DRIVIN - Modeling Drivers' Acceleration and Lane Changing...

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Unformatted text preview: Modeling Drivers' Acceleration and Lane Changing Behavior by Kazi Iftekhar Ahmed B. Sc. Eng. Civil Bangladesh Univ. of Eng. and Technology BUET, Dhaka, Bangladesh 1991 M.S. in Transportation Massachusetts Institute of Technology, Cambridge, MA 1996 Submitted to the Department of Civil and Environmental Engineering in partial ful llment of the requirements for the degree of Doctor of Science in Transportation Systems and Decision Sciences at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1999 c Massachusetts Institute of Technology 1999. All rights reserved. Author Certi ed by Certi ed by Accepted by ............................................ Department of Civil and Environmental Engineering January 8, 1999 ............................................ Moshe E. Ben-Akiva Professor of Civil and Environmental Engineering Thesis Supervisor ............................................ Dr. Haris N. Koutsopoulos Operations Research Analyst Thesis Supervisor ............................................ Andrew J. Whittle Chairman, Departmental Committee on Graduate Studies Modeling Drivers' Acceleration and Lane Changing Behavior by Kazi Iftekhar Ahmed Submitted to the Department of Civil and Environmental Engineering on January 8, 1999, in partial ful llment of the requirements for the degree of Doctor of Science in Transportation Systems and Decision Sciences Abstract This thesis contributes to the development of microscopic tra c performance models which includes the acceleration and lane changing models. It enhances the existing models and develops new ones. Another major contribution of this thesis is the empirical work, i.e., estimating the models using statistically rigorous methods and microscopic data collected from real tra c. The acceleration model de nes two regimes of tra c ow: the car following regime and the free ow regime. In the car following regime, a driver is assumed to follow his her leader, while in the free ow regime, a driver is assumed to try to attain his her desired speed. A probabilistic model, that is based on a time headway threshold, is used to determine the regime the driver belongs to. Heterogeneity across drivers is captured through the headway threshold and reaction time distributions. The parameters of the car following and free ow acceleration models along with the headway threshold and reaction time distributions are jointly estimated using the maximum likelihood estimation method. The lane changing decision process is modeled as a sequence of three steps: decision to consider a lane change, choice of a target lane, and gap acceptance. Since acceptable gaps are hard to nd in a heavily congested tra c, a forced merging model that captures forced lane changing behavior and courtesy yielding is developed. A discrete choice model framework is used to model the impact of the surrounding tra c environment and lane con guration on drivers' lane changing decision process. The models are estimated using actual tra c data collected from Interstate 93 at the Central Artery, located in downtown Boston, MA, USA. In addition to assessing the model parameters from statistical and behavioral standpoints, the models are validated using a microscopic tra c simulator. Overall, the empirical results are encouraging, and demonstrate the e ectiveness of the modeling framework. Thesis Supervisor: Moshe E. Ben-Akiva Title: Professor of Civil and Environmental Engineering Massachusetts Institute of Technology Thesis Supervisor: Dr. Haris N. Koutsopoulos Title: Operations Research Analyst Volpe National Transportation Systems Center Cambridge, MA, USA. 4 To Abbu, Ammu, my son, Sabih, and my wife, Lubna 5 Thesis Committee Moshe E. Ben-Akiva Chairman Professor Department of Civil and Environmental Engineering Massachusetts Institute of Technology Haris N. Koutsopoulos Operations Research Analyst Volpe National Transportation Systems Center Ismail Chabini Assistant Professor Department of Civil and Environmental Engineering Massachusetts Institute of Technology Mithilesh Jha Research Associate Center for Transportation Studies Massachusetts Institute of Technology 6 Acknowledgments I acknowledge with deep sense of gratitude the guidance, invaluable advice, and constant inspiration provided by my supervisors Prof. Moshe Ben Akiva and Dr. Haris Koutsopoulos during the course of my studies. I feel privileged to get the opportunity to work with them for the last ve years. I have learned a lot from them during the course of this research. I am grateful to the other members of my dissertation committee|Prof. Ismail Chabini and Dr. Mithilesh Jha, for their advice, feed back, and inspiration during the course of this research. My special thanks goes to the following individuals without whose contribution this thesis could not be completed: Dr. Qi Yang, Dr. Kalidas Ashok, Prof. Rabi Mishalani, Prof. Michel Bierlaire, Alan Chachich, Dave Cuneo, Masroor Hasan, Dr. Owen Chen, Russel Spieler, Tania Amin, Khwaja Ehsan, Shahnaz Islam, and Prof. Sha qul Islam. I am also thankful to the CA T project at the ITS Research Program for nancially supporting my ve years of studies at MIT. I would like to thank my friends, fellow students, and administrative sta at the CEE Department, CTS, and ITS O ce, that made my life at MIT an enjoyable experience, especially, Adriana, Amalia, Andras, Atul, Bruno, Cheryl, Chris, Cynthia, Denise, Didier, Dinesh, Dale, Deiki, Dong, Hari, Hong, Je , Jessei, John, Jon, Joan, Juli, Krishna, Lisa, Mark, Masih, Nagi, Niranjan, Pat, Paula, Peter, Prodyut Da, Shenoi, Scott, Sreeram, Sridevi, Sudhir, Susan, Tomer, Winston, and Yan. Thanks are due to fellow Bangladeshis Adnan, Fahria and Zeeshan, Minu and Monjur, Oni and Arif, Rima, Rita and Mukul, Rumi and Saquib, Sabah and Mahmood, and Shampa and Sabet, for their friendship and support. Finally, I wish I knew a better way to express my indebtedness to my wife, Lubna, my three year old son, Sabih, for their unconditional support, endless love, to my parents for their encouragement and inspiration throughout my life that helped me outgrow again and again. 7 Contents 1 Introduction 1.1 1.2 1.3 1.4 1.5 The Problem . . . . Motivation . . . . . . Thesis Objectives . . Thesis Contributions Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Literature Review 2.1 Acceleration Models . . . . . . . . . . . . . . . 2.1.1 Car Following Models . . . . . . . . . . 2.1.2 General Acceleration Models . . . . . . . 2.1.3 Estimation of the Brake Reaction Time . 2.2 Lane Changing Models . . . . . . . . . . . . . . 2.2.1 Gap Acceptance Models . . . . . . . . . 2.3 Summary . . . . . . . . . . . . . . . . . . . . . 3 The Acceleration Model 3.1 Introduction . . . . . . . . . . . . . . . . . . 3.2 The Acceleration Model . . . . . . . . . . . 3.2.1 The Car Following Model . . . . . . 3.2.2 The Free Flow Acceleration Model . 3.2.3 The Headway Threshold Distribution 3.2.4 The Reaction Time Distribution . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 18 19 21 22 24 25 25 26 34 37 38 42 44 46 46 48 49 54 56 57 3.3 Likelihood Function Formulation . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Lane Changing Model 4.1 Introduction . . . . . . . . . . . . . . . . 4.2 The Lane Changing Model . . . . . . . . 4.2.1 Conceptual Framework . . . . . . 4.2.2 Model Formulation . . . . . . . . 4.2.3 Likelihood Function Formulation 4.2.4 Discussions . . . . . . . . . . . . 4.3 The Forced Merging Model . . . . . . . . 4.3.1 Conceptual Framework . . . . . . 4.3.2 Model Formulation . . . . . . . . 4.3.3 Likelihood Function Formulation 4.3.4 Discussion . . . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Data Requirements for Estimating Driver Behavior Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 61 63 64 65 65 67 71 73 75 76 77 78 84 84 85 5.1 Methodology for Estimating Instantaneous Speed and Acceleration from Discrete Trajectory Data . . . . . . . . . . . . . . . . . . . . . . 86 5.1.1 The Local Regression Procedure . . . . . . . . . . . . . . . . . 87 5.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2.1 Description of the Data Collection Site . . . . . . . . . . . . . 91 5.2.2 Video Processing Software . . . . . . . . . . . . . . . . . . . . 93 5.2.3 Processing the Tra c Data . . . . . . . . . . . . . . . . . . . 94 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Estimation Results 107 6.1 Estimation Results of the Acceleration Model . . . . . . . . . . . . . 107 6.1.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Estimation Results of the Lane Changing Model . . . . . . . . . . . . 121 9 6.2.1 Estimation Results of the Discretionary Lane Changing Model 6.2.2 Estimation Results of the Mandatory Lane Changing Model . 6.2.3 Estimation Results of the Forced Merging Model . . . . . . . 6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Model Validation Using a Microscopic Tra c Simulator 7.1 MITSIM: a Microscopic Tra c Simulator . . 7.1.1 The Acceleration Model . . . . . . . 7.1.2 The Lane Changing Model . . . . . . 7.2 Validation Methodology . . . . . . . . . . . 7.2.1 Number of Replications . . . . . . . 7.2.2 Measures of Goodness of t . . . . . 7.3 Case Study . . . . . . . . . . . . . . . . . . 7.3.1 The Network . . . . . . . . . . . . . 7.3.2 Tra c Data . . . . . . . . . . . . . . 7.3.3 O D Estimation from Tra c Counts 7.3.4 MITSIM Modi cations . . . . . . . . 7.3.5 Experimental Design . . . . . . . . . 7.3.6 Validation Results . . . . . . . . . . 7.4 Conclusions . . . . . . . . . . . . . . . . . . 8 Conclusions and Future Research Directions 8.1 Summary of Research . . . . . . . . . 8.1.1 The Acceleration Model . . . 8.1.2 The Lane Changing Model . . 8.1.3 Validation by Microsimulation 8.2 Contributions . . . . . . . . . . . . . 8.3 Future Research Directions . . . . . . 8.3.1 Modeling . . . . . . . . . . . 8.3.2 Estimation and Validation . . 8.4 Conclusion . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 133 140 144 146 147 148 149 153 153 154 157 157 159 160 163 164 164 170 173 173 173 175 176 176 178 178 179 180 A Speci cation of the Random Utility Model Appropriate for Panel Data 181 B Calibration of the Simulation Model Parameters 183 Bibliography 185 11 List of Figures 2-1 The subject and the front vehicle. . . . . . . . . . . . . . . . . . . . . 2-2 De nition of reaction time corresponding to the four actions source: Ozaki, 1993. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 The subject, lead, lag, and front vehicles, and the lead and lag gaps. . 25 3-1 The subject and the front vehicle. . . . . . . . . . . . . . . . . . . . . 3-2 Impact of the relative speed on drivers' acceleration decision. . . . . . 47 50 4-1 The lane changing model structure. . . . . . . . . . . . . . . . . . . . 4-2 The subject, lead, lag, and front vehicles, and the lead and lag gaps. . 4-3 The lane changing decision tree for a driver driving in a two lane roadway and possible states of the driver. . . . . . . . . . . . . . . . . . . 4-4 De nition of the adjacent gap. . . . . . . . . . . . . . . . . . . . . . . 4-5 The forced merging model structure. . . . . . . . . . . . . . . . . . . 4-6 Initial state of the driver for the forced merging model for di erent cases. F 4-7 De nition of n M t for the forced merging model. . . . . . . . . . . . 65 69 5-1 An example of estimation of instantaneous speed and acceleration from discrete position measurements. . . . . . . . . . . . . . . . . . . . . . 5-2 The weight function and the tted curve for an observation at time period 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 Schematic diagram of the I 93 southbound data collection site  gure not drawn to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Flow, density, and average speed of the I 93 southbound trajectory data. 12 32 40 74 75 76 79 83 90 91 92 95 5-5 Histograms of the absolute values of the position estimation error using di erent window sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6 Estimated speed and acceleration pro les using di erent window sizes. 5-7 Examples of curve tting by local regression. . . . . . . . . . . . . . . 5-8 Histograms of the acceleration, subject speed, relative speed, time and space headway, and density in the data used for estimating the acceleration model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9 The subject and the front, lead, and lag vehicles. . . . . . . . . . . . 5-10 The subject, lead, lag, and front vehicles, and the lead and lag gaps. . 6-1 The likelihood function as a function of  . . . . . . . . . . . . . . . . 6-2 Sensitivity of di erent factors on the car following acceleration and deceleration decisions. . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3 Comparison between the car following acceleration and deceleration estimated in this thesis with those obtained by Subramanian 1996. . 6-4 The headway threshold distribution and the probability of car following as a function of time headway. . . . . . . . . . . . . . . . . . . . . . . 6-5 Comparison between the estimated mean headway threshold and the 61 meters threshold suggested by Herman and Potts 1961. . . . . . 6-6 The probability density function and the cumulative distribution function of the reaction time. . . . . . . . . . . . . . . . . . . . . . . . . . 6-7 Schematic diagram of the I 93 southbound data collection site  gure not drawn to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8 The decision tree for a driver considering a discretionary lane change with the current and the left lanes as choice set. . . . . . . . . . . . . 6-9 The subject and the front, lead, and lag vehicles. . . . . . . . . . . . 6-10 The estimated probability of acceptance of gaps that were acceptable and merging were completed. . . . . . . . . . . . . . . . . . . . . . . 6-11 The median lead and lag critical gaps for discretionary lane change as a function of relative speed. . . . . . . . . . . . . . . . . . . . . . . . 13 97 98 99 100 102 104 109 114 115 117 118 119 122 123 126 131 132 6-12 The decision tree for a driver merging from an on ramp to the adjacent mainline lane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13 The subject, lead, lag, and front vehicles, and the lead and lag gaps. . 6-14 The probability of responding to MLC as a function of delay. . . . . 6-15 The estimated probability of acceptance of gaps that were acceptable and merging were completed. . . . . . . . . . . . . . . . . . . . . . . 6-16 The mean lag critical gap for mandatory lane change as a function of lag relative speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17 Comparison between the estimated critical gap lengths under DLC and MLC situations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18 Remaining distance versus explanatory variable remaining distance impact, the utility function, and the estimated probability of being in state M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1 7-2 7-3 7-4 7-5 7-6 7-7 7-8 7-9 7-10 133 137 137 138 139 140 143 The network used in the validation exercise. . . . . . . . . . . . . . . 157 Schematic diagram of the on ramp and Storrow Drive merging area. . 158 Flow of tra c entering the network. . . . . . . . . . . . . . . . . . . . 160 O D estimation from tra c counts for the case study. . . . . . . . . . 161 Comparison of average speeds obtained from di erent versions of MITSIM for p = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Comparison of average speeds obtained from di erent versions of MITSIM for p = 85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Comparison of average speeds obtained from di erent versions of MITSIM for p = 70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Comparison of the real tra c counts with those obtained from di erent versions of MITSIM for p = 100. . . . . . . . . . . . . . . . . . . . 170 Comparison of the real tra c counts with those obtained from di erent versions of MITSIM for p = 85. . . . . . . . . . . . . . . . . . . . . 171 Comparison of the real tra c counts with those obtained from di erent versions of MITSIM for p = 70. . . . . . . . . . . . . . . . . . . . . 172 14 B-1 Model parameter calibration approach. . . . . . . . . . . . . . . . . . 183 15 List of Tables 2.1 Estimation results of the model developed by Gazis et al. 1959. . . 2.2 Estimation results of the GM Model by May and Keller 1967. . . . 2.3 Estimation results of the GM Model by Subramanian 1996. . . . . . 28 31 37 4.1 Possible decision state sequences of observing a lane change by forced merging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.1 Description of the collected tra c video. . . . . . . . . . . . . . . . . 5.2 Statistics of the data used for estimating the acceleration model. . . . 5.3 Statistics of the discretionary lane changing model data corresponding to the gaps that the drivers merged into. . . . . . . . . . . . . . . . . 5.4 Statistics of the mandatory lane changing model data corresponding to the gaps that the drivers merged into. . . . . . . . . . . . . . . . . 5.5 Statistics of the data used for estimating the forced merging model. . 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 93 101 103 104 105 Estimated likelihood function for di erent values of h ; h ; and max .108 min max Estimation results of the acceleration model. . . . . . . . . . . . . . . 110 Estimation results of the acceleration model for  = 1. . . . . . . . . . 112 Comparison between the reaction time distribution parameters obtained from di erent sources. . . . . . . . . . . . . . . . . . . . . . . 120 Estimation results of the discretionary lane changing model. . . . . . 125 Estimation results of the discretionary lane changing model. . . . . . 127 Estimation results of the mandatory lane changing model. . . . . . . 134 Estimation results of the mandatory lane changing model. . . . . . . 136 16 6.9 Estimation results of the forced merging model. . . . . . . . . . . . . 141 6.10 Estimation results of the forced merging model. . . . . . . . . . . . . 142 7.1 The cumulative distribution of speed that is added to the posted speed limit to obtain the desired speed. . . . . . . . . . . . . . . . . . . . . 149 7.2 Maximum acceleration m s2. . . . . . . . . . . . . . . . . . . . . . . 150 7.3 Summary statistics of the comparison of the eld observed counts with those obtained from di erent versions of MITSIM using three di erent O D sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 17 Chapter 1 Introduction 1.1 The Problem Tra c congestion in and around the urban areas of the world is a major problem. Congestion during peak hours extends for longer periods each day. Congestion adversely a ects mobility, safety, and air quality. These cause direct economic losses due to delays and accidents, and indirect economic losses due to environmental impact. In most cases, the capacity of the existing roadway systems cannot be increased by adding additional lanes due to space, resource, or environmental constraints. Potential ways to address the congestion problem are to improve the utilization of the existing systems through better tra c management and operations strategies, and improve the geometric design of roads and highways. Tra c operations in the congested sections of roadways is very complex, since di erent drivers employ di erent techniques to travel through such sections while interacting with other drivers. To understand the occurrence of bottlenecks and to devise solutions for it, a comprehensive analysis of vehicle to vehicle interactions is essential. This requires the development of tra c theories to explain driver behavior at the microscopic level, the main elements of which are the acceleration and lane changing dimensions. Drivers' acceleration behavior, when they are in the car following regime, has been studied extensively since the 1950s. In this regime, drivers are assumed to 18 follow their leaders. However, estimation of these models using microscopic data, for example, speed of a subject and its leader, gap length, acceleration applied by the subject, collected from real tra c has not received much attention. On the other hand, researchers started paying attention to the acceleration behavior in the free ow regime beginning early 1980s. In the free ow regime, drivers are not close to their leaders and therefore, have the freedom to attain their desired speed. The parameters of the general acceleration model, that captures drivers' acceleration decision in both the car following and free ow regimes, have not been estimated. The principal focus of research in modeling drivers' lane changing behavior has been on modeling the gap acceptance behavior at stop controlled T intersections. The gap acceptance phase is a part of the lane changing process. Researchers started paying attention to the lane changing model as microscopic tra c simulation emerged as an important tool for studying tra c behavior and developing and evaluating different tra c control and management strategies. However, the existing lane changing models are rule based and do not explicitly capture variability within driver and between drivers. Furthermore, the model parameters have not been estimated formally. In this thesis, we present a comprehensive framework for modeling drivers' acceleration and lane changing behavior. This includes enhancing existing models, developing new ones, providing framework for model estimation, and nally, estimating the models using statistically rigorous method and microscopic data collected from real tra c. 1.2 Motivation Research in Intelligent Transportation Systems ITS is being performed to develop tra c management and operations strategies to deal with problems associated with congestion. The number of strategies needed to be tested for a transportation system may be large and eld testing would be prohibitively expensive. For this purpose, `microscopic tra c simulation' is a suitable tool. An important element of a tra c simulator is the set of driver behavior models that is used to move vehicles in the 19 network. This includes the acceleration1 and lane changing models. Reliability of simulation results depends heavily on these underlying driver behavior models. Near on and o ramps or weaving sections, drivers often change to the lanes that are connected to their destinations. These areas are potential locations for bottleneck formation when the fraction of drivers trying to change lanes is high. Lane changing operations are critical in selecting geometric con guration of such areas AASHTO 1990. Drivers' lane changing behavior has direct in uence on the capacity and safety of such areas HCM 1985. Therefore, a detailed understanding of drivers lane changing behavior is necessary. Tra c engineers use the mean of the minimum acceptable gap length at intersections to estimate the capacities of and delays at intersections and pedestrian crossings. Therefore, the mean has to be estimated as accurately as possible; this requires a thorough understanding of the gap acceptance process. Gap acceptance behavior also a ects the design of the length of an acceleration lane which is an important design element from capacity and safety perspectives. Microscopic driver behavior models play a very important role in the analysis of tra c ow characteristics in the presence of ITS technologies, such as lane use sign, variable message signs, tra c control, and route guidance. Macroscopic speed ow density relationship assumes homogeneous speed and density for a given freeway segment and treats capacity as an exogenous parameter. In the presence of ITS technologies, these assumptions may not be realistic Yang 1993. Capacity can be in uenced not only by drivers' acceleration pattern, but also by the number of lane changes taking place. A better understanding of driver acceleration and lane changing behavior is, therefore, essential to model the impact of the ITS Technologies on the tra c ow relationships. Rear end collision accounted for 2.2 million automobile crashes in 1990, which was 19 of the total number of crashes in the US in that year NSC 1992. NSC 1992 also reported that nearly half of these crashes were due to drivers following their leaders too closely. In such cases, drivers are not able to decelerate fast enough 1 Acceleration refers to both acceleration and deceleration unless deceleration is mentioned. 20 when their leaders decelerate at unexpectedly high rates. Studying safety in the car following situations is, therefore, very important to the design of an Automated Highway Systems and Intelligent Cruise Control Chen 1996. To evaluate safety in car following situations, a detailed understanding of drivers car following behavior and braking reaction time is required. In conclusion, there is a need for improving the current understanding of drivers' acceleration and lane changing behavior at a microscopic level. 1.3 Thesis Objectives The main objective of this thesis is to advance the state of the art in modeling drivers' acceleration and lane changing behavior. The models need to be estimated using real driver data and have to be assessed from statistical and behavioral standpoints. The acceleration model should capture drivers acceleration behavior in both the car following and free ow regimes. In the car following regime, drivers follow their leaders and try to match their leaders' speed, whereas, in the free ow regime, they try to attain their desired speed. The headway threshold, that is used to determine the regime a driver belongs to, should be modeled as a random variable to capture variability between drivers. In addition, the reaction time or the time lag of response to stimulus should be modeled to be sensitive to the tra c conditions. Furthermore, the sensitivity of di erent factors on the car following acceleration and deceleration decisions may not be same, di erent set of parameters should be allowed while estimating the models. Modeling a lane changing decision process is very complex due to its latent nature and the number of factors a driver considers before reaching a decision. The only observable part is a successful lane change operation. The exact time at which a driver decides to change lanes cannot be observed except in a few specialized situations, for example, turning left right at an intersection. In addition, the in uence of past lane changes, such as time elapsed since the most recent lane change, on the current lane changing behavior further complicates the modeling of such a process. Therefore, the 21 modeling e ort should nd a balance between simplicity in modeling and representing reality. 1.4 Thesis Contributions This thesis advances the state of the art in modeling drivers' acceleration and lane changing behavior. It enhances the existing models and develops new ones. Another major contribution of this thesis is the empirical work, i.e., estimating the models using statistically rigorous methods and microscopic data collected from real tra c. More speci cally, Contribution to the modeling framework: The car following model, which captures drivers' acceleration behavior when they are following their leader, is extended by assuming that the stimulus is a nonlinear function of the lead relative speed and capturing the impact of tra c conditions ahead of the driver. These are signi cant improvements over the existing models that restrict the impact of the lead relative speed the stimulant on the acceleration response to be linear and do not model the impact of tra c conditions ahead of the driver except for the position and speed of the leader. The existing models restricts the lead relative speed the stimulant and other factors such as subject speed, gap in front of the subject that a ect the acceleration decision to be observed at the same time. This corresponds to an assumption that drivers base their decisions on the tra c environment at the time they were stimulated into action. We relax this assumption by allowing drivers to update their perception of the tra c environment during the decision making process. A headway threshold distribution is introduced that allows any driver behavior to be captured aggressive or conservative. The headway threshold 22 de nes whether a driver is following its leader or trying to attain its desired speed. An individual driver speci c reaction time is introduced which is allowed to be sensitive to the tra c situations under consideration. A probabilistic lane changing model is developed that captures drivers' lane changing behavior under both the mandatory and discretionary lane changing situations. This is a signi cant improvement over the existing deterministic rule based lane changing models. The proposed lane changing model allows for di erent gap acceptance model parameters for mandatory and discretionary lane changing situations. It also captures the variability within driver and between drivers in the lane changing decision process. A forced merging model is proposed that captures merging in heavily congested tra c by gap creation either through force or through courtesy yielding. Contribution to model estimation: A methodology to estimate instantaneous speed and acceleration that is required for model estimation from discrete trajectory data that can be obtained from real tra c is developed. All the components of the acceleration model are estimated jointly using real microscopic tra c data. The component models are the car following acceleration and deceleration models, the free ow acceleration model, and the headway threshold and reaction time distributions. Estimation results demonstrate the robustness of the modeling framework. Separate car following model parameters under acceleration and deceleration situations are allowed in the estimation. This captures the fact that, the sensitivity of di erent factors on drivers' acceleration behavior may not be same under these two situations. 23 Separate gap acceptance models for the mandatory and discretionary lane changing situations are estimated. The proposed lane changing model and the forced merging model are estimated using the maximum likelihood estimation method. 1.5 Thesis Outline In Chapter 2, a literature review of the existing acceleration and lane changing models is presented. The acceleration and the lane changing models are presented in chapters 3 and 4 respectively. In Chapter 5, data needs of this research is presented. First, a methodology to estimate instantaneous speed and acceleration from discrete trajectory data is presented. Then, the data source and the data extracted from this source to estimate di erent driver behavior models are presented. Estimation results of all the models described in chapters 3 and 4 are presented in Chapter 6. In Chapter 7, validation of the acceleration model and a part of the lane changing model, using a microscopic tra c simulator, is presented. Conclusions and directions for future research are presented in Chapter 8. 24 Chapter 2 Literature Review In this chapter, a literature review of the acceleration and lane changing models is presented. Findings from this review are summarized at the end of the chapter. 2.1 Acceleration Models The models capturing drivers' acceleration behavior can be classi ed as: Car following models, General acceleration models. The car following models capture acceleration behavior in the car following regime. In this regime, the drivers are close to their leaders and follow their leaders see Figure 2-1. The general acceleration models capture acceleration behavior in both subject front vehicle or leader space headway Figure 2-1: The subject and the front vehicle. 