The traffic condition likelihood extraction using incomplete observation in distributed traffic loop detectors

Abstract

Forecasting and extracting traffic state likelihood function is a challenging statistical issue in advanced traveler information systems and here new approaches will be explored to determine maximum likelihood of key measurement observed in a traffic sensor network. This problem can be seen as a nonlinear maximum-likelihood or maximum a posteriori (MAP) estimation problems. Practically every urban traveler information or traffic management system is based on traffic collected by point e.g. single or double loop detectors, overhead video cameras [1]. Real-time traffic includes mean speed, traffic density and traffic flow. Traffic management and control with high performance require the estimation/prediction of Maximum Likelihood related to traffic conditions in congestion or accident situations for large spatial and temporal coverage [2]. The traffic condition models are continuously subject to changes over time due to traffic conditions (traffic compositions, incidents …) and environmental factors (dense fog, strong wind, snow …) and missing regarding to problems in distributed sensor network and communication links. Each sensing node in the traffic detector network senses traffic conditions and subsequently, a large number of measurements spatially distributed over traffic network environment will be available. These large streams can not be fully handled at a central processing unit in real time application due to communication links capability and power source limitation. So, as an alternative in distributed strategy, traffic network can be described as a mixture of some elementary geographically distributed conditions. The measurements are thus statistically modeled with a mixture of Gaussians, where each Gaussian component corresponds to one of the elementary conditions. To solve the mentioned distributed maximum likelihood estimator, some distributed methods based on Expectation-Maximization (DEM)-type algorithms have been developed. In the DSN context, [3] has reported an incremental (I-) DEM scheme, while [4] has investigated a Gossip-based (G-) DEM alternative. The first distributed approach will be used here is fully DEM algorithms which can be viewed as an application and adaptation of the incremental EM algorithm [5]. The distributed EM algorithm used here aims to reduce the number of iterations and, hence, the number of communications required. The incremental EM algorithm was shown to converge to a fixed point in [5], and more recently, it was shown that the incremental EM algorithm and the standard EM algorithm have the same fixed points [6]. The second distributed approach, is termed consensus based (CB-) DEM algorithm which implemented for nonlinear ML or MAP parameter estimation based on collected across spatially distributed traffic sensors. The E-step in CBDEM relies on local (per sensor) information. The key difference lies in re-formulating the M-step, where the average log-likelihood of EM's complete data is maximized. CB-DEM lends itself naturally to a general distributed clustering scheme where class-conditional pdfs are even allowed to be non-Gaussian. In addition, CB-DEM relies on bridge sensors offering a more desirable tradeoff between robustness and overhead. The paper is organized as follows. In Section two, we explain how the EM algorithm can be implemented over distributed sensor networks and the general formulations are defined. The decentralized EM is outlined leading to the development of the CB-DEM algorithm are given in Section three. Section four presents the particle filter using the distributed EM algorithm. Section five reviews the case study and simulation results for implementation of standard and fully distributed EM algorithm, Consensus-Based DEM and Particle Filter algorithm based on DEM for mixture density estimation.