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Solitary wave solution to Aw-Rascle viscous model of traffic flow 被引量:1
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作者 吴春秀 张鹏 +2 位作者 S.C.WONG 乔殿梁 戴世强 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2013年第4期523-528,共6页
A traveling wave solution to the Aw-Rascle traffic flow model that includes the relaxation and diffusion terms is investigated. The model can be approximated by the well-known Kortweg-de Vries (KdV) equation. A nume... A traveling wave solution to the Aw-Rascle traffic flow model that includes the relaxation and diffusion terms is investigated. The model can be approximated by the well-known Kortweg-de Vries (KdV) equation. A numerical simulation is conducted by the first-order accurate Lax-Friedrichs scheme, which is known for its ability to capture the entropy solution to hyperbolic conservation laws. Periodic boundary conditions are applied to simulate a lengthy propagation, where the profile of the derived KdV solution is taken as the initial condition to observe the change of the profile. The simulation shows good agreement between the approximated KdV solution and the numerical solution. 展开更多
关键词 hyperbolic conservation law higher-order traffic flow model traveling wave solution conservative scheme
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A Diffusion Model for Traffic Data Imputation 被引量:1
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作者 Bo Lu Qinghai Miao +5 位作者 Yahui Liu Tariku Sinshaw Tamir Hongxia Zhao Xiqiao Zhang Yisheng Lv Fei-Yue Wang 《IEEE/CAA Journal of Automatica Sinica》 2025年第3期606-617,共12页
Imputation of missing data has long been an important topic and an essential application for intelligent transportation systems(ITS)in the real world.As a state-of-the-art generative model,the diffusion model has prov... Imputation of missing data has long been an important topic and an essential application for intelligent transportation systems(ITS)in the real world.As a state-of-the-art generative model,the diffusion model has proven highly successful in image generation,speech generation,time series modelling etc.and now opens a new avenue for traffic data imputation.In this paper,we propose a conditional diffusion model,called the implicit-explicit diffusion model,for traffic data imputation.This model exploits both the implicit and explicit feature of the data simultaneously.More specifically,we design two types of feature extraction modules,one to capture the implicit dependencies hidden in the raw data at multiple time scales and the other to obtain the long-term temporal dependencies of the time series.This approach not only inherits the advantages of the diffusion model for estimating missing data,but also takes into account the multiscale correlation inherent in traffic data.To illustrate the performance of the model,extensive experiments are conducted on three real-world time series datasets using different missing rates.The experimental results demonstrate that the model improves imputation accuracy and generalization capability. 展开更多
关键词 Data imputation diffusion model implicit feature time series traffic data
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A Comparative Study of Optimized-LSTM Models Using Tree-Structured Parzen Estimator for Traffic Flow Forecasting in Intelligent Transportation 被引量:1
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作者 Hamza Murad Khan Anwar Khan +3 位作者 Santos Gracia Villar Luis Alonso DzulLopez Abdulaziz Almaleh Abdullah M.Al-Qahtani 《Computers, Materials & Continua》 2025年第5期3369-3388,共20页
Traffic forecasting with high precision aids Intelligent Transport Systems(ITS)in formulating and optimizing traffic management strategies.The algorithms used for tuning the hyperparameters of the deep learning models... Traffic forecasting with high precision aids Intelligent Transport Systems(ITS)in formulating and optimizing traffic management strategies.The algorithms used for tuning the hyperparameters of the deep learning models often have accurate results at the expense of high computational complexity.To address this problem,this paper uses the Tree-structured Parzen Estimator(TPE)to tune the hyperparameters of the Long Short-term Memory(LSTM)deep learning framework.The Tree-structured Parzen Estimator(TPE)uses a probabilistic approach with an adaptive searching mechanism by classifying the objective function values into good and bad samples.This ensures fast convergence in tuning the hyperparameter values in the deep learning model for performing prediction while still maintaining a certain degree of accuracy.It also overcomes the problem of converging to local optima and avoids timeconsuming random search and,therefore,avoids high computational complexity in prediction accuracy.The proposed scheme first performs data smoothing and normalization on the input data,which is then fed to the input of the TPE for tuning the hyperparameters.The traffic data is then input to the LSTM model with tuned parameters to perform the traffic prediction.The three optimizers:Adaptive Moment Estimation(Adam),Root Mean Square Propagation(RMSProp),and Stochastic Gradient Descend with Momentum(SGDM)are also evaluated for accuracy prediction and the best optimizer is then chosen for final traffic prediction in TPE-LSTM model.Simulation results verify the effectiveness of the proposed model in terms of accuracy of prediction over the benchmark schemes. 展开更多
