Modeling Driving Decisions with Latent Plans

Transcription

1 Modelig Drivig Decisios with Late Plas by Charisma Farhee Choudhury Bachelor of Sciece i Civil Egieerig Bagladesh Uiversity of Egieerig ad Techology (2002) Master of Sciece i Trasportatio Massachusetts Istitute of Techology (2005) Submitted to the Departme of Civil ad Eviromeal Egieerig i partial fulfillme of the requiremes for the degree of Doctor of Philosophy i The Field of Civil ad Eviromeal Egieerig at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September Massachusetts Istitute of Techology All rights reserved. Sigature of Author Departme of Civil ad Eviromeal Egieerig August 27, 2007 Certified by Accepted by Moshe E. Be-Akiva Edmud K. Turer Professor of Civil ad Eviromeal Egieerig Thesis Supervisor Daiele Veeziao Chairma, Departmeal Committee for Graduate Studes

2

3 Modelig Drivig Decisios with Late Plas by Charisma Farhee Choudhury Submitted to the Departme of Civil ad Eviromeal Egieerig O August 17, 2007 i partial fulfillme of the requiremes for the degree of Doctor of Philosophy i the Field of Civil ad Eviromeal Egieerig Abstract Drivig is a complex task that icludes a series of ierdepede decisios. I may situatios, these decisios are based o a specific pla. The pla is however uobserved or late ad oly the maifestatios of the pla through actios are observed. Examples iclude selectio of a target lae before executio of the lae chage, choice of a mergig tactic before executio of the merge. Chage i circumstaces (e.g. reactio of the eighborig drivers, delay i executio) ca lead to updates to the iitially chose pla. These late plas are igored i the state-of-the-art drivig behavior models. Use of these myopic models i the traffic simulators ofte lead to urealistic traffic flow characteristics ad icorrect represeatio of cogestio. A modelig methodology has bee formulated to address the effects of uobserved plas i the decisios of the drivers ad hece overcome the deficiecy of the existig drivig behavior models ad simulatio tools. The actios of the driver are coditioal o the curre pla. The curre pla ca deped o previous plas ad be iflueced by aicipated future coditios. A Hidde Markov Model is used to address the effect of previous plas i the choice of the curre pla ad to capture the state-depedece amog decisios. Effects of aicipated future circumstaces i the curre pla are captured through predicted coditios based o curre iformatio. The heterogeeity i decisio makig ad plaig capabilities of drivers are explicitly addressed. The methodology has bee applied i developig drivig behavior models for four traffic scearios: freeway lae chagig, freeway mergig, urba iersectio lae choice ad urba arterial lae chagig. I all applicatios, the models are estimated with disaggregate trajectory data usig the maximum likelihood techique. Estimatio results show that the late pla models have a sigificaly better goodess-of-fit compared to the reduced form models where the late plas are igored ad oly the choice of actios are modeled. The justificatios for usig the late pla modelig approach are further stregtheed by validatio case studies withi the microscopic traffic simulator MITSIMLab where the simulatio capabilities of the late pla models are compared agaist the reduced form models. I all cases, the late pla models better replicate the observed traffic coditios. Thesis Supervisor: Moshe E. Be-Akiva Title: Edmud K. Turer Professor of Civil ad Eviromeal Egieerig

4

5 Thesis Committee Moshe E. Be-Akiva (Chair) Edmud K Turer Professor Departme of Civil ad Eviromeal Egieerig Massachusetts Istitute of Techology Nigel H. M. Wilso Professor Departme of Civil ad Eviromeal Egieerig Massachusetts Istitute of Techology Patrick Jaillet Departme Head ad Edmud K Turer Professor Departme of Civil ad Eviromeal Egieerig Massachusetts Istitute of Techology Haris N. Koutsopoulos Associate Professor Departme of Civil ad Eviromeal Egieerig Northeaster Uiversity Joa L. Walker Assista Professor Departme of Geography ad Evirome & Ceer for Trasportatio Studies Bosto Uiversity Tomer Toledo Seior Lecturer Departme of Civil ad Eviromeal Egieerig Techio Israel Istitute of Techology 5

6 6

7 Ackowledgemes I would like to express my sicere gratitude to my Advisor Professor Moshe Be-Akiva. His iovative ideas provided the foudatio of this research ad his ecourageme, meticulousess ad pursuit for perfectio greatly ehaced its quality. It has bee a hoor ad life-chagig experiece workig with him. Thaks to Dr. Tomer Toledo who has bee a fried, philosopher ad guide from my first year at MIT. His techical ad practical isights for makig the models tractable were ivaluable for this research. Dr. Joa Walker was a excelle meor ad provided umerous suggestios that led to a complete face lift to this thesis. Special thaks to her for her support ad ispiratio durig the termial stage of this research. Other members of my doctoral committee: Professors Nigel Wilso, Patrick Jaillet ad Haris Koutsopoulous also provided may helpful suggestios from differe perspectives ad I thak them for their advice ad ecourageme. I also thak Professor Joseph Sussma for his coiued ierest i my academic ad research progress. The Trasportatio Egieerig Faculty at MIT have bee excelle meors ad eriched my experiece. I am thakful to the Federal Highway Admiistratio for providig fudig ad data for this research as part of the Next Geeratio Simulatio Model (NGSIM) project. The feedback from the members of the NGSIM Expert Pael was very useful for improvig the models. Some parts of this thesis were joi work with my colleagues at the MIT ITS Lab: Aita Rao, Guwoo Lee, Varu Ramaujam, Vaibhav Rathi ad Maya Abou Zeid. I thoroughly ejoyed workig with them ad ackowledge their coributio i ehacig this research. I would also like to ackowledge the friedship ad support of my other frieds i CEE: Leae Russell, Bhau Mahai, Dr. Ramachadra Balakrisha, Dr. Costaious Aoiou, Yag We, Ashish Gupta, Aa Agarwal, Vikra Vaze ad Emma Frejiger. The momes of joys ad disappoimes that we shared together will be oe of the greatest treasures of my life. I was extremely fortuate to have a amazig group of Bagladeshi frieds who eased stressful times ad exteded their warmth: Mahrukh, Noree, Suja, Nehree, Tavir, Rizwaa, Shakib, Ada, Ummul, Nasree ad Haris - thaks for everythig. Thaks also to Dr. Kauser Jaha, Dr. Tariq Ahmed, Dr. Kazi Ahmed ad Masroor Hasa for their advice regardig my career ad research. I could ot have reached this poi i my life without the care, uedig support ad guidace of my woderful family. My greatest source of ispiratio is my pares Dr. Jamilur Reza Choudhury ad Selia Choudhury. I am idebted to them for their edless love, affectio ad sacrifices. My mother-i-law Justice Ziat Ara s ucoditioal support i all my edeavors was a tremedous source of joy. Thaks also to my brother Kaashif for his ecourageme ad to my Gradmothers for always keepig me i their prayers. O a fial ote, I ca ever thak my husbad Zia Wadud eough for always beig there as my best fried ad meor ad for helpig me follow my dreams. 7

8 8

9 Table of Coes 1 Iroductio Motivatio Plaig i Drivig Decisios Modelig Approach Theoretical Framework Empirical Studies Thesis Coributios Thesis Outlie Literature Review Lae Chagig Models Gap Acceptace Models Acceleratio Models Combied Models Limitatios of Existig Models Summary Modelig Methodology Modelig Plaig Behavior Late Pla Models Late Pla Model without State-depedece Late Pla Models with State-depedece Compariso with Other Discrete Choice Modelig Approaches Summary Freeway Lae Chagig Backgroud Model Structure Choice of Pla: The Target Lae Model Choice of Actio: The Gap Acceptace Model Model Estimatio Data Likelihood Estimatio Results Aggregate Calibratio ad Validatio i MITSIMLab Data Aggregate Calibratio Aggregate Validatio Model Validatio i Other Simulators Summary

10 5 Freeway Mergig Backgroud Model Structure Model Compoes Choice of Pla: Selectig the Mergig Tactic Choice of Actio: Executio of the Merge Model Estimatio Data Likelihood of the Trajectory Estimatio Results Model Validatio Data Aggregate Calibratio Aggregate Validatio Summary Lae Selectio o Urba Arterials Backgroud Model Estimatio Estimatio Data Lae Choice at Iersectio Mailie Lae Chagig Aggregate Validatio Data Aggregate Calibratio Aggregate Validatio Summary Coclusio Summary Coributios Directios for Future Research A Microscopic Traffic Simulatio Laboratory (MITSIMLab) B Calibratio Methodology C Reduced Form Models D Related Publicatios Bibliography

11 List of Figures Figure 1.1: Classificatio of traveler behavioral algorithms Figure 1.2: Geeral framework of drivig behavior Figure 1.3: Framework of choice of pla Figure 1.4: Model developme framework Figure 2.1: Structure of the lae chagig model proposed by Ahmed (1999) Figure 2.2: Structure of the lae shift model proposed by Toledo et al. (2003) Figure 2.3: Relatio betwee subject, lead ad lag vehicles Figure 2.4: The forced mergig model structure proposed by Ahmed (1999) Figure 2.5: The mergig model structure proposed by Wag et al. (2005) Figure 2.6: Coceptual framework for the drivig behavior process (Toledo, 2002) Figure 2.7: Structure of the drivig behavior model (Toledo 2002) Figure 3.1: Framework for reward ad actio costs (Boutilier et al. 1999) Figure 3.2: First-order Hidde Markov Model (adapted from Bilmes 2002) Figure 3.3: Geeral decisio structure Figure 3.4: Late pla model without state-depedece Figure 3.5: Basic model framework (without state-depedece) Figure 3.6: Model framework of late pla models with state-depedece Figure 3.7: First-order Hidde Markov Model Figure 3.8: Model framework with state-depedece Figure 3.9: Cross-ested logit model Figure 4.1: Illustratio of myopic behavior i existig lae chagig models Figure 4.2: Examples of the structure of the proposed lae chagig model Figure 4.3: Defiitios of the lead ad lag vehicles ad the gaps they defie Figure 4.4: The I-395 data collectio site Figure 4.5: Schematic diagram of the I-395 data collectio site Figure 4.6: The subject, fro, lead ad lag vehicles ad related variables Figure 4.7: Distributios of speed, acceleratio, desity ad time headway Figure 4.8: Distributios of relative speed with respect to fro, lead ad lag vehicles Figure 4.9: Distributios of spacig with respect to the fro, lead ad lag vehicles Figure 4.10: Defiitio of path-pla variables Figure 4.11: Structure of the lae-shift model (Toledo et al. 2003) Figure 4.12: Variatio of lae utilities depedig o the curre lae of the driver Figure 4.13: Impact of path-pla lae chages o the utility of a lae Figure 4.14: Combied effects of path-pla ad lae-specific attributes Figure 4.15: Media lead ad lag critical gaps as a fuctio of relative speed Figure 4.16: Schematic diagram of the I-80 data collectio sectio (ot to scale) Figure 4.17: Calibratio results for the target lae model Figure 4.18: Compariso of ed lae distributio of vehicles Figure 4.19: Compariso of umber of lae chages by vehicles Figure 4.20: Compariso of lae chages From ad To laes Figure 4.21: Compariso of flow o HOV lae Figure 5.1: Structure of the combied mergig model

12 Figure 5.2: Decisio tree for ormal iitial state Figure 5.3: Decisio tree for courtesy iitial state Figure 5.4: Decisio tree for forced iitial state Figure 5.5: Vehicle relatioships i a mergig situatio Figure 5.6: Estimatio data collectio site Figure 5.7: Schematic of the estimatio data collectio site (ot i scale) Figure 5.8: Defiitio of merge poi Figure 5.9: Distributios of speed, acceleratio, desity, ad distace to MLC poi Figure 5.10: Distributios of lead relative speed ad spacig i the full dataset Figure 5.11: Distributios of lag relative speed ad spacig i the full dataset Figure 5.12: Distributios of lead relative speed ad spacig for the accepted gaps Figure 5.13: Distributios of lag relative speed ad spacig for the accepted gaps Figure 5.14: Distributio of umber of merges with distace to MLC poi Figure 5.15: Framework of the sigle level mergig model (Lee 2006) Figure 5.16: Lead critical gap as a fuctio of relative average speed i the mailie Figure 5.17: Lead critical gap as a fuctio of relative lead speed Figure 5.18: Lag critical gap as a fuctio of relative lag speed Figure 5.19: Lag critical gap as a fuctio of lag vehicle acceleratio Figure 5.20: Lead critical gap as a fuctio of remaiig distace to MLC poi Figure 5.21: Lag critical gap as a fuctio of remaiig distace to MLC poi Figure 5.22: Media critical aicipated gap as a fuctio of desity i target lae Figure 5.23: Distributio of aicipatio time Figure 5.24: Validatio data collectio site Figure 5.25: Compariso of merge locatios Figure 6.1: Lakershim Boulevard arterial sectio Figure 6.2: A schematic represeatio of the arterial stretch Figure 6.3: Distributio of directios Figure 6.4: Defiitios of the lead ad lag vehicles ad the gaps they defie Figure 6.5: Defiitios of the lead ad lag gaps i absece of lead ad/or lag vehicles 159 Figure 6.6: Iersectio lae selectio Figure 6.7: Structure of the iersectio lae selectio model Figure 6.9: Perspective of drivers who pla-ahead Figure 6.10: Example of a situatio whe the target lae is blocked Figure 6.11: Simple model structure Figure 6.12: Effect of aicipated delay Figure 6.13: Effect of path-pla ad driver class Figure 6.14: Heterogeeity i immediate lae choice Figure 6.15: Framework for withi sectio lae chagig model Figure 6.16: Defiitios of the lead ad lag vehicles ad the gaps they defie Figure 6.17: Structure of lae-shift model (Toledo et al. 2003) Figure 6.18: Trade-off betwee curre lae iertia ad path-pla effect Figure 6.19: Trade-off betwee curre lae iertia ad path-pla effect Figure 6.20: Variatio of lead critical gap with lead speed ad aggressiveess Figure 6.21: Variatio of lag critical gap with relative lead speed ad aggressiveess. 189 Figure 6.22: Locatios of syhetic sesors Figure 6.23: Locatios of the sesors

13 Figure 6.24: Compariso of lae distributios (Sectio 1) Figure 6.25: Compariso of lae distributios (Sectio 2) Figure 6.26: Compariso of lae distributios (Sectio 3) Figure 7.1: Framework of choice of pla Figure 7.2: Estimated model framework for freeway lae selectio model Figure 7.3: Estimated model framework for urba arterial lae selectio model

14 List of Tables Table 4.1: Lae-specific variables Table 4.2: Statistics of variables related to the subject vehicle Table 4.3: Statistics of relatios betwee the subject ad the fro vehicle Table 4.4: Distributio of lae chages by directio ad destiatio Table 4.5: Statistics describig the lead ad lag vehicles Table 4.6: Estimatio results of the target lae chagig model Table 4.7: Statistics for the model with explicit target lae ad the lae shift model Table 4.8: Estimatio results of the target lae selectio model Table 4.9: Estimatio results of the gap acceptace model Table 4.10: Iitial ad calibrated values of the parameters of the target-lae model Table 4.11: Goodess of fit statistics for the traffic speed compariso Table 4.12: Compariso of flows (vph) Table 4.13: Compariso of speeds (mph) Table 5.1: Statistics of variables related to the subject vehicle Table 5.2: Statistics for the lead ad lag vehicles of mergig vehicles Table 5.3: Estimatio results of the mergig model Table 5.4: Model compariso Table 5.5: Estimatio results of the ormal gap mergig model Table 5.6: Estimatio results of the iitiate courtesy model Table 5.7: Estimatio results of the courtesy gap acceptace model Table 5.8: Estimatio results of the iitiate forced merge model Table 5.9: Estimatio results of the forced mergig model Table 5.10: Calibratio results of the combied model Table 5.11: Compariso of lae-specific speeds Table 5.12: Compariso of lae-specific cous Table 6.1: Aggregate lae-specific statistics Table 6.2: Distributio of locatios of lae chage pois for through vehicles Table 6.3: Vehicle observatios without lead/lag vehicle i adjace lae Table 6.4: Statistics describig the lead ad lag vehicles Table 6.5: Estimatio results of the target lae chagig model Table 6.6: Model compariso Table 6.7: Iersectio lae choice: Target lae model Table 6.8: Iersectio lae choice: Immediate lae model Table 6.9: Estimatio results of the target lae chagig model Table 6.10: Model compariso Table 6.11: Estimatio results of the target lae selectio model Table 6.12: Estimatio results for the gap acceptace model Table 6.13: Estimatio results of the executio model Table 6.14: Calibratio parameters Table 6.15: Calibratio results Table 6.16: Compariso of lae-specific cous Table 6.17: Compariso of lae- specific speeds

15 Chapter 1 Iroductio 1.1 Motivatio Traffic cogestio is a major problem i urba areas that adversely affects mobility, air quality ad safety. Accordig to the Urba Mobility Report (Schrak ad Lomax 2005), cogestio caused 3.7 billio vehicle-hours of delay ad 2.3 billio gallos of wasted fuel i major US cities aloe, resultig a total loss more tha $63 billio. Califoria Air Resources Board estimates that emissios are 250% higher uder cogested coditios tha durig free-flow coditios (Schiller 1998). Icreased drivig stresses resultig from cogestio have led to aggressive drivig ad usafe drivig behaviors (NHTSA 1997). All these factors cause direct ecoomic losses due to delays ad accides, ad idirect ecoomic losses due to icreased stress, health ad eviromeal impacts. Moreover, with the rapid growth of populatio ad car owership, the exte of traffic cogestio is spreadig both spatially ad temporally. These cocers make cogestio alleviatio a major trasportatio priority. Cogestio reductio primarily ivolves icreasig the roadway capacity: either through buildig ew roads to icrease the physical capacity or by improvig the operatioal capacity of the existig etwork by adaptig optimum traffic maageme ad corol strategies. Additioal cogestio maageme mechaisms iclude demad maageme techiques ad plaig measures to reduce urba sprawl. The optimum strategy ofte icludes the combiatio of multiple measures of cogestio reductio ad is difficult to deduce theoretically. Field tests of these cogestio maageme techiques are also geerally prohibitively expesive ad ot feasible. Microscopic traffic simulatio tools, which mimic idividual drivers to deduce real world traffic situatios, are ideal tools to aalyze ad test differe cogestio 15

