Stochastic modeling of equilibrium speed-density relationship for non-lane-based heterogeneous traffic

Abstract

In traffic flow theory, fundamental diagrams (FDs) address the relationship among three variables: flow, speed and density. Among them, equilibrium speed-density relationship explains the speed dynamics in road incorporating two important parameters, i.e., free-flow speed and jam density. This speed-density relationship is widely used for designing strategies related to traffic control and management. The analysis of real time traffic dynamics largely depends upon the output from the fundamental diagram where the congestion occurs. Fundamental diagram is the relation between traffic speed and density which comes from the driver’s speed choices under capricious car following distances. About eighty-five years ago Greenshields proposed a seminal linear relationship between speed and density. Later, a number of researchers have devoted to reviewing or improving such a simplified relationship. But these models are mainly deterministic in nature which cannot addresses the randomness of traffic behavior i.e. driver’s behavior. Therefore, the randomness makes speed-density relation to view as a stochastic process. Additionally, most of the existing speed-density model are applicable for traffic having strict lane discipline along with homogeneous traffic stream. They are not being formulated, calibrated and validated for non-lane-based heterogeneous traffic (prevails in many south Asian countries) with unpredictable randomness. Moreover, these models cannot calibrate the jam density or shockwave speed in case of an incomplete dataset (e.g. collected traffic data includes only free-flow traffic regime). Furthermore, human decisions about lane change, gap acceptance, and acceleration and deceleration process affect the equilibrium speed–density relationship. Thus, a highly adaptive and robust FD model is required, which can incorporate non-lane-based heterogeneous behavior of traffic and randomness. This study aims to develop such a stochastic speed-density model to better represent empirical observations from non-lane based heterogeneous traffic and provide a base for a stochastic prediction of traffic dynamics. It will be more acceptable if such a model is formulated with both mathematical elegance and empirical accuracy. The mathematical elegance of the model mainly depends upon a single equation (single-regime) with physically meaningful parameters and empirical accuracy comes from the model fitness. For this study, video data of several days have been collected using high-resolution camera from the study corridor i.e. Tongi Diversion Road, a section of the Dhaka-Mymensingh Highway (N3), containing non-lane based heterogeneous traffic at five locations (mainline, on-ramp close to mainline, off-ramp close to mainline, on-ramp and off-ramp) considering the geometric variation. The collected traffic flow data is then extracted by using pixel-based heterogeneous traffic flow measurement technique. Lane based homogeneous traffic data is also collected from I-80 Berkeley at five different locations to prove the robustness of the proposed model. This data is used to develop the stochastic speed-density model choosing the best transfer function from various transfer function i.e hardlimit, heaviside, saturation, pureline. Among them the best fit saturation function is chosen based on the goodness of fit (R2). The data is split in different composition like 50-50, 60-40, 70-30, 75-25, 80-20, 85-15, 90-10 in training and testing data set respectively. Among the above composition the 80-20 split division gives better results in terms of goodness of fit (R2=0.96259). Four types of hidden layer number (2, 3, 4, 5, 6) are considered since it gives flexibility of the speed-density curve for fitting this curve properly. It is seen from the analysis hidden layer number 5 gives better results. Three types of optimization algorithm named Bayesian regularization, Scaled conjugate gradient, Levenberg-Marquardt (LM) is exercised. Among them LM algorithm preforms well. Additionally, it takes less memory and time to fit. This study also analyses the effect of time aggregation 20 seconds, 1, 2, 3, 5, 6 minutes and it can be concluded that 5 min minutes yields better performance. The proposed model is then compared with other prominent existing model i.e. ANFIS, 5 PL model. The study analysis dictates that ANN model performs well than other model. From this study, three new equations have been developed for free-flow speed (Vf), jam density (ρj) and capacity (C). The model is further used to develop a tool named Fundamental DiagRam CalibrAtion using Machine LEaring (FRAME) which can calibrate the speed-density relationship automatically.