关键词 Short-term traffic prediction sequential time series prediction TPE tree-structured parzen estimator LSTM hyperparameter tuning hybrid prediction model
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Multi⁃Step Short⁃Term Traffic Flow Prediction of Urban Road Network Based on ISTA⁃Transformer Model
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作者 Leyao Xiao Qian Chen 《Journal of Harbin Institute of Technology(New Series)》 2025年第6期1-14,共14页
Short⁃term traffic flow prediction plays a crucial role in the planning of intelligent transportation systems.Nowadays,there is a large amount of traffic flow data generated from the monitoring devices of urban road n... Short⁃term traffic flow prediction plays a crucial role in the planning of intelligent transportation systems.Nowadays,there is a large amount of traffic flow data generated from the monitoring devices of urban road networks,which contains road network traffic information with high application value.In this study,an improved spatio⁃temporal attention transformer model(ISTA⁃transformer model)is proposed to provide a more accurate method for predicting multi⁃step short⁃term traffic flow based on monitoring data.By embedding a temporal attention layer and a spatial attention layer in the model,the model learns the relationship between traffic flows at different time intervals and different geographic locations,and realizes more accurate multi⁃step short⁃time flow prediction.Finally,we validate the superiority of the model with monitoring data spanning 15 days from 620 monitoring points in Qingdao,China.In the four time steps of prediction,the MAPE(Mean Absolute Percentage Error)values of ISTA⁃transformers prediction results are 0.22,0.29,0.37,and 0.38,respectively,and its prediction accuracy is usually better than that of six baseline models(Transformer,GRU,CNN,LSTM,Seq2Seq and LightGBM),which indicates that the proposed model in this paper always has a better ability to explain the prediction results with the time steps in the multi⁃step prediction. 展开更多
关键词 urban road network traffic flow prediction spatio⁃temporal feature ISTA⁃transformer model
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THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE NON-ISENTROPIC IMPROVED AW-RASCLE-ZHANG MODEL
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作者 蒋伟峰 陈停停 +1 位作者 李彤 王振 《Acta Mathematica Scientia》 SCIE CSCD 2023年第1期237-258,共22页
In this paper,we study the Radon measure initial value problem for the nonisentropic improved Aw-Rascle-Zhang model.For arbitrary convex F(u)in this model we construct the Riemann solutions by elementary waves andδ-s... In this paper,we study the Radon measure initial value problem for the nonisentropic improved Aw-Rascle-Zhang model.For arbitrary convex F(u)in this model we construct the Riemann solutions by elementary waves andδ-shock waves using the method of generalized characteristic analysis.We obtain the solutions constructively for initial data containing the Dirac measure by taking the limit of the solutions for that with three piecewise constants.Moreover,we analyze different kinds of wave interactions,including the interactions of theδ-shock waves with elementary waves. 展开更多
关键词 Riemann problem non-isentropic improved aw-rascle-Zhang model δ-shock wave wave interactions traffic flow
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Wave Interactions of the Aw-Rascle Model for Generalized Chaplygin Gas
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作者 Yujin Liu 《Journal of Applied Mathematics and Physics》 2021年第2期317-327,共11页
In this paper, we investigate the elementary wave interactions of the Aw-Rascle model for the generalized Chaplygin gas. We construct the unique solution by the characteristic analysis method and obtain the stability ... In this paper, we investigate the elementary wave interactions of the Aw-Rascle model for the generalized Chaplygin gas. We construct the unique solution by the characteristic analysis method and obtain the stability of the corresponding Riemann solutions under such small perturbations on the initial values. We find that the elementary wave interactions have a much more simple structure for Temple class than general systems of conservation laws. It is important to study the elementary waves interactions of the traffic flow system for the generalized Chaplygin gas not only because of their significance in practical applications in the traffic flow system, but also because of their basic role for the general mathematical theory. 展开更多
关键词 Wave Interaction aw-rascle model Generalized Chaplygin Gas Riemann Problem Delta Shock Hyperbolic Conservation Laws
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Decision model and algorithm for traffic rescue resource dispatching on expressway 被引量:6
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作者 柴干 朱苍晖 +1 位作者 万水 濮居一 《Journal of Southeast University(English Edition)》 EI CAS 2009年第2期252-256,共5页
