Bayesian Variable Selection in a Large Vector Autoregression for Origin-Destination Traffic Flow Modelling

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.

Appendices

Appendix 1

Recall that the joint posterior is

$$ p\left(\mathrm{B},\Sigma, \gamma \Big|Y\right)\propto p\left(\mathrm{Y}\Big|\mathrm{B},\Sigma, \gamma \right)\cdot p\left({\mathrm{B}}_{\gamma}\Big|\Sigma, \gamma \right)\cdot p\left(\Sigma \right)\cdot p\left(\gamma \right) $$

To integrate out B

$$ \begin{array}{l}p\left(\Sigma, \gamma \Big|Y\right)={\displaystyle \underset{\mathrm{B}}{\int }}p\left(\mathrm{B},\Sigma, \gamma \Big|Y\right)d\mathrm{B}\\ {}\propto {\displaystyle \underset{\mathrm{B}}{\int }}p\left(\mathrm{Y}\Big|\mathrm{B},\Sigma, \gamma \right)\cdot p\left({\mathrm{B}}_{\gamma}\Big|\Sigma, \gamma \right)\cdot \left[\,p\left(\Sigma \right)\cdot p\left(\gamma \right)\right]d\mathrm{B}\\ {}\propto {\displaystyle \underset{\mathrm{B}}{\int }}p\left(\mathrm{Y}\Big|\mathrm{B},\Sigma, \gamma \right)\cdot p\left({\mathrm{B}}_{\gamma}\Big|\Sigma, \gamma \right)d\mathrm{B}\cdot p\left(\Sigma \right)\cdot p\left(\gamma \right)\\ {}\propto {\displaystyle \underset{\mathrm{B}}{\int }}{\left(2\pi \right)}^{-\frac{NT}{2}}\cdot {\left|\Sigma \right|}^{-\frac{T}{2}}\cdot exp\left\{-\frac{1}{2}{\left(Y-{\mathrm{X}}_{\gamma }{\mathrm{B}}_{\gamma}\right)}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}\left(Y-{\mathrm{X}}_{\gamma }{\mathrm{B}}_{\gamma}\right)\right\}\\ \quad\cdot{\left(2\pi \right)}^{-\frac{q_{\gamma }}{2}}\cdot {g}^{-\frac{q_{\gamma }}{2}}\cdot {\left|{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right|}^{\frac{1}{2}}\\[6pt] \quad\cdot exp\left\{-\frac{1}{2}{\mathrm{B}}_{\gamma}^{\prime}\left[\left(\frac{1}{g}\right){\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]{\mathrm{B}}_{\gamma}\right\}d\mathrm{B}\cdot \left[\,p\left(\Sigma \right)\cdot p\left(\gamma \right)\right]\end{array} $$

Complete the squares and divide and multiply

$$ \begin{array}{l}{\displaystyle \underset{\mathrm{B}}{\int }}p\left(\mathrm{B},\Sigma, \gamma \Big|Y\right)d\mathrm{B}\\ {}\propto {\left|\Sigma \right|}^{-\frac{T}{2}}\cdot exp\left\{-\frac{1}{2}{Y}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y\right\}\cdot {\displaystyle \underset{\mathrm{B}}{\int }}{\left(2\pi \right)}^{-\frac{q_{\gamma }}{2}}\cdot {\left|{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right|}^{\frac{1}{2}}\cdot {\left(\frac{g}{1+g}\right)}^{-\frac{q_{\gamma }}{2}}\\ {}\cdot exp\left\{-\frac{1}{2}\left[\left(\frac{1+g}{g}\right){\mathrm{B}}_{\gamma}^{\prime }{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma }{\mathrm{B}}_{\gamma }-2{\mathrm{B}}_{\gamma}^{\prime }{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y\right.\right.\\ \qquad\left.\left.{}+{Y}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma }{\left[\left(\frac{1+g}{g}\right){\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]}^{-1}{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y\right]\right\}d\mathrm{B}\\[9pt] {}\cdot exp\left\{\frac{1}{2}{Y}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma }{\left[\left(\frac{1+g}{g}\right){\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]}^{-1}{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y\right\}\\ {}\cdot {\left(1+g\right)}^{-\frac{q_{\gamma }}{2}}\cdot \left[\,p\left(\Sigma \right)\cdot p\left(\gamma \right)\right]\end{array} $$

Let

$$ {\widetilde{\mathrm{B}}}_{\gamma }=\left(\frac{g}{1+g}\right){\left[{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]}^{-1}{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y $$

and let

$$ \left({\mathrm{B}}_{\gamma }-{\widetilde{\mathrm{B}}}_{\gamma}\right) \sim N\left(0,\left(\frac{g}{1+g}\right){\left[{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]}^{-1}\right) $$

then

$$ \begin{array}{l}{\left({\mathrm{B}}_{\gamma }-{\widetilde{\mathrm{B}}}_{\gamma}\right)}^{\prime}\left[\left(\frac{1+g}{g}\right){\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]\left({\mathrm{B}}_{\gamma }-{\widetilde{\mathrm{B}}}_{\gamma}\right)\\ {}=\left[\left(\frac{1+g}{g}\right){\mathrm{B}}_{\gamma}^{\prime }{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma }{\mathrm{B}}_{\gamma }-2{\mathrm{B}}_{\gamma}^{\prime }{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y\right.\\ \qquad\left.{}+{Y}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma }{\left[\left(\frac{1+g}{g}\right){\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]}^{-1}{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y\right]\end{array} $$