16 maageme strategies i a corolled evirome. These tools aalyze traffic pheomea through explicit ad detailed represeatio of the behavior of idividual drivers. Drivig behavior models are thus a importa compoe of the microscopic traffic simulatio tools. These models iclude route choice models, speed/acceleratio models ad lae chagig models. Speed/acceleratio models describe the movemes i the logitudial directio ad lae chagig models describe drivers lae selectio ad gap acceptace behaviors. Drivig decisios are iflueced by a wide rage of factors. These iclude eighborhood coditios, features of the vehicle ad characteristics of the driver, attributes of the etwork, overall traffic situatio etc. The relative speed, positio ad type of vehicles i the viciity of the driver have a direct effect o the lae chagig ad acceleratio decisios. The features of the vehicle like acceleratio ad deceleratio capabilities ad the characteristics of the driver, such as the path-pla ad schedule, the etwork kowledge ad drivig capabilities ca also sigificaly ifluece drivig behavior. The speed ad acceleratio of the driver ca also be affected by the etwork attributes: grade, curvature, surface quality ad speed limit for example. Further, i the same etwork, drivers ca behave differely i differe traffic situatios. I particular, the level of cogestio ca have a sigifica impact o drivig decisios. For example, i heavily cogested situatios, there ca be sigifica cooperatio amog the drivers; they are likely to be more alert ad coscious about their actios, ad their drivig decisios ca ivolve substaial plaig ad aicipatio. It is esseial to address these factors i the correspodig drivig behavior models for proper simulatio of cogested traffic. The existig drivig behavior models address may of these factors: either fully or partially. The effects of eighborhood coditios o the decisios of the driver i particular have received cosiderable atteio from researchers. However, i most cases the models do ot adequately capture the sophisticatio of driver behavior ad the causal mechaism behid their observed decisios. Specifically, the existig models represe istaaeous decisio-makig ad assume drivers to be myopic. These shortcomigs are more evide i cogested ad icide affected scearios where the observed drivig behavior is actually the result of a coscious plaig process. These plas may evolve 16

17 dyamically ad a iitially chose pla may ot be executed i the ed. The plas are however uobserved ad oly the actios (e.g. maeuvers like acceleratio, lae chages, route choice etc.) are observed. The behavioral predictios based oly o myopic cosideratios are therefore boud to coai sigifica oise as a result of the models structural iability to ucover uderlyig causal mechaisms. Implemeatio of these models i traffic micro-simulatio tools ca lead to urealistic traffic flow characteristics: uderestimatio of bottleeck capacities ad icorrect represeatio of cogestio (Abdulhai et al. 1999, DYMO 1999). This was reflected i the fidigs of the Next Geeratio Simulatio (NGSIM) study o Ideificatio ad Prioritizatio of Core Algorithm Categories where cogested, oversaturated ad flow breakdow scearios have bee ideified by the users as weak pois of traffic micro-simulatio tools (Alexiadis et al. 2004). Usig these tools to evaluate cogestio maageme plaig ad policy scearios ca result i bias i the aalysis. Therefore, i order to properly simulate cogested scearios i a microscopic simulator, it is esseial to develop more realistic drivig behavior models that will capture the complexity of huma decisio makig processes. 1.2 Plaig i Drivig Decisios Accordig to the NGSIM Core Algorithm Aalysis Report (Hraac et al. 2004a), travel decisios ca be classified io the followig categories based o the time scale of applicatio (show i Figure 1.1): 1. Pre-trip traveler decisios: These strategic decisios are take before startig a trip ad costitute the pre-trip pla of the traveler. Examples iclude, decidig whether or ot to travel, selectig the time of departure, destiatio, mode of trasportatio ad route etc. 2. Strategic e-route traveler decisios: Oce the pre-trip decisios are made, the traveler either executes the origially selected pla without ay chage, or makes oe or more modificatios to the iitial pla. This category of decisios icludes modificatio of destiatio, mode or route, parkig choice etc. The decisios i category 1 ad 2 take over 30 secods (ad i most cases much loger) to make ad execute. 17

18 3. Tactical route executio decisios: This category deals with traveler decisios that take betwee 5 ad 30 secods to make ad execute. While executig a route from a origi to a destiatio, a series of tactical maeuvers are performed by drivers based o sub-goals geerated from a variety of factors. Examples iclude, maiaiig a desired travel speed, makig up lost time from a previous delay, avoidig large trucks, prepositioig to get io the appropriate lae, etc. These broad set of route executio decisios result i a combiatio of lower-level tactical plas. 1. Pre-trip 2. Strategic E-route 30 sec 3. Tactical Route Executio 5 sec 4. Operatioal Drivig 5. Vehicle Corol ε Figure 1.1: Classificatio of traveler behavioral algorithms (adapted from NGSIM Core Algorithm Aalysis Report, 2004) 4. Operatioal drivig decisios: The operatioal behaviors of travelers iclude decisios to corol their vehicle at a time scale of less tha five secods. These iclude lae shiftig, gap acceptace for executig a lae chage or for maeuver at a usigalized iersectio, acceleratio/deceleratio, queue discharge behavior etc. 5. Vehicle corol decisios: This category deals with driver decisios related to corollig the vehicle at a aoscopic time-scale level, steerig the wheel of the vehicle or pressig the accelerator for example. Drivig behavior models ecompass the tactical route executio ad operatioal drivig decisios. It should be oted that oly the actios associated with the operatioal 18

19 drivig decisios ad sometimes the vehicle corol decisios are observed. The strategic ad tactical plas that lead to that actio are geerally uobserved or late. Figure 1.2: Geeral framework of drivig behavior A geeral framework of the drivig behavior model is preseed i Figure 1.2. As see i the figure, i the iitial positio, the driver makes a pla: selectig a target lae for example. Depedig o the traffic situatio ad the driver characteristics, the pla ca cosist of various additioal levels: the choice of target gap, the choice of tactic for executio of the lae chage, choice of gaps for makig a passig maeuver etc. The choice of actio depeds o the choice of pla ad cosists of lae choice ad acceleratio decisios. The chose actio is reflected i the updated positio of the driver. A example of choice of plas of the driver is show i Figure 1.3. The pre-trip ad e-route strategic plas of the driver (illustrated i Figure 1.1) may lead to the tactical pla to reach a target lae to take a exit for example. The subseque actios of the driver ivolve lookig for a acceptable gap to maeuver to the target lae i order to execute the pla. I this process, the driver may also target forward or backward gaps ad adjust the acceleratio to avail those gaps. I cogested situatios, where ormally acceptable gaps may ot be available, the chose pla ca also ivolve selectio of a alterate lae chagig tactic (e.g. courtesy or forced gap acceptace). The chose pla is 19

20 uobserved ad maifests itself through the chose lae actios ad acceleratios. However, the plas may be updated due to situatioal costrais ad coextual chages ad the observed actios may ot be the oes that were origially ieded. Failure to chage to the target lae, for example, may lead to a observatio of o chage from the curre lae. Figure 1.3: Framework of choice of pla Further, the strategic ad tactical plas ad actios ca take place i a dyamic evirome where a driver s goals, resultig plas, ad exteral coditios are all subject to chage. The driver may cosider several alteratives to come up with a pla, but the actios that he/she eds up executig might be differe from those iitially plaed. This evolutio i plas could be due to several factors. First, situatioal costrais or coextual chages might lead to revisio of the pla. For example, a uusual level of cogestio might lead a driver to revise the plaed time of travel or route. Or o-cooperatio of a driver i the target lae may lead to reevaluatio of the lae chagig tactic to that lae. Secod, the driver s curre plas are iflueced by the past experieces so that as the history evolves, the pla ca also evolve. For example, the choice of a actio with a ufavorable outcome might lead oe to abado the pla that led to this actio i future choice situatios. Third, drivers might eveually adapt to 20

21 coditios i their evirome so that they might exhibit iertia i the choice of their plas ad actios. For istace, drivers may have a preferece to stay i the curre lae. There ca be cosiderable differece i aggressiveess, drivig skills, ielligece ad plaig ability of drivers. Drivers may also have differe levels of familiarity with the etwork. These driver-specific characteristics (geerally uobserved) ca have sigifica impact o the late plas. The strategic ad tactical choices comprisig the late plas ca also be iflueced by the geometric ad traffic attributes. The effect of late path-pla for example may be more evide i a urba arterial with closely spaced turs compared to a freeway etwork where exits are far apart. Similarly, there ca be higher propesity to target a dista lae if there is a large differece i level of service (LOS) amog differe laes. Agai, the uderlyig pla for executig a lae chage i a cogested freeway ca differ sigificaly from the choice of pla i a ucogested situatio where acceptable gaps are readily available. Thus the iclusio of the effect of plas i the behavioral framework is more importa i certai scearios. Examples iclude urba arterials, traffic situatios with sigifica cogestio ad/or high differeial i level of service, work zoes, icide spots etc. 1.3 Modelig Approach The models preseed i this thesis address the plaig behaviors described i the previous sectio i the behavioral framework of drivers to icrease the reliability of microscopic traffic simulatios. The methodology for modelig behaviors with uobserved or late plas is developed first ad the demostrated through empirical studies of lae chagig behaviors of drivers i differe scearios. The overall model developme approach is summarized i this sectio Theoretical Framework Drivers are assumed to coceive plas that are uobserved (late) ad execute actios based o the plas (as show i Figure 1.2). These late plas are defied by the chose target/tactic of the driver. The actios are represeed by drivig maeuvers. The 21

22 ierdepedecies ad causal relatioships betwee the choice of pla ad choice of actio of the same driver are captured through idividual-specific late variables. The plas deped o past decisios as well as aicipated future coditios. The ierdepedecies betwee successive plas lead to state-depedece i the decisios. A Hidde Markov Model (HMM) based methodology is adapted to capture the dyamics of the plas. The heterogeeity i plaig capability ad aggressiveess of the drivers is also captured i the model framework. Two differe approaches: a discrete late class based techique ad a coiuous late pla-ahead distace based approach, have bee proposed ad demostrated to address the heterogeeity amog drivers i terms of plaig. The aggressiveess of the driver is captured through coiuous late variables that eer successive decisios across all choice dimesios of the same driver (age effect) Empirical Studies As discussed i the previous sectios, the decisios leadig to the selectio of pla, ad the choice of actio give the selected pla, differ depedig o the drivig sceario ad the effect of plaig is more evide i urba arterials, cogested ad icide affected traffic situatios, traffic streams with high differeial i level of service etc. This was also reflected i the fidigs of the NGSIM study o Ideificatio ad Prioritizatio of Core Algorithm Categories (Alexiadis et al. 2004), where the urba arterial lae selectio, oversaturated freeway behavior, freeway lae chagig ad weavig sectio behaviors topped the list of prioritized scearios chose for improveme. Based o the priority rakig of this NGSIM study ad guided by data availability (Hraac et al. 2004b), four lae selectio scearios have bee selected to demostrate empirically the effect of late plaig i observed drivig decisios. These selected scearios are as follows: Freeway lae chagig, Freeway mergig, Urba iersectio lae choice, ad Urba arterial lae chagig withi sectios. 22

23 The geeral decisio framework is the same i all cases: late plas followed by observed actios. However, the type of pla ad the causal relatioship amog plas ad actios of drivers ca differ depedig upo the sceario ad is ofte dictated by the level of cogestio. For example, i a relatively ucogested freeway lae chagig situatio, if acceptable gaps are readily available, the target gap is always the adjace gap ad the lae chagig tactic is always ormal. Therefore, the target gap choice ad lae chagig tactic selectio levels are reduda ad the late pla is maifested oly through the selectio of target laes. O the other had, i freeway o-ramp merges i cogested situatios, the target lae is always the rightmost lae of the mailie ad the target gap is restricted to the adjace gap (due to maeuverability costrais). The late pla i such situatios thus costitutes oly the choice of mergig tactic. Agai, i urba iersectio lae choice ad lae chagig i urba arterial sectios, the motivatio behid the lae selectio ad the implemeatio of the late plas differ sigificaly from the freeway scearios. The models i all scearios have bee developed usig the process show i Figure 1.4, which ivolves usig both disaggregate ad aggregate data. Disaggregate data, which are detailed vehicle trajectories at a high time resolutio are used i the model estimatio phase. I this phase, the model is specified ad explaatory variables, such as speeds ad relatios betwee the subject vehicle ad other vehicles are geerated from the vehicle coordiates extracted from the trajectory data. Parameters of all model compoes: the pla selectio, the pla trasitio (for the state-depede case) ad the actio choice are estimated joily usig a maximum likelihood techique to match observed lae chages of the drivers that occurred i the trajectory data (pael data). I this study, the statistical estimatio software GAUSS (Aptech Systems 2003) has bee used to program the log-likelihood for the model estimatio. The likelihood fuctio is ot globally cocave. For example, if the sigs of all the coefficies of the idividual-specific error term are reversed, the solutio is uchaged due to its symmetric distributio fuctio. To avoid obtaiig a local solutio, differe startig pois are used i the optimizatio procedure. Statistical tests are performed to refie the models ad to determie the best model specificatios. It may be oted that the estimatio 23

24 approach does ot ivolve the use of ay traffic simulator, ad so the estimated models are simulator idepede. Data collectio Model estimatio Model refieme Specificatio testig Implemeatio ad verificatio Aggregate calibratio of simulatio model Aggregate validatio Calibrated ad validated simulatio model Figure 1.4: Model developme framework The value of iclusio of the late plas is demostrated i two ways: Goodess-of-fit of the estimated model Model validatio usig simulatio rus The late pla models are compared agaist correspodig reduced form models that have o late pla mechaisms. Both models are estimated with the same data. These reduced form models however caot be viewed as ested withi the late pla models. Therefore adjusted goodess-of-fit measures are used to statistically compare the o-ested models. I model validatio, the simulatio capabilities of the late pla models are compared agaist the replicatios of the reduced form models. The validatio results demostrate the beefits that ca be derived from usig the modified models. For this, the 24

25 improvemes must be demostrated withi a microscopic traffic simulator usig data that has ot bee used for model estimatio. The microscopic traffic simulator icorporates ot oly the lae chagig models beig studied, but also other drivig behavior models, such as acceleratio models. MITSIMLab (Yag ad Koutsopoulos, 1996) has bee used for validatio of the models preseed i this thesis. A brief descriptio of MITSIMLab ad its model compoes is preseed i Appedix A. I the validatio case studies, aggregate data has bee used. The key parameters of the behavior models of the simulator eed to be adjusted before the validatio rus. These parameters are ofte ideified through sesitivity aalysis where the impact of a idividual factor o the overall predictive quality of the simulator is measured by allowig the correspodig parameter to chage while keepig all other parameters at their origial values. Part of the aggregate data is first used for aggregate calibratio of behavioral parameters of MITSIMLab as well as for estimatig the travel demad o the case study etwork. This aggregate calibratio problem is formulated as a optimizatio problem, which seeks to miimize a fuctio of the deviatio of the simulated traffic measuremes from the observed measuremes ad of the deviatio of calibrated values from their a-priori estimates (Toledo ad Koutsopoulos 2004). The formulatio is detailed i Appedix B. The remaiig part of the aggregate data (ot used for calibratio of the model) is used for the validatio rus. The measures of performaces are calculated from the remaiig validatio data ad compared with the correspodig outputs from the simulator for both the proposed ad the reduced form models. The measures of performaces iclude sesor speeds ad flows, the distributio of vehicles amog the laes, frequecy ad locatios of lae chages etc. 1.4 Thesis Coributios The objective of the thesis is to improve the simulatio of cogested traffic situatios by developig more realistic drivig behavior models that capture the uobserved plas behid the observed drivig maeuvers. A late pla based modelig approach for drivig behaviors is proposed that differs sigificaly from the state-of-the art drivig 25

26 behavior modelig procedures which adopt a black-box approach based o a limited field of view ad istaaeous decisio makig of drivers. The effectiveess of the ew approach has bee demostrated i the thesis through modelig lae chagig behaviors i differe scearios (freeway lae selectio, freeway mergig, urba iersectio lae choice ad urba arterial lae chagig). The usefuless of capturig the uderlyig causal mechaism i each sceario has bee preseed through compariso of goodess-of-fit of estimatio results ad validatio case studies withi traffic simulators. I both cases, the late pla models outperform the correspodig reduced form models that do ot have ay late mechaism establishig the supremacy of the approach. The developed lae chagig models have bridged some of the sigifica gaps i the existig simulatio tools. The specific coributios of each empirical study are listed below: I freeway lae chagig sceario, the ew lae chagig model with explicit choice of target lae gives the flexibility to accommodate lae chagig behavior with exclusive laes (e.g. High Occupacy Vehicle Laes, High Occupacy Tolled Laes, ad Heavy Vehicle Laes etc.). These laes are characterized by high level of service differeial. Traditioal modelig approaches ted to fail i such situatios. The ew model, with its agility to address choice of dista targets, performs substaially better. I the freeway mergig model, late plas i terms of lae chagig tactics of the driver (ormal, courtesy ad forced) are iegrated i a combied decisio framework for the first time. The combied decisio framework gives the flexibility to model the trasitio betwee the three mergig tactics. This eables the model to better capture merges that occur earlier i the merge sectio. The urba iersectio lae choice ad arterial lae chagig models costitute the first rigorously estimated behavior models for urba arterials. These models replace the existig rule-based lae assigme models used for modelig urba arterial lae choices. 26

27 Thus, implemeatio of the ew models i micro-simulatio tools ca coribute to simulatio of more realistic traffic flow ad better represeatio of cogestio, ad hece result i better plaig ad policy aalysis tools. 1.5 Thesis Outlie The remaider of this thesis is orgaized i six chapters. I Chapter 2, a literature review o state-of-the-art drivig behavior models is preseed. Chapter 3 provides the geeric model structure for late pla models ad preses the modelig methodology. The applicatio of the late pla models i differe scearios: freeway lae chagig, freeway mergig ad lae selectio i urba arterials (both iersectio lae choice ad lae chagig withi sectios) are preseed i Chapter 4, Chapter 5 ad Chapter 6 respectively. Each chapter preses the detailed model structure, descriptio of the data used for model developme, ad the model estimatio ad validatio results. Compariso of the late pla models agaist reduced form models are also show i each chapter: both i terms of goodess-of-fit of model estimatio ad i terms of simulatio capabilities withi MITSIMLab. Fially, coclusios ad directios for further research are summarized i Chapter 7. 27

