Bayesian hierarchical non-stationary hybrid modeling for threshold estimation in peak over threshold approach

The peak over threshold (POT) approach in extreme value theory is widely used for crash risk estimation, but the reliability is often undermined by the subjective and arbitrary selection of the conflict threshold, which can lead to biased outcomes. This study advances the hybrid modeling method for objective threshold determination by developing a non-stationary framework and comprehensively comparing five distinct model structures. The framework allows the threshold to vary with real-time traffic covariates, while the comparison identifies the optimal distribution for general conflicts. The Bayesian hierarchical structure is used to combine traffic conflicts from different sites, incorporating covariates and site-specific unobserved heterogeneity. Five non-stationary BHHM models, including Normal-GPD, Cauchy-GPD, Logistic-GPD, Gamma-GPD, and Lognormal-GPD models, were developed and compared. Traditional graphical diagnostic and quantile regression approaches were also used for comparison. Traffic conflicts collected from three signalized intersections in the city of Surrey, British Columbia were used for the study. The Bayesian approach is employed to estimate the threshold and other parameters in the non-stationary BHHM models. The results show that the proposed BHHM approach could estimate the threshold parameter objectively. The non-stationary BHHM models capture how the threshold varies dynamically across signal cycles in response to changing traffic status. The Lognormal-GPD model is superior to the other four BHHM models in terms of crash estimation accuracy and model fit. The crash estimates using the threshold determined by the BHHM outperform those estimated based on the graphical diagnostic and quantile regression approaches, indicating the superiority of the proposed threshold determination approach. The findings of this study contribute to enhancing the existing EVT methods for providing a threshold determination approach as well as producing reliable crash estimations.