ESTIMATION OF HIGHWAY TRAFFIC FROM SPARSE SENSORS: STOCHASTIC MODELING AND PARTICLE FILTERING A. Pascale 1 , G. Gomes 2 , M. Nicoli 1 1 Dip. Elettronica e Informazione, Politecnico di Milano, Italy, {pascale,nicoli}@elet.polimi.it 2 PATH, UC Berkeley, CA, USA, gomes@path.berkeley.edu ABSTRACT Traffic control is essential for the achievement of a sustain- able and safe mobility. Monitoring systems deployed over the roads collect a great amount of traffic data that must be effi- ciently processed by statistical methods to draw traffic macro- parameters that are needed for control operations. In this pa- per we propose a particle filtering approach to estimate the density over a road network starting from noisy and sparse measurements provided by road-embedded sensors. We pro- pose a new Bayesian framework based on the link-node cell transmission model to take into account the stochastic behav- ior of traffic and the hysteresis phenomenon that are typically observed in real data. Numerical tests show that the estima- tion method is able to reliably reconstruct the traffic field even in case of very sparse sensor deployments. Index Terms— ITS, Statistical Modeling, Particle Filter- ing, Traffic Densities Reconstruction. 1. INTRODUCTION Intelligent transport systems (ITS) are expected to improve quality, safety and sustainability of mobility by integrating in- formation and communication technologies with transport en- gineering. ITS rely on a capillary network of sensors, either road-embedded or mobile probes, that are on roads provid- ing measurements of traffic macro-parameters, such as speed, flow, and density [1]. Modern traffic management systems rely on estimates and predictions of the overall state of traffic based on sparse and often noisy measurements. The objective of this paper is to develop a statistical al- gorithm able to accurately estimate the evolution of traffic variables. Several approaches have been presented in the literature to reach this goal, from historical trends [2] to non-parametric methods, auto-regressive and moving average models [3, 2], Kalman filtering, and neural networks [4, 5, 6]. Recent studies have proposed Bayesian filtering to track the evolution of traffic on a real-time basis. In particular, non lin- ear Bayesian models [7, 8], Bayesian networks (BN), graph theory [9, 10] and particle filtering (PF) [11, 8] have been proved to be promising solutions for traffic estimation. The original contribution of this paper is a Bayesian framework for the estimation of the densities in highways us- ing PF [12]. Our effort is devoted to the proposal of a new Bayesian model that extends previous approaches so as to ac- count for the stochastic behavior of traffic, and also for phe- nomena of hysteresis and capacity drop that are typically ob- served in flow-density scatter plots [13, 14]. Although micro- scopic models have long incorporated random driver behav- ior [14, 15, 16], stochastic extensions to the cell transmission model (CTM) are relatively recent, in particular [17] contains an overview of the most relevant contributions. In this paper we add randomness to the link-node CTM (LN-CTM [18]) an application of the Godunov scheme to network topolo- gies. Based on the modified LN-CTM, we develop a Bayesian method for the estimation of the space-time traffic evolution from sparse sensor observations. Since the traffic model is non-linear and non-Gaussian, we propose a PF approach. A numerical analysis is carried out on a realistic highway sce- nario where only a subset of road links is monitored by loop sensors while the rest are not observed (e.g., due to sparse sen- sor deployments or loop failures). The results show that the estimation method is able to provide a good reconstruction of the traffic field over all road links, as for a virtual sensor deployment with higher spatial density (increased by a factor ∼ 7 in the specific scenario). The estimate accuracy outper- forms the loops’ accuracy up to 40% on monitored links. 2. PROBLEM FORMULATION AND MODELING We model the road as a set of N L links (road segments) in- terconnected by nodes (road junctions), as depicted in Fig. 1 1 . A subset S ⊆{1, 2,...,N L } of N S ≤ N L links is monitored by traffic sensors, e.g., loops installed on the links. The length of the n-th link is denoted as l n , time evolution is sampled with time interval Δt. The variable that describes the traffic state in link n at time k is the density ρ n (k), de- fined as the number of vehicles per space unit [veh/mile]. The N L × 1 state vector for the whole road is defined as x k =[ρ 1 (k) ...ρ N L (k)] T and the N S × 1 measurement vec- tor as y k = [˘ ρ s1 (k) ... ˘ ρ s Ns (k)] T where ˘ ρ s (k) is the density measured by sensor s ∈ S at time k. Traffic is modelled as a hidden Markov model ruled by the following equations: x k = g (x k−1 , w k ) y k = h (x k , r k ) (1) where g (·, ·) is the function - defined in next section - describ- ing how densities evolve over space and time. The function 1 Networks were built with the TOPL Network Editor [19]. Traffic data was obtained by PeMS [20].