28 Chapter 2 Literature Review Existig literature o drivig behavior models focus o several key aspects: logitudial maeuvers or acceleratio ad lateral moveme decisios ivolvig lae selectio ad gap acceptace. These behaviors have bee modeled both as disjoi models ad iegrated models combiig multiple aspects. The sigifica disjoi ad iegrated drivig behavior models are described below with their overall limitatios highlighted i the ed. 2.1 Lae Chagig Models The first lae chagig model ieded for micro-simulatio tools was iroduced by Sparma (1978). I this model, a distictio is made betwee the desire to chage laes ad the executio of the lae chage. The model also distiguishes betwee chages to the earside (i the directio of the exit) ad to the offside (i the directio away from the exit). Chages to the earside are motivated by ot havig obstructios i that lae. Chages to the offside are motivated by a obstructio i the curre lae (e.g. slow vehicles) ad/or better coditios o the offside lae. The model implemes psychophysical thresholds o the relative speed ad spacig to defie obstructios to which drivers will respod. The possibility of executio of a lae chage is determied by the space available i the selected lae. Gipps (1986) developed a rule based zoe depede model that addresses the ecessity, desirability ad safety of lae chages. Drivers behavior is govered by two basic cosideratios: maiaiig a desired speed ad beig i the correct lae for a ieded turig maeuver. The distace to the ieded tur defies which zoe the 28

29 driver is i ad which of the cosideratios are active. Whe the tur is far away it has o effect o the behavior ad the driver cocerates o maiaiig a desired speed. I the middle zoe, lae chages are oly cosidered to the turig laes or laes that are adjace to those. Close to the tur, the driver focuses o beig i the correct lae ad igores other cosideratios. The zoes are defied determiistically, igorig heterogeeity amog drivers ad variatios i the behavior of a driver over time. Whe more the oe lae is acceptable, the coflict is resolved determiistically by a priority system cosiderig locatios of obstructios, presece of heavy vehicles ad poteial speed gai. The limitatio of the rule based models is that the lae selectio rules are evaluated sequeially, ad therefore less importa cosideratios are oly evaluated if more importa oes did ot yield a lae choice. The determiistic rule priority system thus igores trade-offs amog the cosideratios (e.g. drivers would always avoid laes with heavy trucks ad avoid laes away from their exit, eve if these laes offer immediate speed advaage ad overtakig provisios etc.). No framework for rigorous estimatio of the model parameters has bee proposed. Several micro-simulators impleme lae chagig behaviors based o Gipps' model. I CORSIM (Halati et al. 1997, FHWA 1998) lae chages are classified as either madatory (MLC) or discretioary (DLC). MLC is performed whe the driver must leave the curre lae (e.g. i order to use a off-ramp or avoid a lae blockage). DLC is performed whe the driver perceives that drivig coditios i the target lae are better, but a lae chage is ot esseial. A similar distictio betwee MLC ad DLC is also cosidered by SITRAS (Hidas ad Behbahaizadeh 1999), Yag ad Koutsopoulos (1996), Ahmed (1999) ad Zhag et al. (1998). I SITRAS (Hidas ad Behbahaizadeh 1999), dowstream turig movemes ad lae blockages may trigger either MLC or DLC, depedig o the distace to the poi where the lae chage must be completed. I this model, MLC is also performed i order to obey lae-use regulatios. DLC is performed i a attempt to obtai speed or queue advaage, defied as the adjace lae allowig faster travelig speed or havig a shorter queue. Model parameters were ot rigorously calibrated ad o framework to perform this task has bee proposed. 29

30 Ulike the determiistic rule based models, i Yag ad Koutsopoulos model (1996), lae selectio is based o a radom utility, which captures trade-offs betwee the various factors affectig this choice (e.g. speed advaage, the presece of heavy vehicles ad mergig traffic). I Ahmed s model (1999), a more rigorous discrete choice framework is used to model the lae chagig decisios i three steps: decisio to cosider a lae chage, choice of a lae ad acceptace of gaps i the chose lae. The model framework is preseed i Figure 2.1 with uobserved decisios show i ovals. Figure 2.1: Structure of the lae chagig model proposed by Ahmed (1999) Whe a MLC situatio applies, the decisio whether or ot to respod to it depeds o the time delay sice the MLC situatio arose. DLC is cosidered whe MLC coditios do ot apply or the driver chooses ot to respod to them. The driver s satisfactio with coditios i the curre lae depeds o the differece betwee the curre ad desired speeds. If the driver is ot satisfied with drivig coditios i the curre lae, eighborig laes are compared to the curre oe ad the driver selects the most desirable lae. Lae utilities are affected by the speeds of the lead ad lag vehicles i these laes relative to the curre ad desired speeds of the subject vehicle. Gap acceptace models (detailed i the ext sub-sectio) are used to model the executio of the lae chages. The parameters of this model are estimated usig secod-by-secod 30

31 vehicle trajectory data. The model however does ot explai the coditios that trigger MLC situatios ad the parameters of the MLC ad DLC compoes of the model have bee estimated separately. The MLC model has bee estimated for the special case of vehicles mergig to a freeway, uder the assumptio that all vehicles are i MLC state. The DLC model has bee estimated with offside lae chagig data collected from a freeway sectio (to esure that the lae chages are discretioary). Zhag et al. (1998) use similar defiitios of MLC ad DLC ad the gap acceptace logic. The authors validate the model but do ot suggest a framework for its calibratio. The separatio betwee MLC ad DLC i the above meioed models imply that there are o trade-offs betwee madatory ad discretioary cosideratios. For example, a vehicle o a freeway that ieds to take a off-ramp will ot overtake a slower vehicle if the distace to the off-ramp is below a threshold, regardless of the speed of that vehicle. Furthermore, i order to impleme MLC ad DLC models separately, rules that dictate whe drivers begi to respod to MLC coditios eed to be defied. This poi is however uobservable, ad judgme based heuristic rules, which are ofte defied by the distace from the poi where the MLC must be completed, are used. Toledo et al. (2003) developed a iegrated lae shift model that allows joi evaluatio of madatory ad discretioary cosideratios. I this model, the relative importace of MLC ad DLC cosideratios vary depedig o explaatory variables such as the distace to the off-ramp. This way the awareess to the MLC situatio is more realistically represeed as a coiuously icreasig fuctio rather tha a step fuctio. The structure of the model is show i Figure 2.2. The model cosists of two levels: choice of lae shift ad gap acceptace decisios for executio of the lae chage. Variables that capture the eed to be i the correct laes ad to avoid obstacles ad variables that capture the relative speed advaages ad ease of drivig i the curre lae ad i the laes to the right ad to the left are all icorporated i a sigle utility model that captures the trade-offs amog these variables. Estimatio results idicate that path-pla related variables play a importa goal i the lae chagig behavior of drivers. Path-pla effects are captured by a group of variables like the distace to the poi where drivers have to be i specific laes ad the umber of lae chages that are eeded i order to be i these laes. The parameters of the lae shift 31

32 ad gap acceptace models have bee estimated joily usig secod by secod trajectory data collected from a freeway situatio. Figure 2.2: Structure of the lae shift model proposed by Toledo et al. (2003) Most of the existig lae chagig models have bee developed for freeway scearios. Wei et al. (2000) developed a determiistic rule based model for a two-lae urba arterial based o observatios from Kasas City, Missouri. Lae selectio is determied by the locatio ad directio of ieded dowstream turs ad classified as madatory, preemptive or discretioary. Drivers who ied to tur at the ext iersectio are i a MLC situatio ad try to move to the correct lae. Drivers who ied to tur farther dowstream try to move to the lae that coects to their plaed path ad attempt preemptive lae chages. Vehicles already i the correct lae may udertake a discretioary passig maeuver (double lae chage to the other lae ad back) i order to gai speed advaage oly if the maeuver is perceived to be possible. The model requires that both the adjace gap i the other lae ad the gap i the curre lae betwee the subject s leader ad its leader are acceptable for passig maeuvers to take place. Hu ad Lyos (1994) used eural etworks as a alterative method of modelig driver behavior withi road traffic systems. Their mai approach makes use of a learig vector quaizatio classificatio type of eural etwork. A driver is assumed to make a decisio based o vehicle movemes withi a zoe of ifluece, i.e., the activity withi a certai distace behid the vehicle ad a certai distace i fro. Their model uses visual patter based iput to describe the drivig evirome aroud the vehicle about to make 32

33 a lae chage. The model is calibrated by exposure to a large umber of represeative example iputs ad correspodig decisios or aswers. 2.2 Gap Acceptace Models Gap acceptace models have bee studied i the coext of iersectio crossig ad withi mergig ad lae chagig models. The defiitios of terms used i this sectio are illustrated i Figure 2.3 with a example of a lae chagig sceario. Adjace gap Lag vehicle Lag gap Lead gap Lead vehicle Subject vehicle Traffic directio Figure 2.3: Relatio betwee subject, lead ad lag vehicles Gap acceptace models are formulated as a biary choice problem. The driver either accepts or rejects the available gap, based o compariso of the gap with a uobserved critical gap (miimum acceptable gap). This ca be expressed as follows: Y 1 if G G = 0 if G < G cr cr Where, Y = choice idicator variable with value 1 if the gap is accepted ad 0 otherwise G = available gap G =critical gap cr ( 2.1) The defiitio of critical gap varies amog differe models. I Highway Capacity Maual (1997), the critical gap for a two-way stop corolled iersectio, is defied as the miimum time ierval i the major-street traffic stream that allows iersectio ery to oe mior-stream vehicle. I CORSIM (Halati et al. 1997), critical gaps are defied through risk factors. The risk factor is defied by the deceleratio a driver will have to apply if the leader brakes to a stop. The risk factors are calculated for every lae chage based o the relative speed ad positio of the lead ad lag vehicles ad compared to a 33

34 acceptable risk factor, which depeds o the type of lae chage to be performed ad its urgecy. Yag ad Koutsopoulos (1996) ad Ahmed (1999) defie critical gaps as miimum space gaps. For critical gap, Herma ad Weiss (1961) assume a expoeial distributio, Drew et al. (1967) assume a log-ormal distributio, ad Miller (1972) assumes a ormal distributio. Dagazo (1981) proposes a framework to capture critical gap variatio i the populatio as well as i the behavior of a sigle driver over time. He uses a multiomial probit formulatio appropriate for pael data to estimate parameters of the distributio of critical gaps. Mahmassai ad Sheffi (1981) assume that the mea critical gap is a fuctio of explaatory variables, ad so could capture the impact of various factors o gap acceptace behavior. They estimate the model for a stop corolled iersectio ad fid that the umber of rejected gaps (or waitig time at the stop lie), which captures drivers impatiece ad frustratio has a sigifica impact o critical gaps. Madaat et al. (1993) use total queuig time to capture impatiece. Cassidy et al. (1995) differeiate the first gap from subseque gaps, ad gaps i the ear lae from gaps i the far lae. These variables sigificaly improve the fit of the model. Other parameters that may affect critical gaps iclude the type of maeuver, speeds of vehicles o the major road, geometric characteristics ad sight distaces, the type of corol i the iersectio, the presece of a pedestria, police activities, ad daylight coditios (e.g. Brilo 1988, 1991, Adebisi ad Sama 1989, Saad et al. 1990, Hamed et al. 1997). However, most of the discussio is qualitative ad addresses macroscopic characteristics rather tha microscopic driver behavior. I cogested situatios, acceptable gaps are ofte ot available ad more complex gap acceptace pheomea may be observed. For example, drivers may chage laes through courtesy of the lag driver i the target lae or decide to force their way i ad compel the lag driver to slow dow. Existig microscopic traffic simulators, such as AIMSUN, Paramics ad VISSIM, use basic or modified versios of their ormal gap acceptace models to model freeway mergig behavior (TSS 2004, Quadstoe 2004, PTV 2004). These models cosider gaps created by adjace vehicles, ad i some cases model reduced gap acceptace thresholds durig cogested coditios, but they do ot explicitly cosider the aicipatory aspect of cooperatio amog drivers ad aggressive 34

35 merges by impatie drivers. Further discussio about gap acceptace i mergig coditios is preseed i Sectio Acceleratio Models Acceleratio models ca be broadly classified io two groups: car followig models ad geeral acceleratio models. Car followig models describe the behavior of drivers reactig to the behavior of their leaders ad the geeral acceleratio models iclude behaviors i both car followig ad o car followig situatios. The cocept of car followig was first proposed by Reuschel (1950) ad Pipes (1953). Pipes assumes that the follower wishes to maiai safe time headway of 1.02 s from the leader. This value was derived from a recommedatio i the Califoria Vehicle Code. Usig Laplace trasformatios, he develops theoretical expressios for the subject s acceleratio give a mathematical fuctio that describes the leader s behavior. Researchers at the GM Research Laboratories iroduced the sesitivity-stimulus framework that is the basis for most car followig models to date. Accordig to this framework a driver reacts to stimuli from the evirome. The respose (acceleratio) the driver applies is lagged to accou for reactio time ad is give as follows: () () ( ) respose t = sesitivity t stimulus t τ ( 2.2) Where, t=time of observatio τ =reactio time for driver The reactio time icludes perceptio time (time from the preseatio of the stimulus uil the foot starts to move) ad foot moveme time. The GM models assume that the stimulus is the leader relative speed (the speed of the leader less the speed of the subject vehicle) ad the respose is liear. Over the years, several extesios to the GM model were proposed to overcome its limitatios (Chadler et al. 1958, Gazis et al. 1959, 1961, May ad Keller 1967, Ozaki 1993). Herma ad Rothery (1965) ad Bexelius (1968) hypothesized that drivers follow vehicles i fro of their leader as well as the immediate leader ad assumed differe sesitivities to the relative speed with respect to each of these leaders. 35

36 Lee (1966) developed a variatio of the GM model that takes io accou the past observatios of the driver i the curre acceleratio decisio by meas of cosiderig the relative leader speed over a period of time rather tha the istaaeous speeds. The mathematical model is expressed as follows: t ( 2.3) 0 fro () = ( ) ( ) a t M t t' V t' dt' Where, () fro V t =relative speed of fro vehicle at time t M (). =memory (or weightig) fuctio, which represes the way the driver acts o iformatio that has bee received over time. Lee proposed several fuctioal forms of the memory fuctio ad aalyzed the stability of the resultig respose to periodic chages i the leader speed. Darroch ad Rothery (1972) empirically estimated the shape of the memory fuctio usig spectral aalysis. Helly (1961), Bekey et al. (1977), Gabard et al. (1982), Koshi et al. (1992) developed acceleratio models assumig that the driver tries to attai some desired measure, for example: miimizig both the leader relative speed ad the differece betwee the actual ad desired space headway. Gipps (1981) developed the first geeral acceleratio model that applies to both car followig ad free flow coditios. The maximum applicable acceleratio is determied based o two costrais: the desired speed may ot be exceeded ad a safe headway must be kept. Models with similar structure are developed by Beekohal ad Treiterer (1988) ad Hidas (2002). Yag ad Koutsopoulos (1996), Ahmed (1999) ad Zhag (1998) exteded these models by icludig additioal drivig regimes (e.g. emergecy regime, ucomfortable car followig regime etc.). Multiple drivig regimes require defiitio of boudaries to determie which regime the driver is i. For example, headway thresholds are used to determie whether a vehicle is i the car followig or free-flow regimes. However, i most of the above models (except Ahmed 1999), these thresholds are modeled determiistically. Similarly, reactio time is explicitly represeed i acceleratio models, but is ofte assumed to be determiistic ad assiged arbitrary values. 36

37 Moreover, may of these model developmes do ot ivolve rigorous estimatio of model parameters. Most models either completely igore the issue of estimatio or assume values for some parameters ad use ad-hoc procedures to determie values for others. 2.4 Combied Models Several models have bee developed that icorporate multiple model compoes i a sigle framework ad capture the plaig behavior of the drivers to some exte. Hidas (2002) developed a mergig model with compoes esseial for lae chagig uder cogested traffic coditios. I this model, if a vehicle caot merge by ormal gap acceptace, it evaluates the flow coditios i the target lae, ad attempts to set a acceleratio which may lead to a more favorable situatio for lae chagig. These decisios costitute the lae chagig pla of the driver. Hidas (2005) exteded this model ad icluded cooperative mergig by explicit modelig vehicle ieractios usig iellige age cocepts. I the exteded model, drivers i a mergig sceario have idividual goals ad they ieract ad cooperate with each other to solve the coflictig goals. Lae chage maeuvers are classified as free, forced ad cooperative based o the relative gaps betwee the leader ad follower. I free lae chages there is o oticeable chage i the relative gap betwee the leader ad follower durig the whole process, idicatig that there is o ierferece betwee the subject ad the followig vehicle. I forced lae chage, the gap betwee the leader ad follower is either costa or arrowig before the merge, but starts to wide after the subject vehicle eers, idicatig that the subject vehicle has forced the follower to slow dow. I cooperative lae chage the gap betwee the leader ad follower is icreasig before the ery poi ad starts to decrease afterwards, idicatig that the follower has slowed dow to allow the subject vehicle to eer. However, it is postulated i this model that each vehicle ivolved i a lae chagig maeuver has perfect iformatio about the lae chagig plas of other vehicles ad vehicles are able to commuicate with each other i order to cooperate, coordiate ad resolve coflicts. Video data was used to develop the model, but details of the calibratio methodology were ot available. 37

38 Several other models have bee developed specifically to model the cooperative ad/or forced lae chagig plas of the driver (Ahmed 1996, Wag et al. 2005). Ahmed (1996) estimated a forced mergig model that captures drivers lae chagig behavior i heavily cogested traffic as show i Figure 2.4. A driver is assumed to evaluate the traffic evirome i the target lae to uderstad whether the driver s right of way is established ad a forced merge is possible. If a driver ieds to merge i fro of the lag vehicle ad right of way is established the decisio process eds ad the driver gradually moves io the target lae. Oce the forced mergig has started the driver is assumed to remai i this state, persistig till the merge to the target lae is completed. However, the model assumes that oce a driver iitiates a forced merge, he/she completes it. There is o gap acceptace level after the decisio to iitiate a forced merge is take. I other words, the probability of completio of the merge is 1 if the driver has iitiated a force merge. Normal lae chage ad voluary cooperatio amog drivers is igored. Figure 2.4: The forced mergig model structure proposed by Ahmed (1999) Wag et al. (2005) cosider the mergig pla of the driver with the possibility of courtesy from the lag driver i the mailie. The model framework is preseed i Figure 2.5. The probabilities of the lag driver providig courtesy are draw from biomial distributios with parameters calibrated usig video observatios. The mergig vehicle selects a target gap ad accelerates or decelerates to adjust speed ad positio with respect to that gap. The merge is executed if the target gap is acceptable. The model however igores the possibility to force merge ad if the mergig vehicle has ot foud a acceptable gap before reachig the ed of the mergig lae, the vehicle is removed 38