In order to solve the problems of potential incident rescue on expressway networks, the opportunity cost-based method is used to establish a resource dispatch decision model. The model aims to dispatch the rescue reso... In order to solve the problems of potential incident rescue on expressway networks, the opportunity cost-based method is used to establish a resource dispatch decision model. The model aims to dispatch the rescue resources from the regional road networks and to obtain the location of the rescue depots and the numbers of service vehicles assigned for the potential incidents. Due to the computational complexity of the decision model, a scene decomposition algorithm is proposed. The algorithm decomposes the dispatch problem from various kinds of resources to a single resource, and determines the original scene of rescue resources based on the rescue requirements and the resource matrix. Finally, a convenient optimal dispatch scheme is obtained by decomposing each original scene and simplifying the objective function. To illustrate the application of the decision model and the algorithm, a case of the expressway network is studied on areas around Nanjing city in China and the results show that the model used and the algorithm proposed are appropriate. 展开更多
关键词 dispatch decision model scene decomposition algorithm traffic rescue resource EXPRESSWAY
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FEM tire model oriented to virtual experiment of off-road vehicle trafficability 被引量:2
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作者 庞罕 张为公 王霞 《Journal of Southeast University(English Edition)》 EI CAS 2007年第4期540-544,共5页
In order to estimate the trafficability of off-road vehicles, the linear relationships between the pressure and the stiffness of the tire and the action of the vertical tire force with the viscoelasticity are analyzed... In order to estimate the trafficability of off-road vehicles, the linear relationships between the pressure and the stiffness of the tire and the action of the vertical tire force with the viscoelasticity are analyzed. The method to improve the precision of the model by the coefficients is presented. The constitutive equation of the three-parameter linear model and the stiffness matrix of four-node isoparametric elements are derived to construct the FEM (finite element method) tire model in plan stress. A demarcation and verification system is designed based on the six-dimensional wheel force transducer and the vertical tire force is measured under different velocities. The results show that the model and the method proposed are reasonable. 展开更多
关键词 trafficABILITY tire model VISCOELASTICITY six-dimensional wheel force transducer
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Expressway traffic flow prediction using chaos cloud particle swarm algorithm and PPPR model 被引量:2
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作者 赵泽辉 康海贵 李明伟 《Journal of Southeast University(English Edition)》 EI CAS 2013年第3期328-335,共8页
Aiming at the real-time fluctuation and nonlinear characteristics of the expressway short-term traffic flow forecasting the parameter projection pursuit regression PPPR model is applied to forecast the expressway traf... Aiming at the real-time fluctuation and nonlinear characteristics of the expressway short-term traffic flow forecasting the parameter projection pursuit regression PPPR model is applied to forecast the expressway traffic flow where the orthogonal Hermite polynomial is used to fit the ridge functions and the least square method is employed to determine the polynomial weight coefficient c.In order to efficiently optimize the projection direction a and the number M of ridge functions of the PPPR model the chaos cloud particle swarm optimization CCPSO algorithm is applied to optimize the parameters. The CCPSO-PPPR hybrid optimization model for expressway short-term traffic flow forecasting is established in which the CCPSO algorithm is used to optimize the optimal projection direction a in the inner layer while the number M of ridge functions is optimized in the outer layer.Traffic volume weather factors and travel date of the previous several time intervals of the road section are taken as the input influencing factors. Example forecasting and model comparison results indicate that the proposed model can obtain a better forecasting effect and its absolute error is controlled within [-6,6] which can meet the application requirements of expressway traffic flow forecasting. 展开更多
关键词 expressway traffic flow forecasting projectionpursuit regression particle swarm algorithm chaoticmapping cloud model
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Gaussian mixture models for clustering and classifying traffic flow in real-time for traffic operation and management 被引量:1
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作者 孙璐 张惠民 +3 位作者 高荣 顾文钧 徐冰 陈鲤梁 《Journal of Southeast University(English Edition)》 EI CAS 2011年第2期174-179,共6页