is easily recognized as the kernel of the integral over B. Thus

$$ \begin{array}{l}p\left(\Sigma, \gamma \Big|Y\right)\\ {}\propto {\left|\Sigma \right|}^{-\frac{T}{2}}\cdot {\left(1+g\right)}^{-\frac{q_{\gamma }}{2}}\cdot exp\left\{-\frac{1}{2}{Y}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y\right\}\\ {} \cdot exp\left\{\frac{1}{2}{Y}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma }{\left[\left(\frac{1+g}{g}\right){\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]}^{-1}{\mathrm{X}}_{\gamma}^{\prime }{\left(\Sigma \otimes {I}_T\right)}^{-1}Y\right\}\\ {} \cdot \left[\,p\left(\Sigma \right)\cdot p\left(\gamma \right)\right]\end{array} $$

Finally, substitute the prior density of

$$ p\left(\Sigma \right)=iW\left(\overline{\Sigma},\ \alpha \right) = \frac{{\left|\overline{\Sigma}\right|}^{\frac{\alpha }{2}}}{{\left|\Sigma \right|}^{\frac{\alpha +N+1}{2}}\cdot {2}^{\frac{\alpha N}{2}}\cdot {\Gamma}_N\left(\frac{\alpha }{2}\right)}\cdot exp\left\{-\frac{1}{2}tr\left({\overline{\Sigma}\Sigma}^{-1}\right)\right\} $$

Into the above, Eq. (10.19)

$$ \begin{array}{l}p\left(\varSigma, \gamma \Big|Y\right)\propto {\left(1+g\right)}^{-\frac{q_{\gamma }}{2}}\cdot \left|\varSigma \right|{}^{-\frac{T+\alpha +N+1}{2}}\cdot exp\left\{-\frac{1}{2}tr\left(\overline{\varSigma}{\varSigma}^{-1}\right)\right\}\cdot \\ {} exp\left\{-\frac{1}{2}{Y}^{\prime}\left\{{\left(\varSigma \otimes {I}_T\right)}^{-1}-{\left(\varSigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma }{\left[\left(\frac{1+g}{g}\right){\mathrm{X}}_{\gamma}^{\prime }{\left(\varSigma \otimes {I}_T\right)}^{-1}{\mathrm{X}}_{\gamma}\right]}^{-1}\right.\right.\\ \qquad\left.\left.{}{\mathrm{X}}_{\gamma}^{\prime }{\left(\varSigma \otimes {I}_T\right)}^{-1}\phantom{\frac{a_X\frac{\int_a^x}{x}}{x_2^X}}\right\}Y\right\}\cdot p\left(\gamma \right)\end{array} $$

is obtained.

Appendix 2

Define ${q}_{\gamma_i^0}$ as the size of γ excluding element i, one can write

$$ p\left({\gamma}_i=1\Big|{\gamma}_{j\ne i}\right)=\frac{p\left({q}_{\gamma_i^0}+1\right)}{p\left({q}_{\gamma_i^0}\right)+p\left({q}_{\gamma_i^0}+1\right)} $$

Using combinatorics, as defined for the same prior in Cui and George (2008), one could write

$$ \begin{array}{l}p\left({\gamma}_i=1\Big|{\gamma}_{j\ne i}\right)=\frac{p\left({q}_{\gamma_i^0}+1\right)}{p\left({q}_{\gamma_i^0}\right)+p\left({q}_{\gamma_i^0}+1\right)}\\ {}=\frac{\left(\frac{1}{q+1}\right){\left(\begin{array}{c}\hfill q\hfill \\ {}\hfill {q}_{\gamma_i^0}\hfill \end{array}\right)}^{-1}}{\left(\frac{1}{q+1}\right){\left(\begin{array}{c}\hfill q\hfill \\ {}\hfill {q}_{\gamma_i^0}\hfill \end{array}\right)}^{-1}+\left(\frac{1}{q+1}\right){\left(\begin{array}{c}\hfill q\hfill \\ {}\hfill {q}_{\gamma_i^0}+1\hfill \end{array}\right)}^{-1}}\end{array} $$

which, after some algebra, yields

$$ p\left({\gamma}_i=1\Big|{\gamma}_{j\ne i}\right)=\frac{q_{\gamma_i^0}+1}{q+1} $$

which implies

$$ p\left({\gamma}_i=0\Big|{\gamma}_{j\ne i}\right)=\frac{q-{q}_{\gamma_i^0}}{q+1} $$

It is straightforward to show that Eq. (10.19)

$$ \frac{p\left({\gamma}_i=1\Big|{\gamma}_{j\ne i}\right)}{p\left({\gamma}_i=0\Big|{\gamma}_{j\ne i}\right)}=\frac{q_{\gamma_i^0}+1}{q-{q}_{\gamma_i^0}} $$

holds.