25 the car following and free ow regimes. In the free ow regime, drivers are not close to their leaders and therefore, have the freedom to attain their desired speed. Drivers' acceleration behavior, when they are in the car following regime, has been studied extensively since the 1950s. Estimation of these models using microscopic data, for example, speed of a subject and its leader, gap length, acceleration applied by the subject, has not received much attention. Simple correlation analysis was used to estimate the models in most cases. Researchers started paying attention to the acceleration behavior in the free ow regime in the early 1980s as microscopic simulation emerged as an important tool for studying tra c behavior and developing and evaluating di erent tra c control and management strategies. However, the parameters of a general acceleration model, that captures drivers' acceleration behavior in both the car following and free ow regimes, have not been estimated. Previous research on each of these categories and the estimation of the brake reaction time is presented next. 2.1.1 Car Following Models The general form of the car following models developed in the late 1950s is as follows: responsent = sensitivityn t , n   stimulusnt , n  2.1 where, t = time of observation, n = reaction time for driver n, responsent = acceleration applied at time t. The reaction time, n, includes the perception time time from the presentation of the stimulus until the foot starts to move and the foot movement time. The front 26 relative speed1 see Figure 2-1 is generally considered as the stimulus and sensitivity is a proportionality factor that may be a function of factors such as subject speed, space headway. Chandler et al. 1958 developed the rst car following model that is a simple linear model. Mathematically, the model can be expressed as ant = Vnfrontt , n  2.2 where, an t = = Vnfront t , n  = Vnt , n  = Vnfront t , n  = acceleration applied by driver n at time t, constant, Vnfront t , n , Vnt , n : stimulus, subject speed at time t , n, leader or front vehicle speed at time t , n . A driver responds to the stimulus at time t , n  by applying acceleration at time t. The same sensitivity terms are used for both the acceleration and deceleration situations. They estimated the model using the correlation analysis method and microscopic car following data. The data was collected from a sample of 8 drivers driving test vehicles in a two lane two way road in real tra c for 20 to 30 minutes. For each driver, the data included discrete measurements of the acceleration, speed, space headway, and relative speed over the time of observation. For di erent values of and , correlations between the observed and the estimated accelerations were computed. The values of and that yielded the highest correlation were used as the estimates of and for each driver. The estimated and averaged over all samples were 1.5 seconds and 0.37 second,1 respectively. A major limitation of the above model is the assumption of a constant sensitivity In this thesis, relative speed with respect to another vehicle is de ned as the speed of that vehicle less the speed of the subject. 1 27 for all situations. Gazis et al. 1959 address it by incorporating the space headway see Figure 2-1 between the two vehicles in the sensitivity term. Their model is as follows: ant = X t ,  Vnfront t , n n n 2.3 where, Xnt , n  denotes the space headway at time t , n . The model was estimated using microscopic data collected from the car following experiments in the Holland Tunnel and the Lincoln Tunnel in New York and at the General Motors test track. The parameters and were estimated for each driver of each data set using correlation analysis. For each data set, the values of the parameters averaged over all samples were reported as the estimates. Table 2.1 summarizes the estimation results. Table 2.1: Estimation results of the model developed by Gazis et al. 1959. Data Number collection site of drivers mph second GM Test Track 8 27.4 1.5 Holland Tunnel 10 18.3 1.4 Lincoln Tunnel 16 20.3 1.2 The mean reaction time measured at the test track varied from 1.0 to 2.2 seconds. Edie 1961 pointed out that, the model given by Equation 2.3 su ers from two limitations. First, from a behavioral standpoint, the follow the leader theory is not applicable at low densities. Second, the macroscopic speed density relationship derived from Equation 2.3 yields in nite speed as the density approaches zero. De ne, u = speed of a stream of tra c at density k, c = integration constant, kj = jam density. Assuming that tra c is in a steady state and ignoring the reaction time, integrating 28 both sides of Equation 2.3 yields: Z a dt u u at k = kj ; u = 0 u Z = X V dt = c + ln X  1 = c + ln k  c =  ln! j  k = ln kj k 2.4 In this equation, corresponds the stream speed at maximum ow. This equation is the macroscopic speed density relationship developed by Greenberg 1959. It does not yield free ow speed at zero density. Edie addressed the above mentioned limitations by changing the sensitivity term and the model is as follows: t ant = Vnt, n2 Vnfront t , n Xn , n 2.5 Sensitivity is now proportional to the speed and inversely proportional to the square of the headway. Equation 2.5 can be integrated as was done to obtain the model given by Equation 2.4 to obtain a model that yields free- ow speed as the density approaches zero. This model performed better than the model proposed by Gazis et al. 1959 at low densities. However, the stimulus term is still a function of the front relative speed, which is not realistic at low densities, in particular, when the headways are high. Instead of using the sensitivity stimulus formulation to explain the car following acceleration decision, Newell 1961 suggested the following relationship between the speed and the headway: Vnt = Gn Xnt , n 29 2.6 where, Gn is a function whose form determines the speci cation of the car following models that are presented above. Di erent forms of Gn were assumed for the acceleration and deceleration decisions. Although, the model had the advantage of integrability to obtain di erent macroscopic speed ow density relationships, no attempt was reported to obtain a quantitative result to validate the model. The car following model developed by Gazis et al. 1961, known as the General Motors Nonlinear Model, is the most general one. The model is given by: n ant = XVtt  Vnfront t , n n ,n 2.7 where, , , and are model parameters. The sensitivity is proportional to the speed raised to the power and inversely proportional to the headway raised to the power . The parameter is a constant and the front relative speed is the stimulus. The models developed earlier by Chandler et al. 1958and Gazis et al. 1959 can be derived from this model as special cases. It should be mentioned that the macroscopic ow speed relationship developed by Greenshields 1934 can be derived from the GM Model by setting = 0 and = 2. No rigorous framework for estimating the model was provided. Bexelius 1968 suggested that instead of following only the immediate leader, drivers in a car following situation also follow the vehicles ahead of the leader. Mathematically, the model is given by: ant = N X i=1 i Vnit , n , Vnt , n 2.8 where, i and Vnit , n  are the sensitivity and speed associated with the i-th front vehicle and N is the number of drivers. However, the model was not estimated and validated. May and Keller 1967 estimated the GM Model Equation 2.7 using a macroscopic relationship between speed and density that was derived by Gazis et al. 1961. In addition to using integer values of and , May and Keller 1967 also used non 30 integer values and found higher correlation coe cients for the non integer cases. The estimated parameters are presented in Table 2.2. Since they used a macroscopic relationship between speed and density, reaction time could not be identi ed. Table 2.2: Estimation results of the GM Model by May and Keller 1967. Parameter Estimates with integer and 1.35  10,4 1.0 3.0 48.7 free speed uf , mph jam density kj , vpm 1 optimum speed, mph 29.5 optimum density ko, vpm 60.8 maximum ow, vph 1795 macroscopic model u = uf e,0:5k=ko 2 Estimates with non integer and 1.33  10,4 0.8 2.8 50.1 220 29.6 61.1 1810 1:8 5 u = uf 1 , kkj Leutzbach 1968 proposed a psycho physical spacing model that addresses two limitations of the car following models from a behavioral standpoint. First, drivers do not follow their leaders at large spacings, and second, drivers cannot perceive small di erences in front relative speeds and therefore, do not react to such di erences. Leutzbach introduced the term perceptual threshold" to de ne a relative speed threshold which is a function of the space headway. The threshold is smaller at low space headways and gradually increases with space headway. A driver reacts to the stimulus, the front relative speed, only when the stimulus exceeds the perceptual threshold. At a certain large space headway, the threshold becomes in nity, i.e., a driver no longer follows its leader beyond that space headway. An important nding of his research is that the perceptual threshold for negative relative speed is smaller than that for positive relative speed. This implies that the sensitivity of spacing and front relative speed on drivers' acceleration and deceleration decisions are di erent. Leutzbach, however, did not provide any mathematical formulation of the proposed model, nor provided any direction as to how the perceptual threshold can be estimated. 31 Recently, Ozaki 1993 estimated the GM Model Equation 2.7 parameters. He used regression analysis to estimate a model for drivers' reaction time and correlation analysis to estimate parameters ; and . Ozaki listed four actions to identify reaction time. Figure 2-2 shows the de nition of reaction time corresponding to these actions. The actions are: T T Action D Action A ∆V time an time T Action B T Action C Figure 2-2: De nition of reaction time corresponding to the four actions source: Ozaki, 1993. Action A start of deceleration: time elapsed since the relative speed became zero and the subject, who was accelerating at that instant of time, started decelerating; Action B maximum deceleration: time elapsed since the relative speed reached its minimum value negative and the subject applied the maximum deceleration; Action C start of acceleration: time elapsed since the relative speed became zero and the subject, who was decelerating at that instant of time, started accelerating; 32 Action D maximum acceleration: time elapsed since the relative speed reached its maximum value positive and the subject applied the maximum acceleration. These de nitions of reaction time are not consistent with those suggested by earlier car following model researchers and the Tra c Engineering Literature Gerlough and Huber 1975. These researchers de ned the reaction time as the summation of perception and foot movement times. Depending on the deceleration capability of a vehicle, its driver may start reacting at di erent times. For example, a driver driving a vehicle with powerful brakes may not decelerate, even after realizing that its leader is slower, until the driver gets very close. This does not imply that the driver's reaction time is larger as suggested by Ozaki. He, however, made an important observation: tra c conditions, such as the headway and the acceleration of the leader, in uence the reaction time. To estimate the car following model parameters, he rst identi ed the reaction time using the de nition of reaction time for di erent actions listed above. Then, the correlation between the observed acceleration and estimated acceleration obtained by using the explanatory variables lagged by the reaction time and setting the car following model parameters to present numbers was calculated for di erent values of the parameters. The combination that yielded the maximum correlation was reported as the estimates. Ozaki assumed a di erent set of parameters for the acceleration and the deceleration decisions; this captures the fact that di erent factors, such as subject speed, front relative speed, and headway, may not have the same impact on driver's acceleration and deceleration decisions. The parameters ; and were estimated to be 1.1, -0.2, and 0.2 respectively for the acceleration model, and 1.1, 0.9, and 1.0 respectively for the deceleration model. Aycin and Benekohal 1998 developed a car following model which estimates the acceleration rate at any instant of time. Acceleration for the next time instant is then computed by adding the product of the acceleration rate estimate and the time di erence to the current acceleration. This guarantees continuity in the acceleration pro le for a given driver. Equations of laws of motion are used to compute the 33 acceleration rate required for a driver to attain its leader's speed while maintaining a preferred time headway. The preferred time headway is de ned as a headway the driver wants to maintain under steady state car following conditions. For each driver in the car following data set that traveled at speeds within 5 ft sec of its leader's speed, the discrete time headways measurements over time were averaged. Then, the average was taken as the driver's preferred time headway. The preferred time headway values ranged from 1.1 to 1.9 seconds with a mean of 1.47 seconds. The e ect of reaction time is explicitly modeled. According to this model, drivers are assumed to be in the car following regime if the clear gap see Figure 2-1 is less than 250 feet. This rule ignores variability between drivers. The reaction time was not estimated using a rigorous method. It was assumed to be 80 of the estimated preferred time headway. 2.1.2 General Acceleration Models The models presented above apply to the car following regime only. When the headways are large, drivers do not follow their leader, instead they try to attain their desired speeds. Developing an appropriate acceleration model for the free ow regime is important for microscopic simulation models. Gipps 1981 developed the rst general car following model that is applicable to both the car following and free ow regimes. This model calculates a maximum acceleration for a driver such that the speed would not exceed a desired speed, and the clear gap would be at least a minimum safe distance. Mechanical limitations of vehicles were captured by using the parameters maximum acceleration and most severe deceleration. Equations of laws of motion were used in the above computations. The parameters of the models were not estimated rigorously and the reaction time was set arbitrarily for all drivers. Benekohal and Treiterer 1988 developed a car following simulation model, called CARSIM, to simulate tra c in both normal and stop and go conditions. The acceleration for a vehicle is calculated for ve di erent situations and the most binding acceleration is used to update the vehicle's speed and position. These situations are 34 the subject i.e., the following vehicle is moving but has not reached its desired speed; the subject has reached its desired speed; the subject was stopped and starts from a standstill position; the subject's movement is governed by the car following algorithm in which a space headway constraint is satis ed; and, the subject is advancing according to the car following algorithm with a non collision constraint. Equations of laws of motion are used in the above computations. In addition, a comfortable and a maximum allowable deceleration are assumed to limit the output from the acceleration models within a reasonable boundary. The reaction times of drivers are randomly generated, and shorter reaction times are assigned at higher densities. No rigorous framework for parameter estimation was presented and the reaction time distribution parameters were adopted from Johansson and Rumer 1971 which is presented in Section 2.1.3. Yang and Koutsopoulos 1996 developed a general acceleration model that is used in MITSIM, a microscopic tra c simulator. Based on headway, a driver is assigned to one of the three following regimes: the emergency regime, if the current headway is less than a lower threshold; the car following regime, if the current headway is greater than the lower threshold but less than an upper threshold; and nally, the free ow regime, if the current headway is greater than the upper threshold. In the emergency regime, a driver applies the necessary deceleration to avoid colliding with its leader and increase headway. The GM Model Equation 2.7 is used to determine the acceleration rate in the car following regime. Di erent set of parameters are used for positive and negative relative speed cases. In the free ow regime, a driver 35 tries to attain its desired speed by applying a maximum acceleration if the current speed is less than the desired speed or a normal deceleration otherwise. The model parameters were not estimated using eld data. Subramanian 1996 developed a general acceleration model that captures drivers' acceleration behavior in both the car following and free ow regimes. A space headway threshold distribution was assumed that determines which regime a driver is in at any instant of time. In the car following regime, drivers are assumed to follow their leader, and in the free ow regime, they are assumed to try to attain their desired speed. He, however, estimated only the car following model parameters using data that was collected in 1983 from a section of Interstate 10 Westbound near Los Angeles Smith 1985. His speci cation of the car following model is an extension of the GM Model Equation 2.7 and is given by: Vn , ant = Xtt , n   Vnfront t , n  + cf t n n n 2.9 where, cf t is the random term associated with driver n at time t. He modeled n the reaction time as a random variable to capture the variability within driver and between drivers. Variables cf t and n are assumed to be distributed normal and n truncated lognormal respectively. He estimated separate models for acceleration and deceleration observations. The estimation results are presented in Table 2.3. The estimated mean reaction time was larger than those reported by Johansson and Rumer 1971 and Lerner et al. 1995, and the mean reaction time estimate for the deceleration decision was higher than that for the acceleration decision|a counter intuitive result. He also estimated the GM Model using di erent headway thresholds and concluded that the headway threshold has signi cant impact on the parameter estimates. 36 Table 2.3: Estimation results of the GM Model by Subramanian 1996. Model for Model for acceleration deceleration parameter estimate estimate 9.21 15.24 -1.67 1.09 -0.88 1.66 std. dev cf  0.780 0.632 mean , sec. 1.97 2.29 std. dev  1.38 1.42 2 , speed in ft sec, space in feet. Note: acceleration in ft sec 2.1.3 Estimation of the Brake Reaction Time In this section, we present the studies that were conducted to obtain the brake reaction time of drivers driving in real tra c. Johansson and Rumer 1971 estimated the distribution of the brake reaction time from a sample of 321 drivers traveling in a real tra c. The subjects were instructed to apply the brake pedal as soon as they hear a sound. The time elapsed from the moment the sounds were made to the moment the drivers' brake light turned on were recorded as the brake reaction time. The brake reaction time varied from 0.4 to 2.7 seconds with a median, mean, and standard deviation of 0.89, 1.01, and 0.37 seconds respectively and a 90 percentile value of 1.5 seconds. These numbers may be biased downwards, since the sound, to which the drivers reacted, might have reduced the perception time, and hence the reaction time. Recently, Lerner et al. 1995 estimated the reaction time distribution from a sample of 56 drivers driving in real tra c. To estimate the brake reaction time for unexpected situations to mimic real driving conditions, subjects were not informed that they were participating in a brake reaction time study. When a subject reached the test site at 40 mph speed, a large yellow highway crash barrel was released approximately 200 ft in front of the vehicle. The barrel was chained so that it was held within the median. The time elapsed since a barrel is released to the instant a driver 37 applies brake was recorded as the driver's reaction time. The brake reaction time varied from 0.7 to 2.5 seconds with a median, mean, and standard deviation of 1.44, 1.51, and 0.39 seconds respectively. 2.2 Lane Changing Models In this section, a literature review of the lane changing models is presented followed by a literature review of the gap acceptance models. The principal focus of research in modeling drivers' lane changing behavior has been on modeling the gap acceptance behavior at stop controlled T intersections. The gap acceptance phase is a part of the lane changing process. Gipps 1986 presented a lane changing decision model to be used in a microscopic tra c simulator. The model was designed to cover various urban driving situations where tra c signals, obstructions, and the presence of heavy vehicles for example, bus, truck, semi trailer a ect a driver's lane selection decision. Three major factors were considered in the lane changing decision process: necessity, desirability, and safety. Di erent driving conditions were examined including the ones where a driver may face con icting goals. However, di erent goals were prioritized deterministically, and inconsistency and non homogeneity in driver behavior were not modeled. The terms inconsistency implies that a driver may behave di erently under identical conditions at di erent times, while the term non homogeneity implies that di erent drivers behave di erently under identical conditions. The model parameters were not estimated formally. CORSIM FHWA 1998 is a microscopic tra c simulator that uses FREESIM to simulate freeways and NETSIM to simulate urban streets. In CORSIM, a lane change is classi ed as either mandatory MLC or discretionary DLC. A driver performs an MLC when the driver must leave the current lane and performs a DLC when the driver perceives the driving conditions in the target lane to be better, but, a lane change is not required. The necessity or desirability of changing lanes is determined by computing a risk factor that is acceptable to a driver which is a function of a 38 driver's position relative to the object that gives rise to the need for a lane change. A default set of model parameters are provided with the exibility of using user provided parameters. The gap acceptance behavior is not modeled in a systematic manner. Minimum gap lengths for di erent situations are listed and all drivers are assumed to have identical gap acceptance behavior. Yang and Koutsopoulos 1996 developed a rule based lane changing model that is applicable only for freeways. Their model is implemented in MITSIM. A lane change is classi ed as either mandatory MLC or discretionary DLC. Unlike Gipps 1986, they used a probabilistic framework to model drivers' lane change behavior when they face con icting goals. A driver considers a discretionary lane change only when the speed of the leader is below a desired speed, and checks neighboring lanes for opportunities to increase speed. Two parameters, impatience factor and speed indi erence factor, were used to determine whether the current speed is low enough and the speeds of the other lanes are high enough to consider a DLC. They also developed a gap acceptance model that captures the fact that the critical gap length de ned as the minimum acceptable gap length under an MLC situation is lower than that under a DLC situation. They pointed out that, for a case of merging into a tra c parallel to the current lane, a gap is acceptable only when both the lead and lag gaps are acceptable. However, no formal parameter estimation was done and a framework to do so was not developed. Recently, Ahmed et al. 1996 developed a framework for a general lane changing model that captures lane changing behavior under both the MLC and DLC situations. Lane change is modeled as a sequence of four steps: decision to consider a lane change, choice of a target lane, acceptance of gaps in the target lane, and performing the lane change maneuver. A discrete choice framework is used to model these decision elements that allows for modeling impact of di erent tra c and roadway environment on driver behavior. From a model estimation view point, the utilities capturing the rst and the fourth steps cannot be uniquely identi ed in the absence of any indicator available to the analyst di erentiating these two steps. They estimated parameters of the model only for a special case: merging from a freeway on ramp. They used 39 the data collected in 1983 from a site at Interstate 95 northbound near the Baltimore Washington Parkway Smith 1985. In this case, it was assumed that drivers have already decided to change to the adjacent freeway and therefore, the decision process involved acceptance of a gap and the actual lane change maneuver. Following Yang and Koutsopoulos 1996, a gap is considered acceptable only when both the lead and lag gaps are acceptable. Figure 2-3 shows the de nition of the lead and lag gaps. The lead and lag critical X total clear gap + vehicle length lag gap Y lead gap lag vehicle lead vehicle subject front vehicle X Y Figure 2-3: The subject, lead, lag, and front vehicles, and the lead and lag gaps. gap lengths were assumed to be lognormally distributed and whether a lane change will take place immediately, given the gap is acceptable, was modeled using a binary logit model. The estimated lead critical gap for driver n at time t is Gcr;leadt = exp 2:72 , 0:055 n + n lead t n 2.10 where, Gcr;leadt = lead critical gap feet, n n = driver speci c random term that is constant for a given driver, assumed distributed standard normal, lead t = random term that varies across di erent components of a gap n for a given driver, across di erent gaps for a given individual, 40 as well as across drivers, lead t  N 0; 1:612: n The estimated lag critical gap for driver n at time t is Gcr;lag t = n exp ,9:32 + 0:1170 minVnlag t; 10 + 0:1174 maxVnlag t , 10; 0 + 1 1:57 nstGapt + 1:88 lnLremt + 1:90 n + lnag t 2.11 n where, Gcr;lag t = lag critical gap feet, n Vnlag t = lag vehicle speed subject speed mph, 8 1 if delaynt = 0 1stGap t = n : 0 otherwise. delaynt = time elapsed since MLC conditions apply seconds, Lremt = remaining distance to the point at which lane change must be n completed feet, lag t  N 0; 1:312: n The estimated model of changing lanes, given that both the lead and lag gaps are acceptable, is: Pn change lanes at timet j gap acc: = 1 2.12 1 + exp1:90 , 0:52 delaynt The gap acceptance model, however, cannot be applied to a case of forced merging or merging through courtesy yielding. In this case, gaps of acceptable lengths may not exist due to high congestion level, and in order to merge gaps have to be created. 41 2.2.1 Gap Acceptance Models Di erent gap acceptance models were developed in the 1960s and 1970s based on the assumption on the distribution of the critical gap length. Herman and Weiss 1961 assumed the critical gap to be exponentially distributed, Drew et al. 1967 assumed a lognormal distribution, and Miller 1972 assumed a normal distribution. They, however, did not capture the e ect of previously rejected gaps on the critical gap. In general, data collected for estimating gap acceptance models is panel in nature, i.e., it contains one or more observations from each individual. Di erent observations from a given sample are likely to be correlated which may introduce bias in the parameter estimates. Daganzo 1981 used a probit model formulation appropriate for panel data to estimate the gap acceptance model parameters for drivers merging from the minor leg of a stop controlled T intersection to the major leg. The critical gap for driver n at time t is assumed to have the following functional form: Gcr t = Gn + cr t n n 2.13 where, Gn = component of critical gap attributable to driver n, cr t = random term that varies across di erent gaps for a given driver as n well as across di erent drivers. Gn, and cr t are assumed to be mutually independent. Further, he assumed Gn  n 2 N G; G and cr t  N 0; 2. The individual speci c random term, Gn, captures n the correlation between di erent observations from driver n. The model has the exibility to incorporate the impact of other factors on a driver's gap acceptance behavior by varying the mean of the distribution of Gn. However, he had estimability problems and the estimated critical gap lengths were not guaranteed to be non negative. Mahmassani and She 1981 used the data that Daganzo 1981 used and ad42 dressed the estimability problem by ignoring panel data formulation, i.e., they treated the data as cross sectional data. They assumed the critical gap to be normally distributed. The mean of the critical gap was allowed to be a function of explanatory variables, a framework that allows for incorporating the impact of di erent factors on a driver's gap acceptance behavior. The variable number of gaps rejected, capturing the impatience factor, was found to have a signi cant impact on drivers gap acceptance behavior. The Highway Capacity Manual HCM 1985, or HCM, uses the mean critical gap length of drivers at an intersection to estimate the delay at and the capacity of that intersection. The HCM de ned the critical gap for a two way stop controlled intersection as the median of all acceptable gap lengths. A major limitation of this de nition is that an observation of a large gap accepted by a driver provides no information about the minimum acceptable gap length. In the revised HCM procedure, the critical gap is de ned as the largest observed rejected gap length. This de nition is again awed, since one very conservative driver can greatly increase the estimate. In addition, Cassidy et al. 1995 listed other de ciencies of this approach. First, only a subset of the data is used and all accepted gaps shorter than the largest one is not included in the estimation. Second, inconsistency in driver behavior accepting a gap smaller than a previously rejected gap is addressed either by discarding or by modifying the data. However, the bene t of using the HCM de nition of a critical gap is ease in estimation. Kita 1993 used a logit model to estimate the gap acceptance model for the case of merging from a freeway on ramp. The impact of di erent factors on drivers' gap acceptance behavior was modeled by using a random utility model. Although he used panel data, he did not use an appropriate panel data formulation. In addition to the gap length, relative speed of the subject with respect to the mainline vehicles and the remaining distance of the acceleration lane were found to have impact on drivers' gap acceptance behavior. Cassidy et al. 1995 used Kita's approach to model the gap acceptance behavior at a stop controlled T intersection. They, too, ignored the panel data formulation 43 and found that a gap acceptance function with disaggregate factors have signi cantly more predictive power than a function that includes only the mean gap length. 2.3 Summary A summary of the ndings from the literature review is presented below. Modeling acceleration behavior: Primary attention of the research has been on modeling drivers' acceleration behavior in the car following regime. The impact of stimulus the front relative speed on the car following acceleration was assumed to be linear. The reaction time was modeled but not estimated rigorously in most cases. Variability within driver and between drivers were not captured in most cases. The headway threshold, that determines whether a driver is in the car following regime or in the free ow regime, is modeled deterministically in most cases. A general acceleration model was proposed by Subramanian 1996. The model captures acceleration behavior in both the car following and free ow regimes. It also captures the inconsistency in driver behavior. A probabilistic framework was used to model variability in headway threshold and reaction time. However, only the car following part of the model was estimated. Modeling lane changing behavior: Modeling drivers' gap acceptance behavior has been the primary focus of the research in modeling drivers' lane changing behavior. 44 A majority of the research in modeling drivers' gap acceptance behavior used panel data. However, model formulation appropriate for panel data was not used while estimating the parameters. A model capturing drivers' lane change decision process was developed by Ahmed et al. 1996. However, the model is not applicable to mandatory lane changing situations in a heavily congested tra c where gaps of acceptable lengths are hard to nd. The parameters of the discretionary lane changing model have not been estimated. The impact of an acceleration decision, which determines drivers speed, on the lane changing decision is modeled by using speed as an explanatory variable in the lane changing model. The acceleration model proposed in this thesis builds on the earlier work by Subramanian 1996 and extends his model. The impact of the stimulus on the car following acceleration is allowed to be a nonlinear function of the lead relative speed and the sensitivity term is extended to capture the impact of tra c conditions ahead of the subject and its leader. In addition, all the components of the acceleration model are estimated jointly using microscopic data collected from real tra c. On the other hand, the lane changing model proposed in this thesis extends the model proposed by Ahmed et al. 1996 to capture merging behavior in heavily congested tra c and the model is estimated using statistically rigorous methods and real driver data. 45 Chapter 3 The Acceleration Model In this chapter, a rigorous framework for estimating the parameters of the acceleration model is presented that builds on the previous work by Subramanian 1996 and extends it. The proposed model consists of two components: the car following model and the free ow acceleration model. The car following model is applied when a driver follows its leader i.e., the vehicle in front. The free ow acceleration model is applied when a driver tries to attain its desired speed and is not following its leader. This chapter starts with a presentation of the conceptual framework and specication of the model. Next, the likelihood function that is necessary for estimating the model is formulated. 