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

Extreme value theory (EVT) has been utilized to estimate crash risk from traffic conflicts with the peak over threshold approach. However, it's challenging to determine a suitable threshold to distinguish extreme conflicts in an objective way. The subjective and arbitrary selection of the threshold in the peak over threshold approach can result in biased estimation outcomes. This study proposes a Bayesian hierarchical hybrid modeling (BHHM) framework for the threshold estimation in the peak over threshold approach. Specifically, BHHM is based on a piecewise function to model the general conflicts with specific distribution while model the extreme conflicts with generalized Pareto distribution (GPD). 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 results show that the proposed BHHM approach could estimate the threshold parameter objectively. 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.