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. Accepted by Transportation Research Part C on 16th Feb 2023.

Understanding the inter-relationships between traffic flow, density, and speed through the study of the fundamental diagram of road traffic is critical for traffic modelling and management. Consequently, over the last 85 years, a wealth of models have been developed for its functional form. However, there has been no clear answer as to which model is the most appropriate for observed (i.e. empirical) fundamental diagrams and under which conditions. A lack of data has been partly to blame. Motivated by shortcomings in previous reviews, we first present a comprehensive literature review on modelling the functional form of empirical fundamental diagrams. We then perform fits of 50 previously proposed models to a high quality sample of 10,150 empirical fundamental diagrams pertaining to 25 cities. Comparing the fits using information criteria, we find that the non-parametric Sun model greatly outperforms all of the other models. The Sun model maintains its winning position regardless of road type and congestion level. Our study, the first of its kind when considering the number of models tested and the amount of data used, finally provides a definitive answer to the question “Which model for the functional form of an empirical fundamental diagram is currently the best?’‘. The word “currently” in this question is key, because previously proposed models adopt an inappropriate Gaussian noise model with constant variance. We advocate that future research should shift focus to exploring more sophisticated noise models. This will lead to an improved understanding of empirical fundamental diagrams and their underlying functional forms. Accepted by IEEE Transactions On Intelligent Transportation Systems on 14th Dec 2021