39 ad a merge failure is registered. Moreover, heterogeeity amog drivers is ot explicitly cosidered i this model. Mergig Vehicle Existece of Lead or Lag No Yes Merge I Yes Closig of Lead or Lag Acceleratio Adjustme No Acceleratio Adjustme Car-followig Model Figure 2.5: The mergig model structure proposed by Wag et al. (2005) Toledo (2002) preseed a framework based o the cocepts of a short-term goal ad short-term pla for a driver. Drivig behavior cosists of three mai elemes: the shortterm goal, the short-term pla ad the driver s actios. The short-term goal is defied by the driver s target lae. The driver costructs a short-term pla, which is defied by the target gap i the target lae that the driver wishes to use i order to accomplish the goal. The acceleratios ad lae chages are the driver s actios used to execute the short-term pla. The coceptual framework of the model is illustrated i Figure 2.6. Short term Goal Lae Choice (Target lae) Lae Chagig (Gap acceptace) Pla Gap Choice (Target gap) Acceleratio Actios Figure 2.6: Coceptual framework for the drivig behavior process (Toledo 2002) 39

40 Whe the adjace gap is rejected by the driver, the driver creates a short-term pla by choosig a target gap i the target lae traffic. The alteratives i the target gap choice set iclude available gaps i the viciity of the subject vehicle. A gap which may ot be acceptable at the time of the decisio may still be chose i aicipatio of becomig acceptable i the future. However, due to the computatioal difficulty of modelig all possible combiatios of states of the short-term goal ad short-term pla (which are uobserved), a partial short-term pla was hypothesized. It is assumed that the driver executes oe step of the short-term pla, re-evaluates the situatio ad decides the ext actio to be take. Thus, it is assumed that a driver formulates a pla at every ista ad the effect of previous plas is ot fully captured. The structure of the combied lae chagig ad acceleratio model proposed by Toledo is preseed i Figure 2.7. Target Lae Left Curre Right Gap acceptace No Chage Chage Left Chage Right No Chage Target Gap Gap L1 Gap Gap LM R1 Gap RK Acceleratio Acc.... Acc. Acc. Acc. Acc. Acc.... Acc. Figure 2.7: Structure of the drivig behavior model (Toledo 2002) The model captures both lae chagig ad acceleratio behaviors. The driver selects the best lae amog the curre ad adjace laes ad if a lae shift is required, looks for a acceptable gap to make the lae chage. Drivers who wish to chage laes but caot chage laes immediately, select a short-term pla to perform the desired lae chage. Short-term plas are defied by the various gaps i traffic i the target lae. Drivers adapt their acceleratio behavior to facilitate the lae chage usig the target gap. The scope of the partial short-term pla thus oly captures variables associated with the immediate 40

41 surroudigs. For example, the choice set for lae selectio oly icludes the curre ad adjace laes, ad laes beyod the adjace laes do ot affect the lae selectio. Similarly, the choice set for target gap selectio oly icludes the adjace ad immediate forward ad backward gaps. The model therefore does ot address the sequece of maeuvers to achieve a dista target ad ca fail if there are sigifica differeces i level of service amog differe laes. Rao (2006) formulated a theoretical framework for a dyamic programmig based approach to modelig lae chagig decisios where expectatios of future coditios are explicitly addressed. The solutio of the dyamic program takes the form of a optimal decisio rule that specifies drivers optimal utility based decisios as a fuctio of their curre iformatio. The computatioal complexity of applyig such a model however prohibited model estimatio. Webster et al. (2007) proposed a tactical lae chage model usig the forward search algorithm. The completed forward search tree eumerates a complete set of subject vehicle maeuver sequeces, ad each sequece is evaluated i terms of how it improves the distace traversed over the plaig horizo. The model however makes several simplifyig assumptios. For example, the decisios are based oly o distace traversed ad effects of path-pla; the iertia i the decisio makig process ad effects of other variables are igored. Also, it imposes restrictios o lateral movemes of other vehicles ad igores the heterogeeity of the plaig horizo of drivers. It is meioed that the model parameters are calibrated with trajectory data usig simulatio rus but the computatioal burde associated with the forward search is ot detailed. 2.5 Limitatios of Existig Models It is appare from the critique i the previous sectios that although there have bee may advaces i drivig behavior models over the years, the existig models still have sigifica limitatios as described below: Tactical ad strategic plaig Most models assume that drivers make istaaeous decisios based o curre traffic coditios. I reality, drivers may coceive a pla ad perform it over a legth of 41

42 time. The otio of plaig is igored i most of the existig models. The few models that address the effect of plaig i drivig decisios have a limited exte ad/or make simplifyig assumptios. For example, as described i Sectio 1.2, the plaig process is likely to be affected by strategic trip plaig ad avigatio decisios such as selected trip schedule ad path. Drivers may adjust their speeds accordig to the trip schedule; they may pre-positio themselves i correct laes to follow their path. The effect of pathpla is cosidered i some of the lae selectio models but the effect of the trip schedule has ot bee icorporated i the existig models. Aicipatio Aicipatio of future coditios has a sigifica effect o the plas ivolvig the drivig decisios. Drivers ted to aicipate the dowstream traffic coditios, the behavior of other vehicles etc. ad make their decisios to facilitate their plas. Drivers familiar with the etwork ca pre-positio i specific laes i order to avoid delays caused by turig or mergig traffic. Drivers may avoid followig a bus or delivery vehicle that is likely to make freque stops. This is more evide i cogested ad icide affected traffic coditios where cosideratio of the aicipated coditios ca substaially miimize travel delays. The effect of aicipatio i strategic drivig decisios has ot bee adequately represeed i most of the existig models. Ierdepedece The decisios of a driver over time ad choice dimesios are ierdepede. For example, a driver s gap acceptace ad acceleratio decisios ca deped o his/her iitial decisio to chage laes. Ierdepedecies amog decisios, particularly over the time dimesio for the same driver are ot captured i detail i most of the existig models. For example, the persistece of drivers to follow their origially chose plas, which ca lead to state-depedece, has bee igored i the state-of-the-art models. Choice set I most cases, existig models explai drivig behaviors usig variables related to the subject s immediate drivig eighborhood, such as the relative speeds ad positios of 42

43 eighborig vehicles i the adjace laes. But i reality drivers are ot myopic ad are likely to select their targets based o a broader set of factors. Mixed traffic Mixed traffic streams, with vehicles havig distict differeces i size ad speed sharig the same right of way, exhibit behavior sigificaly differe from homogeeous, lae based traffic streams ad are geerally characterized by weak lae disciplie. The state-of the art drivig behavior models have focused o modelig homogeeous lae based traffic coditios ad are ot applicable i heterogeeous traffic coditios. Heterogeeity amog drivers The heterogeeity i driver behavior is igored i most of the existig models, mostly due to data limitatios. The heterogeeity i aggressiveess ad reactio time of the drivers has bee cosidered i some of the models through estimated distributios. But heterogeeity exists i may other aspects of drivig ad icludes traits of the driver like ielligece, plaig capability, risk averseess etc. The effects of socio-ecoomic characteristics of the driver (e.g. age, educatio, drivig experiece etc.) o drivig behavior have also ot bee explored. 2.6 Summary With these limitatios, applicatio of the state-of-the-art models i a simulatio evirome ca result i urealistic traffic flow characteristics. This ca result i errors i the correspodig aalysis ad bias plaig ad policy decisios. Accordig to the NGSIM study for Ideificatio ad Prioritizatio of Core Algorithm Categories (Alexiadis et al. 2004), the scearios with highest priority iclude urba arterial lae selectio, oversaturated freeway behavior, freeway lae distributio ad decisios at a weavig sectio. As discussed i Sectio 1.2, a commo lik betwee all these scearios is that the decisios i all these cases ivolve sigifica plaig ad aicipatio by the drivers. Success i bridgig the existig gaps i the traffic simulators therefore depeds o a efficie modelig techique to address the plas behid the observed decisios. 43

44 Chapter 3 Modelig Methodology This chapter preses a geeral methodology ad framework for modelig behaviors with uobserved or late plas. The plaig behavior of decisio makers have bee modeled by researchers i may differe fields. A short review of these research methodologies are also preseed i this chapter. The chapter is structured as follows: we first prese a overview of approaches that are used i differe fields to capture the plaig behavior of idividuals. The features of late pla models are the preseed. The geeral model frameworks are preseed ext: first for a basic case with oly serial correlatio ad o state-depedece, ad the exteded to iclude state-depedece. The chapter cocludes with comparisos of the modelig methodology with the state-of-the-art discrete choice modelig approaches Modelig Plaig Behavior The problems regardig modelig plaig ad decisio makig uder ucertaiy have bee addressed by researchers i may differe fields, icludig artificial ielligece, ecoomic aalysis, operatios research ad corol theory. Artificial ielligece plaig algorithms are cocered with fidig the course of actio (plas or policies) to be carried out by some age (decisio maker) to achieve its goals. I the classical case, the aim is to produce a sequece of actios that targets to guaraee the achieveme of certai goals whe applied to a specified startig state. Decisio-theoretic plaig (DTP) (Feldma & Sproull 1977) is a attractive extesio of the classical artificial ielligece plaig paradigm that selects courses of actio that 1 Earlier versios of parts of this chapter have bee preseed i Be-Akiva et al. (2006, 2007a ad 2007b) 44

45 have high expected utility. These models capture the risks ad tradeoffs of differe plas rather tha guaraeeig the achieveme of certai goals. However, i may practical cases, calculatio of expected utility ivolves evaluatio of umerous possible plas ad it is usually ot feasible to search the eire space of plas to fid the maximum utility pla. With icreasig plaig horizo, computig the expected utility of a sigle pla ca also be prohibitively expesive sice the umber of possible outcomes from the pla ca be very large (Blythe 1999). Some other assumptios i artificial ielligece plaig algorithms such as complete kowledge of the iitial state ad completely predictable effects of actios have also bee challeged by researchers, for istace, i coditioal plaig (Peot ad Smith 1992) ad probabilistic plaig (Kushmerick et al. 1994). Dyamic programmig techiques have bee applied to model the plaig behavior i partially observable settigs (Smallwood ad Sodik 1973). I cases with partially observable curre states, past observatios ca provide iformatio about the system's curre state ad decisios are based o iformatio gleaed i the past. The optimal policy thus depeds o all previous observatios of the age. These history-depede policies ca grow i size expoeially with the legth of the plaig horizo. While history-depedece precludes dyamic programmig, the observable history ca ofte be summarized adequately with a probability distributio over the curre state, ad policies ca be computed as a fuctio of these distributios (Astrom, 1965). Markov Decisio Processes (MDP) (Bellma 1957) assume that curre state trasitios ad actios deped oly o the curre state ad are idepede of all previous states. This sigificaly improves the computatioal tractability. MDP have two kids of variables: state variables s t ad corol variables a t. Accordig to Rust (1994) a decisio-maker ca be represeed by a set of primitives ( U, p, β ) where U( s, a ) is a utility fuctio represeig the prefereces at time t, p( s s a ) t t t t t, + 1 is a Markov trasitio probability represeig the subjective beliefs about ucertai future states, ad β ( 0,1) is the rate at which the idividual discous utilities i future periods. Rece research o DTP has explicitly adopted the MDP framework as a uderlyig model (Barto et al. 1995, Boutilier ad Dearde 1994, Boutilier et al. 1995, 45

46 Dea et al. 1995, Simmos ad Koeig 1995, Tash ad Russell 1994), allowig the adaptatio of existig results ad algorithms for solvig MDPs from the field of operatios research to be applied to plaig problems. The tradeoffs usig MDP based utility discouig methods have bee reviewed i detail by Rao (2006). I the artificial ielligece coext, the utility of a pla is based o the reward ad cost values associated with the actios costitutig the pla (Boutelier et al., 1999). Boutelier et al. describe two approaches for calculatig the utility fuctio: the timeseparable approach ad the additive approach. I the time-separable approach, the utility is take to be a fuctio of costs ad rewards at each stage, where the costs ad rewards ca deped o the stage t, but the fuctio that combies these is idepede of the stage, most commoly a liear combiatio or a product (see Lueberger 1973 for details). The additio of rewards ad actio costs i a system with time-separable value is illustrated i Figure 3.1, where at time t, the cost (C t ) is a fuctio of the previous state (s t-1 ) ad previous actio (a t-1 ) ad the reward R t is a fuctio of the curre state (s t ). A value fuctio is additive if the combiatio fuctio is a sum of the rewards ad cost fuctio values accrued over the history of stages. Thus, i both cases, the derivatio of the utility fuctios associated with the plas ad actios do ot ivolve ay rigorous calibratio framework. a t-1 st-1 s t Ct R t Figure 3.1: Framework for reward ad actio costs (Boutilier et al. 1999) Baum ad Petrie (1966) proposed the Hidde Markov Model (HMM) framework where the system beig modeled is assumed to be a Markov process with ukow parameters. The challege i this framework is to determie the hidde parameters from the observable parameters. This is illustrated i Figure 3.2 where late plas l affect observed actios j ad evolve over time t. 46

47 l 0 l 1 l 2 l T j 1 j 2 j T Figure 3.2: First-order Hidde Markov Model (adapted from Bilmes 2002) The HMM framework has bee used i various applicatios icludig speech recogitio (Rabier 1989, Baker 1975, Jeliek 1976), machie traslatio (Vogel et al. 1996), bioiformatics (Koski 2001), ad the evolutio of health ad wealth i elderly people (Ribeiro 2002, Ribeiro et al. 2003). However, its use i these applicatios has geerally bee to model certai processes that do ot ivolve behavioral states. I other words, these applicatios do ot ivolve choice or decisiomakig of idividuals. To summarize, plaig models i differe research fields address the dyamics of plaig through various approaches. While the assumptios ad perspectives adopted i these areas differ i substaial ways, Markovia approaches are widely used to capture the model dyamics i a tractable maer. However, these models do ot focus much o the behavioral aspect of choice or decisio makig ad the methods reviewed i this sectio are ot directly applicable to modelig the evolutio of the uobserved drivig decisios. But they form the basis of the modelig methodology proposed i the ext sectio. 3.2 Late Pla Models The geeral framework of late pla models is schematically show i Figure 3.3. At ay ista, the decisio maker makes a pla based o his/her curre state. The choice of pla is uobserved ad maifested through the choice of actios give the pla. The actios are reflected i the updated states. 47

48 Figure 3.3: Geeral decisio structure The key features of the late pla model are as follows: 1. Idividuals choose amog distict plas (target/tactic). Their subseque decisios are based o these choices. The chose plas ad iermediate choices are late or uobserved ad oly the fial actios (maeuvers) are observed. 2. Both the choice of pla ad the choice of actio coditioal o the pla ca be based o the theory of utility maximizatio. The ierdepedecies ad causal relatioships betwee the successive decisios of a idividual result i serial correlatio amog the observatios. 3. The observed actios of the idividuals deped o their late plas. The utility of actios ad the choice set of alteratives may differ depedig o the chose pla. 4. The choice of the pla at a particular time may deped o previous plas. For example, persistece ad iertia effects may affect the choice whether or ot to coiue to follow the origial pla or to shift to a alterative oe. Thus, the choice of plas ca lead to state-depedece i the decisio process. 5. The curre pla ca also deped o aicipated future coditios ad may iclude expected maximum utility (EMU) derived from the decisios ivolved with the executio of the pla. I the followig subsectios, we first prese the basic late pla model that is applicable for cases without state-depedece (oly serial correlatio). These iclude situatios ivolvig oe-time decisios, as well as pael observatios where the subseque choices of plas (coditioal o idividual-specific characteristics) are 48

49 idepede. The basic model is the exteded to explicitly capture the state-depedece betwee subseque plas ad actios Late Pla Model without State-depedece I this sectio the basic late pla model framework is preseed. This framework oly addresses the serial correlatio amog the decisios of the idividual across time ad choice dimesios but do ot address the state-depedece amog subseque plas. That is, coditioal o idividual-specific characteristics, the successive plas of idividuals are assumed to be idepede. The overall model formwork is preseed i Figure 3.4. Variables or choices i rectagles are observable, while those i ovals are uobservable or late. ( X ) ( υ ) ( l ) ( j ) Figure 3.4: Late pla model without state-depedece The pla of a idividual at ay ista t ( l ) is iflueced by explaatory variables ad idividual-specific characteristics. The attributes of the alteratives ( X ) are geerally observed but the idividual-specific characteristics associated with the idividual ( υ ) are geerally uobserved or late. For example, i case of lae selectio behavior, attributes of the alteratives (target laes) like average speed, desity, lead ad lag vehicle characteristics etc. are observed ad driver characteristics like aggressiveess, drivig skills, plaig horizo etc. are late. These late variables ca be discrete or coiuous. Characteristics of the driver such as plaig capability, for example, ca be 49

50 represeed by discrete classes of drivers (e.g. drivers who pla-ahead ad drivers who do ot). Coiuous late variables iclude attitudes, perceptios ad persoality traits of the idividual (e.g. impatiece, aggressiveess, plaig horizo etc.). The actios of the idividuals deped o the chose pla as well as the observed ad late explaatory variables. These idividual specific variables remai the same for all decisios of the same idividual across time ad choice dimesios (age effect). However, it is assumed that actios ( j ) ad plas ( l ) of idividual (coditioal o υ ) are idepede over time. This assumptio is relaxed i Sectio The geeral model framework is preseed i Figure 3.5. This framework cosists of two levels: choice of pla ad choice of actio coditioal o the pla. The selectio of the pla (idexed by l) i the upper level drives the selectio of a actio (idexed by j). The actio choice sets ad correspodig utilities, show i the lower level, may vary depedig o the pla. Pl ( t υ) l P( jt lt, υ) Figure 3.5: Basic model framework (without state-depedece) Probability of a Trajectory The trajectory of a idividual icludes a series of observed actios. For drivig behavior models, this correspods to a series of lae actios ad acceleratio decisios of the driver. Let, ( ) P l υ = probability of idividual selectig pla l at time t coditioal o t idividual-specific characteristics (, ) P j l υ = probability of idividual selectig actio j at time t give pla l t t coditioal o idividual-specific characteristics 50

