A finite mixture model of vehicle-to-vehicle and day-to-day variability of traffic network travel times

Jiwon Kim, Hani S. Mahmassani*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


This study proposes an approach to modeling the effects of daily roadway conditions on travel time variability using a finite mixture model based on the Gamma-Gamma (GG) distribution. The GG distribution is a compound distribution derived from the product of two Gamma random variates, which represent vehicle-to-vehicle and day-to-day variability, respectively. It provides a systematic way of investigating different variability dimensions reflected in travel time data. To identify the underlying distribution of each type of variability, this study first decomposes a mixture of Gamma-Gamma models into two separate Gamma mixture modeling problems and estimates the respective parameters using the Expectation-Maximization (EM) algorithm. The proposed methodology is demonstrated using simulated vehicle trajectories produced under daily scenarios constructed from historical weather and accident data. The parameter estimation results suggest that day-to-day variability exhibits clear heterogeneity under different weather conditions: clear versus rainy or snowy days, whereas the same weather conditions have little impact on vehicle-to-vehicle variability. Next, a two-component Gamma-Gamma mixture model is specified. The results of the distribution fitting show that the mixture model provides better fits to travel delay observations than the standard (one-component) Gamma-Gamma model. The proposed method, the application of the compound Gamma distribution combined with a mixture modeling approach, provides a powerful and flexible tool to capture not only different types of variability-vehicle-to-vehicle and day-to-day variability-but also the unobserved heterogeneity within these variability types, thereby allowing the modeling of the underlying distributions of individual travel delays across different days with varying roadway disruption levels in a more effective and systematic way.

Original languageEnglish (US)
Pages (from-to)83-97
Number of pages15
JournalTransportation Research Part C: Emerging Technologies
StatePublished - Sep 2014


  • Expectation-Maximization algorithm
  • Finite mixture model
  • Gamma-Gamma distribution
  • Travel time reliability
  • Travel time variability

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Automotive Engineering
  • Transportation
  • Computer Science Applications


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