Fitting fundamental diagrams to LWR using flow data Jeremie Coullon, Yvo Pokern, Benjamin Heydecker, and Eugeny Buldakov July 14, 2017 The Lighthill Whitham Richards (LWR, in [2] and [3]) model of traffic flow was developed over 50 years ago to model the propagation of shock waves on motorways. Although it has been extended in many ways to model different phenomena and mechanisms in traffic (capacity drop, vehicle acceleration, flow instability), a nonparametric statistical approach to inferring the underlying partial differential equation is so far absent from the literature. To use the LWR model, a suitable fundamental relationship must be chosen and fitted to data. This relationship gives rise to the so-called fundamental diagram (FD). Choosing the relationship is usually done based on theoreti- cal properties of the FD and fit to flow-density traffic data. Such theoretical properties include the concavity (or non-concavity) of the FD (see [1]) and the interpretability of its parameters. There are two limitations to this approach of fitting the FD. Firstly, quanti- fying the uncertainty in the estimate of FD parameters in the flow-density plane would not include a time component. As flow-density data is generated from a process that varies in space and time, the fit or misfit of a FD should rather be evaluated by solving LWR and comparing the output to traffic data in both space and time. Secondly, density must be estimated from traffic data, which leads to different results depending on the estimation procedure used. For ex- ample, density can be estimated from flow and speed data or from occupancy data, yielding different values of flow for a given density value. Furthermore, each of these different procedures have disadvantages; for example, one needs either to estimate the average vehicle length if one uses occupancy, and one needs to convert time mean speed to space mean speed if one uses speed and flow. To remedy the first of these limitations, we estimate the parameters in two FDs: one due to Underwood ([4]) and another due to del Castillo ([1]). We estimate these parameters by solving the inverse problem for LWR using a Bayesian framework. We elicit a prior for the FD parameters, and construct a likelihood by solving LWR and comparing it to flow data on a stretch of M25 motorway using a Poisson statistical model. As we consider traffic data in both time and space, the uncertainty quantification for the parameters is then justified, as discussed above. 1