51 ( ) P j υ = probability of actio j by idividual at time t coditioal o t idividual-specific characteristics L = the set of plas i the choice set of idividual T = umber of cosecutive observatios of idividual At time t for idividual, the probability of observig a particular actio j is the sum of probabilities that he/she is observed to execute actio j give that the selected pla is l, over all plas i the choice set of the idividual. ( υ ) (, υ ) ( υ ) P j = P j l P l ( 3.1) t t t t l L Assumig that actios ( j ) ad plas ( l ) of idividual (coditioal o υ ) are idepede over time (relaxed i ext sectio), the probability of observig his/her sequece of decisios ca be expressed as follows: T L ( 1, 2,..., υ ) (, υ ) ( υ ) P j j j = P j l P l ( 3.2) T t t t t= 1 l= 1 The ucoditioal choice probabilities of observig the sequece of decisios by idividual are give by the followig equatio: (,,.. ) (,,.. ) ( ) P j j j = P j j j υ f υ dυ ( 3.3) 1 2 T 1 2 T υ Where, ( ) f υ aggressiveess). is the distributio of the idividual-specific radom term (e.g. Specificatio The probabilities of choice of pla ad actio ca be calculated usig a utility-based choice framework. The specificatios of these utilities are discussed below. Choice of Pla The choice of a pla ca be based o utility maximizatio ad may iclude expected maximum utility (EMU) derived from the decisios ivolved with executig that pla. The utility of late pla l for idividual at time t ca be expressed as follows: 51

52 (,, υ, ε ) ( ( )) U = U X I l l l l I = E max U,U,...,U,...,U l 1 l 2 l j l Jl l Where, (3.4) X I U υ ε l l l jl =attributes of pla l for idividual at time t, a subset of X =expected maximum utility from actios associated with pla l of idividual at time t =utility of actio j = idividual-specific radom effect uder pla l to idividual at time t =radom utility compoe of pla l for idividual at time t Choice of Actio The observed choices/actios deped o the chose pla. The choice set, as well the fuctioal form of the utility of a actio j may vary depedig o the chose pla. The utility of actio j uder pla l ca be expressed as follows: U (, υ, ) = U X ε (3.5) jl jl jl Where, X = atttributes of actio j ad pla l at time t, a subset of X jl υ = idividual-specific radom effect ε jl = radom utility compoe of actio j ad pla l at time t The coditioal probabilities of selectig pla ( P ( lt υ ) ) ad actio ( P ( jt lt, υ ) ) are based o the utilities discussed above ( U ad U jl respectively). The specificatio of the probabilities will deped o the assumptios made regardig the distributio of the radom utility compoes of U l ad U jl. For example, if the radom compoes are idepedely ad ideically extreme value distributed, the the kerel of the choice model will be logit Late Pla Models with State-depedece I the model with explicit cosideratio of state-depedece, the previous assumptio regardig idepedece of successive plas of idividuals (coditioal o idividual- l 52

53 specific characteristics) is relaxed. Selectio of pla l by idividual at time t i this case is iflueced by his/her previously chose plas ad actios leadig to state-depedece i the choice process. The overall framework of late pla models with state-depedece is preseed i Figure 3.6. As show i the figure, i the geeral case, the pla at time t is iflueced by previous plas ( l 1, l 2,..., l t -1 ) ad previous actios ( j 1, j2,..., j t-1) i additio to the curre attributes of the alteratives ad idividual-specific characteristics. The observed choices/actios deped o the previously chose plas ad actios as well as the curre pla, attributes of the alteratives ad idividual-specific characteristics. ( X ) ( υ ) ( l, l,..., l ) 1 2 t-1 ( l ) ( j, j,..., j ) 1 2 t-1 ( j ) Figure 3.6: Model framework of late pla models with state-depedece Probability of Trajectory As i the case preseed before, the trajectory of a idividual icludes a series of observed actios. But i this case the coditioality of curre plas ad actios o previous plas ad actio are cosidered. Let, ( 1: 1, 1: 1, ) P lt l t j t υ = coditioal probability of idividual selectig pla l at time t ( 1:, 1: 1, ) P j l j υ t t t ( ) = coditioal probability of idividual selectig actio j at time t P j υ = coditioal probability of actio j by idividual at time t t L = plas i the choice set of idividual Where, 1: t is shorthad for 1,2,, t-1, t. 53

54 At time t for idividual, the probability of observig a particular actio j is the sum of probabilities that he/she is observed to execute actio j give that the selected pla is l, over all sequeces of plas that could have led to pla l. ( 1: -1, υ ) ( 1:, 1: 1, υ ) ( 1: 1, 1: 1, υ ) P j j = P j l j P l l j ( 3.6) t t t t t t t t ( l1,, lt ) The umber of possible sequeces i the summatio of Equatio 3.6 is l T, where l deotes the maximum cardiality of the set of discrete plas over all decisio istaces. Except for degeerate cases with a very small choice set of plas or a very short observatio period, modelig all possible sequeces is thus prohibitively expesive. Applicatio of a first order Hidde Markov Model (HMM) (Baum ad Petrie 1966, Baum 1972) based solutio approach simplifies the problem of estimatig the model with a large umber of late plas ad/or observatio periods. HMM is represeed graphically i Figure 3.7, i which the upper level represes the evolutio of the plas from a iitial pla at time 0 (deoted as l 0 ) to a fial pla at time T deoted as l T. The pla at every time period is determied oly by the pla at the previous time period (firstorder Markov model) ad may be affected by the actio take i the previous time period (experiece). The lower level represes the observed actios. A actio at a give time period is determied oly by the pla durig the same time period. Also, the dyamics i the observed actios are explaied by the dyamics i the uderlyig late or uobserved plas (Hidde Markov Model). l 0 l 1 l 2 l T j 1 j 2 j T Figure 3.7: First-order Hidde Markov Model (late plas l affect observed actios j ad evolve over time t) 54

55 The first order HMM assumptio thus eables us to simplify the choice of pla ad choice of actio. This ca be expressed as follows: Plas: The pla at a give time period depeds oly o the pla of the previous time period ad all previous actios. The expressio for the choice probability of a pla i the curre time period, uder the above assumptios, is as follows: ( 1: -1, 1: -1, υ ) ( 1, 1: 1, υ ) P l l j = P l l j ( 3.7) t t t t t t Actios: The dyamics i the observed actios are caused by the dyamics i the late plas. That is, the effects of past plas ad past actios affect the curre actios through the choice of curre pla ad there is o direct causal effect of past plas ad past actios o the curre actios. Therefore, coditioal o the pla, the actio observed at a give time period is idepede of the plas ad actios observed at previous time periods; it is oly depede o the curre pla. ( 1:, 1: 1, υ ) (, υ ) P jt l t j t = P jt lt ( 3.8) The model framework is preseed i Figure 3.8. Figure 3.8: Model framework with state-depedece 55

56 Uder these assumptios, the probability of observig a particular actio j at time t ca be expressed as follows: ( 1: -1, υ ) (, υ ) ( 1, 1: 1, υ ) P j j = P j l P l l j ( 3.9) t t t t t t t ( l1,, lt ) The joi probability of a sequece of actios of a idividual over a time horizo T ca be expressed as follows: ( 1,, υ ) ( 1 1, υ ) (, υ ) ( 1 0, υ ) ( 1, 1: -1, υ ) P j j P j l P j l P l l P l l j = ( 3.10) ( l1,, l ) T T T T T T T (, υ ) (,, ) (, 1 1: ) (,, ) (, 1 1 ) (, 1 0 ) υ υ υ υ υ = P j l P l l j P j l P l l j P j l P l l T T T T T T T lt l T 1 l1 Where, the iitial pla l 0 is assumed to be fixed or, if radom, ca be assumed to be hadled through specific methods desiged for dealig with iitial coditios problems i this coext (see for example Wooldridge 2003). The above simplificatio reduces the T order of complexity for computig the probability from O( l ) to O( lt ), where l deotes the maximum cardiality of the set of discrete plas over all decisio istaces. The ucoditioal choice probabilities of observig the sequece of decisios are give by: (,,.. ) (,,.. ) ( ) P j j j = P j j j υ f υ dυ ( 3.11) 1 2 T 1 2 T υ Where, f ( υ) deotes the distributio of the idividual-specific radom effect. Specificatio The probabilities of choice of pla ad actio ca be calculated usig a utility-based choice framework. The specificatios of these utilities are discussed below. Choice of Pla With HMM assumptios, the choice of the pla at time t i the state-depede case depeds o the choice of pla i the previous time period ( l -1 ) ad all previous 56

57 actios( j 1: t-1 ). As i the case without state-depedece, the choice of the pla ca be a fuctio of attributes of the plas ad idividual-specific characteristics, ad may iclude expected maximum utility (EMU) derived from the decisios ivolved with executig that pla. The utility of late pla l for idividual at time t ca therefore be expressed as follows: (, -1, j,, υ, ε 1: t-1 ) ( ( l )) U = U X l I l l t t l l I = E max U,U,...,U,...,U l 1 l 2 l j l Jl Where, X U l l jl =attributes of pla l for idividual at time t =utility to idividual from actio j at time t uder pla l I =expected maximum utility from actios associated with pla l of idividual l at time t υ = idividual-specific radom effect ε =radom utility compoe of pla l for idividual at time t ( 3.12) Choice of Actio Accordig to the HMM assumptio, the actio observed at a give time period depeds o the curre pla. The pla ad actio of previous time periods affect the curre actio through the curre pla. The utility of actio j uder pla l ca therefore be expressed as follows: (,,, ) U = U X l υ ε ( 3.13) jl jl t jl Where, X jl= atttributes of actio j uder pla l at time t υ = idividual-specific radom effect ε = radom utility compoe of actio j ad pla l at time t jl ( ) The specificatio of the coditioal probabilities of pla P (,, lt lt 1 j1: t 1 υ) actio ( ( t t, ) ) ad P j l υ will deped o the assumptios made regardig the distributio of the radom utility compoes of U l ad U jl. For example, if the radom compoes 57

58 are idepedely ad ideically extreme value distributed, the the kerel of the choice model will be logit. 3.3 Compariso with Other Discrete Choice Modelig Approaches The late pla choice model preseed i the previous sectio have similarities with existig discrete choice models that are commoly used to model choice behavior from multidimesioal choice sets (see Be-Akiva ad Lerma, 1985 ad the rece update i Be-Akiva ad Bierlaire, 2003). From the structural poi of view, the late pla models resemble the cross-ested logit (CNL) model (McFadde 1978), where a alterative ca share uobserved utility compoes from differe ests (Figure 3.9). Pl () P ( ) jl Figure 3.9: Cross-ested logit model For the two-dimesioal case preseed i Figure 3.9, the probability of selectig a alterative i the lower level ca be expressed as follows: L ( ) ( ) ( ) P j = P j l P l ( 3.14) l= 1 Where, () = ( ) P l probability of choosig l P j l = probability of choosig j give l L=umber of alteratives i upper level 58

59 A CNL model for the pla ad actio case thus assumes that the margial probability of choosig a particular actio ca be obtaied by summig the joi probabilities of that actio ad each pla leadig to the actio over all plas. However, i CNL models, the systematic utilities of a alterative at the lower level are idepede of the upper est. That implies that the utility associated with alterative j give l ca be expressed as follows: (,, ) U = U X X ε ( 3.15) jl j jl Where, X = attributes of alterative j j X = characteristics of decisio-maker ε = radom utility compoe jl Thus i CNL, the choice of a actio is uaffected by the chose pla that led to that particular actio. I late pla models, o the other had, the utilities of the alteratives at the executio level deped o the pla that led to that decisio. Moreover, CNL models caot capture the choice of idividuals i complex situatios where observable choices are affected by dyamic plaig. Aother existig discrete choice model that is similar to the late pla model is the Late Class Choice Model (LCCM) where the factors geeratig the heterogeeity amog idividuals ca be coceptualized as discrete or categorical costructs (Kamakura ad Russell 1989, Gopiath 1995). The late class choice model ca be expressed as follows: L ( ) ( ) ( ) P j = P j l P l ( 3.16) l= 1 Where, () = ( ) P l class-membership model P j l = class-specific choice model L=umber of classes The class-specific choice models are characterized by heterogeeity i taste variatio ad/or choice sets associated with the class. If idividual belogs to class l, his/her utility associated with alterative j is as follows: 59

60 (,,, ) U = U l X X ε ( 3.17) jl j jl Where, X = vector of attributes of alteratives j X = vector of characteristics of decisio-maker However, the class-membership models are based oly o characteristics of the idividuals ad ot o other variables that ifluece their attitude. The utility associated with the probability of class-membership ca be expressed as follows: (, ) U = U X ε ( 3.18) l l The membership of a idividual i a class is thus static ad do ot chage over time with chage i situatios. The late pla models o the other had, are estimated with pael data ad the uobserved factor (the late pla) ca vary dyamically with chage i situatio based o eighborhood variables. The late pla models thus have a more flexible structure ad ca therefore be iferred as a extesio of LCCM that is applicable i a dyamic case. 3.4 Summary A geeral methodology ad framework for modelig behaviors with uobserved or late plas has bee preseed i this chapter. The actio at ay time depeds o the pla at that time. For situatios where the subseque choices of plas coditioal o idividual-specific characteristics are idepede, the pla at ay time ca be affected oly by the attributes of plas, expected utilities of executig the pla ad the characteristics of the idividual. However, i the state-depede case, the curre pla ca also depeds o previous plas ad actios as well as attributes of differe plas, expected utilities of executig the plas ad the characteristics of the idividual. The computatioal tractability of the state depede model is attaied by usig the HMM approach. The HMM assumptios imply that the curre pla depeds oly o the pla ad actio of the previous time step, the attributes of alterative plas ad the characteristics of the idividual. 60

61 Structurally, the proposed late pla model has similarities with CNL ad LCCM. The model compariso reveals that late pla models ca be viewed as a hybrid of these models exteded to a dyamic settig. 61

62 Chapter 4 Freeway Lae Chagig I this chapter, the late pla ivolvig the lae chagig decisio of a driver i a freeway is preseed. The overall decisio framework cosists of the two stages preseed i the iroductory chapters: choice of late plas followed by selectio of actio to execute the pla. However, the detailed structure is formulated based o the geometric cofiguratio ad traffic attributes that characterize a typical freeway lae selectio sceario. The chapter is orgaized as follows: the backgroud of the research is preseed i Sectio 4.1. I Sectio 4.2, the structure of the late pla lae chagig model is proposed. The details of the model estimatio are preseed i Sectio 4.3. This sectio icludes descriptio of the data used to estimate the model parameters, the likelihood fuctio ad the estimatio results. This sectio also icludes statistical compariso of the goodess-of-fit of the late pla model ad a reduced form model (estimated with the same data). The aggregate validatio results are preseed i Sectio 4.5. The calibratio ad validatio exercises withi the microscopic traffic simulator MITSIMLab are preseed i this sectio followed by a summary of the validatio results withi the commercial simulators. The chapter cocludes with a summary of the fidigs. 2 2 The model preseed i this chapter has bee developed as part of the NGSIM program of FHWA. The results preseed i this chapter have bee reported i Choudhury (2005), Toledo et al. (2005) ad Choudhury et al. (2006, 2007). The validatio exercises i AIMSUN, Paramics ad VISSIM have bee performed by TSS (Barceló et al. 2006), Quadstoe (Speirs 2006) ad PTV (Vortisch ad Rössel 2006) respectively. 62

63 4.1 Backgroud A driver i a freeway is likely to choose the lae that he/she perceives to be the best ad costruct a teative pla to move to that target lae. However, because of the eighborig vehicles, it may ot be possible to execute this pla immediately. A lae chage occurs i the directio implied by the chose target lae oly if the available gaps are acceptable. The pla that is the choice of the target lae is therefore uobserved ad the observed actios are the gap acceptace decisios i the directio of the target lae. I highly cogested situatios, where acceptable adjace gaps are ot readily available, the pla may also iclude selectio of target gaps ad ivolve alterative lae chagig tactics (e.g. courtesy/forced gap acceptace). However, the focus of this chapter is modelig a freeway lae chagig sceario with moderate cogestio where the target gap is always the adjace gap ad the lae chagig tactic is ormal gap acceptace. I such situatios, the lae chagig maeuver of drivers is a two stage process: Choice of target lae (pla) Decisio to accept available gaps ad make the lae chage (actio) This is illustrated with a hypothetical sceario of a four lae road i Figure 4.1. I this example, Lae 1 is a High Occupacy Vehicle (HOV) lae with sigificaly higher level of service compared to the other laes. The lae utilities may be affected by various variables but for simplicity it has bee assumed i this example that the lae utilities are fully captured by the average speed. It is further assumed that the subject driver (driver A), is eligible to eer the HOV lae. Driver A is therefore likely to choose Lae 1 as the target lae ad look for gaps i Lae 2 to reach Lae 1 eveually. If the available gap is acceptable, the driver is observed to make a lae chage to Lae 2. If the gap is ot acceptable, he/she is still observed i the curre lae (Lae 3). Therefore, a observatio of lae chage to Lae 2 ca result from the pla to move to either Lae 2 or Lae 1. A observatio of o lae chage ca be due to the fact that Lae 3 is ideed the best available lae or aother lae is the target lae but maeuver i that directio is ot possible. Thus the observed lae actio ca result from may possible plas. 63