3.1 Introduction Based on a headway threshold, a driver is assumed to be in one of the two following regimes: the car following regime and the free ow regime. If the current headway is less than the threshold, the driver is assumed to be in the car following regime and follow its leader see Figure 3-1. Speed selection and hence the acceleration decision is governed by the speed of the leader. Otherwise, the driver is assumed to be in the free ow regime in which case speed selection is governed by its desired speed. The existing car following models for example, Gazis et al. 1961, Subramanian 46 subject front vehicle or leader space headway Figure 3-1: The subject and the front vehicle. 1996 restrict the stimulus of the acceleration to be a linear function of the front relative speed and do not capture the impact of tra c conditions ahead of the driver except the speed and position of its leader. The front relative speed is de ned as the speed of the leader less the subject speed. In this chapter, we use the terms front relative speed and relative speed interchangeably. In the car following regime, the sensitivity of di erent factors, such as speed, headway, and front relative speed, on drivers' acceleration decision under acceleration and deceleration situations may be di erent. For example, consider two cases: one with a positive relative speed with a certain magnitude and the other with a negative relative speed with the same magnitude, and all other factors are identical. The acceleration in case one is likely to be less than the deceleration in absolute terms in case two due to safety concerns involved. The model proposed in this thesis relaxes the restriction that the car following stimulus is a linear function of the front relative speed and captures the impact of tra c conditions ahead of the subject on the car following sensitivity by using as explanatory variable the density ahead of the subject. Separate car following model parameters under acceleration and deceleration situations are allowed in the estimation. The model proposed in this thesis, however, does not explicitly capture the impact of lane changing decisions on the acceleration decision. For example, a driver may accelerate or decelerate to t into a gap in an adjacent lane. Instead, random terms are used in all component models that capture the e ect of unobserved factors. This is left as a subject for further research and is discussed in Chapter 8. 47 3.2 The Acceleration Model A driver is assumed to be in the car following regime if the headway is less than a threshold, and in the free ow regime otherwise. Mathematically, the acceleration model can be expressed as: 8 ant = acf t if hnt , n   h n n : aff t otherwise n 3.1 where, t= n= ant = acf t = n aff t = n hnt , n = h = n time of observation, reaction time of driver n, acceleration at time t, car following acceleration at time t, free ow acceleration at time t, time headway1 at time t , n, unobserved headway threshold for driver n. Reaction time refers to the delay in a driver's response to a stimulus, or the response lag. It includes both the perception time from the presentation of the stimulus until the foot starts to move and foot movement times. Since these two cannot be identi ed uniquely from the observed data, the term reaction time is used to designate the summation of the two. We de ne the headway threshold, h , in terms of time headway as opposed to n space headway for two reasons. First, previous research for example, Winsum and 1 Time headway is de ned as: hn t = Xntt ; Vn  Vn t 0 where, Vn t and Xn t denote the subject speed and the space headway see Figure 3-1 at time t respectively. In this research, headway is used to designate time headway unless otherwise mentioned. 48 Heino 1996, Aycin and Benekohal 1998 indicates that drivers maintain certain time headways independent of speed in a steady state car following situations. And second, equal space headways have identical acceleration regimes car following versus free ow although speeds may be very di erent, while the time headway does not su er from this limitation. Speci cation of the car following and free ow acceleration models and the distributions of the headway threshold and reaction time are presented next. 3.2.1 The Car Following Model Since, it is hypothesized that the expected value of the acceleration distribution is greater than zero when the relative speed is positive, the model corresponding to a positive relative speed is called the car following acceleration model. Similarly, the model corresponding to a negative relative speed is called the car following deceleration model. The model can be expressed as follows: cf;g acf;g t = s Xn t ,  n f Vnt , n + n cf;g t n 3.2 where, g 2 facc; decg cf;g cf s Xn t ,  n = sensitivity, a function of Xn t ,  n, cf;g Xn t ,  n = vector of explanatory variables a ecting the car following acceleration sensitivity observed at time t ,  n,  2 0; 1 ; a parameter for sensitivity lag, f Vnt , n = stimulus, a function of relative speed, Vnt , n, Vnt , n = Vnfront t , n , Vnt , n  ; Vnt , n = subject speed at time t , n , Vnfront t , n = front vehicle speed at time t , n , 49 cf;g t n = random term associated with the car following acceleration of driver n at time t. The acceleration or deceleration applied by driver n at time t is proportional to the stimulus, a function of the front relative speed at time t , n. The reaction time, n, varies from driver to driver, and therefore, is modeled as a random variable. The sensitivity term is the proportionality factor, a function of explanatory variables discussed below observed  n seconds earlier. The parameter for the sensitivity lag,  , varies between 0 and 1. The stimulus term is a function of the relative speed. Figure 3-2 a shows the expected e ect of the relative speed on drivers' acceleration decision. At low relative acc or |dec| acc or |dec| V V1 V V2 (a) Effect of relative speed on acceleration/deceleration (b) linear impact of relative speed acc or |dec| acc or |dec| V V (c) nonlinear impact of relative speed, parameter < 1 (d) nonlinear impact of relative speed, parameter > 1 Figure 3-2: Impact of the relative speed on drivers' acceleration decision. speeds, drivers' acceleration response may not be signi cant as they may not be able to perceive a small magnitude of the relative speed. For relative speeds beyond a certain threshold, jV1j, drivers get a better sense of the stimulus and therefore, acceleration increases at an increasing rate. Beyond another threshold, jV2 j, the acceleration applied by a driver is limited by the acceleration capacity of the vehicle 50 and hence, acceleration increases at a decreasing rate until it reaches the maximum acceleration. The e ect of the relative speed on the car following acceleration discussed above can be captured by assuming a piecewise nonlinear function of the relative speed of the following form: g g g f Vnt , n = V 1nt , n + V 2nt , n + V 3nt , n 1 2 3 3.3 where, V 1nt , n = minjVnt , nj; jV1j j j = absolute value, V 2nt , n = max0; minjVnt , n j , jV1j; jV2j , jV1j V 3nt , n = max0; jVnt , nj , jV2j The breakpoints, jV1 j and jV2j, should be chosen such that they are reasonable from a behavioral standpoint. In order to replicate the impact of the relative speed on the acceleration as shown by the curve in Figure 3-2 a, both g and g should 1 2 g should be less than one. In addition, g should be less be greater than one while 3 1 than g . 2 Figure 3-2 b shows the linear approximation of the impact of the relative speed on acceleration which has been used by existing models. This implies the following: f Vnt , n = jVnt , nj 3.4 The functional form given by Equation 3.3 can be simpli ed by allowing only one parameter: f Vnt , n = jVnt , n jg 3.5 Figures 3-2 c and d show the e ect of the relative speed on the acceleration using 51 this speci cation for g 1 and g 1 respectively. We apriori expect g to be less than one for both the acceleration and deceleration due to the existence of a maximum value for acceleration and deceleration that a driver can apply in reality. The parameter g can be tested statistically to determine whether it is signi cantly di erent from one. Note that, g = 1 corresponds to the speci cation given by Equation 3.4. The GM Model Equation 2.7 assumed the sensitivity term to be a nonlinear function of the subject speed at time t and the space headway at time t , n . It allowed a single set of parameters for both the acceleration and deceleration decisions. Mathematically, this is given by: cf s Xn t ,  n = Vnt Xnt , n 3.6 where, ; ; and are constant parameters, and  is set to 0 and 1 for speed and space headway respectively. We extend the GM Model by allowing di erent sets of parameters for the car following acceleration and deceleration sensitivities, by incorporating the density of tra c as explanatory variable into the sensitivity term, and by allowing the time at which the explanatory variables are observed to be a parameter to be estimated: cf;g s Xn t ,  n Vnt ,  n g k t ,   g = X t ,   g n n n n g 3.7 where, g; g; g; g = constant parameters, knt ,  n = density of tra c ahead of the subject within its view at time t ,  n. The parameter  captures the fact that drivers may update their perception of the tra c environment during the acceleration decision making process. Restricting  to be equal to one implies that drivers do not update their perception of the tra c 52 environment and react accelerate decelerate based on the tra c conditions at the time they observe the stimulus. In other words,  = 1 implies that the lag for sensitivity and stimulus are equal, while,  1 implies that lag for sensitivity is smaller than that for stimulus. There are apriori expectations regarding the signs of the various parameters. The constant g in the sensitivity term should be positive and negative for the acceleration and deceleration models respectively. In the car following regime under acceleration situations, drivers are likely to apply a lower acceleration at high speeds compared to low speeds and therefore, the corresponding parameter acc should be negative. On the other hand, under deceleration situations, drivers are likely to apply a higher deceleration at high speeds compared to low speeds which implies that dec should be positive. The sign of the parameter acc can either be negative or positive. Under acceleration situations, drivers may apply a higher acceleration when space headways are larger, implying a negative headway parameter, acc i.e., the space headway should be in the numerator of the sensitivity. However, as the space headway increases, drivers may tend to follow the speed of the lead vehicle less and if this is the case, acc would be positive. Under deceleration situations, drivers are likely to apply smaller decelerations at larger headways, implying a positive headway parameter, dec . Tra c conditions ahead of the subject and its leader are likely to change more rapidly at high densities than at low densities. Due to this, higher uncertainty is involved in predicting the position and speed of the leader in the near future. In addition, high tra c density represents lack of maneuverability compared to low tra c density for both the subject and its leader. As a result, drivers are expected to be more conservative at high densities than at low densities. Hence, at high densities the subject is likely to accelerate at a lower rate, while decelerate at a higher rate. These imply that, acc and dec are expected to be negative and positive respectively. The random term captures the e ect of omitted variables. It is assumed to be independent for di erent decisions of a given driver as well as for di erent drivers. The correlation between di erent acceleration decisions of a given driver is assumed 53 to be captured through the reaction time and headway threshold distributions. This implies: cov cf;g t; n cf;g t n cf;g0 t0  n0  N 0; 8 = 2cf;g  2cf;g if g = g 0 ; t = t0 ; n = n0 : 0 otherwise 3.8 3.2.2 The Free Flow Acceleration Model When the headway is greater than the threshold, the driver has the freedom to attain its desired speed. Hence, the acceleration applied by a driver in this regime is assumed to have the following functional form: aff t = ff Vn t , n , Vnt , n + n f f t n 3.9 where, ff Vnt , n  Vnt , n  , Vnt , n f f t n = constant sensitivity, = desired speed of the driver, = stimulus, = random term associated with the free ow acceleration of driver n at time t. The desired speed of a driver is de ned as the speed the driver wants to maintain after considering the speed limit of the section it is traveling, vehicle's mechanical capability, the e ect of surrounding tra c, the roadway and weather conditions, and the geometry of the roadway section. The desired speed is assumed to have the following functional form: 54 DS Vn t , n = Xn t , n DS 3.10 where, DS Xn t , n = vector of explanatory variables a ecting the desired speed DS , DS = constant parameters. Replacing the speci cation of Vnt , n in Equation 3.9, the free ow acceleration model becomes h DS aff t = ff Xn t , n n i ff n t , n  + n t DS , V 3.11 In this model, the acceleration at time t is assumed to be proportional to the stimulus| the di erence between the driver's desired speed and current speed at time t , n. The sensitivity term is assumed to be a constant. If the desired speed is higher than the current speed, drivers are expected to accelerate and vice versa. The magnitude of the applied acceleration deceleration depends on the di erence between the current and the desired speeds. Important explanatory variables a ecting the desired speed of a driver include geometry of the roadway curvature, grade, lane width, pavement surface quality roughness, presence of pot holes, weather conditions, the speed limit of the roadway section, density of tra c ahead of the subject, speed of the vehicles ahead of the subject which is also a proxy for density and maneuverability, type of the vehicle, and characteristics of the driver. For example, in a curved road or roadway with grades particularly upgrade or in a roadway with rough pavement, vehicles tend to slow down thus the desired speed of drivers reduce even when there is no lead vehicle. Similarly, drivers often set their desired speed relative to the speed limit of the roadway section. To estimate models with these site speci c factors, data from di erent sites is necessary. 55 High density of tra c ahead of a driver within the driver's view, or a lower speed of the lead vehicle reduces desired speed, as might be expected. And nally, heavy vehicles for example, bus, truck, semi trailer etc. that have length greater than 9.14 meters or 30 ft AASHTO 1990 have lower acceleration and deceleration capability and hence respond slowly to free ow conditions. We further assume that, f f t is normally distributed with zero mean and a n variance  2f f , i.e., f f t  N 0; 2f f , and f f t is independent of the random n n cf;acct and cf;dec t for a given driver2 . terms n n Note that, since the desired speed of a driver cannot be observed, extending the free ow acceleration model to have di erent sensitivity under acceleration and deceleration situations cannot be done without increasing the complexity of the current framework. 3.2.3 The Headway Threshold Distribution The headway threshold, h , is assumed to be truncated normally distributed with n truncation on both sides. This distribution is given by: 8 f h  = n : 1 h , nh h h  max  h ,h , hmin ,h if h  h  h min n max 0 otherwise h h 3.12 where, h; h = constant mean and standard deviation of the untruncated distribution, h ; h = minimum and maximum value of h parameters to be estimated, min max n  = probability density function of a standard normal random variable, As mentioned above, the correlation between the car following and the free ow acceleration decisions is assumed to be captured through the reaction time and headway threshold distributions. 2 56  = cumulative distribution function of a standard normal random variable. The advantage of using a truncated normal distribution with mean, variance, and the truncation ends h and h  as parameters is that the distribution is not min max restricted to be skewed to a particular direction. For instance, a distribution skewed to the left implies that, the probability of a driver being aggressive is higher than that of being conservative, since, an aggressive driver is expected to have a shorter headway threshold compared to a conservative driver. The above treatment of the headway distribution is a generalization over Subramanian 1996 who used a shifted truncated lognormal distribution that restricts the distribution to be skewed to the left. Using Equation 3.12, the probability that driver n, who is hnt behind its leader, is in the car following regime is given by: Pncar following at time t = P hnt  h  n 8 1 1, = : 0   hn th,h , hminh,h h , h , max h min h  h , h if hnt  h min  if h min hn t  hmax 3.13 otherwise At very large headways, it is unlikely that a driver would be in a car following regime. Hence, the corresponding probability is zero for headways greater than h . Simimax larly, at very low headways it is unlikely that a driver would be in a free ow regime and the corresponding probability of car following is one for headways less than h . min 3.2.4 The Reaction Time Distribution The reaction time is assumed to be truncated log-normally distributed i.e., skewed to the left as suggested by Subramanian 1996. This implies that the probability of 57 a driver having a smaller reaction time is higher than that of having a larger reaction time. This was also supported by Johansson and Rumer 1971 and Lerner et al. 1995. Truncation is assumed since reaction time is nite. The distribution is as follows: 8  ln f  n = : 1 max  ,  n pe 2 , 1 ln n ,  2 2 if 0 0 n  max 3.14 otherwise where, = reaction time of driver n,  = mean of the distribution of ln n, n max = standard deviation of the distribution of ln n, = upper bound of the distribution of n parameter to be estimated. The mean, median, and variance of the above distribution are as follows:  ln max  ,  , mean = exp + 0:5 2  ln max ,   ,1 median = exp  + variance = e2 +  0:5 ln ln  e ,1 2  3.15 !!! max  ,  max  ,   ln ,2 max  ,  3.16 3.17 The mean of the distribution of ln n,  , is assumed to be a function of explanatory variables:  = Xn 58 3.18 where, Xn = vector of explanatory variables, = model parameters. Important factors a ecting the reaction time include age, mental condition, visibility, weather conditions, roadway geometry, vehicle characteristics, vehicle speed, and tra c conditions. Older drivers are expected to have longer reaction times. Poor visibility increases driving di culty and drivers are expected to be more alert. This implies a reduction in reaction time. During rain or snow drivers are expected to be more alert compared to good weather conditions. Roadway sections with high curvature and or high grade make driving more di cult and hence would make drivers more alert. At high speeds drivers are expected to be more alert compared to low speeds due to safety reason. Tra c conditions such as the density of tra c and the gap in front of the subject may also a ect reaction time. Drivers may be more alert in congested tra c compared to free ow tra c due to higher uncertainty involved in predicting future tra c conditions. 3.3 Likelihood Function Formulation Using Equation 3.2 and the hypothesis that a driver in the car following regime will accelerate if the leader is faster and vice versa, the distribution of the car following acceleration, conditional on n, is given by: f acf t j n n = f acf;acct j n n Vn t, n  f acf;dect j n n 1, Vn t, n   3.19 where, 8 Vnt , n = : 1 if Vnt , n  0 0 otherwise 59 Using Equations 3.2 and 3.8, the distribution of the car following acceleration, conditional on n , are as follows: f acf;g t j n n = 1 cf;g  cf;g cf;g an t , s Xn t ,  n f Vnt , n cf;g ! 3.20 where, g 2 facc; decg. The free ow acceleration distribution, conditional on n , is given by: f aff t j n n = 1 ff  ff an t , ff DS Xn t , n ff DS , Vn t , n  ! 3.21 Combining Equations 3.19 and 3.21, and using Equation 3.1, the distribution of acceleration for driver n at time t, conditional on h and n , is as follows: n f ant j h ; n = f acf t j n n n hn t, n  f aff t j n n 1, hn t, n   3.22 where, 8 hnt , n = : 1 if hnt , n  h n 0 otherwise As mentioned above, the reaction time and the headway threshold capture the correlation between di erent acceleration decisions at di erent times for a given driver. This implies that, conditional on n and h , the Tn di erent observations of driver n n are independent. Therefore, the conditional joint density of observing an acceleration pattern associated with driver n, an1; an2; : : : ; anTn, can be expressed as the product of the conditional densities of each element of the pattern. Mathematically, this can be expressed as follows: f an1; an2; : : : ; anTn  j h ; n n = Tn Y t=1 f ant j h ; n n 3.23 The unconditional distribution that constitutes the likelihood function for driver 60 n is: f an1; an2; : : : ; anTn = Z max Z h max f an1; an2; : : : ; anTn j h ; f h f   dh d  0 hmin 3.24 Finally, assuming that the acceleration observations from di erent drivers in the sample are independent, the log likelihood function is given by: L= N X n=1 ln f an1; an2; : : : ; anTn 3.25 Maximizing the likelihood function would provide the MLE estimate of the model parameters. 3.4 Conclusions In this chapter, a rigorous framework for specifying and estimating the general acceleration model is presented that allows for joint estimation of all component models. The component models are the car following acceleration and deceleration models, the free ow acceleration model, and the headway threshold and reaction time distributions. The proposed model builds on the earlier work by Subramanian 1996 and extends it. First, separate model parameters under acceleration and deceleration situations in the car following regime are allowed in the likelihood function formulation. Second, the sensitivity of the car following acceleration is extended to capture the e ect of tra c conditions ahead of the driver, in addition to the relative position and speed of its leader. Third, it allows the time at which the explanatory variables of the car following acceleration sensitivity are observed to be a parameter to be estimated as opposed to restricting it to be the time at which the stimulus is observed. Fourth, the stimulus of the car following acceleration is extended by making it a nonlinear function of the lead relative speed. And nally, a more general headway threshold 61 distribution is used that allows any driver behavior to be captured aggressive or conservative. 62 Chapter 4 The Lane Changing Model In this chapter, the lane changing model is presented. Lane changes are classi ed as either mandatory or discretionary. When a lane change is required due to, for example, a lane drop, the operation is called a mandatory lane change MLC. On the other hand, when lanes are changed by a driver to improve perceived driving conditions, the operation is called a discretionary lane change DLC. The proposed mandatory lane changing model extends the work by Ahmed et al. 1996 by developing a new model for heavily congested tra c. Under heavily congested tra c, gaps of acceptable lengths are hard to nd. Hence, a forced merging model is proposed which captures merging by gap creation either through courtesy yielding of the lag vehicle in the target lane or through the subject forcing the lag vehicle to slow down. This chapter begins by presenting the conceptual framework of the proposed lane changing model. The model along with the likelihood function formulation is presented next. Then, the complexities associated with modeling the impact of past lane changing decisions on the current lane changing decision are discussed. This chapter concludes by presenting the conceptual framework, the model, and the likelihood function formulation of the forced merging model. 63 4.1 Introduction A lane change decision process is assumed to have the following three steps: decision to consider a lane change either a DLC or an MLC, choice of a target lane, and acceptance of a gap in the target lane. Modeling such a process is extremely complicated. First, the entire lane change decision process is latent in nature. All that is observed is the execution of the lane change decision|the nal acceptance of a gap. Second, the time at which a lane change decision is made cannot be observed in general1. Furthermore, once a decision to change lanes is made, a driver may continue to search for gaps or may change its mind|all of which are unobserved. Finally, the lane changing decision is continuous in nature. To simplify the modeling, time is discretized. Furthermore, drivers are assumed to make decisions about lane changes at every discrete point in time irrespective of the decisions made during earlier time periods. In other words, we do not explicitly model the impact of past lane changing decisions on the current lane changing decision. The complexities associated with capturing such behavior are discussed in Section 4.2.4. The impact of past decisions on the current decision, however, is captured in the proposed forced merging model. Due to their di erent structures, the lane changing and forced merging models are presented separately. Merging from an on ramp to a freeway is a notable exception, since as soon as a driver arrives at the merging point, the driver would recognize the necessity of performing a mandatory lane change. 1 64 4.2 The Lane Changing Model 4.2.1 Conceptual Framework The lane changing model structure is shown in Figure 4-1. As mentioned above, except for the completion of the execution of the lane change, the whole decision process is latent in nature. The latent and observable parts of the process are represented by ovals and rectangles respectively. Start MLC MLC driving conditions not satisfactory driving conditions satisfactory other lanes Left Lane Right Lane Left Lane current lane Right Lane Gap Accept Gap Reject Gap Accept Gap Reject Gap Accept Gap Reject Gap Accept Gap Reject Left Lane Current Lane Right Lane Current Lane Left Lane Current Lane Right Lane Current Lane Current Lane Current Lane Figure 4-1: The lane changing model structure. The MLC branch in the top level corresponds to the case when a driver decides to respond to the MLC condition2. Explanatory variables that a ect such decision include remaining distance to the point at which lane change must be completed, the number of lanes to cross to reach a lane connected to the next link, delay time elapsed since the MLC conditions apply, and whether the subject vehicle is a heavy When a mandatory lane changing situation does not apply, the probability of responding to MLC is set to zero. 2 65 vehicle bus, truck, semi trailer etc.. Drivers are likely to respond to the MLC situations earlier if it involves crossing several lanes. A longer delay makes a driver more anxious and increases the likelihood of responding to the MLC situations. And nally, due to lower maneuverability and larger gap length requirement of heavy vehicles as compared to their non heavy counterparts, they have a higher likelihood of responding to the MLC conditions. The MLC branch corresponds to the case where either a driver does not respond to an MLC condition, or that MLC conditions do not apply. A driver then decides whether to perform a discretionary lane change DLC . This comprises of two decisions: whether the driving conditions are satisfactory, and if not satisfactory, whether any other lane is better than the current lane. The term driving conditions satisfactory implies that the driver is satis ed with the driving conditions of the current lane. Important factors a ecting the decision whether the driving conditions are satisfactory include the speed of the driver compared to its desired speed, presence of heavy vehicles in front and behind the subject, if an adjacent on ramp merges with the current lane, whether the subject is tailgated etc. If the driving conditions are not satisfactory, the driver compares the driving conditions of the current lane with those of the adjacent lanes. Important factors a ecting this decision include the di erence between the speed of tra c in di erent lanes and the driver's desired speed, the density of tra c in di erent lanes, the relative speed with respect to the lag vehicle in the target lane, the presence of heavy vehicles in di erent lanes ahead of the subject etc. In addition, when a driver considers DLC although a mandatory lane change is required but the driver is not responding to the MLC conditions, changing lanes opposite to the direction as required by the MLC conditions may be less desirable. If a driver decides not to perform a discretionary lane change i.e., either the driving conditions are satisfactory, or, although the driving conditions are not satisfactory, the current is the lane with the best driving conditions the driver continues in the current lane. Otherwise, the driver selects a lane from the available alternatives and assesses the adjacent gap in the target lane. The lowest level of ovals in the decision tree shown in Figure 4-1 corresponds 66 to the gap acceptance process. When trying to perform a DLC , factors that a ect drivers' gap acceptance behavior include the gap length, speed of the subject, speed of the vehicles ahead of and behind the subject in the target lane, and the type of the subject vehicle heavy vehicle or not. For instance, a larger gap is required for merging at a higher travel speed. A heavy vehicle would require a larger gap length compared to a car due to lower maneuverability and the length of the heavy vehicle. In addition to the above factors, the gap acceptance process under the MLC conditions is in uenced by factors such as remaining distance to the point at which lane change must be completed, delay which captures the impatience factor that would make drivers more aggressive etc. Note that, delay cannot be used as an explanatory variable except for very specialized situations, for example, merging from an on ramp. This is because the very inception of an MLC condition is usually unobserved. The speci cation of the complete model is presented next. 4.2.2 Model Formulation The decisions in the hierarchy shown in Figure 4-1 can be modeled using the random utility approach Ben-Akiva and Lerman 1985. The model formulation must explicitly capture the fact that, the available data for lane changing model estimation is panel data. Model formulation appropriate for panel data is presented in Appendix A. The Lane Selection Model As mentioned above, the lane selection process consists of the top four levels of the decision hierarchy shown in Figure 4-1. The top level, whether to respond to a mandatory lane change MLC  condition or not MLC , can be modeled using a discrete choice model, for example, a binary logit model. Using the formulation of random term appropriate for panel data see Appendix A, the probability that driver n at time t will respond to MLC , conditional on the individual speci c random term, 67 n, is given by: Pt MLC j n = 1 + exp,X MLC t1 n M LC , M LC n  4.1 where, MLC Xn t = vector of explanatory variables a ecting decision to respond to the MLC conditions discussed in Section 4.2.1, M LC = vector of parameters, n = individual speci c random term assumed to be distributed standard normal, M LC = parameter of  . n The individual speci c random term, n, is introduced to capture the correlation between di erent observations from a given driver. If the correlation is not captured it may introduce bias in the parameter estimates. The larger the product of the parameter M LC and the value of individual speci c random variable, n, the higher is the probability that the driver would respond to an MLC condition earlier. If a driver decides not to respond to an MLC condition, or MLC conditions do not apply, a discretionary lane change DLC  may be considered. The binary decision, whether the driving conditions are satisfactory or not, can be modeled using a binary logit model, 1 Pt DCNS j n = 1 + exp,X DCNS t n DCNS , DCNS n  4.2 where, superscript DCNS denotes driving conditions not satisfactory. Generally, we expect M LC and DCNS to have opposite signs, or, the two corresponding utilities should have a negative correlation see Equation A.4. This implies that, a driver postponing a response to an MLC condition to be an aggressive driver and hence, may have a higher propensity to perform a discretionary lane change. 