Abstract
In this paper, a traffic system is specified as a vector autoregressive (VAR) model. Specifically, traffic flow is defined as a function of temporally and/or spatially lagged traffic flows in the system. Particular attention is paid to the origin-destination nature of traffic, differentiating between the impacts of upstream neighbours and downstream neighbours, as well as accounting for contemporaneous correlations within the system. A Bayesian method will be proposed to estimate this traffic flow model. As the number of equations corresponds to the number of traffic flows (N) in the system, and the number of potential predictors grows geometrically with N, this is a large VAR model. To deal with the issue of ‘overfitting’, knowledge of the spatial configuration of the transportation network will be used to impose zero-restrictions on the coefficient matrices. Moreover, Bayesian variable selection method will be implemented to judiciously select only the significant predictors for each traffic flow of the system. Specification of the priors and the MCMC sampling procedure will be discussed at length. A simulation study will be presented. It will be shown that the estimated posterior distribution over the model space corresponds closely to the true model, and the estimated marginal posterior distribution of the effects vector B centres around their ‘true’ values and largely avoids the problems of ‘overfitting’. It will be argued that this Bayesian VAR approach offers a flexible and powerful alternative to modelling traffic flow.
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References
Alvarez JL, Ballabriga FC (1994) BVAR models in the context of cointegration: A Monte Carlo experiment. Bank of Spain Working Paper No. 9405 (Madrid)
Berger JO, Pericchi L (2001) Objective Bayesian methods for model selection: introduction and comparison. In: Lahiri P (ed) Model selection. Institute of Mathematical Statistics, Hayward, pp 135–193
Bolduc D, Dagenais MG, Gaudry MJ (1989) Spatially autocorrelated errors in origin-destination models: A new specification applied to urban travel demand in Winnipeg. Transp Res B 23(5):361–372
Bolduc D, Laferriere R, Santarossa G (1992) Spatial autoregressive error components in travel flow models. Reg Sci Urban Econ 22:371–385
Brown PJ, Vannucci M (1998) Multivariate Bayesian variable selection and prediction. J R Stat Soc Ser B 60(3):627–641
Chen H, Schmeiser B (1993) Monte Carlo estimation for guaranteed-coverage nonnormal tolerance intervals. Proceedings of the winter simulation conference, 509–515
Cui W, George EI (2008) Empirical Bayes vs fully Bayes variable. J Stat Plan Inference 138:888–900
Deng M, Athanasopoulos G (2010) Modelling Australian Domestic and International Inbound Travel: a spatial-temporal approach. In: Tourism management, vol 32(5). Pergamon, Oxford, pp 1075–1084
Ding Q, Wang X, Zhang X, Sun Z (2011) Forecasting traffic volume with space-time ARIMA model. Adv Mater Res 156–157:979–983
Doan T, Litterman R, Sims C (1984) Forecasting and conditional projections using realist prior distributions. Econ Rev 3(1):1–100
George EI, McCulloch RE (1993) Variable selection via Gibbs sampling. J Am Stat Assoc 88:881–889
Giacomini R, Granger, Clive WJ (2004) Aggregation of space-time processes. J Econ 118(1/2):7–26
Griffith DA (2007) Spatial structure and spatial interaction: 25 years later. Rev Reg Stud 37(1):28–38
Geweke J (1996) Variable selection and model comparison in regression. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 5: proceedings of the fifth Valencia international meeting. Oxford University Press, Oxford, pp 609–620
Kadiyala KR, Karlsson S (1997) Numerical methods for estimation and inference in Bayesian VAR models. J Appl Econ 12:99–132
Kamarianakis Y, Kanas A, Prastacos P (2005) Modelling traffic volatility dynamics in an urban network. Transportation Research Record: Journal of the Transportation Research Board No. 1923. Transportation Research Board of the National Academies, Washington, pp 18–27
Kamarianakis Y, Prastacos P (2005) Space-time modelling of traffic flow. Comput Geosci 31:119–133
