985. [Google Scholar]
- Geman, S.; Geman, D. Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Trans. Pattern Anal. Mach. Intell. 1984, PAMI-6, 721–741. [Google Scholar] [CrossRef]
- Carter, C.K.; Kohn, R. On Gibbs Sampling for State Space Models. Biometrika 1994, 81, 541–553. [Google Scholar] [CrossRef]
- Frühwirth-Schnatter, S. Data augmentation and dynamic linear models. J. Time Ser. Anal. 1994, 15, 183–202. [Google Scholar] [CrossRef]
- Hastings, W.K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 1970, 57, 97–109. [Google Scholar] [CrossRef]
- Bureau of Public Roads. Traffic Assignment Manual for Application with a Large, High Speed Computer; U.S. Departmnet of Commerce, Bureau of Public Roads, Office of Planning, Urban Planning Division: Washington, DC, USA, 1964. [Google Scholar]
Link | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
CL | 0.5500 | 0.8940 | 0.2482 | 0.2535 | 0.6378 | 0.6473 | 0.2481 | 0.2542 | 0.8914 | 0.5685 |
HPD | HPD | MSE (OD Flows) | |||
---|---|---|---|---|---|
0.7 | 0.4941 | [0.3265, 0.6659] | 0.3340 | [0.1475, 0.5007] | 137.27 |
0.8 | 0.4960 | [0.3178, 0.6600] | 0.3380 | [0.1674, 0.5288] | 82.84 |
0.9 | 0.4898 | [0.3121, 0.6606] | 0.3313 | [0.1480, 0.5057] | 33.07 |
Observed Links | HPD | HPD | MSE (OD Flows) | ||
---|---|---|---|---|---|
1 | 0.6088 | [0.0065, 1.1342] | 0.3904 | [0.0014, 0.8312] | 471.67 |
2 | 0.6283 | [0.3897, 0.8247] | 0.4707 | [0.2509, 0.6827] | 256.53 |
9 | 0.5877 | [0.3697, 0.7972] | 0.3644 | [0.1608, 0.5752] | 233.34 |
2 and 5 | 0.5115 | [0.3194, 0.6817] | 0.3681 | [0.1692, 0.5422] | 246.70 |
1 and 9 | 0.5728 | [0.3608, 0.7456] | 0.3640 | [0.1760, 0.5546] | 175.43 |
2, 5 and 9 | 0.5023 | [0.3241, 0.6838] | 0.3323 | [0.1383, 0.5039] | 86.51 |
1, 7 and 9 | 0.5933 | [0.3947, 0.7936] | 0.3676 | [0.1870, 0.5870] | 59.82 |
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Pitombeira-Neto, A.R.; Loureiro, C.F.G.; Carvalho, L.E. Bayesian Inference on Dynamic Linear Models of Day-to-Day Origin-Destination Flows in Transportation Networks. Urban Sci. 2018, 2, 117. https://doi.org/10.3390/urbansci2040117
Pitombeira-Neto AR, Loureiro CFG, Carvalho LE. Bayesian Inference on Dynamic Linear Models of Day-to-Day Origin-Destination Flows in Transportation Networks. Urban Science. 2018; 2(4):117. https://doi.org/10.3390/urbansci2040117
Chicago/Turabian StylePitombeira-Neto, Anselmo Ramalho, Carlos Felipe Grangeiro Loureiro, and Luis Eduardo Carvalho. 2018. "Bayesian Inference on Dynamic Linear Models of Day-to-Day Origin-Destination Flows in Transportation Networks" Urban Science 2, no. 4: 117. https://doi.org/10.3390/urbansci2040117