68 If the driving conditions are not satisfactory, drivers are assumed to compare the driving condition of the current lane with the better among the left and right adjacent lanes. The utilities of perceiving the driving conditions unsatisfactory and selecting the other lanes over the current lane are expected to be positively correlated. Since there is an e ort hassle associated with changing lanes which is not explicitly captured, the utility of the adjacent two lanes are likely to be correlated. The nested logit model Ben-Akiva and Lerman 1985 is a natural choice to capture such phenomenon. First, the utilities of the two adjacent lanes are compared the `left lane' versus `right lane' decision under the `other lanes' oval in Figure 4-1. Then, the utility of the `other lanes' is compared to the utility of the `current lane' to decide if the current lane is the desired lane. The output from the lane selection model is the probability of selecting each of the three lanes in question. If the left or right lane is chosen, a driver seeks an acceptable gap in the target lane. The gap acceptance model is presented next. The Gap Acceptance Model The gap acceptance model captures drivers assessment of gaps as acceptable or unacceptable. Drivers are assumed to consider only the adjacent gap. An adjacent gap is de ned as the gap in between the lead and lag vehicles in the target lane see Figure 4-2. For merging into an adjacent lane, a gap is acceptable only when both X total clear gap + vehicle length lag gap Y lead gap lag vehicle lead vehicle subject front vehicle X Y Figure 4-2: The subject, lead, lag, and front vehicles, and the lead and lag gaps. lead and lag gaps are acceptable. 69 Drivers are assumed to have minimum acceptable lead and lag gap lengths which are termed as the lead and lag critical gaps respectively. These critical gaps vary not only among di erent individuals, but also for a given individual under di erent tra c conditions. The critical gap for driver n at time t is assumed to have the following functional form3: g Gcr;g t = expXn t g + g n + g t n n 4.3 where, g 2 flead; lagg; g = parameter of  for g 2 flead; lag g, n g t = generic random term that varies across all three dimensions, i.e., n g; t; and n: The exponential form of the critical gap guarantees that the estimated critical gap will always be non-negative. The individual speci c random term, n, and its parameter capture the correlation between the lead and lag critical gaps for a driver. This correlation, especially under MLC conditions, is expected to be positive. A conservative driver is expected to have a larger lead lag critical gaps compared to its aggressive counterpart. A larger product of g and n , g 2 flead; lagg implies a larger critical gap length requirement, and hence, represents a conservative driver. The lead lag critical gaps are expected to be positively correlated with the utility of responding to an MLC condition and negatively correlated with the utilities of perceiving the driving conditions as unsatisfactory and selecting the other lanes over the current lane. Assuming g t  N 0; 2g , i.e., the critical gap lengths are lognormally disn tributed, the conditional probability of acceptance of a gap is given by: 3 Adopted from Ahmed et al. 1996. 70 Pt gapAcc j n = Pt lead gap acceptablej n Pt lag gap acceptablej n = PGlead t Gcr;leadt j n PGlag t Gcr;lag t j n n n n n = PlnGleadt lnGcr;leadt j n PlnGlag t lnGcr;lag t j n n n  n lead !n lead lnGn t , Xn t lead , lead n  = lead  lag lag  lnGn t , Xn t lag lag , lag n ! 4.4 where, Gleadt and Glag t denote the lead and lag gaps see Figure 4-2 respectively n n and  denotes the cumulative distribution function of a standard normal random variable. In addition to 's and 's, lead and lag are parameters that can be identi ed. Normalization of lead or lag is not necessary since the variables `lnGlead t' n lag t' in Equation 4.4 do not have any coe cient. and `lnGn 4.2.3 Likelihood Function Formulation Drivers are assumed to consider the entire lane change decision process Figure 4-1 at every discrete point in time, for example, every second. Let the sequence of lane changes performed by driver n be denoted as follows: Jn1; Jn2; : : : ; JnTn where, J L R C Tn 2 fL; R; C g = change to the left lane, = change to the right lane, = continue in the current lane, = number of time periods driver n is observed. 71 4.5 As mentioned above, the individual speci c random term captures the correlation between di erent decisions at di erent times for a given driver. Therefore, conditional on the individual speci c random term, the probability of observing a pattern for a given driver can be expressed as the product of probabilities of observing each element of the pattern. Mathematically, this can be expressed as follows: PJ1n; J2n; : : : ; JTnn j n = = Tn Y t=1 Tn Y t=1 PJtn j n L R LR Pt L j n tn Pt R j n tn Pt C j n1, tn , tn 4.6 where, 8 J tn = : 1 if driver n changes to J at time t J 2 fL; Rg 0 otherwise. 4.7 The unconditional probability of observing a pattern for a given driver is given by: PJ1n ; J2n; : : : ; JTnn = Z1 ,1 PJ1n; J2n; : : : ; JTnn j  f  d where, f   denotes the distribution of  . Assuming that the observations from di erent drivers in the sample are independent, the likelihood function for all drivers is: L= N X n=1 ln PJ1n ; J2n; : : : ; JTnn 4.8 where, N denotes the number of drivers. The probability of staying in the current lane or changing to the left or right lanes can be formulated using the decision tree of Figure 4-1. A driver may change to the left lane when he she: responds to MLC conditions, the left lane is chosen, and the lead and lag gaps in the left lane are acceptable; or, 72 does not respond to MLC conditions or MLC conditions do not apply, perceives the driving conditions as unsatisfactory, selects the other lanes over the current lane, selects the left lane, and the lead and lag gaps in the left lane are acceptable. Therefore, the probability of an observation of change to the left lane, conditional on n, is: Pt L j n  = Pt gap acceptable j left lane chosen; MLC; n  Pt left lane chosen j MLC; n Pt MLC j n + Pt gap acceptable j left lane chosen; other lanes; driving conditions not satisfactory; MLC; n Pt left lane chosen j other lanes; driving conditions not satisfactory; MLC; n Pt other lanes j driving conditions not satisfactory; MLC; n Pt driving conditions not satisfactory j MLC; n Pt MLC j n Similarly, the conditional probability of changing to the right lane or continuing in the current lane can be formulated. 4.2.4 Discussions Complexities Associated with Capturing the Impact of Past Lane Changing Decisions in the Lane Changing Model In this section, the complexities associated with modeling the impact of past lane changing decisions on the current lane changing decision are discussed with the help of a simple example. Consider a vehicle that is observed in a two lane roadway for three consecutive time periods during which time it did not change lanes and mandatory lane changing conditions do not apply. To simplify the discussion further, we combine 73 the two levels, `driving conditions satisfactory or not' and `other lane' or `current lane', of the decision tree shown in Figure 4-1 into one level, DLC versus DLC . Here, DLC implies that the driving conditions of the current lane is not satisfactory and another lane the left lane is better than the current lane. Therefore, for this driver, the lane changing decision tree shown in Figure 4-1 reduces to the one shown in Figure 4-3 a. Since the driver did not change lanes, he she may be in state `DLC and gap reject given DLC ' or in state `DLC ' during these three time periods see Figure 4-3 b. As shown in Figure 4-3 b, there are 23 = 8 possible state sequences that can explain the three observations from the driver. If the driver is observed for Tn time periods, the number of state sequences becomes 2Tn , i.e., the number of possible state sequences increases exponentially with the number of times the driver is observed. Therefore, the number of state sequences to explain a particular pattern of lane time lane period change Start possible states (arrows show state to state transitions) 1 DLC no DLC & gapRej | DLC DLC 2 no DLC & gapRej | DLC DLC 3 no DLC & gapRej | DLC DLC DLC gap Acc gap Rej Target Lane Same Lane Same Lane (b) possible states during the three successive time periods (a) decision tree at any instant of time Figure 4-3: The lane changing decision tree for a driver driving in a two lane roadway and possible states of the driver. changing by a driver is prohibitively large from an estimation point of view. Further research with various modeling approaches and approximations is necessary to capture the impact of past lane changing decisions on the current lane changing decision. 74 Limitations of the Proposed MLC Model Vehicles in heavily congested tra c travel at low speeds, with low space headways. In such situations, it is likely that a driver, trying to change lanes, will not nd a gap that is larger than the driver's minimum acceptable gap length. In order to merge, gaps have to be created either through the lag vehicle's courtesy yielding or through the subject forcing the lag vehicle to slow down. The mandatory lane changing model presented above, however, assumes that drivers would ultimately nd an acceptable gap. A forced merging model that captures driver decisions leading to a gap creation is proposed in the following section. 4.3 The Forced Merging Model We assume that a driver has decided to change to the adjacent lane see Figure 4-4. The merging process involves the driver's decision as to whether he she intends to merge into the adjacent gap and perception as to whether his her right of way is established, and nally moving into the target lane. An adjacent gap is de ned as the gap behind the lead vehicle in the target lane. Establishment of right of way means that an understanding between the subject and the lag vehicle in the target lane has } lag vehicle adjacent gap for the subject target lane lead vehicle subject Figure 4-4: De nition of the adjacent gap. been reached such that the lag vehicle would allow the subject to be in front of it. The conceptual framework of the proposed model is presented next. 75 4.3.1 Conceptual Framework The tree diagram in Figure 4-5 summarizes the proposed structure of the forced merging model. As before, the ovals correspond to the latent part of the process that involves decisions and the rectangles correspond to the events that are directly observable. MLC do not start forced merging (M) start forced merging (M) Same Lane Target Lane Same Lane Figure 4-5: The forced merging model structure. At every discrete point in time, a driver is assumed to a evaluate the tra c environment in the target lane to decide whether the driver intends to merge in front of the lag vehicle in the target lane and b try to communicate with the lag vehicle to understand whether the driver's right of way is established. If a driver intends to merge in front of the lag vehicle and right of way is established, the decision process ends and the driver gradually moves into the target lane. We characterize this instant by state M , where M denotes start forced merging. This process may last from less than a second to a few seconds. This is shown by the arrow below the left `same lane' box. If right of way is not established, the subject continues the evaluation communication process i.e., remains in state M  during the next time instant. 76 4.3.2 Model Formulation Let, Snt denote the state of driver n at time t. Using a binary logit model and the random utility speci cation appropriate for panel data see Appendix A, the probability of switching to state M from state M , conditional on n, is given by: PfSn t = M j Snt , 1 = M; ng = 1 + e,Xn 1 FM t F M , F M n 4.9 where, superscript FM implies forced merging. Important explanatory variables include: lead relative speed only when the lead vehicle is slower: when the lead vehicle is slower, the subject is more likely to slow down to match its speed with the speed of the lead vehicle rst so as to focus exclusively on the interaction with the lag vehicle; this reduces the probability of being in state M ; lag relative speed: when the lag vehicle is faster, the subject is more likely to speed up before attempting to establish right of way and hence this reduces the probability of being in state M ; remaining distance to the point at which lane change must be completed by: as the remaining distance decreases, drivers become more concerned about merging and hence more aggressive. As a result, the probability of being in state M also increases; delay time elapsed since the mandatory lane change conditions apply: higher delay makes a driver more frustrated and hence more aggressive, i.e., the probability of being in state M increases with additional delays4 . total clear gap equal to the sum of the lead and lag gaps, see Figure 4-2: a large clear gap makes merging relatively easier and hence increases the probability of being in state M ; As explained in Section 4.2.1, delay can be used as an explanatory variable only when the starting point is well de ned, for example, merging from an on ramp to the freeway. 4 77 indicator for heavy vehicles for example, bus, truck, semi trailer: due to lower maneuverability and larger gap length requirement of heavy vehicles as compared to their non heavy counterparts, they have a higher probability of being in state M under similar conditions. The likelihood function formulation is presented next. 4.3.3 Likelihood Function Formulation At any discrete point in time, a driver may be in state M or M see Figure 4-5. Once a vehicle is in state M , by de nition, the decision process ends and the remaining process is placing the vehicle in front of the lag vehicle, and the state of the driver cannot return to M . The time taken in placing the vehicle in front of the lag vehicle is captured by the arrow below the left `same lane' rectangle in Figure 4-5. This implies the following: PfSnt0 = M j Snt = M; n g = 1 8 t0 t PfSnt0 = M j Snt = M; n g = 0 8 t0 t 4.10 4.11 We also assume that the initial state of the driver is M . Di erent cases in which forced merging can occur are shown in Figure 4-6. Time period 1 denotes the rst time period considered in the forced merging model and time period 0 denotes the preceding time period. In the rst two cases Figures 4-6 a and b, i.e., merging from an on ramp and exiting, at time period 0 it is practically impossible to start merging. Therefore, the initial state is M . On the other hand, in the last two cases Figures 4-6 c and d, the driver could be in state M or M at time 0. If a driver is already in state M , then the sequence observed for the driver does not involve any decision and therefore, the probability of the observing the sequence is not a function of the model parameters to be estimated. In such cases, assuming drivers' state to be M would be reasonable only if the length of the section measured from the diverging point toward the upstream direction is large enough. A reasonable way to de ne a section large enough is 200 meters or more. Note that, in the last two cases delay 78 merging point merging point Target Lane 0 1 1 0 0 (a) merging from on-ramp initial state (at time 0): M upstream end of the data collection site 0 Target Lane (b) exiting, 1 lane change required initial state (at time 0): M upstream end of the data collection site Target Lane Target Lane 1 0 (c) exiting, 2 lane changes required initial state (at time 0): M or M 1 (d) connecting to the next link initial state (at time 0): M or M Figure 4-6: Initial state of the driver for the forced merging model for di erent cases. cannot be used as an explanatory variable since the time instant at which the driver's state became M for the rst time cannot be observed. Since, n is assumed to capture the correlation between the utilities of di erent states at di erent times, conditional on n, the probability of being in state M at time t, given all earlier states were M , is also given by Equation 4.9. Mathematically, PfSnt = M j Snt0 = M; t0 = 0; 1; : : : ; t , 1; ng = 1 FM 1 + e,Xn t F M , F M n 4.12 The impact of being in state M during the earlier time periods on the probability of being in state M at time t is captured through the explanatory variable delay. Let, Tn be the number of time periods driver n was observed in the original lane. There are Tn possible state sequences that may lead to observing driver n in the target lane at time Tn + 1. These sequences are listed in Table 4.1. Sequence 1 in Table 4.1 implies that driver n reached state M at time period 1 and it took another 79 Table 4.1: Possible decision state sequences of observing a lane change by forced merging. Time Observed Possible state sequences Period lane 1 2 3 ::: t ::: 1 SL MMM M 2 SL MMM M 3 SL MMM M ... ... ... ... ... ... t,1 SL MMM M t SL MMM M ... ... ... ... ... ... Tn , 1 SL MMM M Tn SL MMM M Tn + 1 TL Note: SL = same lane, TL = target lane Tn M M M ... M M ... M M Tn , 1 seconds to execute the lane changing process. Sequence 2 corresponds to the case where driver n was in state M at time period 1, and in state M during the time interval 2 to Tn. Similarly, Sequence t corresponds to the case that the driver was in state M during the time interval 1 to t , 1, and in state M during the time interval t to Tn. Note that, these sequences are mutually exclusive. As mentioned above, the individual speci c random term, n, captures the correlation between di erent decision elements at di erent times for a given driver. Therefore, conditional on n, the probability of observing a particular state sequence for a given driver can be expressed as the product of probabilities of observing each state of the sequence. The conditional probability of observing the tth state sequence for driver n is, 80 Pn fstate sequencet j ng = PfSnTn = M j Snt0 = M; 8t0 = t; : : : ; Tn , 1; Snt00  = M; 8t00 = 0; 1; : : : ; t , 1; ng PfSnTn , 1 = M j Snt0  = M; 8t0 = t; : : : ; Tn , 2; Snt00 = M; 8t00 = 0; 1; : : : ; t , 1; ng : : : PfSnt = M j Snt0 = M; 8t0 = 0; 1; : : : ; t , 1; ng : : : PfSn2 = M j Sn1 = M; Sn0 = M; ng PfSn1 = M j Sn0 = M; ng 4.13 Using Equations 4.11 and 4.12, Pn fstate sequencet j ng = PfSnt = M j Snt0 = M; 8t0 = 0; 1; : : : ; t , 1; ng : : : PfSn2 = M j Sn1 = M; Sn0 = M; ng PfSn1 = M j Sn0 = M; ng = PfSnt = M j Snt0 = M; 8t0 = 0; 1; : : : ; t , 1; ng  tY ,1 t0 =1 PfSnt0  = M j Snt00  = M; 8t00 = 1; : : : ; t0 , 1; ng 4.14 Since, an observed lane change by a driver can be explained by any one of the mutually exclusive state sequences listed in Table 4.1, the conditional likelihood function is the sum of the probabilities of observing all the sequences. This is given by: 81 Ln = = FM; FM j   n Tn X Pnfstate sequencet j ng t=1 Pnfstate sequence1 j n g + Pnfstate Pnfstate sequenceTn j ng sequence2 j ng + : : : + = PfSn1 = M j Sn0 = M; ng + PfSn2 = M j Sn1 = M; Sn0 = M; ngPfSn1 = M j Sn0 = M; ng + : : : + PfSnTn = M j Snt0 = M; 8t0 = 0; 1; : : : ; Tn , 1; ng TY1 n, t0 =1 PfSnt0 = M j Snt00  = M; 8t00 = 1; : : : ; t0 , 1; ng Let us now introduce variable 8 FM n t = : FM tn 4.15 de ned as: 1 if the adjacent gap at time t is the same gap driver n ultimately merged into 0 otherwise 4.16 F Figure 4-7 illustrates the meaning of the variable n M t with the help of an example. The subject vehicle C was observed for 4 time periods in the original lane the right lane. Only during the 3rd and 4th time periods, vehicle C was adjacent to the gap between vehicles A and B that it ultimately merged into. Therefore, the sequence of F M for this driver is f0,0,1,1g. Since, the driver was not adjacent to the gap between vehicles A and B during the rst two time periods, communication with vehicle B cannot be established during these time intervals. Therefore, the driver cannot be in state M during time periods 1 and 2. This implies that, the rst two sequences listed in Table 4.1 do not apply to this driver. While forming the likelihood function for this driver, the rst two sequences must be taken out of the likelihood function Equation 4.15. A convenient way of incorporating this into the likelihood function for a general case is as follows: Ln FM; FM j n  = Tn X t=1 Pnfstate sequencet j ng  82 FM n t 4.17 time = 1 delta = 0 A B C subject time = 2 delta = 0 B A C time = 3 delta = 1 B A C time = 4 delta = 1 A B C time = 5 Lane change completed B Figure 4-7: De nition of FM n t C A for the forced merging model. The unconditional likelihood function for driver n is Ln FM; FM = Z 1 X Tn ,1 t=1 Pn fstate sequencet j  g  FM n t ! f  d 4.18 where, f   denotes, as before, the probability density function of the random variable . Assuming that the observations from di erent drivers in the sample are independent, the log likelihood function for all observations is given by: L FM; FM = Z  Tn N 1X X n=1 ln ,1 t=1 Pnfstate sequencet j  g  83 !  F M t f  d n 4.19 4.3.4 Discussion As mentioned above, in order to merge in heavily congested tra c, drivers must create gaps either through force or through courtesy yielding. A reasonable way to de ne heavy tra c congestion is level of service F de ned by the Highway Capacity Manual HCM 1985. The HCM characterizes this level of service as tra c conditions in which a breakdown of ow occurs and queues form behind breakdown points. At this level of congestion, the probability of nding acceptable gaps is very low and in order to merge gaps have to be created. For level of services A through E, drivers are assumed to merge by the gap acceptance process presented in Section 4.2.2. The boundary level of services F versus A through E that is used to determine whether to apply the usual gap acceptance process or the forced merging process is rather arbitrary. Although the boundary can also be estimated formally e.g. like the headway threshold in the acceleration model, the process the drivers actually follow may be di erent. For example, drivers rst search for acceptable gaps and consider forced merging only when they perceive the probability of nding acceptable gaps to be very low. Further research is necessary to combine the mandatory lane changing and forced merging models into a single framework which would apply to all level of services. We leave this as a subject for future research. 4.4 Conclusions In this chapter, a framework for modeling drivers' lane changing behavior was developed. A signi cant enhancement to the state of the art is the development of the forced merging model that captures merging behavior under heavy tra c congestion. This model is based on the assumption that in heavily congested tra c, gaps of acceptable lengths are rare, and therefore, for a vehicle to merge, gaps must be created either through courtesy yielding of the lag vehicle in the target lane or through the subject forcing the lag vehicle to slow down. 84 Chapter 5 Data Requirements for Estimating Driver Behavior Models In this chapter, the data required to estimate the acceleration and lane changing models and the data that was obtained from real tra c are presented. In addition, a methodology for estimating instantaneous speed and acceleration that are required for model estimation from discrete trajectory data that can be obtained from the eld is developed. Data required to estimate the acceleration and lane changing models include the position, speed, acceleration, and length of a subject vehicle and the vehicles ahead of and behind the subject in the current lane as well as in adjacent lanes. Data on gap lengths, headways, density of tra c, etc. can be extracted from the above mentioned data by simple addition and subtraction operations. In addition, to capture the impact of site speci c factors, such as the speed limit of a section, geometry curvature, grade, and lane con guration and whether the roadway section is a tunnel or not, data from di erent sites is required. Typically, such data is collected using photographic and video equipment see, for example, Smith 1985. The raw data collected through such devices is processed to obtain useful information such as vehicle location at discrete points in time. Instantaneous speed and acceleration data, that is required for estimation of the models, have to be inferred from the trajectory data. 85 This chapter begins with a description of the method that is used to estimate instantaneous speed and acceleration from discrete trajectory data. Then the data collection strategy and the actual processing of the data is presented. 5.1 Methodology for Estimating Instantaneous Speed and Acceleration from Discrete Trajectory Data As mentioned above, the data usually available includes discrete measurements of vehicle positions over time. A continuous function describing the vehicle trajectory can be estimated from the discrete position observations using the local regression procedure developed by Cleveland and Devlin 1988. Once the trajectory function, X t, is estimated, the rst and second derivatives of the estimated trajectory function at time t, provide estimates of the speed and acceleration at time t respectively. Mathematically, V t = dX t dt d2X t at = dt2 5.1 where, V t = speed at time t, at = acceleration at time t. In general, vehicles frequently stop in congested tra c, often for signi cant durations. Whether a vehicle is stopped or not, cannot be ascertained from the observed trajectory, as there are measurement errors while collecting and processing the data. A very high order polynomial would be necessary to t a curve to the trajectories of such vehicles. This gives rise to computational problems as the objective function of 86 such problems becomes nearly singular1. Furthermore, even a high order polynomial may not t the data well during the instances when a vehicle is stopped. The local regression procedure addresses some of these problems by tting local curves using the observations around the time period of interest which is described in more detail in Section 5.1.1. Local regression can be used to estimate a wide class of functions. Three major uses of local regression were listed by Cleveland and Devlin 1988. First, a local regression estimate can be used as a graphical exploratory tool to study the structure of the data. This would help in choosing an appropriate functional form that ts the data. Second, it can be used to validate an already estimated model that used a parametric class of models. And nally, local regression estimates can be used instead of regular regression estimates when dealing with data that require very exible functional form. The application in this research falls in the third category. 5.1.1 The Local Regression Procedure The local regression procedure has three basic elements: weight assignment, function speci cation, and neighborhood or window size. A unit weight is assigned to the trajectory observation at the time period of interest t and a gradually decreasing weight is assigned to the other points, depending on their distance from the tth observation. The window size around time t determines the number of points that are used for tting a polynomial curve of a suitable degree. Weight Assignment. A tricube weight function with the following functional form is used: wto; t = 1 , uto; t33 5.2 1 For polynomials of order 10 or above, the hessian of the objective function becomes nearly singular as the independent variables polynomial of time vary from tens and twenties to billions. This also depends on the precision of the computer. Nearly singular hessian makes the estimation process computationally intensive and time consuming as the convergence rate reduces signi cantly. 87 where, t = time period at which speed acceleration estimates are desired, wto; t = weight for observation at time to , to 2 f1; 2; : : :; t; : : :g, uto; t = distance function for an observation at to = j t , to j ; d dt = distance from t to the farthest point + constant. A small constant is added to d so that the distance function, u, is less than one for the observation farthest from t. This guarantees a non-zero weight for that observation. Note that, 0  uto; t 1 and 0 wto; t  1 8to. Function speci cation. As mentioned above, the trajectory function, X t, is assumed to be a polynomial of time. The parameters of a polynomial are uniquely identi ed if the order of the polynomial is at most one less than the number of observations trajectory points. A perfect t is obtained when the order of the polynomial is one less than the number of points. Window Size. The window size determines the number of points used in each local regression. For example, a window size of 7 implies that the 7 closest in terms of time of observation position measurements including the measurement at the time period of interest t are considered for local tting of data. The bias variance of the estimated position increases decreases with increasing window size Cleveland et al. 1988. Depending on the type of application, the window size should be selected such that either the bias or the variance or the mean square error of the estimates is minimized. Note that, the mean square error is the sum of the bias squared and the variance. Since the proposed curve t algorithm uses inequality constraints discussed below, a close form solution for estimating the bias or variance does not exist. Instead, a sensitivity analysis can be conducted to evaluate the impact of the window size on the quality of the results e.g. magnitude of the position estimation errors, the speed and acceleration pro les etc.. 88 Mathematically, the curve t problem using the local regression can be stated as: tminm ;s 2 X t; s , T t; s t;s 0 W t; s X t; s , T t; s t;s 5.3 where, t = time period at which speed acceleration estimates are desired, s = window size, X t; s = vector of discrete trajectory observations corresponding to time period t and window size s, T t; s = matrix of independent variables, constant; time; time2 ; : : : ; timem , t;s = vector of parameters corresponding the tth time period and window size s, m = order of the polynomial, W t; s = weight function, a diagonal matrix. The ith diagonal element of W t; s corresponds to the weight assigned to the ith trajectory observation obtained by using Equation 5.2. However, it is very common that in the data due to measurement errors, the measured position of a vehicle at two successive time periods may be decreasing. Hence, a curve tted to these points may yield an unrealistic negative speed and or acceleration estimates. To guarantee that the speed estimates are non negative and acceleration estimates are within the acceleration and deceleration capacities, Equation 5.3 has to be minimized subjected to the following set of constraints over the range of time periods considered in a particular local regression: speed  0 acceleration  maximum deceleration acceleration  maximum acceleration 89 5.4 The curve t algorithm Equation 5.3 subjected to constraints 5.4 is repeated for each driver for each instant of time at which position, instantaneous speed and or acceleration are desired. Then for each instant of time, the tted value of the polynomial is used as an estimate of position and the rst and second derivatives of the polynomial as the speed and acceleration respectively. Figure 5-1 shows an example of estimating the speed and acceleration at time 220 200 original position estimated trajectory function 180 Position of vehicle, m 160 140 120 100 80 measured position at time period 8 = 91.0480 m estimated position at time period 8 = 90.8427 m 60 estimated speed at time period 8 = 0.82 m/s estimated acceleration at time period 8 = −2.03 m/s2 40 20 0 5 10 15 time period, seconds 20 25 30 Figure 5-1: An example of estimation of instantaneous speed and acceleration from discrete position measurements. period 8 from discrete position measurements of a vehicle using the local regression methodology. A window of size 9 was used in this exercise. Figure 5-2 shows the weight function used in this exercise and tted curve. The trajectory function was tted from the discrete position measurements around time period 8 time periods 4 to 12. As shown Figure 5-2, a very good t of the data was obtained except for time periods 8 and 9. Vehicle position at time periods 8 and 9 was measured at 91.05 and 91.04 meters respectively. Since, vehicle position cannot decrease which implies an unrealistic negative average speed, the position measurement either at time period 90 1 weight 0.8 0.6 0.4 0.2 0 4 5 6 7 8 9 time period, seconds 10 11 12 Position of the vehicle, meters 100 90 80 70 60 original position estimated position 4 5 6 7 8 9 time period, seconds 10 11 12 Figure 5-2: The weight function and the tted curve for an observation at time period 8. 8 or at time period 9 must be erroneous. When the local regression procedure is used, the speed non negativity constraint Equation 5.4 takes care of this problem. Using the local regression procedure, the position at time period 8 was estimated to be 90.84 meters. Observe that, tting a single curve to the observations in Figure 5-1 would require a polynomial of a very high degree. 5.2 Data Collection Data was collected using standard video equipment. The video tapes were analyzed using VIVA, an image processing software specially designed for tra c application described in Section 5.2.2. 