Understanding bus delay patterns under different temporal and weather conditions: A Bayesian Gaussian mixture model

Abstract

In public transit systems, bus delays significantly impact service reliability and passenger satisfaction. Causal delays, consisting of link running and stop dwell delays, are critical factors contributing to overall bus delay patterns. This paper develops a Bayesian probabilistic model to analyze bus delay patterns with a focus on causal delays under varying weather and temporal conditions, which can help to understand how the underlying causal delay patterns contribute to arrival delay patterns. Employing a Gaussian mixture model integrated with a topic model approach, the study analyzes causal delays as multivariate random variables, capturing the influence of temporal and weather conditions on bus service reliability. For model inference, we propose a Markov Chain Monte Carlo (MCMC) sampling method to estimate the model parameters. The analysis is conducted using real-world data from a bus route in Calgary, Canada. We categorize the identified delay patterns into four on-time categories: extreme earliness, moderate earliness, extreme lateness, and moderate lateness. Results indicate that adverse weather significantly influences extreme delay patterns in particular, suggesting the necessity for transit agencies to consider these factors in schedule optimization. Beyond pattern identification, the proposed model offers probabilistic delay estimation, enabling accurate forecasting of future delays based on current conditions and observations. Validation results demonstrate that our probabilistic estimates align closely with observed data, proving the model’s practical applicability in real-time operations and offering actionable insights to enhance the punctuality and efficiency of urban bus services.