Based on Gaussian mixture models(GMM), speed, flow and occupancy are used together in the cluster analysis of traffic flow data. Compared with other clustering and sorting techniques, as a structural model, the GMM ... Based on Gaussian mixture models(GMM), speed, flow and occupancy are used together in the cluster analysis of traffic flow data. Compared with other clustering and sorting techniques, as a structural model, the GMM is suitable for various kinds of traffic flow parameters. Gap statistics and domain knowledge of traffic flow are used to determine a proper number of clusters. The expectation-maximization (E-M) algorithm is used to estimate parameters of the GMM model. The clustered traffic flow pattems are then analyzed statistically and utilized for designing maximum likelihood classifiers for grouping real-time traffic flow data when new observations become available. Clustering analysis and pattern recognition can also be used to cluster and classify dynamic traffic flow patterns for freeway on-ramp and off-ramp weaving sections as well as for other facilities or things involving the concept of level of service, such as airports, parking lots, intersections, interrupted-flow pedestrian facilities, etc. 展开更多
关键词 traffic flow patterns Gaussian mixture model level of service data mining cluster analysis CLASSIFIER
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Research on a non-linear chaotic prediction model for urban traffic flow 被引量:4
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作者 黄鵾 陈森发 +1 位作者 周振国 亓霞 《Journal of Southeast University(English Edition)》 EI CAS 2003年第4期410-413,共4页
In order to solve serious urban transport problems, according to the proved chaotic characteristic of traffic flow, a non linear chaotic model to analyze the time series of traffic flow is proposed. This model recons... In order to solve serious urban transport problems, according to the proved chaotic characteristic of traffic flow, a non linear chaotic model to analyze the time series of traffic flow is proposed. This model reconstructs the time series of traffic flow in the phase space firstly, and the correlative information in the traffic flow is extracted richly, on the basis of it, a predicted equation for the reconstructed information is established by using chaotic theory, and for the purpose of obtaining the optimal predicted results, recognition and optimization to the model parameters are done by using genetic algorithm. Practical prediction research of urban traffic flow shows that this model has famous predicted precision, and it can provide exact reference for urban traffic programming and control. 展开更多
关键词 traffic flow chaotic theory phase reconstruction non linear genetic algorithm prediction model
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A NOTE ON THE INTERACTIONS OF ELEMENTARY WAVES FOR THE AR TRAFFIC FLOW MODEL WITHOUT VACUUM 被引量:4
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作者 孙梅娜 《Acta Mathematica Scientia》 SCIE CSCD 2011年第4期1503-1512,共10页
In this note, we consider the interactions of elementary waves for the traffic flow model proposed by Aw and Rascle when the vacuum is not involved. The solutions are obtained constructively and globally when the init... In this note, we consider the interactions of elementary waves for the traffic flow model proposed by Aw and Rascle when the vacuum is not involved. The solutions are obtained constructively and globally when the initial data consist of three pieces of constant states. Furthermore, it can be found that the Riemann solutions are stable with respect to such small perturbations of the initial data in this particular situation by investigating the limits of the solutions as the perturbed parameter ε goes to zero. 展开更多
关键词 interaction of elementary wave aw-rascle model Riemann problem traffic flow hyperbolic conservation laws
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Model predictive control for hybrid vehicle ecological driving using traffic signal and road slope information 被引量:11
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作者 Kaijiang YU Junqi YANG Daisuke YAMAGUCHI 《Control Theory and Technology》 EI CSCD 2015年第1期17-28,共12页
This paper presents development of a control system for ecological driving of a hybrid vehicle. Prediction using traffic signal and road slope information is considered to improve the fuel economy. It is assumed that ... This paper presents development of a control system for ecological driving of a hybrid vehicle. Prediction using traffic signal and road slope information is considered to improve the fuel economy. It is assumed that the automobile receives traffic signal information from intelligent transportation systems (ITS). Model predictive control is used to calculate optimal vehicle control inputs using traffic signal and road slope information. The performance of the proposed method was analyzed through computer simulation results. Both the fuel economy and the driving profile are optimized using the proposed approach. It was observed that fuel economy was improved compared with driving of a typical human driving model. 展开更多
关键词 Ecological driving model predictive control intelligent transportation systems traffic signal optimal control
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Stabilisation analysis of multiple car-following model in traffic flow 被引量:8
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作者 彭光含 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第5期434-441,共8页