5.2.1 Description of the Data Collection Site Video data of tra c ow was collected on Interstate 93 at the Central Artery, located in downtown Boston the rectangle area in Figure 5-3. The video was processed using the VIVA software package described in Section 5.2.2. The manual and automatic features of VIVA were used to process congested and uncongested to semi congested 91 lane 1 lane 2 lane 3 lane 4 200 m I-93 SB 402 m (1/4 mile) South Station Exit 402 m (1/4 mile) China Town Exit Mass. Pike Exit Figure 5-3: Schematic diagram of the I 93 southbound data collection site  gure not drawn to scale. tra c respectively to obtain discrete measurements of vehicle lengths and positions over time. The processed data was then used to obtain vehicle trajectories. The section has a three lane mainline lanes 1 to 3 and a weaving lane lane 4. The mainline lanes continue into an underground tunnel. The weaving section leads to Exit 22 The South Station Exit. There are two more exits further downstream from this section. The rst exit is 1 4 mile away Exit 21, The Kneeland Street and Chinatown Exit and the second exit is 1 2 mile away Exit 20, The Massachusetts Turnpike and Albany Street Exit. Trajectory and vehicle length data was extracted for vehicles only when they were within the rectangular area shown in Figure 5-3. The length of the recorded section varied from 150 to 200 meters as di erent zooms were used during the lming process see Table 5.1. Data was collected for 2 hours starting at 10:26 a.m. tape 1 on 92 Table 5.1: Description of the collected tra c video. Date tape 1 tape 2 tape 3 tape 4 tape 5 8 12 12 12 12 9 95 10 97 10 97 10 97 10 97 Time Length of the hrs Section meters 10:26 to 12:26 200 12:30 to 13:00 165 13:09 to 13:39 190 13:47 to 14:17 150 14:25 to 14:55 180 August 9, 1995, and for 30 minutes each starting at 12:30 p.m. tape 2, 1:09 p.m. tape 3, 1:47 p.m. tape 4, and 2:25 p.m. tape 5 on December 10, 1997. On both days of recording, the sky was overcast with periodical sunshine. Vehicles that traveled in the mainline lanes and made no lane changes provide samples for estimating the acceleration model. Vehicles from lanes 2 or 3, that changed to the left adjacent lane or did not change lanes within the data collection site, provide samples for estimating the discretionary lane change model see discussion on page 121. Vehicles that traveled from the on ramp and merged with the mainline provide samples for estimating the mandatory lane change and forced merging models. 5.2.2 Video Processing Software VIVA2 Video Tra c Analysis System is an image processing software developed at Universitat Kaiserslautern, Germany. It is capable of measuring positions of vehicles from video images. It has both an automatic and a manual feature. The automatic feature extracts positions of all vehicles within the video image in real time. However, in heavily congested tra c, due to lack of spacing between vehicles, the software runs into di culty in identifying front and rear bumpers of closely spaced vehicles and hence the position estimates become unreliable. In such situations, the manual feature can be used to identify vehicle positions by clicking on the screen and the software generates the coordinates of vehicle positions. The manual process, however, Information about this software package may be found in the World Wide Web at the URL http: transport.arubi.uni kl.de ViVAtra c English. 2 93 is very time consuming. An initial testing indicated that the manual feature requires approximately 30 person hours to process one minute of video data. The accuracy associated with the position measurements from the video images depends on the sharpness of the images and the scale of the images. In the automatic feature, VIVA uses the contrast between the image of a vehicle and that of the underlying pavement to identify the vehicle. In the manual process, the user identi es the vehicle. Therefore, a sharper image compared to a blurred image and a larger scale3 compared to a smaller scale would increase the accuracy with which the bumpers of the vehicles can be identi ed on which the accuracy of position measurement depends. The position measurement error for the video that we collected was estimated to be 1 meter. 5.2.3 Processing the Tra c Data Description of the Trajectory Data The trajectory data obtained from processing the video data with VIVA included vehicle position recorded at discrete time points. The methodology described in Section 5.1 was used to develop vehicle trajectories and subsequently speed and acceleration pro les for each vehicle. The rst row of plots in Figure 5-4 shows minute by minute tra c ow at the upstream end of the data collection site, and the second and the third row of plots show second by second density and average speed of all vehicles of the mainline lanes 1 to 3 respectively. The rst column of plots corresponds to tape 1 data, the second column of plots corresponds to tape 2 data and so on. Nine minutes of trajectory data was extracted from tape 1 using the manual feature of VIVA as tra c was extremely congested and at times stopped and the software's automatic data extraction feature would not work in such tra c conditions. Using the automatic feature, for the other four tapes that had less congestion, 30, 30, 21, and 18 minutes of trajectory data were extracted. 3 A larger scale compared to a smaller scale implies that a vehicle would appear larger. 94 Note: 1350 hrs implies 1:50pm; tape 1 implies data collected from tape 1 tape 1 tape 2 tape 3 tape 4 tape 5 main line flow in 100, veh/hr/lane 20 20 20 20 20 15 15 15 15 15 10 10 10 10 10 5 5 5 5 5 0 1025 1030 0 1035 1230 1240 1250 time in hrs 0 1310 1320 1330 time in hrs tape 1 0 time in hrs tape 2 1350 1360 0 time in hrs tape 3 tape 4 tape 5 80 80 80 80 70 70 70 70 70 60 60 60 60 60 50 50 50 50 50 40 40 40 40 40 30 30 30 30 30 20 20 20 20 20 10 density, veh/lane−km 1440 time in hrs 80 10 10 10 10 0 1025 1030 0 1035 1230 1240 1250 time in hrs 0 1310 1320 1330 time in hrs tape 1 0 time in hrs tape 2 1350 1360 0 time in hrs tape 3 1430 1440 time in hrs tape 4 tape 5 35 35 35 35 35 30 30 30 30 30 25 25 25 25 25 20 20 20 20 20 15 15 15 15 15 10 10 10 10 10 5 average speed, m/s 1430 5 5 5 5 0 1310 1320 1330 0 0 1025 1030 time in hrs 0 1035 1230 1240 1250 time in hrs time in hrs 1350 1360 time in hrs 0 1430 1440 time in hrs Figure 5-4: Flow, density, and average speed of the I 93 southbound trajectory data. Although in the rst 9 minutes of data, ow was in the order of 800 to 1300 vehicles hr lane compared to a capacity of approximately 2000 vehicles hr lane, due to conditions downstream of the data collection site, tra c moved very slowly. This may have been in part due to a high volume of tra c trying to perform a lane change to take the two exits a quarter mile and a half mile downstream. The density, as shown in the 1st plot of row 2 in Figure 5-4, was always above 41 vehicles km lane which corresponds to level of service F HCM 1985. The average speed of tra c across the mainline lanes varied from 3 to 10 m s meters per second which is also 95 indicative of the heavy congestion. Tra c in the last four tapes varied from free ow to semi congested with level of service between A and E. The last four columns of the rst row of plots in Figure 5-4 show that the ow varied from 1000 to almost 2000 vehicles hr lane. Density, shown in the last four plots of the second row, varied from 0 to 40 vehicles km lane and never exceeded 41 vehicles km lane. The average speed of the mainline tra c varied from 15 to 33 m s. Estimation Results using the Local Regression Procedure Window Size Selection As mentioned above, depending on the type of application, the window size should be selected such that either the bias or the variance or the mean square error of the estimates is minimized. However, a close form solution for estimating the bias or variance does not exist since the curve t algorithm uses inequality constraints Equation 5.4. We, instead, conducted a sensitivity analysis to evaluate the impact of window size on the quality of the results. For this analysis we have used odd number of window sizes e.g. 7, 9, 11 etc. to make the number of observations before and after the time period of interest equal. The minimum window selected was 7. The reason for this choice is the following: with window size equal to 5, the order of the polynomial cannot exceed 4. As a result, the order of the polynomial representing the acceleration pro le would be 2 since the acceleration is obtained by taking the second derivative of the trajectory function. This implies that, the curvature of the acceleration pro le i.e., its second derivative is restricted to be a constant. Windows of size 7 and above do not su er from such a limitation. The curve t procedure was repeated for a subset of vehicles using window sizes 7, 9, 11, 13, and 15. Figure 5-5 shows the histograms of the absolute values of the position estimation error corresponding to the measured position of a driver using different window sizes. Although, in this case the mean of the absolute errors increased with the window size, such phenomenon was not observed when the sensitivity analysis was performed on the trajectories of other vehicles. Considering the 1 meter 96 Pecentage 0.4 window size 7 0.2 Pecentage 0 Pecentage 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 window size 9 0.2 mean error = 0.250 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 window size 11 0.2 0 Pecentage 0 0.4 0 mean error = 0.316 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 window size 13 0.2 0 Pecentage mean error = 0.223 m mean error = 0.346 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 window size 15 0.2 0 mean error = 0.355 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Absolute position estimation error, m 0.8 0.9 1 Figure 5-5: Histograms of the absolute values of the position estimation error using di erent window sizes. accuracy associated with recording vehicle trajectories, the magnitude of errors for di erent window sizes is within a reasonable range. The estimated speed and acceleration pro les of the vehicle for di erent window sizes are shown in Figure 5-6. Except for the rst and last time periods, the speed and acceleration estimates do not di er signi cantly. Therefore, speed and acceleration estimates at these boundary points should not be used in estimating di erent driver behavior models. On the basis of these plots, any window size between 7 to 15 can be considered acceptable. Of these sizes, 9 was chosen since it strikes the best balance between accuracy and computational e ort. Errors for window size greater than or equal to 9 were very similar especially for the case of stopped vehicles. Examples Figure 5-7 shows two examples of curve tting to the whole trajectory by applying the local regression procedure described above. In the second example second plot of Figure 5-7, the vehicle was stopped for a few seconds. As shown in the gure, a 97 14 12 window size 7 window size 9 window size 11 window size 13 window size 15 Speed, m/s 10 8 6 4 2 0 0 5 10 15 Time period, seconds 20 25 30 25 30 Acceleration, m/s2 5 0 window size 7 window size 9 window size 11 window size 13 window size 15 −5 0 5 10 15 Time period, seconds 20 Figure 5-6: Estimated speed and acceleration pro les using di erent window sizes. very good t of the model was obtained in both cases. Data for Estimating the Acceleration Model The data required for estimating the acceleration model includes acceleration, speed, headway, and type of the subject vehicle, speed and type of its leader's vehicle, density ahead4 of the subject, roadway curvature, grade, speed limit of the roadway section, and pavement surface quality. The trajectory information described above was used to estimate acceleration, Although the data collection section is fairly straight, a visibility distance of 100 meters ahead of the driver was used while computing the explanatory variable density ahead. If, however, the distance from a vehicle' current position to the downstream end of the data collection site is less than 100 meters, the density of tra c ahead computed while this distance was greater than 100 meters is used as the density for this case. 4 98 220 200 180 Position of vehicle, meters 160 140 120 100 80 original position estimated position 60 40 20 0 5 10 15 time period, seconds 20 25 30 220 200 180 Position of vehicle, meters 160 140 120 100 80 60 40 original position estimated position 20 0 5 10 15 20 25 time period, seconds 30 35 40 Figure 5-7: Examples of curve tting by local regression. speed, and space headways. The tapes were also used to collect data on vehicle length. Since the data was collected only from one site, site speci c explanatory variables for example, geometric characteristics cannot be used. Observations for a driver were recorded from the instant the driver reached the upstream end of the data collection site. A sample of 1647 observations from 402 drivers was used for estimation. An acceleration observation was recorded at a time instant such that the data on the tra c conditions max seconds the maximum reaction time earlier can be obtained. We adopted the maximum of the range of max to be considered while estimating the model equal to 4 seconds. This is because, 4 seconds is the most conservative value suggested in the literature see, for example, Johansson and Rumer 99 percent of cases 1971, Lerner et al. 1995, and Homburger and Kell 1988. Therefore, the rst acceleration observation was recorded at the 5th second. Furthermore, since reaction time cannot vary for a given driver observed over a short period of time, explanatory variables for the reaction time model for example, average front vehicle speed to be used as a proxy for vehicle travel speed were obtained by averaging the observations recorded during the rst ve seconds, i.e., before the rst acceleration observation recorded for the driver. Figure 5-8 shows the histograms of the acceleration, subject speed, front relative 20 10 percent of cases 0 −4 percent of cases 0 acceleration, m/s/s 1 2 3 4 0 5 10 15 20 25 speed, m/s 40 20 −5 −4 −3 20 −2 −1 0 1 2 front vehicle speed − subject speed, m/s 3 4 5 10 0 percent of cases −1 60 0 0 2 4 40 6 time headway, sec 8 10 12 20 0 percent of cases −2 10 0 percent of cases −3 20 5 15 25 35 45 55 space headway, meters 65 75 85 95 5 15 25 35 45 55 density, veh/km/lane 65 75 85 95 30 20 10 0 Figure 5-8: Histograms of the acceleration, subject speed, relative speed, time and space headway, and density in the data used for estimating the acceleration model. speed, time and space headways, and density of tra c ahead of the subject of all 100 vehicles in the data. The second plot of Figure 5-8 shows two regimes: regime one represents heavily congested tra c with speeds varying between 0 and 12 m s and the tra c density varying between 40 and 102 veh km lane, and regime two represents semi congested to uncongested tra c with speeds varying between 12 and 32 m s and density below 40 veh km lane. Table 5.2 shows more statistics of the data. Table 5.2: Statistics of the data used for estimating the acceleration model. accefront time space density leration speed Vehicle headway headway ahead m s s m s speed m s sec m veh km lane maximum 7.28 27.1 32.3 15.1 152.1 102.0 minimum -5.73 1.01 0.4 0.1 0.1 0.0 mean 0.12 7.2 7.5 2.4 17.2 64.4 median 0.19 5.7 5.8 2.2 12.1 66.7 std. dev. 1.29 5.2 6.0 1.3 17.2 19.8 number of drivers = 402 number of observations = 1647 percent of acceleration observations = 56.0  mean and standard deviation of all acceleration observations: 1.02, 0.78 mean and standard deviation of all deceleration observations: -1.02, 0.81 percent of heavy vehicle = 19 The speed of the subject vehicle varied from 1 to 27 m s. The tra c density varied between 7 and 103 vehicles km lane. The time headway varied from a fraction of a second to 15 seconds while the space headway varied from less than 5 meters to 152 meters. 54.0 of the observations were acceleration observations while the rest were deceleration observations. 19 of the vehicles were heavy vehicles length greater than 9.14 meters or 30 feet. The data, therefore, represents a wide range of tra c conditions. Data for Estimating the Discretionary Lane Changing Model The data used for estimating the discretionary lane changing model consists of observations from 843 drivers. The total number of gaps observed was 4335, and the number of discretionary lane changes was 75 101 For each gap and driver, the data provides information on the lead, lag, and front gaps, vehicle length, speed, and acceleration of the subject, front, lead and lag vehicles see Figure 5-9 for de nition of di erent vehicles and gaps, density of tra c in the total clear gap + vehicle length lag gap lead gap lag vehicle lead vehicle subject front vehicle Figure 5-9: The subject and the front, lead, and lag vehicles. current and target lanes, and whether the driver merged into this gap. In some cases, there was no lead vehicle within the data collection site rectangle area in Figure 5-3. To obtain the lead gap, the trajectory data from earlier time periods was searched to nd the last vehicle that crossed the downstream boundary of the site from the target lane. Assuming that the vehicle continued in the same lane with the same speed, the lead vehicle's position was extrapolated to the time period in question to calculate the lead gap. Similarly, when there was no lag vehicle in the target lane, trajectory data from later time periods was searched to nd the rst vehicle that entered the upstream end of the target lane. Again, assuming that the lag vehicle traveled with the speed of its rst appearance during this time span, the lag vehicle's position was extrapolated backwards to the time period in question to calculate the lag gap. Statistics corresponding to the gaps that the drivers merged into i.e., the gaps that were acceptable to the drivers are shown in Table 5.3. The density of tra c in the target lane varied from 0 to 85 vehicles km lane. Vehicle speeds varied from 2 to 38 m s with a mean around 18 m s. As before, this represents a wide range of tra c conditions. 102 Table 5.3: Statistics of the discretionary lane changing model data corresponding to the gaps that the drivers merged into. current target lead lag front lead lag front lane lane veh. veh. veh. gap gap gap density density speed speed speed speed m m m veh km veh km m s m s m s m s max. 192.5 232.5 166.6 84.9 75.8 28.1 37.9 25.7 37.3 min. 1.7 2.5 0.1 0.0 0.0 3.5 5.4 1.9 2.0 mean 35.1 36.1 29.5 24.8 22.3 16.9 20.1 16.8 17.4 median 22.3 26.9 16.2 18.2 16.9 18.5 22.8 19.4 19.3 std. dev. 41.8 34.5 35.0 21.4 16.1 7.0 8.1 6.5 7.9 number of drivers = 843 number of observations = 4335 number of lane change observations = 75 percent of heavy vehicle vehicle longer than 9.14 m = 22 Data for Estimating the Mandatory Lane Changing Model This data consists of observations from vehicles merging from the on ramp to the mainline. As mentioned in Section 4.3.4, we assumed that drivers merge by gap acceptance when the level of service of the roadway section is between A and E and by gap creation i.e., forced merging when level of service is F. Therefore, to estimate the mandatory lane changing model, observations were recorded only when the level of service was between A and E, i.e., density was less than 41 vehicles km lane. A total of 500 observations was recorded from 202 drivers. For each driver, the observation includes a series of gaps. The last gap observed by each driver before he she changed lanes, was considered acceptable, since at the next time period the driver was observed in the target lane. The variables of interest for each gap and driver include the lead, lag, and front gaps, vehicle length, speed, and acceleration of the subject, front, lead and lag vehicles, delay or time elapsed since the subject crossed the merging point between the on ramp and the freeway section X X in Figure 5-10, remaining distance to point at which the lane change must be completed section Y Y in Figure 5-10, and density of tra c in the target lane. 103 X total clear gap + vehicle length lag gap Y lead gap lag vehicle lead vehicle subject front vehicle X Y Figure 5-10: The subject, lead, lag, and front vehicles, and the lead and lag gaps. Similar to the discretionary lane change data, in some cases there was no lead and or lag vehicle within the data collection site. The technique described in Section 5.2.3 was applied here as well to infer the lead lag gaps for such cases. Statistics corresponding to the gaps that the drivers merged into i.e., the gaps that were acceptable to the drivers are shown in Table 5.4. The maximum and minimum delays in merging were 5 and 0 seconds respectively with a mean and Table 5.4: Statistics of the mandatory lane changing model data corresponding to the gaps that the drivers merged into. rem. adjac. lead lag dislane gap gap delay tance density speed m m sec. m veh km m s max. 302.7 188.1 5.0 179.9 33.96 25.90 min 1.2 2.1 0.0 47.6 0.00 9.09 mean 44.8 36.2 1.8 129.5 15.83 18.48 median 28.3 27.4 2.0 135.4 15.92 18.75 std. dev. 44.6 31.4 1.2 26.4 8.36 3.14 number of drivers = 202 number of observations = 500 percent of heavy vehicle vehicle longer than 9.14 m = 4.5 lead veh. speed m s 35.92 6.52 20.86 20.36 4.71 lag veh. speed m s 29.03 7.49 18.10 18.23 3.11 front veh. speed m s 30.95 9.07 20.78 20.94 4.43 standard deviation of 1.8 and 1.2 seconds respectively. The density of tra c in the target lane varied from 0 to 34 vehicles km lane|which is rather low and explains why the drivers experienced lower delay in merging. Vehicle speeds varied from 7 to 104 36 m s with a mean around 20 m s. The remaining distance to the point at which lane change must be completed varied from 48 to 180m with a mean, median, and standard deviation of 130, 135, and 26 meters respectively. This implies that, for a majority of the drivers in this data, remaining distance may not have signi cant in uence on the merging process. Data for Estimating the Forced Merging Model The data consists of observations from vehicles merging from the on ramp to the adjacent mainline. As mentioned in Section 4.3.4, we assumed that drivers merge by gap creation i.e., forced merging when the level of service of the roadway section is F. Therefore, observations were recorded only when the level of service was F, i.e., the tra c density was more than 41 vehicles km lane. A total of 998 observations was recorded from 79 drivers. Descriptive statistics of the data corresponding to the observations for the forced merging model are presented in Table 5.5. Table 5.5: Statistics of the data used for estimating the forced merging model. rem. mainline target lead lag lead lag dislane lane veh. veh. gap gap delay tance density density speed speed speed m m sec. m veh km veh km m s m s m s max. 31.8 56.9 28.0 154 72.5 75.8 15.1 12.8 11.6 min -13.2 -12.2 0.0 23 41.0 20.2 0.0 0.0 0.0 mean 4.3 7.8 9.5 103 57.9 58.8 5.1 5.2 5.0 median 3.7 3.6 8.0 102 59.0 60.7 4.8 5.1 4.8 std. dev. 6.0 11.9 6.3 27 7.9 10.1 2.8 2.5 2.6 number of drivers = 79 number of observations = 566 percent of heavy vehicle vehicle longer than 9.14 m = 5.1 The variables in the data, for each gap and driver, are the lead and lag gaps, vehicle length, speed, and acceleration of the subject, lead and lag vehicles, time elapsed since the subject crossed the merging point between the on ramp and the freeway section X X in Figure 5-10, remaining distance to point at which the lane 105 change must be completed section Y Y in Figure 5-10, and density of tra c in the mainline lanes and the target lane. The lead gap varied from -13.25 to 32 meters with a mean of 4.3 meters. The lag gap varied from -12 to 57 meters with a mean of 8 meters. The mean delay experienced by the drivers was 9.5 seconds compared to 1.8 seconds observed in the mandatory lane changing data Table 5.4. The mainline tra c density varied from 40 to 73 vehicles km lane with a mean of 58 vehicles km lane representing a very congested tra c. The average speed of vehicles was around 5 m s. 5.3 Conclusions In this chapter, a methodology to estimate instantaneous speed and acceleration that are required for model estimation from trajectory data that can be obtained from the eld using video technology is described. The methodology is based on the local regression procedure of Cleveland and Devlin 1988. The main advantage of this procedure, over the conventional regression, is that it allows for estimating position, speed, and acceleration pro les that, otherwise, would require tting polynomials of a very high order. Although local regression estimates are less e cient, the exibility of the method outweighs this disadvantage. The characteristics of the freeway trajectory data collected in the Central Artery, Boston are also described. The data represents a wide range of tra c conditions, from very congested stop and go tra c to free ow. The tra c density varied from no vehicles within the data collection site to 90 vehicles km lane. In addition to the length of each vehicle that traveled in the section, the data contains position, speed, and acceleration of every vehicle for every second. Finally, descriptive statistics of the data used to estimate the acceleration model, the discretionary and mandatory lane changing models, and the forced merging model are presented. In this case, the lead vehicle and the subject were overlapping and the lead vehicle was a heavy vehicle. 5 106 Chapter 6 Estimation Results In this chapter, estimation results of the acceleration and lane changing models, using the data described in Chapter 5, are presented. Along with the estimation results, assessment of the parameter estimates from statistical and behavioral standpoints are also presented. In addition, parameter estimates of the car following model, the headway threshold and reaction time distributions, and the gap acceptance model under mandatory lane changing situations are compared to those estimated by other researchers. No such comparison can be made for parameters of the free ow acceleration model, the discretionary lane changing model, and the forced merging model since these have not been estimated before. 6.1 Estimation Results of the Acceleration Model Given the complexity of the likelihood function, the estimation of the parameters was simpli ed by estimating the values of h , h , and max non parametrically. The min max overall estimation approach was based on the following algorithm: 1. Set fh ; h ; min max max g to reasonable initial values. 2. Using the current values of fh ; h ; max g specify and estimate the model min max using the maximum likelihood method Equation 3.25. 107 3. Estimate the parameters of the model speci cation in step 2 for di erent set of values of fh ; h ; max g. Through this grid search procedure obtain the min max  ; h ; max g, i.e., the one with the highest likelihood value. best fhmin max 4. Iterate between steps 2 and 3 until the same set of fh ; h ; min max max g is obtained. The parameters h ; h ; and max were initially set to 0, 8, and 3 seconds remin max spectively. Using these values, step two was executed. In this step, we investigated different model speci cations and simultaneously varied the parameter  Equation 3.7 between the 0 to 1 range. The likelihood function attained its maximum at  = 0. Step three was performed next by varying h , h , and max . Table 6.1 shows min max the values of the maximized likelihood function at di erent values of h , h , and min max Table 6.1: Estimated likelihood function for di erent values of h ; h ; and min max h h max min = 0 max = 3 -2255.24 max = 3:5 -2256.00 max = 4 -2263.95 =6 h min = 0:5 -2252.17 -2257.49 -2266.09  =8 hmax h = 0 h = 0:5 min min -2258.63 max = 3 -2254.61 -2256.41 max = 3:5 -2257.50 -2274.23 max = 4 -2263.47 max . max . h min =1 -2257.69 -2256.25 -2259.95 h min =1 -2257.69 -2262.38 -2265.05 The likelihood function attained the maximum value for max = 3 seconds, h = 0:5 second, and h = 6 seconds. In the next iteration of step two, the same min max model speci cation was obtained as was used while executing step three. The parameter  was varied again between the range 0 to 1. Figure 6-1 shows the value of the likelihood function as a function of  . As before, the likelihood function attained its maximum value at  = 0. Since, experience with the model estimation indicated that the likelihood function may not be globally concave, we reestimated the model using di erent starting values of the parameters to obtain the best possible local maxima. We obtained the same solution for di erent starting values of the parameters. 108 −2252 −2254 the likelihood function −2256 −2258 −2260 −2262 −2264 −2266 0 0.1 0.2 0.3 0.4 0.5 ξ 0.6 0.7 0.8 0.9 1 Figure 6-1: The likelihood function as a function of  . Table 6.2 summarizes the estimation results. All the parameters, with the exception of the car following acceleration sensitivity parameters, have the expected signs. Note that, a positive sign of the parameters of speed and headway in the car following acceleration or deceleration model Equations 3.2 and 3.7 implies that the variables are in the numerator and denominator respectively. The signs of the explanatory variables speed and density of the car following acceleration model are not what we anticipated. Their t statistics are highly signi cant. This indicates that Boston drivers may behave di erently than the way we anticipated see Section 3.2.1. The high acceleration sensitivity at high speeds and high densities indicate that drivers are more aggressive in this situations. This may be in part due to the drivers e ort not to let anyone infront of them from an adjacent lane. The positive sign of the space headway parameter for the car following acceleration model indicates that drivers tend to follow the speed of the lead vehicle less as the space headway increases. In the car following acceleration and deceleration models, except the constants, all the parameters have signi cant t-statistics at the 1 level of signi cance. The speed parameter for the car following deceleration model had counterintuitive sign with a t statistic of 0.64, and therefore, was dropped from the speci cation. The explanatory variable density has signi cant t-statistic for both the acceleration and deceleration models. Therefore, the proposed enhancement of the sensitivity term was supported by the data for both the acceleration and deceleration models. 109 Table 6.2: Estimation results of the acceleration model. Variable Parameter t stat. Car following acceleration constant 0.0225 1.08 speed m s 0.722 4.67 space headway m 0.242 6.31 density veh km lane 0.682 4.20 relative speed m s 0.600 7.20 ln cf;acc  -0.193 -2.64 Car following deceleration constant -0.0418 -1.20 space headway m 0.151 5.32 density veh km lane 0.804 4.21 relative speed m s 0.682 10.71 ln cf;dec  -0.221 -5.44 Free ow acceleration sensitivity constant 0.309 7.37 constant 3.28 6.83 front veh. speed m s 0.618 10.04 heavy veh. dummy -0.670 -1.54 indicator for density  19 veh km lane 7.60 5.51 ln f f  0.126 1.99 6 Headway threshold distribution, 0:5 h mean sec 3.17 13.90 h 0.870 3.82 Reaction time distribution, 0 3 constant 0.272 7.62 ln  -1.55 -9.59 number of drivers = 402 number of observations = 1647 L0 = -2727.75 Lc = -2561.26 L ^ = -2252.17 2 = 0.167 Note: Density  19 veh km lane implies level of services A through C HCM 1985. 110 The test statistic for the null hypothesis that the stimulus is a linear function of the lead relative speed for the car following acceleration model i.e., acc = 1 is given by: ^ acc , 1 = 0:600 , 1 = ,4:79 0:0834 ^ varacc q 6.1 Therefore, the null hypothesis can be rejected at the 1 level of signi cance. Similar null hypothesis for the car following deceleration model i.e., dec = 1 can also be rejected at the 1 level of signi cance the t-statistic is equal to ,4:99. These imply that the proposed extension of the the stimulus term to be a nonlinear function of the lead relative speed is supported by the data. The free ow acceleration model parameters with one exception have signi cant t statistics at the 1 level of signi cance. The parameter for the heavy vehicle dummy does not have a signi cant t statistic. Both the headway threshold distribution parameters are statistically signi cant at the 1 level of signi cance. The mean and standard deviation of the reaction time distribution are statistically signi cant at the 1 level of signi cance. The adjusted t of the model1 was 0.167. The acceleration model estimation results for the case of  = 1 is presented in Table 6.3.  = 1 implies that the sensitivity is a function of the tra c conditions at the moment the driver perceives the stimulus and decides that he she should respond to it. Hence, the explanatory variables a ecting the acceleration sensitivity are observed at the time instant at which the stimulus is observed. As shown in Figure 6-1, this model has a signi cantly lower t than the one in which has the best t. By relaxing  to be a parameter to be estimated, the likelihood function improved by 11.2 units for the  = 0 case over this model the  = 1 case. It is interesting to note that, although the car following acceleration and deceleration model parameters are of di erent orders of magnitude, the free ow acceleration model and the headway threshold and reaction time distribution parameters are of 12 ^ parameters . = 1 , L ,no. of0 L 111 Table 6.3: Estimation results of the acceleration model for  = 1. Variable Parameter t stat. Car following acceleration constant 0.468 1.56 speed m s 0.129 1.11 space headway m 0.194 5.44 density veh km lane 0.188 1.57 relative speed m s 0.670 10.48 ln cf;acc  -0.253 -5.48 Car following deceleration constant -0.0470 -1.25 space headway m 0.179 5.68 density veh km lane 0.791 4.31 relative speed m s 0.749 11.12 ln cf;dec  -0.235 -6.32 Free ow acceleration sensitivity constant 0.316 8.31 constant 3.12 7.50 front veh. speed m s 0.611 10.80 heavy veh. dummy -0.638 -1.57 indicator for density  19 veh km lane 7.58 5.96 ln f f  0.170 3.59 6 Headway threshold distribution, 0:5 h mean sec 3.28 13.11 1.08 5.00 h Reaction time distribution, 0 3 constant 0.307 8.89 ln  -1.34 -13.68 number of drivers = 402 number of observations = 1647 L0 = -2727.75 Lc = -2561.26 L ^ = -2263.39 2 = 0.163 Note: Density  19 veh km lane implies level of services A through C HCM 1985. 