Hierarchical Bayesian Modeling with IAR Hexagonal Grids for Reconstructing Incomplete OD Matrices

Abstract

Extracting Origin-Destination (OD) matrices in open Bus Rapid Transit systems such as Transjakarta is essential for urban mobility analysis. However, this process is often hindered by incomplete observations, particularly due to missing tap-out data, which leads to extreme sparsity, zero-inflation, and overdispersion in the resulting matrices. This study addresses the problem of probabilistically reconstructing highly sparse OD matrices while accounting for spatial dependencies. To overcome limitations in previous imputation methods—such as ignoring network topology or being affected by the Modifiable Areal Unit Problem (MAUP)—this research proposes a hierarchical Bayesian approach integrating an Intrinsic Autoregressive (IAR) prior within an isotropic hexagonal (H3) tessellation framework. A Negative Binomial distribution is employed to model overdispersed count data, while latent spatial intensities and missing destinations are jointly estimated using Markov Chain Monte Carlo (MCMC). The proposed Spatial IAR model achieves stable convergence with a maximum , whereas the independent non-spatial model fails to converge adequately ( ). Although the independent model produces lower WAIC and LOOIC values (14313.41 and 14313.66) than the Spatial IAR model (14551.88 and 14611.18), the  indicates that the apparent predictive superiority is spurious due to inferential instability. Posterior Predictive Checks further confirm that the spatial model successfully reproduces the overdispersion and zero-inflation characteristics of the observed mobility data. Overall, the results demonstrate that spatial regularization is essential for reconstructing high-dimensional sparse urban mobility data and improving the robustness of transportation mobility analysis.