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Bayesian Dynamic Gamma Models for Route-Level Travel Time Reliability

Vadim Sokolov, Refik Soyer

TL;DR

The paper tackles route-level travel time reliability by modeling each segment with a Gamma distribution conditional on a shared latent environment that evolves dynamically. This common random environment induces cross-segment dependence while preserving conditional independence, reducing route-time prediction to a one-dimensional integral and enabling a closed-form predictive distribution via an $F$-distribution after Gamma moment-matching. The approach supports exact sequential updating and real-time computation of reliability metrics (on-time probability, PTI, and Buffer Index) with $O(1)$ cost, and demonstrates superior calibration on I-55 data compared with independence and copula-based baselines. Its empirical results show 95.4% coverage of nominal 90% predictive intervals, achieved at the same computational cost as simpler methods, highlighting substantial improvements in reliability assessment and potential for real-time traffic management applications.

Abstract

Route-level travel time reliability requires characterizing the distribution of total travel time across correlated segments -- a problem where existing methods either assume independence (fast but miscalibrated) or model dependence via copulas and simulation (accurate but expensive). We propose a conjugate Bayesian dynamic Gamma model with a common random environment that resolves this trade-off. Each segment's travel time follows a Gamma distribution conditional on a shared latent environment process that evolves as a Markov chain, inducing cross-segment dependence while preserving conditional independence. A moment-matching approximation yields a closed-form $F$-distribution for route travel time, from which the Planning Time Index, Buffer Index, and on-time probability are computed instantly -- at the same $O(1)$ cost as independence-based methods. The conjugate structure ensures that Bayesian posterior updates and the full predictive distribution are available in closed form as new sensor data arrives. Applied to 16 sensors spanning 8.26 miles on I-55 in Chicago, the model achieves 95.4% coverage of nominal 90\% predictive intervals versus 34--37% for independence-based convolution, at identical computational cost.

Bayesian Dynamic Gamma Models for Route-Level Travel Time Reliability

TL;DR

The paper tackles route-level travel time reliability by modeling each segment with a Gamma distribution conditional on a shared latent environment that evolves dynamically. This common random environment induces cross-segment dependence while preserving conditional independence, reducing route-time prediction to a one-dimensional integral and enabling a closed-form predictive distribution via an -distribution after Gamma moment-matching. The approach supports exact sequential updating and real-time computation of reliability metrics (on-time probability, PTI, and Buffer Index) with cost, and demonstrates superior calibration on I-55 data compared with independence and copula-based baselines. Its empirical results show 95.4% coverage of nominal 90% predictive intervals, achieved at the same computational cost as simpler methods, highlighting substantial improvements in reliability assessment and potential for real-time traffic management applications.

Abstract

Route-level travel time reliability requires characterizing the distribution of total travel time across correlated segments -- a problem where existing methods either assume independence (fast but miscalibrated) or model dependence via copulas and simulation (accurate but expensive). We propose a conjugate Bayesian dynamic Gamma model with a common random environment that resolves this trade-off. Each segment's travel time follows a Gamma distribution conditional on a shared latent environment process that evolves as a Markov chain, inducing cross-segment dependence while preserving conditional independence. A moment-matching approximation yields a closed-form -distribution for route travel time, from which the Planning Time Index, Buffer Index, and on-time probability are computed instantly -- at the same cost as independence-based methods. The conjugate structure ensures that Bayesian posterior updates and the full predictive distribution are available in closed form as new sensor data arrives. Applied to 16 sensors spanning 8.26 miles on I-55 in Chicago, the model achieves 95.4% coverage of nominal 90\% predictive intervals versus 34--37% for independence-based convolution, at identical computational cost.
Paper Structure (38 sections, 4 theorems, 22 equations, 15 figures, 7 tables)

This paper contains 38 sections, 4 theorems, 22 equations, 15 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Travel Time Distribution Modeling
  4. Route-Level Reliability and the Convolution Problem
  5. Dynamic and Bayesian Models for Traffic
  6. Common Random Environment Models
  7. Research Gap and Contribution
  8. Dynamic Gamma Model
  9. Observation Model
  10. Dynamic Random Environment
  11. Sequential Bayesian Updating
  12. Predictive Distribution
  13. Multivariate Extension: Common Random Environment
  14. Model Specification
  15. Sequential Updating
  16. ...and 23 more sections

Key Result

Proposition 1

If $(\eta_0 \mid D^0) \sim \mathrm{Gam}(a_0, b_0)$, then for all $t \geq 1$:

Figures (15)

  • Figure 1: Locations of 16 loop detector stations on I-55 northbound (Stevenson Expressway), spanning 8.26 miles from sensor 6030 near I-294 to sensor 6045 near Damen Avenue.
  • Figure 2: Segment-level travel time distributions for four representative sensors, ranging from low variability (sensor 6030, $\hat{\alpha} = 34.2$) to high variability (sensor 6037, $\hat{\alpha} = 4.1$). The Gamma distribution (solid) captures the right-skewness present in the data, while the Normal (dashed) cannot accommodate the positive support and asymmetry. Each segment exhibits distinct distributional characteristics, motivating the segment-specific $\lambda_j$ parameters.
  • Figure 3: (a) Route-level travel time distribution with Gamma, Lognormal, and Normal fits. (b) Q-Q plot of empirical quantiles against the fitted Gamma distribution.
  • Figure 4: Cross-segment travel time correlation matrix. Adjacent segments show strong positive correlation (0.6--0.8), and even distant segments remain positively correlated, supporting the common random environment assumption.
  • Figure 5: Gamma mixture decomposition of route travel time. (a) Single Gamma ($K=1$). (b) Two-component mixture ($K=2$, BIC-preferred): a narrow free-flow component ($\mu = 19$ min) and a broad congested component ($\mu = 31$ min). (c) Three components ($K=3$) do not improve BIC. The bimodality reflects regime switching between free-flow and congested states.
  • ...and 10 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Corollary 1: Stochastic properties
  • Remark 1
  • Remark 2: Identifiability
  • Proposition 2: Multivariate compound Gamma
  • proof
  • Remark 3
  • Proposition 3: Route predictive distribution
  • proof