112 the same order of magnitude. We adopt the model presented in Table 6.2 which has a signi cantly higher t. 6.1.1 Discussion The Car Following Models The estimated car following acceleration model is 0:722 acf;acct = 0:0225 Vntt0:242 knt0:682 jVnt , n j0:600 + n X n cf;acc t n 6.2 where, t= n= Vnt = Xnt = knt = Vnt , n  = cf;acct  n current time period, reaction time for driver n, subject speed at time t m s, space headway at time t m, density of tra c ahead of the subject veh km lane, front vehicle speed subject speed m s, N 0; 0:8252: The estimated car following deceleration model is 1 acf;dect = ,0:0418 X t0:151 knt0:804 jVnt , n j0:682 + n n cf;dec t n 6.3 where, cf;dect  N 0; 0:8022. n Figure 6-2 shows the sensitivity of di erent factors on the car following acceleration and deceleration. Acceleration increases with speed, density, and relative speed, and decreases with space headway. On the other hand, deceleration in absolute term increases with density and relative speed in absolute term, and decreases with headway. 113 Note: unless varied, speed = 10 m/s, headway = 20 m, density = 40 veh/km/lane, relative speed = 3 m/s acceleration, m/s 2 3 2.5 2 1.5 1 0.5 5 10 15 20 speed, m/s −1 deceleration, m/s acceleration, m/s 2 2 2.5 2 1.5 1 0 10 20 headway, m 30 −1.2 −1.4 −1.6 −1.8 40 20 headway, m 30 40 30 40 50 density, veh/km/lane 60 2 −0.8 1.6 1.4 1.2 1 0.8 20 30 40 50 density, veh/km/lane −1 −1.2 −1.4 −1.6 20 60 −0.5 deceleration, m/s 2 2 2.5 acceleration, m/s 10 −0.6 deceleration, m/s acceleration, m/s 2 2 1.8 0 2 1.5 1 0.5 1 2 3 4 relative speed, m/s −1 −1.5 −2 −5 5 −4 −3 −2 relative speed, m/s −1 Figure 6-2: Sensitivity of di erent factors on the car following acceleration and deceleration decisions. At low speeds the mean acceleration is lower compared to those for higher speeds. Tra c conditions ahead of the subject and its leader are likely to change more rapidly at high densities than at low densities. Due to this, higher uncertainty is involved in predicting the position and speed of the leader in the near future. As a result, drivers are expected to be more conservative at high densities than at low densities. Although the mean deceleration increases with density, the mean acceleration does not decrease with density as we had expected. The slopes of the acceleration and deceleration curves with respect to the relative speed are decreasing. This captures the fact that, the acceleration deceleration applied by a driver is limited by the acceleration deceleration capacity of the vehicle and acceleration deceleration gradually reaches the capacity as the relative speed 114 increases. Figure 6-3 shows a comparison between the estimated car following acceleration and deceleration at di erent front gaps as a function of subject speed using the models proposed in this thesis with those obtained by Subramanian 19962. The acceleration Note: speed = 10 m/s, density = 40 veh/km/lane 5 0 4.5 −0.5 5 m/s 1 m/s 4 5 m/s −1.5 3 m/s deceleration, m/s2 2 3 m/s −1 3.5 acceleration, m/s 1 m/s 3 2.5 5 m/s 2 3 m/s 1.5 1 −2 3 m/s −2.5 −3 5 m/s −3.5 −4 1 m/s 1 m/s 0.5 0 10 Acc(Estimated) Acc(Subramanian) 15 20 front gap, m 25 −4.5 30 −5 10 Dec(Estimated) Dec(Subramanian) 15 20 front gap, m 25 30 Figure 6-3: Comparison between the car following acceleration and deceleration estimated in this thesis with those obtained by Subramanian 1996. and deceleration estimated in this thesis are generally smaller in magnitude compared to those estimated by Subramanian. At low speeds, his acceleration and deceleration estimates are too high. The acceleration estimated by the model proposed in this thesis is smaller than expected, while the estimated deceleration is reasonable. This may be due to lack of variability in the data with acceleration observations or may be due to the in uence of the geometric characteristics of the Boston data collection site. The di erence between the two models may be due to several reasons. First, Subramanian used data that was collected in 1983 from a section of Interstate 10 Westbound near Los Angeles, whereas, this research used data that was collected in 1995 and 1997 from a section of Interstate 93 Southbound in Boston. The di erent data collection years and sites may have contributed to the di erences in the estimates. 2 The parameters estimated by Subramanian 1996 are presented in Table 2.3. 115 Second, the LA data collection site is a fairly straight section without any ramps, whereas, the Boston data collection site has a weaving section adjacent to the freeway. The geometry of the freeway and the number of lane changes taking place in the Boston data may have an e ect on the estimates obtained in this research. Finally, Subramanian assumed that all the drivers in the data were car following even at large space headways. He further investigated the implication of this assumption and concluded that the assumption on the headway threshold has signi cant impact on the car following model estimates. The estimation results presented in this thesis do not su er from such a limitation. The Free Flow Acceleration Model The estimated free ow acceleration model is h aff t = 0:309 3:28 + 0:618 Vnfrontt , n , 0:670 neavy n +7:60 knt , n  , Vnt , n  + f f t n 6.4 where, Vnfront t , n  = front vehicle speed at time t , n m s, 8 1 if the subject vehicle is a heavy vehicle heavy = vehicle length  9.14 m or 30 ft n : 0 otherwise 8 1 if knt , n   19 veh km lane : 0 otherwise f f t  N 0; 1:132: n knt , n  = The estimated free ow acceleration increases with front vehicle leader speed. A higher acceleration for level of services A through C captures the e ect of higher maneuverability at low densities compared to high densities. The impact of lower maneuverability for the heavy vehicles compared its non heavy counterparts is cap116 tured by an indicator whether the subject vehicle is heavy. The standard deviation of the free ow acceleration is high compared to its car following acceleration and deceleration counterparts. The Headway Threshold Distribution The headway threshold seconds, that de nes whether a driver is in the car following regime or in the free ow regime, is distributed as follows see Figure 6-4: 0.5 h* pdf 0.4 0.3 0.2 0.1 0 0 1 2 3 * h , sec. 4 5 6 0 1 2 3 headway, sec. 4 5 6 1 P(car−following) 0.8 0.6 0.4 0.2 0 Figure 6-4: The headway threshold distribution and the probability of car following as a function of time headway. 8 f h = : 1 h,3:17 if 0:5 0:868 0:870 h  6 otherwise 0 6.5 For a given headway, hnt, the probability that driver n is in the car following regime is given by: 8 1 if hnt  0:5 h t,3:17 1 Pcar following at time t = 1:00 , 0:998  n 0:870 if 0:5 hn t  6 6.6 :0 otherwise 117 The 5, 50, and 95 percentile values of the headway threshold are 1.75, 3.17, and 4.60 seconds respectively. These values are reasonable. Figure 6-5 shows the mean of the headway threshold as a function of subject 100 90 space headway threshold, meter 80 70 60 50 40 30 20 10 estimated mean threshold Herman and Potts (1961) 5 10 15 20 25 30 speed, m/s Figure 6-5: Comparison between the estimated mean headway threshold and the 61 meters threshold suggested by Herman and Potts 1961. speed. In the Tra c Engineering Literature Herman and Potts 19613 a threshold of 61 meters 200 ft is usually used to distinguish the free ow regime. As shown in Figure 6-5, the 61 meters threshold is too high at low speeds while the threshold estimated in this thesis is high at high speeds. The two estimates are in close agreement in the 15 to 23 m s speed range. The Reaction Time Distribution The estimated distribution of reaction time is 8 f  = : 1p 0:212 2 0 ,0 , 1 ln 0.212:272 2 e 2 if 0 3 otherwise 6.7 Herman and Potts 1961 estimated this 61 meters thresholds based on an observation that, the correlation between the observed accelerations and the accelerations estimated by the car following model was low when the space headways were greater than 61 meters. 3 118 Figure 6-6 shows the probability density function and the cumulative distribution function of the reaction time. The median, mean, and standard deviation of the probability density function 1.5 1 0.5 cumulative distribution function 0 0 0.5 1 1.5 reaction time, sec. 2 2.5 3 0 0.5 1 1.5 reaction time, sec. 2 2.5 3 100 80 60 40 20 0 Figure 6-6: The probability density function and the cumulative distribution function of the reaction time. reaction time distribution are 1.31, 1.34, and 0.31 seconds respectively. As discussed in Section 3.2.4, we apriori expect the surrounding tra c conditions to a ect the reaction time of a driver. Explanatory variables capturing tra c conditions include the density of tra c ahead of the driver, the average front vehicle speed that was used as a proxy for average travel speed of the subject, whether the subject vehicle is a heavy vehicle, and an indicator for free ow tra c conditions density  19 veh km lane. However, the t statistics of these explanatory variables indicated that their impact on the reaction time were not signi cant and in some cases the parameters had counterintuitive signs. Therefore, the model with only the constant as an explanatory variable for the mean of the reaction time distribution was adopted. Finally, a comparison between the estimates of the reaction time distribution parameters obtained in this thesis and those obtained by Johansson and Rumer 1971 and Lerner et al. 1995 is presented in Table 6.4. The median and mean estimated in 119 Table 6.4: Comparison between the reaction time distribution parameters obtained from di erent sources. source sample size stimulus median mean std. dev. sec sec sec speed this thesis 402 di erence 1.31 1.34 0.31 Johansson and Rumer 1971 321 sound 0.89 1.01 0.37 unexpected Lerner et al. 1995 56 object 1.44 1.51 0.39 Note: Speed di erence implies di erence between the target speed and the current speed. this research are higher than those obtained by Johansson and Rumer 19714, while, they are lower than those obtained by Lerner et al. 1995. The standard deviation estimated in this research is smaller than those obtained by others. The di erences between the reaction time estimates from di erent studies may be due to the di erences in the time period of study, data collection site, or procedures used in di erent studies. The acceleration and deceleration capacity of vehicles have increased over the past 27 years which may have increased the reaction time of drivers as better vehicle performance may have made driving more relaxing. Driving habits at di erent locations may have also contributed to di erent reaction time estimates. Finally, di erent stimulus were used in di erent studies. In the Johansson and Rumer 1971 study drivers responded to sound, in the Lerner et al. 1995 study drivers responded to visualizing a rolling drum, while in this thesis, drivers responded to the di erence between their target speeds desired speeds or the leaders' speeds depending on the headways and the current speeds. Overall, the parameters of the reaction time distribution estimated in this research are well within the typical range of other studies. In summary, the empirical work suggests that, the sensitivity term of the car following acceleration is a function of the subject speed, the space headway, and the As mentioned in Chapter 2, drivers responded to sound indicating them to press the brake pedal. This may have reduced the perception time, and hence the reaction time. 4 120 density, while the sensitivity term of the car following deceleration is a function of the space headway and the density of tra c. Furthermore, the sensitivity term comprises of explanatory variables observed at the time of applying acceleration deceleration while the stimulus term is lagged by the reaction time of the drivers. The impact of the reaction time on the sensitivity was not supported by the data. The stimulus is a nonlinear function of the front relative speed. The free ow acceleration is a function of the subject's speed, the leader's speed, an indicator whether the subject vehicle is a heavy vehicle, and an indicator whether the density of tra c is low. 6.2 Estimation Results of the Lane Changing Model The discretionary and mandatory lane change models were estimated separately due to lack of data over a long stretch of roadway approximately 1500 to 3000 meters long. The data collection site, shown in Figure 6-7, used in this study has a length of approximately 200m. If a driver in this site changes to the right lane and takes the exit, it is unlikely that the driver is also performing a discretionary lane change. However, if the remaining distance to the exit is 2000 meters as opposed to 200 meters, the probability of performing a discretionary lane change may not be negligible. Therefore, a model that captures discretionary lane changing decision when the driver is in a mandatory lane change situation cannot be estimated using this data. Estimation results of the discretionary and the mandatory lane changing models are presented rst. Then, estimation results of the forced merging model are presented. 6.2.1 Estimation Results of the Discretionary Lane Changing Model The discretionary lane changing model was estimated using observations from drivers in the following two cases see Figure 6-7 for de nition of lanes 1 to 4: drivers that changed from lanes 2 or 3 to the left, and 121 lane 1 lane 2 lane 3 lane 4 200 m I-93 SB 402 m (1/4 mile) South Station Exit 402 m (1/4 mile) China Town Exit Mass. Pike Exit Figure 6-7: Schematic diagram of the I 93 southbound data collection site  gure not drawn to scale. drivers that traveled in lanes 2 or 3 without changing lanes. If drivers from lanes 1 to 3 change to the right and take the exit at the downstream end of the data collection site, the lane changes would be mandatory. Even if they do not take this exit, since there are two exits a quarter mile and a half mile downstream, it is likely that drivers would be changing lanes to take these exits. Since drivers are not observed downstream of the data collection site, the upstream lane changes towards lane 4 cannot be categorized with certainty as discretionary lane changes. Therefore, the choice set for the discretionary lane change subjects includes the left adjacent and the current lanes. 122 Thus, there are two observable states: change to the left lane and continue in the current lane. The discretionary lane changing decision tree then reduces to the decision tree shown in Figure 6-8. For this decision tree, the likelihood function given by Equation 4.8 reduces to: Start driving conditions not satisfactory left lane driving conditions satisfactory current lane Gap Accept Gap Reject Left Lane Current Lane Current Lane Current Lane Figure 6-8: The decision tree for a driver considering a discretionary lane change with the current and the left lanes as choice set. L= N X n=1 ln Z Tn 1 Y L Pt L j   tn Pt C ,1 t=1 L j  1, tn ! f  d  6.8 where, 8 L tn = : 1 if driver n changes to the left lane at time t 0 otherwise. 6.9 Pt L j n = Pt gap acceptable j left lane; driving conditions not satisfactory; n  Pt left lane j driving conditions not satisfactory; n  Pt driving conditions not satisfactory j n Pt C j n  = 1 , Pt L j n  123 6.10 6.11 The expression for the gap acceptance probability is given by Equation 4.4. The conditional probability that the left lane is chosen is given by: Pt left lane j driving conditions not satisfactory; n = 1 LL 1 + exp,Xn t LL , LLn 6.12 where, superscript `LL' denotes left lane. Finally, the conditional probability that the driver is not satis ed with the driving condition of the current lane is given by: Pt driving conditions not satisfactory j n  = 1 DCNS t DCNS , 1 + exp,Xn DCNS n  6.13 where, superscript `DCNS' denotes driving conditions not satisfactory. Estimation Results Table 6.5 shows the estimation results of the discretionary lane changing model. At convergence, the hessian of the the likelihood function did not invert since it was nearly singular. The estimates of the standard deviation of the generic random terms of the lead and lag critical gaps  lead;dlc, lag;dlc were close to zero. Nearly singular hessian and zero estimates of the standard deviations indicate identi cation problems of the model. Next, we estimated a restricted version of the likelihood function in which the serial correlation between di erent observations from a given driver is not modeled. In this case, the likelihood function converged with a negative de nite hessian at convergence as desired and the estimates of the standard deviation of the generic random terms were reasonable. Further research is required to address the identi cation problem mentioned above and this is left as a topic for future research. The restriction of no serial correlation implies that, DCNS in Equation 6.13, LL in Equation 6.12, and g ; g 2 flead; lag g in Equation 4.3 are restricted to be zero, and the model formulation becomes a cross sectional one. The test statistic 124 Table 6.5: Estimation results of the discretionary lane changing model. Variable Parameter Desired Speed Model average speed, m s 0.727 Utility of Driving Conditions not Satisfactory constant 0.0343 subject speed desired speed, m s -0.0757 heavy vehicle dummy -3.56 tailgate dummy 0.486 DCNS -1.11 Utility of the Left Lane constant -1.87 lead veh. speed desired speed, m s 0.0328 front veh. speed desired speed, m s -0.158 lag veh. speed subject speed, m s -0.0960 LL -0.246 Lead Critical Gap constant 0.665 min0, lead veh. speed subject speed, m s -0.412 lead 0.727 lead;dlc ln -7.16 Lag Critical Gap constant 1.69 min0, lag veh. speed subject speed, m s 0.172 max0, lag veh. speed subject speed, m s 0.177 lag -0.653 lag;dlc ln -15.5 number of drivers = 843 number of observations = 4335 number of discretionary lane change observations = 75 L0 = -482.92 Lc = -360.05 L ^ = -326.51 2 = 0.282 Note: di erent vehicles and gaps are de ned in Figure 6-9. 125 ,2Lrestricted , Lunrestricted is distributed 2 with degrees of freedom equal to the number of restrictions. The test statistic for the null hypothesis of no serial correlation, i.e., DCNS = LL = lead = lag = 0; 6.14 is given by: ,2Lrestricted , Lunrestricted = ,2  ,330:57 , ,326:51 = 8:12 6.15 The critical value of the 2 distribution with 4 degrees of freedom at the 5 level of signi cance is 9.49. Therefore, the null hypothesis of no serial correlation cannot be rejected and we adopt the model with no serial correlation. Table 6.6 shows the parameter estimates obtained by maximizing the restricted likelihood function. The factors a ecting a driver's decision whether the driving conditions are satisfactory are the di erence between the subject speed and their desired speed, an indicator whether the subject vehicle is a heavy vehicle, and an indicator whether the subject is tailgated. See Figure 6-9 for de nition of di erent vehicles and gaps. total clear gap + vehicle length lag gap lead gap lag vehicle lead vehicle subject front vehicle Figure 6-9: The subject and the front, lead, and lag vehicles. The desired speed model is assumed to have the following functional form: DS Vn t = Xn t 126 DS 6.16 Table 6.6: Estimation results of the discretionary lane changing model. Variable Parameter t stat. Desired Speed Model average speed, m s 0.768 Utility of Driving Conditions not Satisfactory constant 0.225 subject speed desired speed, m s -0.0658 heavy vehicle dummy -3.15 tailgate dummy 0.423 Utility of the Left Lane constant -2.08 lead veh. speed desired speed, m s 0.0337 front veh. speed desired speed, m s -0.152 lag veh. speed subject speed, m s -0.0971 Lead Critical Gap constant 0.508 min0, lead veh. speed subject speed, m s -0.420 ln lead;dlc -0.717 Lag Critical Gap constant 2.02 min0, lag veh. speed subject speed, m s 0.153 max0, lag veh. speed subject speed, m s 0.188 ln lag;dlc -0.642 number of drivers = 843 number of observations = 4335 number of discretionary lane change observations = 75 L0 = -482.92 Lc = -360.05 L ^ = -330.57 2 = 0.282 Note: di erent vehicles and gaps are de ned in Figure 6-9. 127 5.37 0.17 -0.62 -3.18 1.71 -2.53 0.76 -3.12 -1.84 1.53 -3.73 -1.45 5.00 1.29 1.69 -1.67 where, DS Xn t = explanatory variables a ecting the desired speed DS , DS = model parameters. Note that, a constant for the desired speed cannot be estimated since it is absorbed into the constants of the utilities of the decisions `driving conditions not satisfactory' and `left lane'. The explanatory variables used for the desired speed model include the average speed of the vehicles ahead of the subject, the speed of the front vehicle, the density of tra c ahead of the subject, and an indicator whether the subject vehicle is a heavy vehicle. A higher average speed of the vehicles ahead or the front vehicle speed and a lower density of tra c are expected to increase the desired speed of the driver. Due to lack of maneuverability and safety concern, driver of a heavy vehicle is expected to have lower desired speed than its non heavy counterpart. However, except for the average speed, the t statistics of the other explanatory variables were insigni cant and some of parameters had counterintuitive signs. In the nal model only the average speed of the vehicles ahead of the subject was used which had a signi cant t statistic. To capture the e ect of tailgating which cannot be observed from the data, a proxy variable tailgate dummy is de ned as follows: 8 tailgate t = n : 1 if gap behind the subject's rear bumper  10 m and tra c level of service is A, B, or C 0 otherwise 6.17 t where, nailgate t denotes the tailgate dummy. A speed above the desired speed implies satisfaction with the current lane, since in this situation a driver has the exibility to adjust its speed. On the other hand, a speed below the desired speed would motivate a driver to perform a discretionary lane change. The corresponding parameter has the desired negative sign. Although, its t statistic is not signi cant, it is included in the model due to its importance from a behavioral standpoint. Due to lack of maneuverability, heavier vehicles are 128 hesitant toward changing lanes and the corresponding parameter has a signi cant t statistic. Finally, when tailgated, drivers tend to seek discretionary lane change and the corresponding parameter has the desired positive sign and its t statistic is signi cant at the 10 level of signi cance. Factors a ecting the decision whether the left lane is more desirable than the current lane include the di erence between the lead vehicle's speed and the subject's desired speed, the di erence between the front vehicle's speed and the subject's desired speed, and the di erence between the subject speed and the speed of the lag vehicle. A higher lead or front vehicle's speed implies higher exibility for the subject in the corresponding lanes. The lag relative speed captures the e ect of safety concern to perform a lane changing decision and its parameter has a signi cant t statistic at the 10 level of signi cance. The only factor a ecting the discretionary lead critical gap is the lead relative speed only when the lead vehicle is slower. Its parameter is statistically signi cant at the 1 level of signi cance. For the lag critical gap, the lag relative speed is the only important factor. To capture the di erent impact of the lag relative speed depending on whether the lag vehicle is faster or not, a piecewise linear approximation of the lag relative speed with a breakpoint at 0 m s is used. The variable max0, lag vehicle speed - subject speed has a signi cant t statistic at the 10 level of signi cance while the variable min0, lag vehicle speed - subject speed does not have a signi cant t statistic. In spite this, the latter variable is included in the model due to its importance from a behavioral standpoint. Higher sensitivity of the lag critical gap when the lag vehicle is faster is captured by the higher parameter estimates of the variable max0, lag vehicle speed subject speed compared to the variable min0, lag vehicle speed subject speed. The adjusted t of the model was 0.282. To test the null hypothesis that all the parameters except the constants and standard deviations are zero, the likelihood ratio test was used. The test statistic is given by: 129 ^ ,2LC  , L ^ = ,2,330:57 , ,360:05 = 58:96 6.18 The critical value of the 2 distribution with 9 degrees of freedom at the 5 level of signi cance is 16.92. Hence, the null hypothesis can be rejected. The estimated probability that driver n is not satis ed with the current lane driving conditions not satisfactory at time t is given by: Pt driving conditions not satisfactory = 1 ,0:225+0:0658 Vn t,Vn t+3:15 1+e h t neavy ,0:423 nailgate t 6.19 The conditional probability of choosing the left lane over the current lane is given by: Pt left lane j driving conditions not satisfactory = 1 2:08,0:0337 Vnlead t,Vn t+0:152 Vnfrontt,Vn t+0:0971 Vnlag t 1+e 6.20 where, Vnlag t denotes the lag vehicle speed minus the subject speed m s. The estimated lead and lag critical gaps in meters for the discretionary lane change case are Glead;dlct = exp 0:508 , 0:420 min0; Vnlead t + lnead;dlct cr;n Glag;dlct = exp 2:02 + 0:153 min0; Vnlag t + cr;n 0:188 max0; Vnlag t + lnag;dlct where, Vnleadt = lead vehicle speed subject speed m s, lead;dlc t  N 0; 0:4882; n 130 6.21 6.22 lag;dlc t n  N 0; 0:5262: Another way of assessing the estimated parameters is to compute the probability of acceptance of gaps that drivers merged into and hence were acceptable to them. There were 75 such cases. These estimates should be higher than 0.5 and close to 1.0. The estimated probability had a mean of 0.83 and a standard deviation of 0.25. Figure 6-10 shows the histogram and cumulative distribution of the estimated probabilities. For 60 Percentage 50 40 30 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Estimated Probability 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Estimated Probability 0.7 0.8 0.9 1 Cumulative Percentage 100 80 60 40 20 0 Figure 6-10: The estimated probability of acceptance of gaps that were acceptable and merging were completed. the majority of the gaps actually accepted, the estimated probability of acceptance was close to one. On 88 of the cases, the estimated probabilities were greater than 0.5. Finally, Figure 6-11 shows the median lead and lag critical gaps for DLC situations as a function of the lead and lag relative speeds. When both the lead and lag relative speeds are zero, the median lead and lag critical gaps are 1.7 and 7.5 meters respectively. This is intuitive since the lag gap acceptance process is more critical than the lead gap acceptance process. The median lead critical gap decreases from 13.5 to 1.7 meters as the lead relative speed increases from -5 m s to 0 m s. As the 131 Median Lead Critical Gap Length, m 10 Median Lag Critical Gap Length, m 40 8 6 4 2 0 −5 −4 −3 −2 −1 0 lead vehicle speed − subject speed, m/s 1 2 30 20 10 0 −4 −2 0 2 4 lag vehicle speed − subject speed, m/s 6 8 Figure 6-11: The median lead and lag critical gaps for discretionary lane change as a function of relative speed. lag relative speed increases from -5 to 8 m s, the median lag critical gap increases from 3.5 to 49.2 meters. These numbers are realistic from a behavioral standpoint. In summary, drivers' decision to perform a discretionary lane change is modeled as a two step decision process. First, drivers examine their satisfaction with the driving conditions of the current lane. Important factors a ecting such decision include the di erence between the current speed and the driver's desired speed, an indicator whether the subject vehicle is a heavy vehicle, and an indicator whether the subject is tailgated. If the driver is not satis ed with the the driving conditions of current lane, he she compares the driving conditions of the current lane with those of the other lanes. Such a decision is in uenced by the the speeds of the vehicles ahead in di erent lanes compared to the subject's desired speed and the lag relative speed. The lead critical gap is a function of the lead relative speed only when the leader is slower while the lag critical gap is a function of the lag relative speed. The importance on the decision to perform a discretionary lane change of other explanatory variables, such as the relative density of tra c in di erent lanes, whether the lead or the lag vehicle is heavy, and whether the lane is adjacent to an on ramp, was not supported by the data. 132 6.2.2 Estimation Results of the Mandatory Lane Changing Model The mandatory lane changing model parameters are estimated using observations from the drivers who merged from the on ramp lane 4 in Figure 6-7 to the adjacent mainline lane lane 3. The data consists of observations from drivers when the level of service of the roadway section was between A and E. For such drivers, the decision tree shown in Figure 4-1 reduces to the decision tree shown in Figure 6-12. Start MLC MLC Left Lane Gap Accept Gap Reject Left Lane Current Lane Current Lane Figure 6-12: The decision tree for a driver merging from an on ramp to the adjacent mainline lane. In this case, the left and the current lanes are the two choices. Thus, there are two observable states: change to the left lane and continue in the current lane. For this decision tree, the likelihood function given by Equation 4.8 reduces to: L= N X n=1 ln Z  Tn 1Y L Pt L j   tn Pt C ,1 t=1 L j  1, tn ! f  d  6.23 L where, tn is de ned in Equation 6.9 and the conditional probability of an observation of driver n changing to the left lane is given by: 133 Pt L j n = Ptgap acceptable j MLC; nPtMLC j n  6.24 The probabilities on the right hand side of Equation 6.24 are given by Equations 4.4 and 4.1 respectively. The probability of staying in the current lane is given by: PtC j n = 1 , PtL j n  6.25 Estimation Results The maximum likelihood estimation results of the mandatory lane changing model are given in Table 6.7. All the parameters that capture the correlation between di erent Table 6.7: Estimation results of the mandatory lane changing model. Variable Parameter t stat. Mandatory Lane Change Utility constant -0.740 -1.75 rst gap dummy -0.884 -2.36 delay sec 0.749 1.36 M LC 0.685 0.65 Lead Critical Gap constant 0.414 0.79 lead 0.676 1.21 ln lead;mlc -1.07 -0.26 Lag Critical Gap constant 0.663 0.97 min 0, lag veh. speed subject speed m s 0.0457 0.24 max 0, lag veh. speed subject speed m s 0.363 2.59 lag 0.330 0.69 lag;mlc ln 0.0101 0.03 number of drivers = 202 number of observations = 500 L0 = -336.16 Lc = -334.83 L ^ = -288.19 2 = 0.107 134 observations from a given driver i.e., M LC in Equation 4.1, and g ; g 2 flead; lagg in Equation 4.3 are statistically insigni cant. To test the null hypothesis of no serial correlation, i.e., M LC = lead = lag = 0; 6.26 a restricted version of the likelihood function with no serial correlation was estimated. These restrictions make the model formulation a cross sectional one. The test statistic is ,2Lrestricted , Lunrestricted = ,2  ,288:45 , ,288:19 = 0:53 6.27 The critical value of the 2 distribution with 3 degrees of freedom at the 5 level of signi cance is 7.81. Therefore, the null hypothesis of no serial correlation cannot be rejected. The parameters obtained by maximizing the restricted likelihood function are shown in Table 6.8. Note that, except for the parameters that correspond to standard deviation of the random terms, the parameter estimates of the unrestricted and restricted models are of the same order of magnitude. Factors a ecting a driver's decision to respond to mandatory lane change situation MLC  are delay time since the driver crossed the merging point, section X X in Figure 6-13 and the indicator for the rst gap when delay is equal to zero5 . The parameters have the expected signs and signi cant t statistics at the 5 level of signi cance. The lead critical gap was found to be insensitive to the tra c conditions, whereas, the lag critical gap length is sensitive only to the lag relative speed. Similar to the discretionary lag critical gap model, a piecewise linear lag relative speed variable was used with a breakpoint at 0 m s was used. As expected, the parameter for the As explained in Section 4.2.1, delay or the rst gap dummy cannot be de ned for general mandatory lane changing cases unless the time at which the driver is in M LC state is well de ned. 5 135 Table 6.8: Estimation results of the mandatory lane changing model. Variable Parameter t stat. Mandatory Lane Change Utility constant -0.654 -2.27 rst gap dummy -0.874 -2.50 delay sec 0.577 3.85 Lead Critical Gap constant 0.384 0.63 ln lead;mlc -0.152 -0.29 Lag Critical Gap constant 0.587 0.79 min 0, lag veh. speed subject speed m s 0.0483 0.23 max 0, lag veh. speed subject speed m s 0.356 2.39 ln lag;mlc 0.0706 0.14 number of drivers = 202 number of observations = 500 L0 = -336.16 Lc = -334.83 L ^ = -288.45 2 = 0.115 explanatory variable max0, lag vehicle speed subject speed was higher than that for min0, lag vehicle speed subject speed. The remaining distance did not a ect a driver's decision process|not an intuitive result. This may be due to the fact that, in the data the mean and median remaining distances were 130 and 135 meters respectively and level of service HCM 1985 varied between A and C. As a result, the remaining distance may not have a signi cant impact on the merging behavior of the sample drivers in the data. The lead and lag critical gap parameters have low t statistics except for the lag relative speed when the lag vehicle is faster. The adjusted t of the model was 0.115. A likelihood ratio test was conducted to test the null hypothesis that all the parameters except the constants and standard ^ deviations are zero. The test statistic, ,2LC  , L ^, is equal to 92.76 and the critical value with 4 degrees of freedom at the 5 level of signi cance is 9.49. Hence, the null hypothesis can be rejected. 