FitFun: A modelling framework for successfully capturing the functional form and noise of observed traffic flow–density–speed relationships

D. M. Bramich, Mónica Menéndez, Lukas Ambühl

Research output: Contribution to journalArticlepeer-review


Measurements of the average properties of vehicular traffic are inherently noisy. The distributions of flow and speed measurements at any particular density are non-Gaussian with density-dependent variance, skewness, and kurtosis. Previous studies have failed to properly account for these complicated noise properties. In remediation, we present FitFun, a general framework for modelling any observed flow–density–speed relationship. Models specified within FitFun incorporate components for both the functional form and the noise. We define three flexible noise model components and we fit 200 different models to a high-quality sample of 10,150 observed urban flow-occupancy relationships. We compare the fits using information criteria and assess fit quality through analysis of the residuals. We find that the non-parametric Sun model for the functional form component combined with a Skew Exponential Power Type III noise component significantly outperforms all of the other models. Interestingly, we find that the city, country, road topology, and detector location have virtually no impact on model performance and fit quality, which is very convenient for model selection. The only factor of relevance from those that we studied is the effective occupancy coverage of the data. We conclude that certain models specified judiciously within FitFun can successfully capture the functional form and noise of observed flow–density–speed relationships without the need to discard data taken during non-stationary conditions. This is particularly advantageous for urban data where stationary traffic conditions are rarely observed.

Original languageEnglish (US)
Article number104068
JournalTransportation Research Part C: Emerging Technologies
StatePublished - Jun 2023


  • Empirical fundamental diagrams
  • Flow–density relationship
  • Loop detector data
  • Speed–density relationship
  • Statistical modelling
  • Traffic

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Automotive Engineering
  • Transportation
  • Management Science and Operations Research


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