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. 2020 Dec 29;49(6):1421-1448.
doi: 10.1080/02664763.2020.1864814. eCollection 2022.

Bayesian hierarchical models for linear networks

Affiliations

Bayesian hierarchical models for linear networks

Zainab Al-Kaabawi et al. J Appl Stat. .

Abstract

The purpose of this study is to highlight dangerous motorways via estimating the intensity of accidents and study its pattern across the UK motorway network. Two methods have been developed to achieve this aim. First, the motorway-specific intensity is estimated by using a homogeneous Poisson process. The heterogeneity across motorways is incorporated using two-level hierarchical models. The data structure is multilevel since each motorway consists of junctions that are joined by grouped segments. In the second method, the segment-specific intensity is estimated. The homogeneous Poisson process is used to model accident data within grouped segments but heterogeneity across grouped segments is incorporated using three-level hierarchical models. A Bayesian method via Markov Chain Monte Carlo is used to estimate the unknown parameters in the models and the sensitivity to the choice of priors is assessed. The performance of the proposed models is evaluated by a simulation study and an application to traffic accidents in 2016 on the UK motorway network. The deviance information criterion (DIC) and the widely applicable information criterion (WAIC) are employed to choose between models.

Keywords: Bayesian methods; Hierarchical models; linear networks; point processes.

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Conflict of interest statement

No potential conflict of interest was reported by the author(s).

Figures

Figure 1.
Figure 1.
The hierarchical structure of UK motorway network in 2016. Mi denotes motorway i. M1GS1 represents the first grouped segments on M1; M1GS83 represents the eighty third grouped segments on M1 and so on for the rest of the symbols of grouped segments for other motorways. The grouped segments are links joining motorway junctions and they are grouped to incorporate the heterogeneity of the accident intensity across them [44]. (a) Two-level hierarchical model. (b) Three-level hierarchical model.
Figure 2.
Figure 2.
Results from the two-level Bayesian hierarchical model and the three-level Bayesian hierarchical model for accident data on the UK motorways in 2016. In the two-level Bayesian hierarchical model, the following prior distributions are used, αN(0,100) and τ2InvGamma(0.001,0.001). In the three-level Bayesian hierarchical model, the prior distributions are αN(6.65,0.092) and τ2InvGamma(18.36,58.06). Results include the posterior mean and the corresponding 95% credible interval for the intensity of accidents λi=1000×exp(αi) per one kilometre on each motorway and the overall intensity of accidents λ per one kilometre. Square boxes represent posterior means of λi,(i=1,,m). The diamond represents the estimated overall intensity of accident λ and its 95% credible interval. Horizontal lines denote 95% credible intervals and the sold vertical line represents the posterior mean of the overall intensity λ. (a) Two-level Bayesian hierarchical model. (b) Three-level Bayesian hierarchical model.
Figure 3.
Figure 3.
Estimated intensity of traffic accidents per one kilometre on the UK motorway network in 2016. The intensities are estimated using the three-level Bayesian hierarchical model with prior distributions αN(6.65,0.092) and τ2InvGamma(18.36,58.06).
Figure 4.
Figure 4.
Estimated intensity of traffic accidents per one kilometre on the UK motorway network in 2016. The intensities are estimated using the two-level Bayesian hierarchical model with prior distributions αN(0,100) and τ2InvGamma(0.001,0.001).

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