136 X Y total clear gap + vehicle length lag gap lead gap lag vehicle lead vehicle front vehicle subject X Y Figure 6-13: The subject, lead, lag, and front vehicles, and the lead and lag gaps. The estimated probability that a driver would respond to an MLC situation is Pt MLC  = 1 1 1 + exp0:654 , 0:577 delaynt + 0:874 nstGapt 6.28 where, delaynt = time elapsed since an MLC situation arises sec, 8 1 if delay = 0 1stGap t = n : 0 otherwise. Figure 6-14 shows the probability of responding to MLC as a function of delay. The 1 0.9 0.8 Probability of MLC 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 delay, seconds 7 8 9 10 Figure 6-14: The probability of responding to MLC as a function of delay. estimated probability approaches one as delay increases beyond 10 seconds and is 137 higher than expected. This may be due to lack of data with larger delay experienced by the drivers in the data set the median and the maximum delay experienced by the drivers in the data were 2 and 5 seconds respectively. The estimated lead and lag critical gaps in meter for mandatory lane change situations are Glead;mlct = exp0:384 + lnead;mlct cr;n Glag;mlct = exp0:587 + 0:0483 min0; Vnlag t + cr;n 0:356 max0; Vnlag t + lnag;mlct 6.29 6.30 where, Vnlag t denotes the lag relative speed m s, lnead;mlct  N 0; 0:8592, and lag;mlc t  N 0; 1:072. n The probability of acceptance of gaps that drivers merged into and hence were acceptable to them was calculated using the estimated parameters. On 72 of the cases, the estimated probabilities were greater than 0.9. Figure 6-15 shows the histogram and cumulative distribution of the estimated probabilities. The estimated 70 60 Percentage 50 40 30 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Estimated Probability 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Estimated Probability 0.7 0.8 0.9 1 Cumulative Percentage 100 80 60 40 20 0 Figure 6-15: The estimated probability of acceptance of gaps that were acceptable and merging were completed. 138 probability had a mean of 0.93 and a standard deviation of 0.12. These results are satisfactory. The median value of the lag critical gap for the mandatory lane change situation was calculated for di erent values of lag relative speed, and the variation is shown in Figure 6-16. When the subject is faster than the lag vehicle in the target lane, the 40 35 Median Lag Critical Gap Length, m 30 25 20 15 10 5 0 −4 −2 0 2 4 lag vehicle speed − subject speed, m/s 6 8 Figure 6-16: The mean lag critical gap for mandatory lane change as a function of lag relative speed. median lag critical gap is less than 2 meters. The median lag critical gap increases at an exponential rate to 32 meters as the lag relative speed increases to 8 m s. The median lead critical gap was 1.5 meters. Ahmed et al. 1996 estimated the median lead and lag critical gaps to be 4.7 and 15.6 meters respectively assuming a 152 meters remaining distance and the gap is the rst gap observed by the driver6 . Although, this research and Ahmed et al. 1996 used the same methodology to estimate the gap acceptance model, the di erences in the estimates may be due to the di erences in the data collection years or sites. From 1983 to 1995 vehicle characteristics have improved KBB 1998 and driving habits of the drivers of these two areas may be di erent which may have contributed to the Ahmed et al. 1996 used a data collected in 1983 from a site at Interstate 95 Northbound near the Baltimore Washington Parkway Smith 1985. The site is a two lane freeway with an adjacent weaving section on the right. Density of tra c varied from 3 veh km lane to 56 veh km lane with a mean of 27 veh km lane. The median length of the lead and lag gaps corresponding to the gaps that the drivers found acceptable and completed the merge were 21 and 25 meters respectively. 6 139 Median Lead Critical Gap Length, m 10 Median Lag Critical Gap Length, m di erences in the critical gap length estimates. The estimated median critical gap lengths under MLC situations are also compared to their DLC counterparts as shown in Figure 6-17. As expected, the median critical lead lag gaps under MLC situations are smaller than their DLC counterparts. 40 DLC situation MLC situation 8 6 4 2 0 −5 −4 −3 −2 −1 0 lead vehicle speed − subject speed, m/s 1 2 DLC situation MLC situation 30 20 10 0 −4 −2 0 2 4 lag vehicle speed − subject speed, m/s 6 8 Figure 6-17: Comparison between the estimated critical gap lengths under DLC and MLC situations. 6.2.3 Estimation Results of the Forced Merging Model The forced merging model was estimated for the case of merging from an on ramp. Since, the forced merging model is assumed to be applicable only in heavily congested tra c, the data consisted of observations from drivers when the level of service of the roadway section was F. The estimation results are shown in Table 6.9. The parameter F M in Equation 4.9 that captures the correlation between di erent observations from the same driver, was estimated to be 0.0012, which is very small. Its t statistic was 0.001. This implies that, the dynamic aspect of driver behavior may be adequately captured by state dependence and that the random term of the utility speci cation is independent over time, even for the same driver. The estimation results presented in Table 6.10 correspond to the parameter estimates obtained by maximizing the 140 Table 6.9: Estimation results of the forced merging model. Variable Parameter t stat. constant -3.16 -10.59 min0, lead veh. speed subject speed m s 0.313 2.66 remaining distance impact 10 2.05 5.33 total clear gap 10 meters 0.285 2.85 FM 0.0012 0.001 number of drivers = 79 number of observations = 566 L0 = -306.5 Lc = -112.3 L^ = -88.5 2 = 0.695 likelihood function with no serial correlation. The probability that the nth driver will switch from state M to state M at time t is given by: PSnt = M j Snt , 1 = M  = 1 + exp 3:16 , 0:303 1 f Vnld t , 2:05 f Lrem t , 0:285 Gnt 6.31 n where, Snt f Vnldt Vnldt f Lremt n Lrem t n = state of driver n at time t, = min0; Vnldt; = lead vehicle speed subject speed m s, = remaining distance impact; = remaining distance to the point at which lane change must be completed by, Gnt = lead gap plus lag gap m. 141 Table 6.10: Estimation results of the forced merging model. Variable Parameter t stat. constant -3.16 -10.59 min0, lead veh. speed subject speed m s 0.313 2.66 remaining distance impact 10 2.05 5.33 total clear gap divided by 10 meters 0.285 2.85 number of drivers = 79 number of observations = 566 L0 = -306.5 Lc = -112.3 L^ = -88.5 2 = 0.698 State M is de ned as the situation in which a driver intends to merge into the adjacent gap in the adjacent lane and perceives that his her right of way has been established and thus starts merging. The explanatory variable min0, lead vehicle speed subject speed captures the fact that, if the subject is interested in merging into the adjacent gap, it would slow down to match the leader's speed to better focus on the interaction with the lag vehicle. The variable should always be non-positive and its estimate re ects such behavior. The explanatory variable total clear gap has the desired positive sign. The variable remaining distance impact, a function of the remaining distance, is used to capture the fact that the remaining distance does not impact a driver's merging behavior when it is greater than a certain threshold, while at small values, drivers become more concerned and hence more aggressive. The variable remaining distance impact for driver n at time t is assumed to have the following functional form: 1 remaining distance impact = 1 , 1 + eFM Lremt n 6.32 where, FM is parameter. For di erent values of FM the likelihood function Equation 4.19 was maximized and the value of FM that corresponds to the highest 142 maximum likelihood value was adopted. FM was estimated to be -0.027. Figure 6-18 shows how the explanatory variable remaining distance impact, the Rem. Distance Impact 5 4 3 2 1 0 0 20 40 60 80 100 120 Remaining Distance, m 140 160 180 200 0 20 40 60 80 100 120 Remaining Distance, m 140 160 180 200 0 20 40 60 80 100 120 Remaining Distance, m 140 160 180 200 10 utility 5 0 −5 1 P(state = M) 0.8 0.6 0.4 0.2 0 Figure 6-18: Remaining distance versus explanatory variable remaining distance impact, the utility function, and the estimated probability of being in state M . utility, and the estimated probability of being in state M change as a function of the remaining distance assuming a zero lead relative speed and a 5m clear gap. Drivers' increasing desperation to complete the merge as remaining distance decreases is shown in the middle plot of Figure 6-18|the utility increases at an increasing rate as a driver approaches the end of the acceleration lane and the probability of being in state M approaches unity. The variable remaining distance impact, as shown in the rst plot of Figure 6-18, increases from 0 to 5 as the remaining distance decreases from greater than 150 to 0 meters. All parameters have signi cant t-statistics. The adjusted t of the model was 0.698. In addition, a likelihood ratio test was used to test the null hypothesis that the parameters of all the explanatory variables except the constant are zero. The statistic, 2Lc , L^, is equal to 47.54. The chi square critical value for 3 degrees 143 of freedom at the 5 level of signi cance is 7.81. Hence, the null hypothesis can be rejected. Note that, the variable lag relative speed was not included in the model since the sign of the corresponding parameter was counter intuitive. The use of the explanatory variable delay time elapsed since crossing the merging point between the on ramp and the freeway was also not supported by the data may be because the e ect is captured by the explanatory variable remaining distance impact. In addition, the vehicle type heavy vehicle or not did not have any impact on the forced merging behavior. This may be due to lack of observations since only 4 vehicles out of the 79 samples were heavy vehicles. 6.3 Conclusions The estimation work presented in this chapter shows that, the sensitivity term of the car following acceleration is a function of the subject speed, the space headway, and the density of tra c, and the sensitivity term of the car following deceleration is a function of the space headway and the density of tra c. The sensitivity lag was estimated to be zero. In other words, the sensitivity is a function of tra c conditions observed at the time instant at which acceleration is applied. In both cases, the stimulus is a nonlinear function of the front relative speed. Enhancements of the car following sensitivity and stimulus proposed in this thesis were supported by the data. The mean of estimated free ow acceleration increases with front vehicle speeds. Heavier vehicles tend to apply slower acceleration due to physical limitations. In free ow tra c conditions i.e., density  19 veh km lane or level of service A through C, drivers are expected to apply a higher acceleration and the parameter of the indicator for free ow tra c conditions has the desired sign and a signi cant t statistic. The headway threshold distribution has a mean and standard deviation of 3.17 and 0.87 seconds respectively. The median, mean, and standard deviation of the reaction time distribution are estimated to be 1.31, 1.34 and 0.31 seconds respectively. 144 The parameters of the discretionary and mandatory lane changing models were estimated separately due to lack of appropriate data. The estimated median lead and lag critical gap lengths under mandatory lane changing situations were lower their discretionary lane changing situation counterparts. Important factors that a ect forced merging behavior include the lead relative speed only when the lead vehicle is slower, remaining distance to the point at which the lane change must be completed by, and total clear gap reduced by the subject vehicle length. Finally, it must be stated that the estimation results presented in this section were obtained using data from a particular freeway segment. Due to the curvature of the roadway upstream of the data collection site, presence of the weaving section, and two exits downstream, the behavior of drivers, while driving in this area, may be in uenced by these conditions. This can be addressed by estimating the models using data from di erent sites with di erent geometrical con gurations. This will also address the issue of applicability of the estimation results to general networks. 145 Chapter 7 Model Validation Using a Microscopic Tra c Simulator In this chapter, the acceleration and lane changing models are evaluated through their use in a microscopic tra c simulator, MITSIM1 . Some basic information about MITSIM is given in Section 7.1. Two new versions of MITSIM were created: MITSIM with only the acceleration model replaced MITSIM ONE, and MITSIM with both the acceleration and lane changing models replaced MITSIM TWO. Tra c in a small network in Boston was simulated using the original version of MITSIM and MITSIM ONE and TWO. Actual tra c counts at di erent locations of the network were collected during the morning peak hours. The actual counts, aggregated over ve minute intervals, were compared to their simulated counterparts to assess the performance of the estimated models. This chapter begins with a brief description of MITSIM with emphasis on the acceleration and lane changing models implemented in the original version. The validation methodology and the case study are presented next. A detailed description of MITSIM can be found in the World Wide Web at the URL http: its.mit.edu products mitsim mitsim.html, or in Yang and Koutsopoulos 1996 or Yang 1997. 1 146 7.1 MITSIM: a Microscopic Tra c Simulator MITSIM is a microscopic tra c simulation model that represents the road network, surveillance system, tra c signs and signals, and the control logic in detail. Each lane is represented with its geometric characteristics for example, curvature, grade, connectivity, its functional classi cation for example, freeway, ramp, local street, tunnel or at grade, speed limit, and lane use regulations. Loop detectors, lane use signals, and variable message signs are simulated in MITSIM. The control logic supported by MITSIM includes ramp metering, mainline metering, urban control, etc. In addition to the network, a time dependent origin destination trip table and the tra c control and route guidance logic are input to the simulator. Vehicles travel through the network between their origins and destinations. The simulator collects the sensor readings, individual vehicle speci c trajectory and trip information, and average link and path travel times to provide measures of e ectiveness required for system evaluation. The sensor readings include tra c counts, occupancies, and speeds at a given frequency e.g. every 5 or 10 minutes. The travel behavior of the driver is captured by a route choice model. The route choice model captures drivers' route selection process which is in uenced by tra c information through variable message signs, highway advisory radio, on board navigation systems, etc. In the route selection process, drivers take lane use regulations into consideration. Two main driving behavior models are used to simulate vehicle movements in a network: the acceleration model, and the lane changing model. The acceleration model calculates the acceleration that drivers apply in response to various situations and factors. The most restrictive acceleration is implemented. The factors that trigger acceleration include: 147 car following, desired speed, signs and signals, connection to appropriate downstream link, speed limit, incidents, and courtesy yielding. The lane changing model captures lane selection and gap acceptance behavior. A driver rst checks for the necessity desirability of changing lanes. Subsequently, the driver selects a lane from the available choices and assesses the adjacent gap in the target lane. Lane change takes place when the driver perceives the gap in the target lane as acceptable. Description of the acceleration and the lane changing models implemented in the original version of MITSIM that are replaced in the case study with the corresponding models estimated in this thesis are presented next. 7.1.1 The Acceleration Model Based on time headway, a driver is categorized to be in one of the three regimes: emergency regime: if the current headway is less than a lower threshold; car following regime: if the current headway is greater than the lower threshold but less than an upper threshold; and nally, free ow regime: if the current headway is greater than the upper threshold. The default thresholds are 0.5 and 1.36 seconds respectively. They were estimated by using engineering judgment in combination with a sensitivity analysis of these parameters on the simulator performance. 148 In the emergency regime, drivers apply the minimum of the deceleration necessary to avoid collisions with the leader and a normal" deceleration. The normal deceleration depends on the speed of the vehicle and was adopted from ITE 1982. The value is 2.38 m s2 for speeds up to 6.1 m s, 2.0 m s2 for speeds within the range 6.1 to 12.2 m s, and 1.5 m s2 for speeds greater than 12.2 m s. The acceleration in the car following regime is calculated using the GM Model Equation 2.7. Di erent sets of parameters are allowed for positive and negative relative speed cases. These parameters  ; ; and  were adopted from Subramanian 1996 and are presented in Table 2.3. In the free ow regime, a driver does not accelerate decelerate if the current speed is equal to its desired" speed Table 7.1. If the current speed is less than the desired speed , he she applies maximum" acceleration Table 7.2, otherwise, he she applies a normal deceleration. Table 7.1: The cumulative distribution of speed that is added to the posted speed limit to obtain the desired speed. Percentile speed above the speed limit m s 5 -4.67 15 -1.73 25 -0.20 35 0.97 45 1.98 55 3.05 65 4.06 75 5.33 85 6.71 95 8.94 7.1.2 The Lane Changing Model In MITSIM, lane changes are classi ed as either discretionary DLC  or mandatory MLC . The implementation is as follows: 149 Table 7.2: Maximum acceleration m s2. vehicle Speed m s class 6.1 6.1 12.2 12.2 18.3 18.3 24.4 24.4 high performance car 3.05 2.41 1.71 1.22 1.22 low performance car 2.65 1.58 1.35 0.88 0.61 bus 2.13 1.52 1.22 0.46 0.30 heavy single unit truck 0.85 0.76 0.46 0.30 0.15 trailer trucks 0.49 0.44 0.27 0.14 0.12 Source: adjusted based on FHWA 1980, FHWA 1994, and Pline 1992. 1. check if a lane change is desired required and de ne the type of lane change, 2. select a target lane, and 3. check if the gap in the target lane is acceptable. To capture di erent driver behavior under DLC and MLC situations, di erent gap acceptance model parameters are allowed under DLC and MLC situations. The models are presented next. The Discretionary Lane Changing Model As mentioned in Section 2.2, MITSIM Yang and Koutsopoulos 1996 uses a rule based discretionary lane changing model. A driver considers a discretionary lane change DLC  only if the driver cannot accelerate more than 85 of the maximum acceleration Table 7.2 or if the current speed is less than an impatience factor times the driver's desired speed Table 7.1. The impatience factor varies from 0.8 to 1.0. Once a driver decides to to perform a DLC , he she selects a desired lane by comparing the speeds of the left and right adjacent lanes with that of the current lane. A parameter speed indi erence factor 10 is used to check whether the current speed is low enough and the speeds in the adjacent lanes are high enough to consider a lane change and start performing a gap assessment. 150 The Mandatory Lane Changing Model Drivers consider a mandatory lane change in order to: connect to the downstream link of their path, bypass a lane blockage downstream, avoid entering a restricted use lane, or, respond to lane use or variable message signs. At each time step of the simulation, a probabilistic model is used to decide when a driver is in MLC state. The probability is a function of the remaining distance to the point at which lane change must be completed by Lremt, the number of lanes n to cross to be in the target lane mnt, and the tra c density. The probability is given by: 8 PnMLC t = where, M LC t  rem t, exp Ln MLC 97:5 t 1 2 : 2  if Lrem t L0 n otherwise. 7.1 is de ned as follows: M LC t = 402:3 1 + 0:5 mn t + 1:0 knt=kj  7.2 where, knt and kj denote the tra c density of the segment and the jam density 130 veh km lane respectively. Once a vehicle is tagged MLC , it keeps the tag until it performs the lane change operation or moves into a downstream link. An MLC tagged driver then searches for an acceptable gap in the target lane. The Gap Acceptance Model In the gap assessment phase, drivers compare the lead and lag gaps in the target lane to the critical lead and lag gaps respectively. The speci cations for the lead and lag critical gaps under mandatory and discretionary lane changing situations for driver 151 n at time t are: Gcr;lead t = 0:5  max0:914; 0:914 + 0:05 Vnt , 0:10 Vnlead t 7.3 n;DLC Gcr;lag t = 0:5  max1:524; 1:524 + 0:10 Vnt + 0:30 Vnlag t 7.4 n;DLC where, Gcr;lead t n;DLC Gcr;lag t n;DLC Vnt Vnleadt Vnlag t = lead critical gap for DLC m, = lag critical gap for DLC in m, = subject vehicle's speed m s, = lead veh. speed less subject speed m s, = lag veh. speed less subject speed m s. At 10 m s speed, the DLC lead critical gap increases from 0.46 to 0.96 meters as the lead relative speed decreases from 10 m s to -5 m s. Similarly, the DLC lag critical gap increases from 0.76 to 2.76 meters as the lag relative speed increases from -5 m s to 10 m s. These values are rather small from a behavioral standpoint. We expect drivers to be more aggressive under MLC situations compared to DLC situations. To capture this, the lead lag critical gaps under MLC situations are assumed to decrease with decreasing remaining distance to the point at which lane change must be completed by. The speci cations for the MLC lead and lag critical gaps are given by: Gcr;lead t = max0:914; n;MLC n 0:914 + 0:05 Vnt , 0:10 Vnleadt 1 , e,2:5E,5 Lrem t  7.5 Gcr;lag t = max1:524; n;MLC n 1:524 + 0:10 Vnt + 0:30 Vnlag t 1 , e,2:5E,5 Lremt  7.6 152 where, Gcr;lead t = lead critical gap for MLC m, n;MLC Gcr;lag t = lag critical gap for MLC in m. n;MLC 7.2 Validation Methodology 7.2.1 Number of Replications MITSIM is a stochastic simulation model. As a result, the output from one simulation run may be di erent from another. Each output represents a sample and a number of simulations are required to get statistics with a prespeci ed accuracy. Hence, an important aspect of the validation methodology is the determination of the number of replications required to obtain reliable estimates of the measures of interest. s Let, yr be an output from the r th run of the simulator corresponding to a eld s observation y. Therefore, yr is a realization of the random variable ys corresponding to the actual observation y. An unbiased estimator of ys is the mean of the R s observations of yr from R di erent simulation runs. Mathematically, ys = R ^1 R X r=1 yrs 7.7 where, ys = an estimator of ys, ^ R = number of replications. Assume that, the R di erent realizations of the random variable ys are distributed iid normal with an unknown variance2 . Then, the number of replications required to Although, the independence assumption may be violated due to the stochastic nature of the simulator, we still make these assumptions in order to get a closed form solution to estimate the required number of replications. 2 153 obtain a certain accuracy e at a certain level of signi cance   is given by: Rreqd  ! s t =2 2 =s ye ^ 7.8 where, s = an estimate of the standard deviation of ys, e = allowable error, = desired level of signi cance, t =2 = critical value of the t distribution at a level of signi cance . Generally, output from the simulation includes speeds, ows and other quantities that have spatial as well as temporal dimensions. For each of these types of output for each time space point, the required number of replications needs to be calculated. Then, the desired number of replications would be the most conservative value | in other words, the maximum number of replications required by all the output elements. 7.2.2 Measures of Goodness of t In this section, di erent measures of goodness of t to compare the simulated data with their eld counterparts are presented see, for example, Pindyck and Rubinfeld, 1981. Let, yis be a simulation estimate corresponding to a eld observed quantity yi, where, subscript i denotes a time space point. For each time space point i, the percent error, di is given by: , di = yi y yi  100 s i 7.9 A positive percent error represents an overprediction whereas a negative percent error represents an underprediction. To evaluate systemwide performance, a useful measure is the bias or mean percent error over all the time space points. The mean percent 154 error, b, is: I 1X d b= I i i=1 7.10 where, I is the total number of time space points. To identify whether an overprediction or an underprediction dominates the bias, a useful measure is the mean positive and negative percent error. The mean positive percent error, bp, is given by: J 1X bp = J dj 7.11 j =1 where, j = index representing the time space point at which the percent error was positive, J = total number of positive percent error observations. Similarly, the mean negative percent error is de ned for all the observations showing underpredictions. To penalize larger errors at a higher rate, the root mean square error is used. The root mean square error RMS is given by: RMS v u I u1 X = t I yis , yi2 i=1 7.12 The RMS percent error is another measure that takes the scale of yi into account and is given by: RMS percent error v u Is u 1 X yi , yi t =I yi i=1  100 !2 7.13 Another useful measure of t is the Theil's inequality coe cient Pindyck and 155 Rubinfeld 1981, de ned as q U 1 PI y s , yi 2 i = q 1 PI I si=1 q 1 PI 2+ 2 I i=1 yi  I i=1 yi  7.14 The value of U will always fall between 0 and 1. A value of U equal to 0 implies a perfect t. Related to the Theil's inequality coe cient are three proportions: the bias U M , the variance U S , and the covariance U C  proportions. The proportions are given by UM US UC ys , y2 = 1 PI s 2 I i=1 yi , yi  s , 2 = 1 PI s 2 I i=1 yi , yi 1 = 1 2PI , s s 2 I i=1 yi , yi 7.15 7.16 7.17 where, ys; y; s; and are the means and standard deviations of the simulated and the original series respectively, and denotes the correlation between the two series. Basically, these proportions allow us to determine the contribution of the bias and the variance in the simulation error. Note that, UM + US + UC = 1 7.18 The bias proportion U M  re ects the systematic error. The variance proportion U S  indicates how well the uctuation in the original data is replicated by the simulation. Therefore, lower values close to zero of U M and U S are desired. Finally, the covariance proportion U C  measures the unsystematic error. This is the remaining error after the deviations from the average values have been accounted for and it is of less worrisome as we desire smaller U M and U S . In addition to these statistical measures, a plot of the real and simulated quantities over time and space may be useful in identifying any systematic under or overpredictions at any particular time and or space. 156 7.3 Case Study 7.3.1 The Network A small network in the Boston area, for which actual ow observations exist, was used in the case study see Figure 7-1. The network is a 2.7 km 1.68 miles long segment of the Storrow Drive in Boston. The network is a two lane freeway with two section A 390m section B 400m section C 545m section D 770m 600m Note: Figure not drawn to scale Figure 7-1: The network used in the validation exercise. on ramps and two o ramps. Video of tra c during the morning peak 7:30am to 9:15am on February 10, 1998 was recorded at four locations marked sections A to D in Figure 7-1. The part of the network between sections A and D, that are 1.7 km apart, was simulated. Three 30 meters long dummy links were added to the upstream of section A to load vehicles in appropriate lanes based on the O Ds estimated from the counts. The simulated freeway was extended arbitrarily by 100 meters beyond section D. Tra c sensors were placed at all four sections in the simulated network to collect aggregate counts and average speeds. The speed limit for the freeway was 17.9 m s 40 mph. The above network was selected for the following reasons: 1. Tra c was light at the beginning and at the end of the data collection time and was congested in between. Therefore, initial conditions for congestion oc157 currence could be simulated accurately, 2. The bottleneck formation originated due to conditions within the network trafc merging from the on ramp near section C and was not a ected by conditions downstream, and 3. The input ows for the network was not a ected by spillbacks from the bottleneck, and therefore, the input ows represent the demand exactly. A limitation of the network though, is that due to the geometric con guration of the on ramp and the freeway merge area, the mandatory lane changing and forced merging models could not be validated. Although few mandatory lane changes take place between sections A and B, the number of mandatory lane changes is too small to validate the model. Figure 7-2 shows the schematic diagram of the merging area adjacent vehicle X Y mainline Onramp front vehicle subject X Y Figure 7-2: Schematic diagram of the on ramp and Storrow Drive merging area. between the on ramp and Storrow Drive downstream from section C in Figure 7-1 where two lanes merge into one lane and no lane change takes place. In geometric con gurations like this, the mainline vehicles have priority over the on ramp vehicles. Two vehicles can overlap laterally at this location since the lane width is more than that of a single lane see location of the subject and the adjacent vehicle in Figure 7-2 but less than that of two lanes. In this case, vehicles from the two upstream lanes merge without any lane changing taking place. In addition, the subject vehicle is not necessarily following its leader as is assumed in developing the 158 car following model, since the headway is negative. Therefore, the acceleration and lane changing models developed in this thesis or those implemented in the original version of MITSIM do not apply here. In MITSIM, lanes are discrete and such geometry cannot be represented. An ad hoc merging model Yang and Koutsopoulos 1996 is implemented to capture the merging phenomenon in such areas. The on ramp vehicle checks whether there is any vehicle from the adjacent mainline and executes the merge only if the gap is acceptable. 7.3.2 Tra c Data Minute by minute tra c counts were collected from video tapes for 1 hour 40 minutes beginning at 7:33am. At section A, counts were collected for the left two lanes combined and the rightmost lane that directly feeds into the rightmost lane of the o ramp 400 meters away. At section B, the mainline the two leftmost lanes counts and the o ramp the two rightmost lanes counts were collected. At section C, before the merge, the mainline counts and the on ramp counts were collected. Finally, at section D the mainline counts were collected. Figure 7-3 shows the minute by minute ow through the left two lanes and the rightmost lane at section A and through the on ramp near section C. The mean ow values at these locations were 1650, 700, and 900 veh hr lane respectively with a standard deviation of 350, 300, and 250 veh hr lane respectively. Although, the on ramp volume was not high, a bottleneck formed near the on ramp and freeway merging area due to merging. A spillback from the bottleneck reached section B brie y, but never reached section A nor the upstream end of the on ramp at section C. Therefore, the input ows for the network were not a ected by spillbacks from the bottleneck and they represent the demand accurately. 