An improved multiple car-following model is proposed by considering the arbitrary number of preceding cars, which includes both the headway and the velocity difference of multiple preceding cars. The stability conditi... An improved multiple car-following model is proposed by considering the arbitrary number of preceding cars, which includes both the headway and the velocity difference of multiple preceding cars. The stability condition of the extended model is obtained by using the linear stability theory. The modified Korteweg-de Vries equation is derived to describe the traffic behaviour near the critical point by applying the nonlinear analysis. Traffic flow can be also divided into three regions: stable metastable and unstable regions. Numerical simulation is in accordance with the analytical result for the model. And numerical simulation shows that the stabilisation of traffic is increasing by considering the information of more leading cars and there is unavoidable effect on traffic flow from the multiple leading cars information. 展开更多
关键词 traffic flow optimal velocity model numerical simulation
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A Survey of Model Predictive Control Methods for Traffic Signal Control 被引量:14
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作者 Bao-Lin Ye Weimin Wu +4 位作者 Keyu Ruan Lingxi Li Tehuan Chen Huimin Gao Yaobin Chen 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2019年第3期623-640,共18页
Enhancing traffic efficiency and alleviating(even circumventing)traffic congestion with advanced traffic signal control(TSC)strategies are always the main issues to be addressed in urban transportation systems.Since m... Enhancing traffic efficiency and alleviating(even circumventing)traffic congestion with advanced traffic signal control(TSC)strategies are always the main issues to be addressed in urban transportation systems.Since model predictive control(MPC)has a lot of advantages in modeling complex dynamic systems,it has been widely studied in traffic signal control over the past 20 years.There is a need for an in-depth understanding of MPC-based TSC methods for traffic networks.Therefore,this paper presents the motivation of using MPC for TSC and how MPC-based TSC approaches are implemented to manage and control the dynamics of traffic flows both in urban road networks and freeway networks.Meanwhile,typical performance evaluation metrics,solution methods,examples of simulations,and applications related to MPC-based TSC approaches are reported.More importantly,this paper summarizes the recent developments and the research trends in coordination and control of traffic networks with MPC-based TSC approaches.Remaining challenges and open issues are discussed towards the end of this paper to discover potential future research directions. 展开更多
关键词 Autonomous vehicles coordination control mixed integer programming model predictive control system decomposition traffic flow models traffic signal control
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Short-term traffic safety forecasting using Gaussian mixture model and Kalman filter 被引量:6
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作者 Sheng JIN Dian-hai WANG +1 位作者 Cheng XU Dong-fang MA 《Journal of Zhejiang University-Science A(Applied Physics & Engineering)》 SCIE EI CAS CSCD 2013年第4期231-243,共13页
In this paper,a prediction model is developed that combines a Gaussian mixture model(GMM) and a Kalman filter for online forecasting of traffic safety on expressways.Raw time-to-collision(TTC) samples are divided into... In this paper,a prediction model is developed that combines a Gaussian mixture model(GMM) and a Kalman filter for online forecasting of traffic safety on expressways.Raw time-to-collision(TTC) samples are divided into two categories:those representing vehicles in risky situations and those in safe situations.Then,the GMM is used to model the bimodal distribution of the TTC samples,and the maximum likelihood(ML) estimation parameters of the TTC distribution are obtained using the expectation-maximization(EM) algorithm.We propose a new traffic safety indicator,named the proportion of exposure to traffic conflicts(PETTC),for assessing the risk and predicting the safety of expressway traffic.A Kalman filter is applied to forecast the short-term safety indicator,PETTC,and solves the online safety prediction problem.A dataset collected from four different expressway locations is used for performance estimation.The test results demonstrate the precision and robustness of the prediction model under different traffic conditions and using different datasets.These results could help decision-makers to improve their online traffic safety forecasting and enable the optimal operation of expressway traffic management systems. 展开更多
关键词 Forecasting traffic safety Gaussian mixture model Kalman filter
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Small-time scale network traffic prediction based on a local support vector machine regression model 被引量:10
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作者 孟庆芳 陈月辉 彭玉华 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第6期2194-2199,共6页
