![[submission_moon24.pdf]] | Step | Substep | Poisson Process Traffic | | ----------------------- | ------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | 1. Problem to intervene | | Aggregated traffic is not Poisson process in most cases, and even if it is, it has little managerial impact due to the difference between the unit of observation and intervention. | | 2. Cause of 1 | | Unrealistic assumption of traffic being a Poisson process i.e. delay time is exponential distribution, leads to inaccurate prediction of traffic delays. | | | 2.1 Nature | Aggregation over different data generating processes (e.g. delay types) masks underlying patterns and lowers prediction accuracy. | | | 2.2 Agent level use based on 2.1 | Local Policy makers, who are capable of customized intervening (e.g. customized intervention to each delay types, action1 for uniform delay type, action2 for overflow delay, action3 for dwell time delay), intervene on aggregated level, as they don't have access to observed data on more granular delay type of data. | | | 2.3 Institution level use based on 2.1, 2.2 | Global Policy makers (transportation agencies), who are capable of initiating collecting granular level of data (for each delay type), lack incentives to do so as the benefit from added data is not quantified and justified. | | 3. Solution for 2 | | Collect data at the fineness of intervention and use hierarchical Bayesian models when fine data collection is not viable. | | | 3.1 Solution for 2.1 | Aggregation is unavoidable as the level of observation and intervention doesn't match. However, we can incentivize agents and institution to narrow this gap (3.2,3) | | | 3.2 Solution for 2.2 | Provide managers with disaggregated data on delay types for targeted interventions. | | | 3.3 Solution for 2.3 | Provide institutional support for data collection at the level of intervention, persuaded by increased prediction accuracy of delay via disaggregating delay type (contribution of this paper). | | 4. Theory | | The theory focuses on using hierarchical Bayesian models as a temporary solution to decompose the effects of latent layers (delay types) on the observed data (delay time) until the more effective solution of collecting data at the fineness of intervention becomes available. This approach addresses the issue of aggregated delay time not being a Poisson process and improves the understanding of specific delay types for targeted interventions. Mixed Poisson becomes homogeneous Poisson using hierarchical Bayesian theory. We can use hierarchical Bayesian theory to explain the effect of separating sources as decomposing observations whose rate lambda is first sampled from a Gamma distribution (three types: signal delay, dwell delay, overflow delay). Then, the delay rate of each type (lambda1, lambda2, lambda3) is used to generate a Poisson process.<br> 1. Observational model for the data: $(Y_i \mid \lambda_i) \sim \text{Poisson}(\lambda_i), \quad i=1, \ldots, k$ where $Y_i$ represents the observed delay for the $i$-th type (signal delay, dwell delay, or overflow delay), and $\lambda_i$ is the corresponding rate parameter. 2Structural model for the parameters of the likelihood $(\lambda_i \mid \alpha, \beta) \sim \text{Gamma}(\alpha, \beta), \quad i=1, \ldots, k$ where $\alpha$ and $\beta$ are the shape and rate parameters of the Gamma distribution, respectively. 3. Hyperparameter model for the parameters of the structural model: $\alpha \sim \text{Uniform}(0, 0.5)$ $\beta \sim \text{Gamma}(0.1, 1)$ By separating the sources of delay into signal delay, dwell delay, and overflow delay, we are essentially decomposing the inhomogeneous Poisson process into multiple homogeneous Poisson processes, each with its own rate parameter $\lambda_i$. The hierarchical Bayesian model allows us to account for the uncertainty in the rate parameters by assigning them a Gamma prior distribution, which is a conjugate prior for the Poisson likelihood. The hyperparameters $\alpha$ and $\beta$ are given their own prior distributions to complete the hierarchical structure. This hierarchical Bayesian approach helps to explain how separating the sources of delay can lead to a more homogeneous Poisson process for each source, as the rate parameters are modeled as being drawn from a common Gamma distribution. The posterior distribution of the rate parameters can be inferred using Bayesian inference techniques, such as Markov Chain Monte Carlo (MCMC) methods. | | 5. Action | | Build map between delay type and intervention actions so that the bottleneck delay can be mitigated by Implementing corresponding actions. |