Dependence patterns associated with the fundamental diagram: a copula function approach

Randomness plays a major role in the interpretation of many interesting traffic flow phenomena, such as hysteresis, capacity drop and spontaneous breakdown. The analysis of the uncertainty and reliability of traffic systems is directly associated with their random characteristics. Therefore, it is beneficial to understand the distributional properties of traffic variables. This paper focuses on the dependence relation between traffic flow density and traffic speed, which constitute the fundamental diagram (FD). The traditional model of the FD is obtained essentially through curve fitting. We use the copula function as the basic toolkit and provide a novel approach for identifying the distributional patterns associated with the FD. In particular, we construct a rule-of-thumb nonparametric copula function, which in general avoids the mis-specification risk of parametric approaches and is more efficient in practice. By applying our construction to loop detector data on a freeway, we identify the dependence patterns existing in traffic data. We find that similar modes exist among traffic states of low, moderate or high traffic densities. Our findings also suggest that highway traffic speed and traffic flow density as a bivariate distribution is skewed and highly heterogeneous.