64 Lae 1 (HOV) Avg. Speed 70 mph Lae 2 Avg. Speed 40 mph A Lae 3 Avg. Speed 45 mph Lae 4 Avg. Speed 50 mph Traffic directio Figure 4.1: Illustratio of myopic behavior i existig lae chagig models As meioed i the literature review i Chapter 2, most lae chagig models (e.g., Gipps 1986, Yag ad Koutsopoulos 1996, Zhag et al. 1998, Ahmed 1999, Hidas ad Behbahaizadeh 1999, Hidas 2002, Toledo et al. 2003) are based o the assumptio that drivers evaluate the curre ad adjace laes ad choose a directio of chage (or ot to chage) based o the attributes of these laes oly. The lae choice set is therefore dictated by the curre positio of the vehicle, ad i multi-lae facilities would be restricted to a subset of the available laes. Thus, existig models lack a explicit tactical choice of a target lae, which may require a sequece of lae chages from the curre lae. Istead, these myopic models ca oly explai oe lae chage at a time. The eed to improve the existig freeway lae selectio model is also reflected i the fidigs of the NGSIM study o Ideificatio ad Prioritizatio of Core Algorithm Categories, where developme of freeway lae selectio model was raked as third i importace by model developers ad users (Alexiadis et al. 2004). This deficiecy of existig models is most evide i situatios where there are large differeces i the attributes of the available laes. A example of this is facilities with HOV laes or other types of exclusive laes, where a particular lae may be sigificaly more attractive compared to other laes. Eligible vehicles may make several lae chages i order to get to the exclusive lae. However, i existig models sice oly the adjace laes are cosidered for each lae chage, the ifluece of a o-adjace exclusive lae may ot be captured. To illustrate this, cosider the hypothetical situatio preseed i Figure 4.1. With existig models, the driver oly compares the curre lae (Lae 3) with 64

65 the left lae (Lae 2) ad the right lae (Lae 4). Based o the lae speeds, Lae 4 is the most desirable of the three ad the model will idicate that the driver will try to chage to this lae. However, a more plausible model would be that based o the average lae speeds the driver chooses the HOV lae (Lae 1) as the most desirable lae. Thus, driver A is likely chage to Lae 2 to reach Lae 1 eveually. I other words, the driver is likely to move to a worse adjace lae (Lae 2) as the meas of gettig to a lot better target lae further away (Lae 1). 4.2 Model Structure The discussio i the previous sectio demostrates the eed to iroduce a explicit choice of target lae i the lae chagig model framework. The target lae is the lae the driver perceives as the best lae to be i cosiderig a wide rage of factors ad goals. These factors may iclude attributes of specific laes as well as variables that relate to the spatial relatios betwee the subject vehicle ad eighborig vehicles, the driver s path-pla ad driver-specific characteristics. The choice of the immediate directio for chagig laes is determied by the directio from the curre lae to the target lae. Examples of the structure of this lae chagig model are show i Figure 4.2. The decisio structure show o the top (Figure 4.2a) is for the driver of a vehicle that is currely i the third lae (Lae 3) i a four-lae road. Laes 1 ad 2 are o its left, ad Lae 4 is o its right. At the highest level, the driver chooses the target lae. I corast with existig models, the choice set costitutes all four laes i the road (Laes 1, 2, 3 ad 4). If the target lae is the same as the curre lae (Lae 3 i this case), o lae chage is required (No Chage). Otherwise, the directio of chage is to the right if the target lae is Lae 4, ad to the left if the target lae is Lae 1 or Lae 2. If the target lae choice dictates a lae chage, the driver evaluates the gaps i the adjace lae correspodig to the directio of chage ad either accepts the available gap ad moves to the adjace lae (Chage Right or Chage Left) or rejects the available gap ad stays i the curre lae (No Chage). The bottom decisio structure (Figure 4.2b) is for the driver of a vehicle i Lae 1 i a similar settig. 65

66 The model hypothesizes two levels of decisio-makig: the target lae choice ad the gap acceptace. The target lae choice ad the directio of immediate lae chage that is implied by the selected target lae are late. Oly completed lae chages (or No Chages) are observed. I the figure late choices are show as ovals ad observed choices are represeed as rectagles. Figure 4.2: Examples of the structure of the proposed lae chagig model We ow describe i detail the specificatio of the models to explai the two choices drivers make withi the late pla lae chagig model: the target lae choice ad the gap acceptace. 66

67 4.2.1 Choice of Pla: The Target Lae Model At the highest level of lae chagig, the driver chooses the lae with the highest utility as the target lae. The target lae choice set costitutes all the available laes i the roadway. The total utility of lae l as a target lae to driver at time t ca be expressed as follows U = V + ε l L ( 4.1) l l l Where, V = systematic compoe of the utility ε L l l = radom utility compoe of target lae l for idividual at time t = choice set of target lae of driver The systematic utilities ca be expressed as follows: l V = V( X, βα,, υ) l L ( 4.2) l l Where, X l = explaatory variables that affect the utility of lae l β = correspodig vector of parameters υ = idividual-specific radom effect (e.g. aggressiveess): υ ~N(0,1) l α = parameter correspodig to idividual specific radom effect for lae l The choice of the target lae implies whether or ot the curre lae of the driver is the most preferred lae ad if ot, which adjace lae the driver eeds to move to get to the target lae. The target lae utilities of a driver may be affected by the followig: Lae attributes Neighborig vehicle attributes Path-pla Geeral lae attributes, such as the desity ad speed of traffic i the lae, traffic compositio (e.g. perceage of heavy vehicles) etc. ca affect the target lae utilities. Apart from these, particular laes may have special lae-specific attributes that eer the utility fuctio of that particular lae. For example, the exclusive lae-specific variables are icluded i the utility if the lae i cosideratio is a exclusive lae. If the driver is eligible to eer the lae, the exclusive lae is likely to have a very high utility for that 67

68 driver. O the other had, if the driver is ot eligible to move to a particular lae, a very high disutility is likely to be associated with that particular lae for that specific driver. Thus, for a sigle occupacy vehicle, the HOV lae is likely to have a high disutility capturig the pealty associated with movig to that lae violatig the law. Similarly, for high occupacy tolled (HOT) laes, the associated value of tolls ca eer the utility of the exclusive lae for drivers of sigle occupacy vehicles. The variables associated with the surroudig vehicles, such as speed, spacig ad type of the eighborig vehicles may affect the driver s target lae choice. For example, if the fro vehicle i the curre lae has a very low speed compared to the driver s desired speed, the curre lae is likely to be less preferred by the driver, eve if the average speed i that lae is higher tha that of the other laes. It may be oted that the value of these eighborig variables is deoted by the curre positio of the vehicle. The driver usually has a pre-defied destiatio ad schedule (e.g. desired arrival time) for the trip ad chooses a path accordigly. These path-pla variables have a importa effect o target lae choice. Variables i this group may iclude distace to a poi where the driver eeds to be i a specific lae ad the umber of lae chages required from the target lae to the correct laes. For example, if the driver is very close to the exit that he/she eeds to take to follow the path, he/she is less likely to choose a lae further away from the rightmost lae as the target lae. Drivers have differe irisic prefereces, aggressiveess ad level of iertia for example. All else beig equal, driver heterogeeity ca lead to differe target lae choices by differe drivers. Thus the systematic utility of a lae ca have up to five compoes at ay ista: Utility compoe comprisig the geeric characteristics of the lae; Utility compoe comprisig the exclusive/special characteristics of the lae; Utility derived from the relative positio of the lae with respect to the curre lae; Utility compoe derived from the path-pla of the driver; Utility compoe derived from the idividual-specific characteristics of the driver which ca have differe specificatios: liear or o-liear (e.g. ieractio with other variables i the utility). 68

69 Assumig a liear specificatio of the idividual-specific characteristics, the total systematic utility of lae l for idividual at time t ca be expressed as follows: V = V + V + V + V + αυ l L ( 4.3) g s c p l l l l l l Where, g V l= geeral systematic utility compoe of the lae l s V l= exclusive/special lae-specific utility compoe c V = utility compoe of lae l that depeds o the curre lae of the vehicle l V p l = utility compoe of lae l from the path pla p of the vehicle It may be oted that V s l is equal to zero if lae l is ot a exclusive lae, has a positive value if driver is eligible to use the exclusive lae ad a egative value if there is a cost associated with usig that lae (amou of toll, pealty associated with movig to that lae violatig the law etc.). Differe choice models are obtaied depedig o the assumptio made about the distributio of the radom termε l. Assumig that these radom terms are idepedely ad ideically extreme value distributed, choice probabilities for target lae l, coditioal o the idividual-specific error term ( υ ) are give by a logit model: exp( V υ ) P ( l υ ) = l, l' L t l exp( Vl' υ) l' L ( 4.4) The choice of the target lae dictates the directio of lae chage, if oe is required. If the curre lae is chose as the target lae, o chage is eeded. Otherwise, the chage will be i the directio from the curre lae to the target lae. For example, i Figure 4.2a, the curre lae is Lae 3. If the target lae is Lae 3, o chage is eeded. If the target lae is Lae 4, a lae chage to the right is eeded. If the target lae is Lae 1 or Lae 2, a lae chage to the left is eeded Choice of Actio: The Gap Acceptace Model The directio of immediate lae chagig is determied as a cosequece of the chose target lae idicated by the target lae selectio model. Next, the driver evaluates the gaps i the correspodig adjace lae to decide whether or ot the desired lae 69

70 chage ca be udertake. Coditioal o the target lae choice, the gap acceptace model idicates whether a lae chage is possible or ot usig the existig gaps. The adjace gap i the target lae is defied by the lead ad lag vehicles i that lae as show i Figure 4.3. The lead gap is the clear spacig betwee the rear of the lead vehicle ad the fro of the subject vehicle. Similarly, the lag gap is the clear spacig betwee the rear of the subject vehicle ad the fro of the lag vehicle. It may be oted that oe or both of these gaps may be egative if the vehicles overlap. Adjace gap Lag vehicle Lag gap lag G l Lead gap lead G l Lead vehicle Subject vehicle Traffic directio Figure 4.3: Defiitios of the lead ad lag vehicles ad the gaps they defie The structure of the gap-acceptace model is based o the oe proposed, estimated ad validated by Ahmed (1999) ad later by Toledo (2002).The model assumes that if the adjace gap i the target lae is acceptable the driver performs the lae chage ad does ot cosider ay other gaps. This assumptio is cosiste with satisficig behavior theory (Simo 1955), which states that huma behavior is ot optimizig, but is satisficig: if a available optio (i.e. usig the adjace gap to chage to the target lae) is satisfactory the driver does ot try to fid a better oe. The driver therefore compares the available lead ad lag gaps to the correspodig critical gaps, which are the miimum acceptable space gaps. A available gap is acceptable if it is greater tha the critical gap. Critical gaps are modeled as radom variables. Their meas are fuctios of explaatory variables (Mahmassai ad Sheffi 1981). The idividual-specific error term captures correlatios betwee the critical gaps of the same driver over time. Critical gaps are assumed to follow logormal distributios to esure that they are always o-egative ad have bee expressed as follows (Ahmed 1999, Toledo 2002): gcr T g g g G = exp( β X + α υ + ε ) g { lead, lag} ( 4.5) l l l 70

71 Where, gcr G l = critical gap g i the directio of target lae l, measured i distace uits (e.g. meters) g X l= explaatory variables that affect the critical gap g i the directio of target lae l β = coefficies of explaatory variables g α = coefficies of idividual-specific late variable υ for gap acceptace g g 2 ε = radom term: ε ~ N 0, σ ( ) l l g The gap acceptace model assumes that the driver must accept both the lead gap ad the lag gap to chage laes. The probability of chagig laes at time t, (lae actio j t =1), coditioal o the idividual-specific term υ ad the choice of target lae l t, is therefore give by: P( jt = 1 lt, υ) = P( accept lead lt, υ) P( accept lag lt, υ) ( 4.6) lead cr lead lag cr lag =P Gl G l ( υ ) P Gl Gl ( υ ) Based o the assumptio that critical gaps follow logormal distributios ( ε gl is ormally distributed), the coditioal probabilities that gap g { lead, lag} by: g gcr P Gl > Gl ( υ ) g T g g l ( Gl ) ( β Xl + α υ ) g gcr = P l ( Gl) l ( G l ( υ) ) > =Φ σ g g ε σ 2 l ~ N(0, g ) Φ [ ] deotes the cumulative stadard ormal distributio. is acceptable is give ( 4.7) Probability of o lae chages at time t (j t =0), coditioal o the idividual-specific term υ ad the choice of target lae l t, is therefore give by the followig equatio: P( jt = 0 lt, υ) = 1 P( jt = 1 lt, υ) g T g g l ( Gl ) ( β Xl + α υ ) = 1 Φ σ g ( 4.8) Gap acceptace is affected by the ieractio betwee the subject vehicle ad the lead ad lag vehicles i the adjace lae. This may be captured by variables such as the 71

72 relative speed of the subject vehicle with respect to the lead ad lag vehicles, type of lag vehicle etc. I case of madatory lae chages acceptable gaps ca also be a fuctio of the distace to the madatory lae chagig poi ad/or the associated delay. For example if the driver eeds to take a exit to follow the path, acceptable gaps ca reduce as the driver approaches the exit or has become impatie after waitig for a suitable gap for a cosiderable time. 4.3 Model Estimatio Data Study Area The dataset used i this study was collected i 1983 by FHWA i a four-lae sectio of Ierstate 395 (I-395) Southboud i Arligto, Virgiia (Figure 4.4). Figure 4.4: The I-395 data collectio site It is 997 meters i legth, oe of the logest sites for which trajectory data is available, ad icludes a o-ramp ad two off-ramps. The sectio is show schematically i Figure 4.5. A hour of data at a rate of 1 frame per secod was collected through aerial photography of the sectio. A detailed techical descriptio of the systems ad techologies used for data collectio ad reductio is foud i FHWA (1985). The dataset, smoothed by Toledo (2002) usig the local regressio procedure developed by 72

73 Clevelad (1979) ad Clevelad ad Devli (1988), coais observatios of the positio, lae ad dimesios of every vehicle withi the sectio every 1 secod. This dataset is particularly useful for estimatio of the proposed lae chagig model sice the geometric characteristics of the site, with two off-ramps ad a o-ramp, iitiate a lot of weavig ad lae chagig. Though there are o exclusive laes, the drivers are free to select the lae with the highest utility as the target lae ad make subseque lae chages depedig o availability of gaps alog the stretch of collectio site. The ramps withi the site provide path-pla iformatio for the various drivers. However, the pathpla beyod the sectio is ot observable. Characteristics of the drivers such as aggressiveess ad level of drivig skill are also uobserved. Figure 4.5: Schematic diagram of the I-395 data collectio site (ot to scale) Characteristics of Estimatio Dataset The vehicle trajectory data of the various vehicles i the sectio ad the speeds ad acceleratios derived from these trajectories are used to geerate the required variables. The resultig estimatio dataset icludes 442 vehicles for a total of observatios at a 1 secod time resolutio. O average a vehicle was observed for 35.4 secods (observatios). All the vehicles are first observed at the upstream ed of the freeway sectio. At the dowstream ed, the majority of traffic (76%) remais i the freeway. The 8% ad 16% of vehicles, which exit the sectio usig the first ad secod off-ramps (Figure 4.5) respectively, are useful to capture the effect of the path-pla o drivig behavior. 73

74 Lae-specific variables icludig lae desity, lae speed, ad perceage of heavy vehicle have bee calculated from the raw dataset. The lae-specific variables across the differe laes are summarized i Table 4.1. Table 4.1: Lae-specific variables Variable Lae 4 Lae 3 Lae 2 Lae 1 Segme Average Desity d/s, veh/km/lae Average Desity u/s, veh/km/lae Average Speed, m/sec The same dataset was used by Toledo (2002) i estimatig the iegrated lae drivig behavior model. The detailed characteristics of the dataset documeed by Toledo are summarized below: Speeds i the sectio rage from 0.4 to 25.0 m/sec. with a mea of 15.6 m/sec. Desities rage from 14.2 to 55.0 veh/km/lae with a mea of 31.4 veh/km/lae. The level of service i the sectio is D-E (HCM 2000). The vehicles the subject ieracts with ad the variables related to these vehicles are show i Figure 4.6. Traffic directio Lag vehicle Lag spacig Lead spacig Lead vehicle Subject vehicle Fro spacig Fro vehicle Figure 4.6: The subject, fro, lead ad lag vehicles ad related variables Relative speeds with respect to various vehicles are defied as the speed of these vehicles less the speed of the subject. Tables 4.2 ad 4.3 summarize statistics of the variables related to the subject vehicle ad the vehicle i fro. Table 4.2: Statistics of variables related to the subject vehicle Variable Mea Std Dev Media Miimum Maximum Speed (m/sec) Acceleratio (m/sec2) Positive Negative Desity (veh/km/lae)

75 Table 4.3: Statistics of relatios betwee the subject ad the fro vehicle Variable Mea Std Dev Media Miimum Maximum Relative speed (m/sec) Spacig (m) Time headway (sec) The distributios of speed, acceleratio, desity ad time headway are show i Figure 4.7. Figure 4.7: Distributios of speed, acceleratio, desity ad time headway Lae selectio ad gap acceptace behaviors are captured by observig lae chages performed by the drivers. A importa factor i these behaviors is drivers desire to follow their path. I this dataset drivers have three possible destiatios, each with a correspodig path-followig behavior: Exitig the sectio at the first off-ramp. Exitig the sectio at the secod off-ramp. Stayig i the freeway at the dowstream ed of the sectio. 75

76 The distributio of observed lae chages by directio (right, left) ad by destiatio is described i Table 4.4. It is worth otig that may of the vehicles that exit the sectio through the off-ramps are observed i the right-most lae at the upstream ed of the sectio. This idicates that they may have started cosiderig the path-pla costrai earlier. As a result the coefficies of explaatory variables related to the path-pla may be biased towards aggressive behaviors sice the more timid drivers are discoued i the dataset. Table 4.4: Distributio of lae chages by directio ad destiatio Destiatio Right Left Total Freeway st ramp d ramp 40 3 The relatios betwee the subject ad the lead ad lag vehicles i the right ad left adjace laes affectig the gap acceptace ad gap choice behaviors of the driver are preseed i Table 4.5. This table summarizes statistics of the accepted lead ad lag gaps (i.e. the gaps vehicles chaged laes io) both for the accepted gaps ad for the eire dataset (both accepted ad rejected gaps). Statistics for the eire dataset are preseed i pareheses. Table 4.5: Statistics describig the lead ad lag vehicles Variable Mea Std Dev Media Miimum Maximum Relatios with lead vehicle Relative Speed (m/sec) 0.2 (0.0) 2.6 (2.9) 0.5 (0.1) (-17.5) 8.1 (15.5) Lead spacig (m) 22.2 (19.6) 21.9 (39.9) 14.1 (13.0) 0.04 (-18.1) (268.9) Relatios with lag vehicle Relative Speed (m/sec) -0.4 (0.0) 2.2 (2.7) -0.3 (0.0) -6.7 (-15.0) 5.2 (14.1) Lag spacig (m) 23.1 (18.6) 20.6 (23.0) 16.6 (12.0) 1.7 (-18.1) (232.6) Accepted lead gaps vary from 0.04 to meters, with a mea of 22.2 meters. Accepted lag gaps vary from 1.7 to meters, with a mea of 23.1 meters. No sigifica differeces were foud betwee the right ad left laes. Relative speeds are 76