In this paper we apply the nonlinear time series analysis method to small-time scale traffic measurement data. The prediction-based method is used to determine the embedding dimension of the traffic data. Based on the... In this paper we apply the nonlinear time series analysis method to small-time scale traffic measurement data. The prediction-based method is used to determine the embedding dimension of the traffic data. Based on the reconstructed phase space, the local support vector machine prediction method is used to predict the traffic measurement data, and the BIC-based neighbouring point selection method is used to choose the number of the nearest neighbouring points for the local support vector machine regression model. The experimental results show that the local support vector machine prediction method whose neighbouring points are optimized can effectively predict the small-time scale traffic measurement data and can reproduce the statistical features of real traffic measurements. 展开更多
关键词 network traffic small-time scale nonlinear time series analysis support vector machine regression model
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Cellular automata model for traffic flow with safe driving conditions 被引量:4
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作者 María Elena Lárraga Luis Alvarez-Icaza 《Chinese Physics B》 SCIE EI CAS CSCD 2014年第5期216-226,共11页
In this paper, a recently introduced cellular automata (CA) model is used for a statistical analysis of the inner micro-scopic structure of synchronized traffic flow. The analysis focuses on the formation and dissol... In this paper, a recently introduced cellular automata (CA) model is used for a statistical analysis of the inner micro-scopic structure of synchronized traffic flow. The analysis focuses on the formation and dissolution of clusters or platoons of vehicles, as the mechanism that causes the presence of this synchronized traffic state with a high flow. This platoon formation is one of the most interesting phenomena observed in traffic flows and plays an important role both in manual and automated highway systems (AHS). Simulation results, obtained from a single-lane system under periodic boundary conditions indicate that in the density region where the synchronized state is observed, most vehicles travel together in pla- toons with approximately the same speed and small spatial distances. The examination of velocity variations and individual vehicle gaps shows that the flow corresponding to the synchronized state is stable, safe and highly correlated. Moreover, results indicate that the observed platoon formation in real traffic is reproduced in simulations by the relation between vehicle headway and velocity that is embedded in the dynamics definition of the CA model. 展开更多
关键词 traffic flow models synchronized traffic cellular automata platoons formation
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Feedback control scheme of traffic jams based on the coupled map car-following model 被引量:6
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作者 周桐 孙棣华 +1 位作者 赵敏 李华民 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第9期203-208,共6页
Based on the pioneering work of Konishi et al. [Phys. Rev. E (1999) 60 4000], a new feedback control scheme is presented to suppress traffic jams based on the coupled map car-following model under the open boundary ... Based on the pioneering work of Konishi et al. [Phys. Rev. E (1999) 60 4000], a new feedback control scheme is presented to suppress traffic jams based on the coupled map car-following model under the open boundary condition. The effect of the safe headway on the traffic system is considered. According to the control theory, the condition under which traffic jams can be suppressed is analyzed. The results are compared with the previous results concerning congestion control. The simulations show that the suppression performance of our scheme on traffic jams is better than those of the previous schemes, although all the schemes can suppress traffic jams. The simulation results are consistent with theoretical analyses. 展开更多
关键词 traffic jam coupled map car-following model control scheme safe headway
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A new car-following model with consideration of the traffic interruption probability 被引量:5
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作者 唐铁桥 黄海军 +1 位作者 黄仕进 姜锐 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第3期975-983,共9页
In this paper, we present a new car-following model by taking into account the effects of the traffic interruption probability on the car-following behaviour of the following vehicle. The stability condition of the mo... In this paper, we present a new car-following model by taking into account the effects of the traffic interruption probability on the car-following behaviour of the following vehicle. The stability condition of the model is obtained by using the linear stability theory. The modified Korteweg-de Vries (KdV) equation is constructed and solved, and three types of traffic flows in the headway sensitivity space-stable, metastable, and unstable--are classified. Both the analytical and simulation results show that the traffic interruption probability indeed has an influence on driving behaviour, and the consideration of traffic interruption probability in the car-following model could stabilize traffic flow. 展开更多
关键词 car-following model traffic interruption probability STABILITY
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