Koop G, Korobilis D (2009) Bayesian multivariate time series methods for empirical macroeconomics. Found TrendsR Econ 3(4):267–358
Korobilis D (2013) VAR forecasting using Bayesian variable selection. J Appl Econ 28:204–230
Leonard T, Hsu J (1992) Bayesian inference for a covariance matrix. Ann Stat 20(4):1669–1696
LeSage JP, Fischer MM (2010) Spatial econometric methods for modelling origin–destination flows. In: Fischer MM, Getis A (eds) Handbook of applied spatial analysis: software tools, methods and applications. Springer, Berlin, pp 413–437
LeSage JP, Krivelyova A (1999) A spatial prior for Bayesian vector autoregressive models. J Reg Sci 39(2):297–317
LeSage JP, Pan Z (1995) Using spatial contiguity as Bayesian prior information in regional forecasting models. Int Reg Sci Rev 18:33–53
LeSage JP, Pace RK (2008) Spatial econometric modelling of origin-destination flows. J Reg Sci 48(5):941–967
LeSage JP, Polasek W (2008) Incorporating transportation network structure in spatial econometric models of commodity flows. Spat Econ Anal 3(2):225–245
Liang F, Paulo R, Monila G, Clyde MA, Berger JO (2008) Mixture of g priors for Bayesian variable selection. J Am Stat Assoc 103(148):410–425
Litterman R (1980) Techniques for forecasting with vector autoregressions. PhD Dissertation, University of Minnesota, Minneapolis
Litterman R (1986) Forecasting with Bayesian vector autoregressions: five years of experience. J Bus Econ Stat 4:25–38
Pan Z, LeSage JP (1995) Using spatial contiguity as prior information in vector autoregressive models. Econ Lett 47:137–142
Pfeifer PE, Deutsch SJ (1980a) A three-stage iterative procedure for space–time modelling. Technometrics 22(1):35–47
Pfeifer PE, Deutsch SJ (1980b) Identification and interpretation of first-order space–time ARMA models. Technometrics 22(3):397–403
Pfeifer PE, Deutsch SJ (1981a) Variance of the sample-time autocorrelation function of contemporaneously correlated variables. SIAM J Appl Math Ser A 40(1):133–136
Pfeifer PE, Deutsch SJ (1981b) Seasonal space–time ARIMA modelling. Geogr Anal 13(2):117–133
Pfeifer PE, Bodily SE (1990) A test of space-time ARMA modelling and forecasting of hotel data. J Forecast 9(3):255–272
Porojan A (2001) Trade flows and spatial effects: the gravity model revisited. Open Econ Rev 12:265–280
Schmeiser BW, Chen MH (1991) On Hit-and-Run Monte Carlo sampling for evaluating multidimensional integrals. Technical Report, 91-39, Dept. Statistics, Purdue University
Sims C (1972) Money, income and causality. Am Econ Rev 62(1):540–553
Sims C (1980) Macroeconomics and reality. Econometrica 48:1–48
Smith M, Kohn R (1996) Nonparametric regression using Bayesian variable selection. J Econ 75:317–343
Yang R, Berger JO (1994) Estimation of a covariance matrix using the reference prior. Ann Stat 22(3):1195–1211
Zellner A (1971) An introduction to Bayesian inference in econometrics. Wiley, New York
Zellner A (1986) On assessing prior distributions and Bayesian regression analysis with g-Prior distributions. In: Goel PK, Zellner A (eds) Bayesian inference and decision techniques: essays in Honor of Bruno de Finetti, North-Holland/Elsevier, Amsterdam, pp 233–243
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Appendices
Appendix 1
Recall that the joint posterior is
To integrate out B
Complete the squares and divide and multiply
Let
and let
then
is easily recognized as the kernel of the integral over B. Thus
Finally, substitute the prior density of
Into the above, Eq. (10.19)
is obtained.
Appendix 2
Define \( {q}_{\gamma_i^0} \) as the size of γ excluding element i, one can write
Using combinatorics, as defined for the same prior in Cui and George (2008), one could write
which, after some algebra, yields
which implies
It is straightforward to show that Eq. (10.19)
holds.
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Deng, M. (2016). Bayesian Variable Selection in a Large Vector Autoregression for Origin-Destination Traffic Flow Modelling. In: Patuelli, R., Arbia, G. (eds) Spatial Econometric Interaction Modelling. Advances in Spatial Science. Springer, Cham. https://doi.org/10.1007/978-3-319-30196-9_10
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