77 defied as the speed of the lead (lag) vehicle less the speed of the subject. Statistics for the eire dataset are also show i parehesis. With these statistics, egative spacig values idicate that the subject ad the lead vehicle partly overlap (this is possible because they are i differe laes). As expected, the mea accepted gaps are larger tha the mea gaps i the traffic stream. Similarly, lead relative speeds i accepted gaps are larger tha the mea of the dataset ad lag relative speeds are smaller i the eire dataset (i.e. o average, i accepted gaps the subject vehicle is slower relative to the lead vehicle ad faster relative to the lag vehicle compared to the eire dataset). Figure 4.8: Distributios of relative speed with respect to fro, lead ad lag vehicles 77

78 The distributios of relative speeds ad spacig, with respect to the fro, lead ad lag vehicles are show i Figures 4.8 ad 4.9 respectively. Figure 4.9: Distributios of spacig with respect to the fro, lead ad lag vehicles Likelihood I this sectio, the likelihood fuctio used to model the trajectory of the driver is preseed. Importa explaatory variables affectig the target lae choice are those 78

79 related to the path-pla. For vehicles exitig the freeway withi the data collectio sectio, the remaiig distace to the exit ( d ) is observed. However, for vehicles exitig the freeway dowstream of the observed sectio, this iformatio is ot likely to be observed for some of the vehicles. I order to capture the effect of these variables, a distributio of the distaces from the dowstream ed of the road sectio beig studied to the followig exit pois ( s ) exit is estimated. The alteratives cosidered are the first, secod ad subseque exits. For a driver takig the 1 st dowstream exit, the defiitio of the remaiig distace to the exit is illustrated i Figure exit d Figure 4.10: Defiitio of path-pla variables The probability mass fuctio of the distace beyod the dowstream ed of the sectio to the off-ramps used by drivers is give by the followig expressio: π1 for s = s p ( s) = π 2 for s = s 3 1 π1 π2 for s = s Where, s,, ad subseque exits, respectively π, π = parameters to be estimated s s s = distace beyod the dowstream ed of the sectio to the first, secod 1 = remaiig distace to the exit poi of driver 2 The first ad secod exit distaces ( s 1 ad ( 4.9) 2 s ) were extracted from maps ad a ifiite distace was used for the subseque exits ( s 3 = ). This correspods to a 79

80 assumptio that o the sectio beig studied, drivers that use these subseque exits have path-plas that are ot costraiig. The joi probability of a combiatio of target lae (l) ad lae actio (j) observed for driver at time t, coditioal o the distace to the exit poi ( s ) ad the idividualspecific characteristic ( υ ) is give by: P ( l, j s, υ ) = P ( l s, υ ) P ( j l, υ ) ( 4.10) t t t t t Where, P ( l.) ad P ( j.) are give by Equatios 4.4, 4.6 ad 4.8 respectively. t t Oly the lae chagig actios are observed. The margial probability of the laechagig actio is therefore give by: (, υ ) (,, υ ) P j s = P l j s ( 4.11) t t t l L The behavior of driver is observed over a sequece of T cosecutive time iervals. Assumig that, coditioal o s ad υ, these observatios are idepede, the joi probability of the sequece of observatios is give by: (,,...,, υ ) (, υ ) 1 2 T t t= 1 T P j j j s = P j s ( 4.12) The ucoditioal idividual likelihood fuctio ( L ) is obtaied by iegratig (summig for the discrete variable s ) over the distributios of the idividual-specific variables: ( 1 2 ) ( 1 2 ) L = P j, j,..., jt = P j, j,..., jt s, υ p( s) f( υ) dυ ( 4.13) υ s Assumig that the observatios from differe drivers are idepede, the loglikelihood fuctio for all N idividuals observed is give by: N L = l( L ) ( 4.14) = 1 The maximum likelihood estimates of the model parameters are foud by maximizig this fuctio. I this study, the Broyde-Fletcher-Goldfarb-Shao (BFGS) optimizatio algorithm implemeed i the statistical estimatio software GAUSS (Aptech Systems 2003) has 80

81 bee used. BFGS is a quasi-newto method, which maiais ad updates a approximatio of the Hessia matrix based o first-order derivative iformatio (see, for example, Bertsekas 1999). GAUSS implemes a varia of BFGS due to Gill ad Murray (1972), which updates the Cholesky decompositio of the Hessia (Aptech Systems 1995). The iegrals i the likelihood fuctio were calculated umerically usig the Gauss-Legedre quadrature method (Aptech Systems 2003). The likelihood fuctio is ot globally cocave. For example, if the sigs of all the coefficies of the idividualspecific error term are reversed, the solutio is uchaged due to its symmetric distributio fuctio. To avoid obtaiig a local solutio, differe startig pois have bee used i the optimizatio procedure. It may be oted that the estimatio approach does ot ivolve the use of ay traffic simulator, ad so the estimated models are simulator idepede Estimatio Results All compoes of the model were estimated joily usig a maximum likelihood estimatio procedure as described i the previous sectio. However, i order to simplify the preseatio, estimatio results for the target lae choice ad gap acceptace levels are preseed ad discussed separately. The summary of estimatio results of the proposed lae chagig model is preseed i Table 4.6. Table 4.6: Estimatio results of the target lae chagig model Fial log-likelihood Iitial log-likelihood Number of drivers 442 Number of observatios Number of parameters 31 Adjusted rho-bar square 0.37 To demostrate the eed to iclude the late plas i the freeway lae selectio model by meas of target laes, the estimatio results were compared agaist a reduced form model with restricted late targets (Toledo et al. 2003). I the reduced form model (referred as the lae shift model i the subseque discussio), oly the adjace laes are 81

82 cosidered for the lae shift. The model framework is illustrated i (Figure 4.11) ad detailed i Appedix C.1. Lae shift LEFT CURRENT RIGHT Gap acceptace NO CHANGE CHANGE LEFT NO CHANGE CHANGE RIGHT NO CHANGE Figure 4.11: Structure of the lae-shift model (Toledo et al. 2003) The myopic lae shift model caot be viewed as ested withi the model with explicit target lae choice, ad therefore classic statistical tests caot be applied to select betwee the two. For comparig the goodess-of-fit of o-ested models, the Adjusted 2 Rho-bar square ( ρ ) ad the Akaike Iformatio Criteria (AIC) have bee used. 2 Adjusted Rho-bar square ( ρ ) measures the fractio of a iitial log-likelihood value explaied by the model takig io accou the model complexity. The measure is defied as follows: ρ 2 * L( β ) k = 1 ( 4.15) L( 0) Where, * L( ) β is the maximum log-likelihood value, ( 0) L is the maximum loglikelihood value, k is the umber of estimated parameters. Akaike (1973, 1974) developed the Akaike iformatio criterio (AIC) as a tool for selectig betwee competig model specificatios. The AIC pealizes the maximum likelihood value of each model to accou for model complexity: * ( β ) AIC = L k ( 4.16) 82

83 2 I model selectio, ρ ad AIC are computed for all cadidate models ad the model with the larger AIC is selected (see Be-Akiva ad Lerma 1985 ad Gourieroux ad Mofort 1995 for details). The test statistics are preseed i Table 4.7. Table 4.7: Model compariso Lae Shift Target Lae Statistic (R) (U) Likelihood value Number of parameters (k) Akaike iformatio criteria (AIC) Adjusted rho-bar square ( ρ ) For both statistics, the model with explicit target lae choice has larger values, which idicates that it has a better goodess-of-fit eve after discouig for the icreased umber of parameters. The detailed estimatio results are preseed i the followig sectios. Choice of Pla: The Target Lae Model The driver selects the lae that he/she perceives to be the best as the target lae. A liear utility fuctio is associated with each lae. The choice set of the driver icludes all available laes i the freeway stretch. The utility of lae target lae l of idividual at time t ca be expressed as follows: U = β X + αυ + ε ( 4.17) T l l l l Where, X l= explaatory variables that affect the utility of lae l β = correspodig vector of parameters υ = idividual-specific radom effect (e.g. aggressiveess): υ ~N(0,1) l α = parameter correspodig to idividual specific radom effect for lae l As discussed i Sectio 4.3.2, the target lae choices are affected by the attributes of the alterative laes, the variables related to the path-pla ad the eighborig vehicles as well as driver-specific characteristics. However, ot all of the cadidate variables meioed i Sectio were foud to be statistically sigifica ad/or have iuitive sigs. For example, the perceage of heavy vehicles i the lae ad type of the 83

84 eighborig vehicles were ot foud to be sigifica. I some cases, ieractios of multiple variables have bee used to better capture a particular effect. These ieractio variables have bee icluded oly if there was a improveme i the goodess-of-fit. For example, i case of path-pla effect, ieractio of the remaiig logitudial distace ad lateral distace were foud to yield a improveme i the likelihood ad led to the proposed fuctioal form. The estimatio results are preseed i Table 4.8. Table 4.8: Estimatio results of the target lae selectio model Lae Attributes Neighborhood Variables Iertia Variables Path-pla Heterogeeity Variable Parameter t-stat Lae 2 costa Lae 3 costa Lae 4 costa (right most lae) Lae desity, vehicle/km Average speed i lae, m/sec Fro vehicle spacig, m Relative fro vehicle speed, m/sec Tailgate dummy Curre lae (CL) dummy lae chage from the CL Each additioal lae chage from the CL Path-pla impact, 1 lae chage required Path-pla impact, 2 lae chages required Path-pla impact, 3 lae chages required Next exit dummy, lae chage(s) required MLC Expoe of remaiig distace, θ Probability of takig 1 st exit, π Probability of takig 2 d exit, π lae1 Coefficie of aggressiveess: Lae 1, α lae2 Coefficie of aggressiveess: Lae 2, α lae3 Coefficie of aggressiveess: Lae 3, α lae4 Coefficie of aggressiveess: Lae 4, α The estimated values of the lae-specific costas imply that, everythig else beig equal, the right-most lae is the least desirable. This may be the result of drivers preferece to avoid the mergig ad weavig activities that take place i that lae. I geeral laes that are to the left are more desirable. However, laes 3 ad 4 have oegative costas, which may idicate that the advaage of beig away from the slower right laes is balaced by the disadvaage associated with beig i laes that are further away from the off-ramp, ad by the icreased ieractio with vehicles travelig at higher 84

85 speeds. The results idicate that drivers are more likely to choose laes with higher average speeds ad lower desities, which is iuitive. Some of the lae-specific variables are depede o the curre lae of the driver. For example, the required maeuver to reach a specific lae is a fuctio of the distace of the lae from the curre lae of the driver. The values of the coefficies of the umber of lae chages required from the curre lae to the target lae deote the disutility associated with choosig target laes that require lae chagig maeuvers. This has bee modeled as a step fuctio ad the results idicate that the disutility associated with each additioal lae chage is much higher whe more tha oe lae chagig maeuver is associated. The positive coefficie of the curre lae dummy captures the iertia preferece to stay i the curre lae. As expected, the sig of this coefficie is positive. As appare i Figure 4.12, the lae-specific part of the utility thus chages depedig upo the curre positio of the vehicle, beig the highest for the curre lae ad dimiishig with the distace from the curre lae. Lae Desity= 30 veh/km Lae Speed=15 m/s Lae specific utility CL=Lae 1 CL=Lae 2 CL=Lae 3 CL=Lae 4 Lae 1 Lae 2 Lae 3 Lae 4 Curre Lae Figure 4.12: Variatio of lae utilities depedig o the curre lae of the driver The ieractios betwee the subject vehicle ad the vehicles i fro of it i the curre ad adjace laes, also affect the target lae choice. Results show that lae utilities icrease with the relative fro speed ad the spacig betwee the vehicles. The tailgatig dummy variable captures drivers' tedecy to move out of their curre lae if they are beig tailgated. Tailgatig is ot directly observable i the data but tailgatig behavior is assumed if a vehicle is close behid the subject vehicle whe traffic 85

86 coditios permit a loger headway (i.e. free-flow coditios apply). Mathematically, the tailgate dummy variable is defied by: δ tailgate 1 = 0 gap behid otherwise 10m ad level of service is A, B or C ( 4.18) Levels of service defiitios are based o desities (HCM 2000). The estimated coefficie of the tailgate dummy is egative ad its magitude is large relative to the coefficies of other variables. It implies a strog preferece to avoid these situatios. This result is comparable with those of Ahmed (1999) ad Toledo et al. (2002), who also foud tailgatig to be a importa explaatory variable. The path-pla impact variables idicate that the utility of a lae decreases with the umber of lae chages the driver eeds to perform i order to maiai the desired path. This effect is magified as the distace to the off-ramp d exit decreases. This has bee MLC captured by the egative power of the distace to the off-ramp ( θ = 0.417) that guaraees that at the limits, the path-pla impact approaches 0 whe d exit + ad exit approaches whe d + 0. The disutility associated with beig i a wrog lae is larger whe the driver eeds to take the ext exit. Figure 4.13 shows the impact of pathpla lae chages o the utility of a lae as a fuctio of the distace from the off-ramp. Utility of Lae Distace from off-ramp (km) Lae 3 Lae 2 Lae 1 Figure 4.13: Impact of path-pla lae chages o the utility of a lae The combied effect of path-pla ad lae-specific attributes is show i Figure

87 Lae Desity= 30 veh/km Average Speed= 15 m/s Distace From Exit= 50 km Utility of Lae CL=Lae 1 CL=Lae 2 CL=Lae 3 CL=Lae 4 Curre Lae Lae 1 Lae 2 Lae 3 Lae 4 a. Distace from exit=50 km Lae Desity= 30 veh/km Average Speed= 15 m/s Distace From Exit= 0.5 km Utility of Lae CL=Lae 1 CL=Lae 2 CL=Lae 3 CL=Lae 4 Curre Lae Lae 1 Lae 2 Lae 3 Lae 4 b. Distace from exit=0.5 km Lae Desity= 30 veh/km Average Speed= 15 m/s Distace From Exit= 0.05 km Utility of Lae CL=Lae 1 CL=Lae 2 CL=Lae 3 CL=Lae 4 Curre Lae Lae 1 Lae 2 Lae 3 Lae 4 c. Distace from exit=0.05 km Figure 4.14: Combied effects of path-pla ad lae-specific attributes 87

88 I this example, the exits are o the right, Lae 4 is closest to the exit ad Lae 1 is the farthest. It is ierestig to ote the tradeoff betwee path-pla ad iertia of the driver. Whe the driver is very far from the desired exit, the lae utilities are affected primarily by the positio of the lae with respect to the curre lae of the driver (Figure 4.14a). As the distace to exit decreases, the disutility of beig i a lae far from Lae 4 becomes more ad more proouced ad the relative utility of the laes i the directio of the exit (right i the example) gradually icrease. Whe the driver is very close to the exit the path-pla effect clearly domiates (Figure 4.14c) ad the laes far from Lae 4 have a very high disutility. The heterogeeity coefficies, lae1 α, lae2 α, lae3 α ad idividual-specific error term υ o the target lae choice. α lae1 ad egative compared to α lae3 ad lae4 α lae4 α capture the effects of the lae2 α are more. Hece, υ ca be ierpreted as correlated with aggressiveess implyig aggressive drivers are less likely to choose the right laes over the left oes compared to more timid drivers. I summary, the target lae utility ca be give by: l avg fro CL fro adj / CL Ul = β Dl Vl Xl δl Vl δl tailgate CL CL CL= 1 CL> δ δl δl δl 3.34( CLl 1) δl exit ( 4.19) ext exit + d ( 2.55δl 4.95δl 6.96δl ) 0.872δ Exitl l αυ + ε Where, l l β =costa for lae l D l =desity of lae l,vehicle/km avg V l = average speed i lae l, m/s fro X l =spacig of the fro vehicle i lae l, m fro V l = relative speed of the fro vehicle i lae l, m/s CL δ l = curre lae dummy, 1 if lae l is the curre lae, 0 otherwise adj/cl δ l = curre /adjace lae dummy, 1 if lae l is the curre/adjace lae, 0 otherwise tailgate δ = tailgate dummy, 1 if vehicle is beig tailgated at time t, 0 otherwise 88

89 δ = required chage dummy, 1 if lae l ivolves oe lae chage from the CL=1 l CL>1 l curre lae,0 otherwise δ = required chage dummy, 1 if lae l ivolves more tha oe lae chages CL exit 1k l l from the curre lae,0 otherwise = umber of lae chages required to get from the curre lae to lae l d = distace to the exit driver ieds to take δ = idicator with value 1 if lae l is k (k=0,1,2,3) laes away from the ext exit desired exit of idividual, 0 otherwise δ = idicator with value 1 if lae i is 1 lae away from the desired exit l of driver, 0 otherwise Exit =umber of lae chages required to get from lae l to the exit lae of driver l α = heterogeeity coefficie of lae l Choice of Actio: The Gap Acceptace Model The directio of the target lae idicates the directio of immediate lae chage ad the driver is assumed to evaluate the available adjace gap i the target lae ad decide whether or ot to chage laes immediately. I order for the gap to be acceptable both the lead ad lag gaps, must be acceptable. That is, the available lead ad lag gaps must be larger tha the correspodig critical gaps. As preseed i Equatio 4.5, i order to esure that the critical gaps are always positive, they are assumed to follow logormal distributios: l( G ) = β X + α υ + ε lead cr T lead lead lead l l l l( G ) = β X + α υ + ε Where, lag cr T lag lag lag l l l G,G = lead ad lag critical gap i the directio of target lae l, measured i lead cr l lead l lag cr l lag l distace uits (e.g. meters) X,X = explaatory variables that affect the lead ad lag critical gaps respectively i the directio of target lae l lead lag α α υ, = coefficies of idividual-specific late variable for lead ad lag gap acceptace 2 2 ( ) ( ) lead lag lead lag ε, ε = radom terms: ε ~ N 0, σ, ε ~ N 0, σ l l l lead l lag ( 4.20) 89