159 (a) Two left lanes, Section A flow, veh/hr/lane 2500 2000 1500 1000 500 0 10 20 30 40 50 time, minute 60 70 80 90 100 70 80 90 100 70 80 90 100 (b) Rightmost lane, Section A flow, veh/hr/lane 1500 1000 500 0 0 10 20 30 40 50 time, minute 60 (c) On ramp, Section C flow, veh/hr/lane 1500 1000 500 0 0 10 20 30 40 50 time, minute 60 Figure 7-3: Flow of tra c entering the network. 7.3.3 O D Estimation from Tra c Counts A required input to the simulator is a time dependent O D matrix. The minute by minute O D matrix was estimated from the minute by minute tra c counts at section A, the exit counts at section B, and the on ramp counts at section C by using an ad hoc method. Note that, the method is developed considering the geometry and counts of this particular network, and is not applicable to a general network. It was assumed that, a certain percentage p of the tra c from the rightmost lane at section A the exit only lane takes the exit ramp near section B. Since p is unknown, di erent sets of O D matrices were created by varying p from 70 to 100 to investigate the e ect of the assumption of p on the validation results. The O D matrix for a particular minute say, the tth minute is estimated from the 160 counts of the corresponding minute by using the following equations see Figure 7-4 c: Legends { { D4 { C1 C2 { C3 C: counts O: origin D: destination p: % of rightmost lane traffic exiting D2 (a) traffic counts from video tape, veh/min O3 { O1 (b) origin and desitination definitions O 3D 4 = C 3 - O 3D 2 3 4 O 3D 2 = C 2 - O 1D 2 O 1D 4 = C 1 - O 1D 2 1 O 1D 2 = min(C 1 * p , C 2) 2 (c) O-D estimation Figure 7-4: O D estimation from tra c counts for the case study. O1D2t O1D4t O3D2t O3D4t = minC1t p; C2t + tt = C1t , O1D2 t = C2t + tt , O1D2t 7.19 = C3t , O3D2 t 7.22 161 7.20 7.21 where see Figures 7-4 a and b, t tt C1t C2t C3t O1D2t = time period between minutes t and t + 1, = average travel time between sections A and B, = counts at the rightmost lane at section A, = counts at the exit ramp at section B, = counts at the left two lanes at section A, = number of vehicles from the rightmost lane at section A exiting at section B, O1D4t = number of vehicles from the rightmost lane at section A that continue on the freeway, O3D2t = number of vehicles from the left two lanes at section A exiting at section B, O3D4t = number of vehicles from the left two lanes at section A that continue on the freeway. The count C2 was advanced by tt to take into account that a vehicle counted at section A at time to would reach section B at time to + tt. It was assumed that tt is the travel time between sections A and B for everyone. We conducted a sensitivity analysis on tt to investigate its impact on the O D estimations. The average travel time to reach section B from section A ranged from 19 to 28 seconds observations from simulation. Within this range of tt, C2t + tt di ered by less than 2 vehicles minute for 92 of the cases. Therefore, the assumption on tt does not a ect the O D estimation signi cantly. The variable tt was set to 22.4 seconds which assumes that vehicles traveled at the speed limit. From the rightmost lane at section A, the minimum of a certain percent p of C1 and the exit ramp count at section B C2 is assigned to take the exit ramp at section B O1D2. The remaining C1 is assigned to continue on the freeway O1D4. This will guarantee that at least 1 , p percent of the rightmost lane drivers would perform a 162 mandatory lane change to continue on the freeway. The number of vehicles from the left two lanes exiting section B O3D2 is estimated by deducting O1D2 from the o ramp counts at section B C2. The remaining tra c counted at the left two lanes at section A i.e., C3 , O3D2 continues in the freeway O3D4. Finally, all vehicles entering the network through the on ramp at section C have only one destination, i.e., they travel through section D. 7.3.4 MITSIM Modi cations As mentioned above, two additional versions of MITSIM were created by incorporating the models developed in this thesis. In MITSIM ONE, only the acceleration model was replaced with the one estimated in this thesis. In MITSIM TWO, both the acceleration and lane changing models were replaced with those estimated in this thesis. A version of MITSIM with only the lane changing model replaced with the one estimated in this thesis was not created due to the following reason. Early testing using the original version of MITSIM indicated that, it was not capable of handling the demand used in this case study due to over prediction of congestion and the resulting spillback. Vehicles were queued outside of the network and were loaded only when spaces to load them became available. As a result, vehicles could not be loaded on time according to their O D and departure times. By replacing the default lane changing model with the one developed in this thesis, the problem still persisted. Therefore, the results from such simulation runs would not be reliable, and MITSIM with only the lane changing model replaced was not tested. In addition to replacing the acceleration and lane changing models with those estimated in this thesis, the following changes to MITSIM parameters were introduced see Appendix B for a general approach to calibrate the simulation model parameters: The lower headway threshold of the acceleration model was set to 0.4 seconds and the upper threshold was adopted from Equation 6.5. The lower threshold was set after some trial and error to avoid vehicle to vehicle collisions. Compared to the 1.36 seconds upper threshold used in the original version of MITSIM, the 163 upper threshold estimated in this thesis was much higher. For example, the 5 percentile and the median upper headway thresholds were 1.75 and 3.17 seconds respectively. Since the data collection site used to estimate the models in this thesis and the network used for the validation case study have di erent geometric con gurations, the constant of the car following acceleration model was adjusted to make the model predictions more realistic for the validation network. The constant of the car following acceleration model  acc in Equation 6.2 was increased from 0.023 to 0.040. 7.3.5 Experimental Design The three versions of MITSIM were used under di erent scenarios with respect to the O D ows values of p were set equal to 100, 85, and 70. In order to determine the number of replications required, we need estimates of the mean and standard deviation of the measures of interest discussed in Section 7.2.1. To get estimates of mean and standard deviation, all three versions of the simulator were run 10 times using the three di erent sets of O D matrices. Then the required number of replications for all the cases were estimated using Equation 7.8. The most conservative estimate was 4 and in subsequent computations output from all 10 runs were used. Tra c counts and speeds for each scenario were aggregated over 5 minute periods. The counts were compared to the corresponding real tra c counts. Speeds predicted by di erent versions of the simulators were also compared. The statistics reported in Section 7.2.2 were used to measure the goodness of t of the various simulation runs. 7.3.6 Validation Results Table 7.3 summarizes the comparison of the original MITSIM with the two revised MITSIM versions using three di erent set of O D matrices assuming that 100, 85, and 70 of the drivers from the rightmost lane at section A took the exit at section B. Observations corresponding to the rst ve minutes were not used in computations 164 since vehicles were loaded into an empty network and it took approximately 2 minutes to ll the network. As evident in Table 7.3, by varying the values of p, the percent of the rightmost drivers at section A exiting at section B, the statistics did not show signi cant variation. This may be due to low ow through the rightmost lane at section A and the exit at section B. Therefore, based on the low sensitivity of the results to the value of p, the conclusion drawn should be valid for the actual O D ows as well. For all cases, MITSIM ONE and TWO performed consistently better than the original MITSIM version. Tra c in the original MITSIM got jammed 15 minutes after the beginning of simulation and continued to be jammed throughout the simulation period. The congestion originated near the on ramp merge. Tra c spilled back all the way up to section A and beyond, and a ected vehicle loading into the network. Whereas, in reality, the merging area near section C was congested from 8:04am to 9:05am and the e ect of spillback reached section B brie y but never reached section A. MITSIM ONE and TWO performed much better in this respect. This is also re ected by consistently low average speeds at sections B and C for the original MITSIM as shown in Figures 7-5, 7-6, and 7-7. Due to lack of speed observations from the eld, simulated speeds could not be compared to the eld observations. At section B, the average speeds were around 12 to 14 m s for MITSIM ONE and TWO compared to 5 m s for the original MITSIM. The speed limit of the freeway is 17.9 m s 40 mph. Average speeds from the original MITSIM were signi cantly lower than expected. The RMS Percent Error in counts for the original MITSIM was 9.08 which reduced to 8.09 and 7.53 for MITSIM ONE and TWO respectively. The root mean square error was 28, 24, and 22 vehicles per 5 minutes for the original MITSIM and MITSIM ONE and TWO respectively. The Theil's inequality coe cient was 0.050, 0.042, and 0.039 for the original MITSIM and MITSIM ONE and TWO respectively. Note that, a smaller coe cient implies a better t. The mean percent error contributed signi cantly toward the RMS Percent Error for the original MITSIM and is equal to -5.81 compared to -1.95 and -1.56 for 165 Table 7.3: Summary statistics of the comparison of the eld observed counts with those obtained from di erent versions of MITSIM using three di erent O D sets. Percent vehicles from the rightmost lane taking exit = 100 Revised Revised Statistical Original MITSIM, MITSIM, Measure MITSIM Acc only Acc & LC RMS percent error  9.08 8.09 7.53 RMS error veh. per 5 min 27.82 24.23 22.22 mean percent error  -5.81 -1.95 -1.56 Theil's inequality coe cient 0.050 0.042 0.039 U M bias proportion 0.419 0.073 0.059 U S variance proportion 0.063 0.005 0.011 avg. positive error 4.53 6.78 5.99 no. of positive errors 12 18 20 max. positive error 9.13 20.85 20.49 avg. negative error  -8.56 -5.98 -5.64 no. of negative errors 45 39 37 max. negative error -19.97 -18.99 -16.17 Percent vehicles from the rightmost lane taking exit = 85 Revised Revised Statistical Original MITSIM, MITSIM, Measure MITSIM Acc only Acc & LC RMS percent error  9.08 7.83 7.44 RMS error veh. per 5 min 27.76 23.44 22.02 mean percent error  -5.77 -2.13 -1.96 Theil's inequality coe cient 0.049 0.041 0.038 U M bias proportion 0.415 0.090 0.088 U S variance proportion 0.059 0.008 0.012 avg. positive error 3.90 5.86 5.72 no. of positive errors 14 19 19 max. positive error 10.24 19.46 18.74 avg. negative error  -8.92 -6.12 -5.81 no. of negative errors 43 38 38 max. negative error -20.10 -17.28 -16.10 Percent vehicles from the rightmost lane taking exit = 70 Revised Revised Statistical Original MITSIM, MITSIM, Measure MITSIM Acc only Acc & LC RMS percent error  9.17 8.13 7.71 RMS error veh. per 5 min 28.06 24.21 23.00 mean percent error  -5.83 -2.25 -2.20 Theil's inequality coe cient 0.050 0.042 0.040 U M bias proportion 0.416 0.091 0.099 U S variance proportion 0.060 0.003 0.011 avg. positive error 4.41 6.85 6.65 no. of positive errors 12 17 16 max. positive error 8.97 20.72 19.69 avg. negative error  -8.56 -6.11 -5.79 no. of negative errors 45 40 40 max. negative error -20.51 -17.65 -16.41 166 ORIGINAL MITSIM MITSIM ONE MITSIM TWO 16 speed, m/s 14 12 10 section B 8 6 4 2 4 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 14 speed, m/s 12 10 section C 8 6 4 2 4 15 speed, m/s 14.5 14 section D 13.5 13 12.5 2 4 Figure 7-5: Comparison of average speeds obtained from di erent versions of MITSIM for p = 100. MITSIM ONE and TWO respectively. The bias proportion U M  for the original MITSIM is 0.419 which is very high. The bias proportions for MITSIM ONE and TWO were 0.073 and 0.059 respectively. The variance proportions were 0.063, 0.005, and 0.011 for the original MITSIM and MITSIM ONE and TWO respectively. As a result of the systematic underrepresentation of ow, the number of positive errors for the original MITSIM was small and the corresponding average and maximum positive errors were low compared to those for the two other MITSIMs. The mean positive percent errors were 4.53, 6.78, and 5.99 for the original MITSIM and MITSIM ONE and TWO respectively. The negative mean percent error for the original MITSIM was -8.56 compared to -5.98 and -5.64 for MITSIM ONE and 167 ORIGINAL MITSIM MITSIM ONE MITSIM TWO 16 speed, m/s 14 12 10 section B 8 6 4 2 4 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 14 speed, m/s 12 10 section C 8 6 4 2 4 15 speed, m/s 14.5 14 section D 13.5 13 12.5 2 4 Figure 7-6: Comparison of average speeds obtained from di erent versions of MITSIM for p = 85. TWO respectively. Compared to MITSIM ONE and TWO, the performance of the original MITSIM was poor at all three sections see Figures 7-8, 7-9, and 7-10. Performances of MITSIM ONE and TWO at section D were not as good as they were at sections B and C. This may be due to the fact that, section D is near the simulated network boundary where all tra c exits the network. As a result, vehicles leave the network at speeds higher than the real speed. Therefore, the uctuation in the ow for section D could not be reproduced well. Performance of the original MITSIM improved signi cantly after the acceleration model was replaced with the one estimated in this thesis MITSIM ONE. Due to 168 ORIGINAL MITSIM MITSIM ONE MITSIM TWO 16 speed, m/s 14 12 10 section B 8 6 4 2 4 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 14 speed, m/s 12 10 section C 8 6 4 2 4 15 speed, m/s 14.5 14 section D 13.5 13 12.5 2 4 Figure 7-7: Comparison of average speeds obtained from di erent versions of MITSIM for p = 70. the high congestion level, drivers traveled with low headways most of the time. As shown in Figure 6-3, the deceleration calculated by the deceleration model used in the original MITSIM adopted from Subramanian 1996 is too high at low headways. This may have contributed to vehicles moving slowly in the original MITSIM, and thereby reducing the volume of tra c the network could handle, especially, near the merging area. Finally, MITSIM TWO performed better than MITSIM ONE with respect to all the statistics except the variance proportion U S  and the bias proportion U M  for the p = 70 case. However, the RMS and the mean percent errors for MITSIM TWO were smaller than its MITSIM ONE counterparts. This may be due to the fact that 169 ORIGINAL MITSIM MITSIM ONE MITSIM TWO real data flow, veh/hr/lane 2400 2200 2000 section B 1800 1600 1400 1200 2 4 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 flow, veh/hr/lane 2400 2200 2000 1800 section C 1600 1400 1200 2 4 flow, veh/hr/lane 2400 2200 2000 1800 section D 1600 1400 1200 2 4 Figure 7-8: Comparison of the real tra c counts with those obtained from di erent versions of MITSIM for p = 100. the variance and the bias proportions do not take into account the scale of the errors the di erences between the simulated and the original tra c counts with respect to the original counts. Therefore, MITSIM TWO demonstrates the e ectiveness of the discretionary lane changing model. 7.4 Conclusions The acceleration and discretionary lane changing models were tested using a microscopic tra c simulator, MITSIM. Tra c on a 1.83 km long segment of a freeway with one on and one o ramps was simulated using di erent versions of MITSIM: the original MITSIM, MITSIM with only the acceleration model replaced with the 170 ORIGINAL MITSIM MITSIM ONE MITSIM TWO real data flow, veh/hr/lane 2400 2200 2000 section B 1800 1600 1400 1200 2 4 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 flow, veh/hr/lane 2400 2200 2000 1800 section C 1600 1400 1200 2 4 flow, veh/hr/lane 2400 2200 2000 1800 section D 1600 1400 1200 2 4 Figure 7-9: Comparison of the real tra c counts with those obtained from di erent versions of MITSIM for p = 85. model estimated in this thesis, and MITSIM with both the acceleration and the lane changing model replaced with the corresponding models estimated in this thesis. Simulated counts aggregated over ve minute intervals at di erent locations were compared to the corresponding eld observations. Performance of the original MITSIM signi cantly improved after the acceleration model was replaced with the one estimated in this thesis MITSIM ONE. It improved further when the lane changing model of the original MITSIM was replaced with the one estimated in this thesis in addition to replacing the acceleration model MITSIM TWO. For a full evaluation, a wider set of experiments covering di erent weather, geometry, and congestion conditions is needed. 171 ORIGINAL MITSIM MITSIM ONE MITSIM TWO real data flow, veh/hr/lane 2400 2200 2000 section B 1800 1600 1400 1200 2 4 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 6 8 10 12 time interval 14 16 18 20 flow, veh/hr/lane 2400 2200 2000 1800 section C 1600 1400 1200 2 4 flow, veh/hr/lane 2400 2200 2000 1800 section D 1600 1400 1200 2 4 Figure 7-10: Comparison of the real tra c counts with those obtained from di erent versions of MITSIM for p = 70. 172 Chapter 8 Conclusions and Future Research Directions This chapter summarizes the conceptual framework and estimation results of the proposed acceleration and lane changing models. Major contributions of this thesis are also discussed. Finally, suggestions for future research are presented. 8.1 Summary of Research 8.1.1 The Acceleration Model A comprehensive framework for estimating a general acceleration model is developed that is applicable to both congested and uncongested tra c. The model de nes two regimes based on a time headway threshold: the car following regime and the free ow regime. At headways less than the threshold, a driver is assumed to be in the car following regime trying to match its leader's speed, and at headways larger than the threshold, the driver is assumed to be in the free ow regime trying to attain its desired speed. A headway threshold distribution is assumed to capture the variability among drivers. A reaction time distribution is also assumed which captures the e ect of response lag to stimulus. The mean of the reaction time distribution depends on the 173 tra c environment. Both the car following and free ow acceleration models employ the response equal to the stimulus times the sensitivity structure. The car following model uses the GM Nonlinear Model as a basis and extends it. The original model was modi ed to include the e ect of density in the sensitivity term and allow the stimulus to be a nonlinear function of the front relative speed i.e., leader speed less the subject speed. In addition, the estimation allows for capturing the fact that, drivers may update their perception of the tra c environment after they recognize the stimuli for the car following acceleration the lead relative speed. In the free ow acceleration model, the sensitivity term is a constant and the di erence between the desired speed and the current speed provides the stimulus. The parameters of all the component models were estimated jointly using the maximum likelihood estimation method and microscopic data collected from the video of real freeway tra c. The network is a part of Interstate 93, the Central Artery in Boston. The section has a three lane mainline and a weaving lane. The estimation results show that the impact of speed, space headway, and density of tra c is di erent under acceleration and deceleration situations. The sensitivity term of the car following acceleration is a function of the subject speed, the space headway, and the density of tra c, while that of the car following deceleration is a function of the space headway and the density of tra c. The stimulus is a nonlinear function of the lead relative speed. The free ow acceleration is a function of the subject speed, its leader speed, an indicator whether the subject vehicle is a heavy vehicle i.e., vehicle length greater than 9.14 meters or 30 ft, and an indicator whether tra c density is low level of services A through C. The mean and standard deviation of the headway threshold distribution were estimated to be 3.17 and 0.87 seconds respectively. The median, mean, and standard deviation of the reaction time distribution were estimated to be 1.31, 1.34, and 0.31 seconds respectively. 174 8.1.2 The Lane Changing Model The lane change model is based on a decision that proceeds in the following three steps: decision to consider a lane change, choice of a target lane, and acceptance of a gap in the target lane. Modeling such a process is extremely complicated due to its latent nature. To simplify, drivers are assumed to make decisions about lane changes at every discrete point in time irrespective of the decisions made during earlier time periods. The proposed gap acceptance model recognizes that for merging into an adjacent lane, both the lead and lag gaps must be acceptable. Drivers are expected to be more aggressive under mandatory lane changing situations compared to discretionary lane changing situations. The proposed model captures this behavior by allowing di erent parameters for the gap acceptance model under the two situations. The models were estimated using the same data as in the estimation of the acceleration model. Drivers' decision to perform a discretionary lane change is modeled as a two steps decision process. First, drivers examine their satisfaction with the driving conditions of the current lane. Important factors a ecting such decision include the di erence between the current speed and the driver's desired speed, an indicator whether the subject vehicle is a heavy vehicle, and an indicator whether the subject is tailgated. If the driver is not satis ed with the driving conditions of the current lane, he she compares the driving conditions of the current lane with those of the other lanes. Such decision is in uenced by the the speeds of the vehicles ahead in di erent lanes compared to the subject's desired speed and the lag relative speed. Factors a ecting a driver's decision to respond to the mandatory lane change situation MLC  are delay time elapsed since MLC conditions apply and an indicator for the rst gap when delay is equal to zero. The estimated median lead and 175 lag critical gap lengths under MLC situations are lower than their DLC situations counterparts, as expected. A forced merging model is developed to capture drivers' lane changing behavior in heavily congested tra c where gaps larger than their minimum acceptable length are hard to nd. In such situations, it is assumed that a driver changes lanes either through courtesy yielding the lag vehicle in the target lane or through the subject forcing the lag vehicle to slow down. Important factors that a ect drivers' forced merging behavior include lead relative speed only when the lead vehicle is slower, remaining distance to the point at which the lane change must be completed by, and total clear gap in excess of the subject vehicle's length. 8.1.3 Validation by Microsimulation The acceleration and lane changing models were tested using a microscopic tra c simulator, MITSIM. A 1.83 km long segment of a freeway with one on and one o ramps was simulated using di erent versions of MITSIM: the original MITSIM, MITSIM with only the acceleration model replaced with the one estimated in this thesis, and MITSIM with both the acceleration and lane changing models replaced with the corresponding models estimated in this thesis. Simulated counts at di erent time intervals, aggregated over ve minutes, at di erent locations were compared to the corresponding eld observed counts. Performance of the original MITSIM signi cantly improved after the acceleration model was replaced with the one estimated in this thesis. It improved further when the lane changing model of the original MITSIM was replaced with the one estimated in this thesis in addition to replacing the acceleration model. 8.2 Contributions This thesis contributes to the state of the art in modeling drivers' acceleration and lane changing behavior in two major areas: enhancing existing models and proposing new models, and estimating the models using statistically rigorous methods and real 176 microscopic tra c data. Contributions in each of these two areas are listed below. Contribution to the modeling framework: The car following model is extended by assuming that the stimulus is a nonlinear function of the lead relative speed and that the sensitivity term is also a function of the tra c conditions ahead of the driver. The existing models restricts the stimulus the lead relative speed and other factors such as subject speed, gap in front of the subject that a ect the acceleration decision to be observed at the same time. This corresponds to an assumption that drivers base their decisions on the tra c environment at the time they were stimulated into action. The proposed model relaxes this assumption by allowing drivers to update their perception of the tra c environment during the decision making process. A headway threshold distribution is introduced that allows any driver behavior to be captured aggressive or conservative. An individual driver speci c reaction time is introduced which is allowed to be sensitive to the tra c situation under consideration. A probabilistic lane changing model is developed that captures drivers' lane changing behavior under both the mandatory and discretionary lane changing situations. This is a signi cant improvement over the existing deterministic rule based lane changing models. The proposed lane changing model allows for di erent gap acceptance model parameters for mandatory and discretionary lane changing situations. It also captures the variability within driver and amongst drivers in the lane changing decision process. A forced merging model is proposed that captures merging in a heavily congested tra c by gap creation either through force or through courtesy yielding. 177 Contribution to model estimation: A methodology to estimate instantaneous speed and acceleration that is required for model estimation from discrete trajectory data that can be obtained from real tra c is developed. All the components of the acceleration model are estimated jointly using real microscopic tra c data. The component models are the car following acceleration and deceleration models, the free ow acceleration model, and the headway threshold and reaction time distributions. Estimation results demonstrate the robustness of the modeling framework. Separate car following model parameters under acceleration and deceleration situations are allowed in the estimation. This captures the fact that, the sensitivity of di erent factors on drivers' acceleration behavior may not be same under these two situations. Separate gap acceptance models for the mandatory and discretionary lane changing situations are estimated. This captures the fact that, driver are expected to be more aggressive under mandatory lane changing situations compared to discretionary lane changing situations. The proposed models were estimated using the maximum likelihood estimation method and real microscopic tra c data. 8.3 Future Research Directions 8.3.1 Modeling The proposed acceleration model should be extended to capture the impact of lane changing decisions on the acceleration decision. For example, drivers may need to accelerate or decelerate to t into a gap in the target lane. In such cases, the headway and speed of the lead and lag vehicles in the target lane will in uence drivers' acceleration decisions. 178 The proposed lane changing model does not capture the impact of past lane changing decisions on the current lane changing decision and various modeling approximations should be considered. The forced merging model and the mandatory lane changing model should be combined into a single framework. In reality, drivers consider forced merging only when they perceive the probability of nding an acceptable gap to be very low. Models capturing driver behavior in a merging area where two lanes gradually become one see Figure 7-2 for an example have to be developed. 8.3.2 Estimation and Validation To enhance the ability of the models proposed in this thesis to predict drivers' acceleration and lane changing behavior, the models should be estimated with richer data that has more variability than the one used in this thesis. For example, the car following model proposed in this thesis predicts acceleration that is smaller than expected which should be reestimated using richer data. In addition, from an estimation point of view, a major research activity is the estimation of the models using richer data that provide the required variability to assess the impact of various factors, such as geometric characteristics etc. More speci cally, The impact of geometric characteristics of a roadway, for example, lane width, curvature, grade, pavement surface quality, on driver behavior was not captured due to lack of data. The models should be estimated using data from di erent sites with di erent geometric characteristics. The discretionary lane changing behavior, when mandatory lane changing situations apply, cannot be estimated due to lack of appropriate data. This requires data collected over a long stretch of a roadway 1500  3000 meters long. The discretionary lane changing model was estimated using a data set in which the choice set was the current lane and one adjacent lane. Ideally, data set with 179 two adjacent lanes in the choice set would be preferable. The identi cation problem that arose while estimating the discretionary lane changing model in which serial correlation was captured should be further investigated. Further validation, using more extensive networks, is also required. 8.4 Conclusion A comprehensive framework for modeling drivers' acceleration and lane changing behavior was developed in this thesis. Both the acceleration and lane changing models were estimated using real microscopic tra c data, and validated from a behavioral standpoint as well as using microsimulation. Overall, the empirical results are encouraging and demonstrate the e ectiveness of the modeling framework. 180 Appendix A Speci cation of the Random Utility Model Appropriate for Panel Data Panel data contains one or more observations for each individual driver. Di erent observations from a given driver are likely to be correlated which may introduce bias in the parameter estimates. To capture this correlation, the random disturbance of the utility function used to model the decisions at various levels is assumed to have two components: an individual speci c random term that does not vary for a given individual, and a generic random term Heckman 1981. Hence, the utility formulation associated with a decision d within the hierarchy is given by: d d Un t = Xn t d + dn + d t n where, n t d Un t d Xn t d = individual driver, = time instance, = unobserved utility of responding to decision d at time t, = vector of explanatory variables, = vector of parameters, 181 A.1 n = individual speci c random term assumed to be distributed standard normal, d = parameter of  for decision d, n d t = generic random term that varies across all three dimensions, i.e., n d; t; and n: These assumptions on the random terms imply: 8 cov d t; d00 t0  n n = 2d if t = t0 , n = n0 and d = d0 0 otherwise 0 d covn ; d 0 t = 0; 8t; n; d; n0 ; d0 n 8  d2 + 2d if d = d0; n = n0 and t = t0  d2 if d = d0; n = n0 and 8t 6= t0 d d covUn t; Un00 t0  = d d0 if d 6= d0; n = n0 and 8t :0 otherwise : A.2 A.3 A.4 where, 2d denotes the variance of d t. Conditional on n, di erent discrete choice n models can be obtained by making di erent assumptions on the distribution of d t, n such as a logit or a probit model. 182 Appendix B Calibration of the Simulation Model Parameters Figure B-1 illustrates a systematic approach to calibrate simulation model paramData Collection Estimation of Individual Models Model Refinement Disaggregate Data Evaluation Validation of Simulation Model Aggregate Data Evaluation Calibrated and Validated Simulation Model Figure B-1: Model parameter calibration approach. eters. Data collection involves collecting both disaggregate microscopic and ag183 gregate macroscopic data. Chapter 5 provided a detailed description of the data required to estimate driver acceleration and lane changing behavior and the actual disaggregate data collected from real tra c. The disaggregate data are used to estimate individual models as is done in this thesis presented in Chapter 6. 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