90 Similar to the target lae choice model, ot all cadidate variables were supported by the data. For example, the remaiig distace to the desired exit did ot have ay sigifica effect o critical gaps. The estimatio results of the gap acceptace model are preseed i Table 4.9. Table 4.9: Estimatio results of the gap acceptace model Lead Critical Gap Variable Parameter t-stat Costa lead Relative lead speed positive, Max( V,0),m/sec lead Relative lead speed egative, Mi( V, 0),m/sec lead Heterogeeity coefficie of lead gap, α Stadard deviatio of lead gap, σ lead Lag Critical Gap Costa lag Relative lag speed positive, Max( V,0), m/sec lag Heterogeeity coefficie of lag gap, α Stadard deviatio of lag gap, σ lag The lead critical gap decreases with the relative lead speed, i.e. it is larger whe the subject vehicle is faster relative to the lead vehicle. The effect of the relative speed is strogest whe the lead vehicle is faster tha the subject. I this case, the lead critical gap quickly dimiishes as a fuctio of the speed differece. This result suggests that drivers perceive very little risk from the lead vehicle whe it is gettig away from them. I the gap acceptace model, the lag critical gap icreases with the relative lag speed: the faster the lag vehicle is relative to the subject, the larger the lag critical gap. I corast to the lead critical gap, the lag gap does ot dimiish whe the subject is faster. A possible explaatio is that drivers may maiai a miimum critical lag gap as a safety buffer sice their perceptio of the lag gap is ot as reliable as it is for the lead gap due to the use of mirrors. Media lead ad lag critical gap variatios, as a fuctio of the relative speeds are preseed i Figure

91 Media lead critical gap (m) Relative lead speed (m/sec) Media lag critical gap (m) Relative lag speed (m/sec) Figure 4.15: Media lead ad lag critical gaps as a fuctio of relative speed Estimated coefficies of the uobserved driver characteristics variable, υ, are egative for both lead ad lag critical gaps. This is cosiste with the ierpretatio of υ as beig egatively correlated with aggressive drivers, who require smaller gaps for lae chagig compared to timid drivers. I summary the estimated lead ad lag gaps are give by: lead Max( 0, V ) lead 0.130Mi( 0, V ) ( ) 2 lag 2 ( ) εl N( ) lead cr G l = exp lead υ + ε lag cr lag lag G = exp Max 0, V ( 4.21) ε ( υ ε ) l l l lead l ~ N 0, ad ~ 0,

92 4.4 Aggregate Calibratio ad Validatio i MITSIMLab The aggregate validatio demostrates the beefits that ca be derived from usig the modified models i traffic simulators. For this the estimated model (which is simulator idepede) is implemeed withi the microscopic traffic simulator MITSIMLab (Yag ad Koutsopoulos 1996) ad calibrated ad validated usig data collected from a differe site. The details of the aggregate validatio are preseed below Data The data for aggregate calibratio ad validatio cosists of sesor data ad aggregate trajectory data collected from a highly cogested 1.5 miles sectio of I-80, i Emeryville ad Berkeley, Califoria. This freeway serves approximately 275,000 vehicles daily, ad is oe of the most vital trasportatio liks i the Sa Fracisco Bay Area. South of the study area, I-80 coects to the Bay Bridge ad dowow Sa Fracisco, as well as freeway ierchages to I-880 ad dowow Oaklad, ad I-580 East. To the orth of the study area are resideial East Bay eighborhoods ad I-580 West, leadig to U.S. 101 ad Mari Couy. Most of the drivers travelig i this area are local commuters. The left-most lae is a HOV lae that ca be accessed to ad exited from at ay poi i the sectio. The presece of this ulimited access HOV lae results i high differece i the level of service amog differe laes ad is therefore useful to test the target lae chagig model m 2100 m 1300 m 1600 m Powell Ashby Uiversity Gilma Traffic directio Figure 4.16: Schematic diagram of the I-80 data collectio sectio (ot to scale) 92

93 The selected segme icludes four o-ramps ad three off-ramps (show schematically i Figure 4.16). The dowstream ed of the etwork exteds beyod the I- 80/I-580 split, which is the major bottleeck i this area. These boudaries have bee selected i order to esure that possible queues formig at this bottleeck could be captured ad explaied i the model. The geometry of this sectio is particularly useful for validatio of the lae chagig model for several reasos: The presece of the ulimited access HOV lae ad the high level of service differeial associated with it. The sectio icludes weavig sectios that are required to test the lae chagig model. The multiple ramps that exist i this sectio provide the ability to verify the path-pla based lae pre-positioig ( look ahead ) effects that are icorporated i the model. The selected etwork is a corridor ad therefore complex route choice situatios do ot arise. This is a desirable property for this study sice it elimiates route choice as a source of modelig error, ad so results should be more idicative of the effect of the drivig behavior. The data for aggregate calibratio ad validatio cosist of sesor data at the locatios show schematically i Figure The data is available for two weeks (10 weekdays) at 30-secod iervals ad icludes lae-specific traffic cous, occupacies ad speeds. I additio trajectory data from the part of the corridor betwee the Powell Street ad Ashby Street ierchages (showed by a dotted rectagular i Figure 4.16) is also available for oe day betwee 2.35PM ad 3.05PM. Aggregate statistics derived from these trajectories, which provide richer iformatio compared to the sesor data, are also used i the validatio (referred as trajectory data i the subseque sectios). The traffic cous ad speeds from 2:35PM to 3:05PM are used for calibratio ad validatio. The available sesor data has bee split io two data sets with oe week of data i each. The first week of data is used for aggregate calibratio of the MITSIMLab model (the calibratio methodology is detailed i Appedix B). The secod week of data is used oly for validatio of the calibrated model ad therefore allows idepede validatio. 93

94 4.4.2 Aggregate Calibratio MITSIMLab (Yag ad Koutsopoulos 1996) is used to estimate the OD flows ad calibrate the sesitive behavioral parameters. To ideify the sesitive parameters, each cadidate parameter was allowed to chage over 10 iteratios, while keepig all other parameters fixed ad the parameters that results the larger improvemes i the objective fuctio were ideified. The estimatio data collectio site (I-395) did ot have ay HOV lae ad the effect of the HOV lae has also bee captured durig the aggregate calibratio process by iroducig a HOV dummy. This parameter was calibrated simultaeously with other behavioral parameters. The parameters chose for calibratio ad their iitial ad calibrated values are show i Table 4.10: Table 4.10: Iitial ad calibrated values of the parameters of the target-lae model Parameter Car followig Iitial Value Calibrated Value Acceleratio Costa Deceleratio Costa Mea Desired Speed Variace Lae chagig Rightmost Lae Costa Curre Lae Dummy HOV Dummy As see i Table 4.10, may of the parameters chaged sigificaly durig calibratio. This is expected sice the estimatio dataset (collected from I-395, VA) ad the aggregate calibratio ad validatio dataset (collected from I-80, CA) had substaial differeces i geometry ad level of service as well as driver characteristics. The model fit after calibratio is preseed i Figure The lae shift model was also calibrated with the same aggregate data i a similar maer. 94

95 Calibrated Model: Observed vs. Simulated Cous (2:30 pm to 6:00 pm at 15 mi iervals) Simulated Cous (veh/15mi) Observed Cous (veh/15mi) Simulated Speed (mph) Calibrated Model: Observed vs. Simulated Speeds (2:30 pm to 6:00 pm at 15 mi iervals) Observed Speed (mph) Figure 4.17: Calibratio results for the target lae model Aggregate Validatio The purpose of aggregate validatio is to determie the exte to which the simulatio model replicates the real system. At this step, the behavior parameters obtaied i the aggregate calibratio step are fixed ad the model predictios are compared agaist the secod set of traffic measuremes, which have ot bee used for calibratio. A separate OD matrix is estimated for the validatio measuremes. The followig measures of performace (MOPs) are selected based o their relevace to the evaluatio of the lae chagig model: Ed lae distributio of vehicles with respect to the startig lae Lae-specific sesor speeds Number of lae chages by vehicles Lae chages From ad To laes 95

96 A umber of goodess of fit measures were used to evaluate the overall performace of a simulatio model. Root Mea Square Error (RMSE), Root Mea Square Perce Error (RMSPE), Mea Error (ME) ad Mea Perce Error (MPE). These measures are defied below: 1 N sim obs N = 1 ( ) 2 RMSE = Y Y ( 4.22) 2 N sim obs 1 Y Y RMSPE = obs N = 1 Y ( 4.23) 1 N sim obs N = 1 ( ) ME = Y Y ( 4.24) N sim obs 1 Y Y MPE = N ( 4.25) Y obs Where, = 1 obs Y ad sim Y are the averages of observed ad simulated measuremes at space-time poi, calculated from all available data (i.e. several days of observatios ad/or multiple simulatio replicatios). RMSE ad RMSPE pealize large errors at a higher rate relative to small errors. ME ad MPE idicate systematic uder-predictio or over-predictio i the simulated measuremes. Ed Lae Distributio The distributio of vehicles across laes at the ed of the sectio with respect to the startig lae was extracted from the aggregate trajectory data ad compared with the simulated lae distributios of both the models. Figure 4.18 preses the results of the compariso. 96

97 Startig Lae: Lae 1 Observed Target Lae Lae Shift Fractio of Vehicles Lae1 (HOV) Lae2 Lae3 Lae4 Lae5 Lae 6 (Off-ramp) Lae Startig Lae: Lae 2 Observed Target Lae Lae Shift Fractio of Vehicles Lae1 (HOV) Lae2 Lae3 Lae4 Lae5 Lae 6 (Off-ramp) Lae Startig Lae: Lae 3 Observed Target Lae Lae Shift Fractio of Vehicles Lae1 (HOV) Lae2 Lae3 Lae4 Lae5 Lae 6 (Off-ramp) Lae Figure 4.18: Compariso of ed lae distributio of vehicles 97

98 Startig Lae: Lae 4 Observed Target Lae Lae Shift Fractio of Vehicles Lae1 (HOV) Lae2 Lae3 Lae4 Lae5 Lae 6 (Off-ramp) Lae Startig Lae: Lae 6 Observed Target Lae Lae Shift Fractio of Vehicles Lae1 (HOV) Lae2 Lae3 Lae4 Lae5 Lae 6 (Off-ramp) Lae Startig Lae: Lae 5 Observed Target Lae Lae Shift Fractio of Vehicles Lae1 (HOV) Lae2 Lae3 Lae4 Lae5 Lae 6 (Off-ramp) Lae Figure 4.18: Compariso of ed lae distributio of vehicles (cod.) 98

99 Startig Lae: O-ramp Observed Target Lae Lae Shift Fractio of Vehicles Lae1 (HOV) Lae2 Lae3 Lae4 Lae5 Lae 6 (Off-ramp) Lae Figure 4.18: Compariso of ed lae distributio of vehicles (cod.) Overall, the model with explicit target lae matched the observatios better ad the RMSE was calculated to be for the target lae model ad for the lae shift model deotig a % improveme. A closer look at the results of the lae shift model shows a sigifica proportio of the error is due to icorrect represeatio of the HOV lae. The RMSE of the perceage of vehicles movig to the HOV from all startig laes is 4.8% for the lae shift model ad 2.1 % for the target lae model. This result idicates that the lae shift lae chagig model is uable to correctly capture the attractiveess of the HOV laes ad therefore uderestimated its use. These uderestimatios of the HOV flows is a poteial source of discrepacy i the laespecific traffic speed outputs discussed ext, sice the reduced flow rates o the HOV lae results i icreased speeds o this lae. Lae-specific Speeds A separate set of lae-specific speed measuremes from sesors (ot used for calibratio) has bee used for validatio purpose. Comparisos of the goodess of fit measures are preseed i Table As see i the table, The target lae model cosistely performs better particularly i terms of Mea Error ad Mea Perce Error. The discrepacy i the lae distributio ca be a poteial source of the speed mismatch i the lae shift model sice the erroeously lower calculatios of the flows i the HOV lae result i icreased speed outputs of the HOV laes ad reduced speed outputs i the other laes. 99

100 Table 4.11: Goodess of fit statistics for the traffic speed compariso Statistic Lae Chages by Vehicles Lae Shift Model Target Lae Model Improveme RMSE, m/sec % RMSPE (%) % ME (m/sec) % MPE (%) % The umber of lae chages by vehicles as observed i the trajectory data was compared agaist the simulated results of the target lae model ad the lae shift model with the results preseed i Figure Number of Lae Chages by Vehicles Observed Target Lae Lae Shift Fractio of Vehicles Number of Lae Chages Figure 4.19: Compariso of umber of lae chages by vehicles The lae shift model uder predicted the umber of more-tha-oe lae chages probably due to the lae shift model icludig oly adjace laes i the choice set of the driver ad the higher level of service prevailig i laes further away ot beig take io accou. The target lae model performed much better tha the lae shift model particularly i terms of predictig the higher umber of lae chages. The RMSE for the fractio of vehicles i the lae shift model ad the target lae model were ad respectively idicatig a improveme of %. 100

101 Lae chages From ad To lae The umber of lae chages by From (startig) ad To (edig) laes was also compared. As see i Figure 4.20, the lae shift model has a sigificaly small umber of lae chages to the HOV Lae. The target lae model performs much better i this respect. Lae Chages From Lae Observed Target Lae Lae Shift Fractio of Vehicles Lae 1 (HOV) Lae 2 Lae 3 Lae 4 Lae 5 Lae 6 (Off-ramp) Startig Lae Lae Chages To Lae Trajectory Data Target Lae Lae Shift Fractio of Vehicles Lae 1 (HOV) Lae 2 Lae 3 Lae 4 Lae 5 Lae 6 (Off-ramp) Edig Lae Figure 4.20: Compariso of lae chages From ad To laes 4.5 Model Validatio i Other Simulators As part of the NGSIM project of the FHWA, the ew lae chagig model was also tested idepedely i three commercial simulators by the model developme teams. The results are summarized below. The detailed results are reported i Barceló et al. (2006), Speirs (2006) ad Vortisch ad Rössel (2006) 3. 3 The Target Lae model is referred as NGSIM Freeway Lae Selectio Algorithm i these reports. 101

102 AIMSUN The target lae model was implemeed i AIMSUN (AIMSUN Target Lae) ad compared agaist the default lae chagig model of AIMSUN (AIMSUN Origial). The compariso results are preseed i Table 4.12 ad Table I the tables, the first rows deote the observed flows ad speeds i the trajectory data ad the simulated flows ad speeds of the AIMSUN Target Lae ad the AIMSUN Origial models are preseed i secod ad third rows respectively. The RMSE values for the two models compared with the observed data are preseed i the last colums. Table 4.12: Compariso of flows (vph) Avg. RMSE Observed AIMSUN Target Lae AIMSUN Origial Source: Commercial Validatio of Freeway Lae Selectio Model: Report o Testig the NGSIM Lae Selectio Model with AIMSUN (Barceló et al. 2006). Table 4.13: Compariso of speeds (mph) Avg. RMSE Observed AIMSUN Target Lae AIMSUN Origial Source: Commercial Validatio of Freeway Lae Selectio Model: Report o Testig the NGSIM Lae Selectio Model with AIMSUN (Barceló et al. 2006). As see i these tables, the target lae chagig model performed better tha the base model i terms of both flow ad speed ad has bee selected to be icorporated i the commercial versio. Paramics I the validatio study i Paramics, the geeral fidig was the algorithm works uder a wide rage of scearios achievig its goal of ecouragig drivers to cosider laes other tha those strictly adjace as viable or desirable to travel i. Rigorous comparisos agaist the default Paramics lae chagig model were ot coducted. 102

103 VISSIM I the validatio study coducted withi the simulator VISSIM (Vortisch ad Rössel 2006), the modelig of exclusive laes was felt to be easier ad more straight forward i the target lae chagig model compared to the existig model i VISSIM (Figure 4.21). The RMSE for flow was reported to improve to 252 vph (VISSIM Target lae) from 288 vph (VISSIM Origial). Although the speed was slightly worse though (6.7m/sfor VISSIM Origial ad 7.0 m/s for VISSIM Target Lae). Flow o HOV Lae VISSIM Origial VISSIM Target Lae Flow (vph) Distace (km) Figure 4.21: Compariso of flow o HOV lae Source: Commercial Validatio of the NGSIM Freeway Lae Selectio Algorithm i VISSIM (Vortisch ad Rössel 2006). 4.6 Summary A lae chagig model with explicit choice of a target lae is developed to capture the effect of late plaig i the immediate maeuvers of the driver. This approach differs from existig models that assume that drivers evaluate the curre ad adjace laes ad choose a directio of chage (or ot to chage) based o the relative utilities of these laes. While the proposed model is applicable to ay geeral freeway situatio, it is most useful i cases where there exists a high differece i the level of service amog the laes, e.g. with a HOV lae. The target lae model parameters have bee estimated usig a maximum likelihood estimator ad detailed vehicle trajectory data. Compariso of goodess-of-fit test 103

104 statistics idicates sigifica improveme over the lae shift model that igores the late targets of the driver. The improveme i the model performace was demostrated through a detailed validatio study withi MITSIMLab where the simulatio capability of the target lae model is compared agaist that of the lae shift model. Test statistics calculated i the aggregate validatio stage idicates that the target lae model provides sigificaly better predictio for all measures of performace. The improvemes i the modelig capability are further stregtheed by idepede validatio withi three commercial microscopic simulators AIMSUN, Paramics ad VISSIM. I the target lae model, the choice of target lae i subseque istas is assumed to be idepede of each other. That is, the driver is assumed to re-evaluate the situatio at each time step ad if required, chage the late pla by selectig a differe target lae. This idirectly captures the evolutio of the late plas ad dyamicity of drivig behavior. The data used for estimatig the model was ot from a highly cogested situatio ad the lae chages were assumed to be through ormal gap acceptace ad the target gap were always the adjace gap. I this model, the heterogeeity i plaig capability of the drivers has bee igored ad it is assumed that all drivers are aware of the locatio of their exit from the begiig ad choose laes accordigly. This assumptio has bee relaxed i the arterial lae selectio models discussed i Chapter

105 Chapter 5 Freeway Mergig I this chapter, the late pla ivolvig the mergig decisio of drivers i a freeway o-ramp is preseed. The similar two-stage geeral decisio structure preseed i the previous chapters, that is, choice of late plas followed by selectio of actio to execute the pla, is also applicable i this sceario. However, the geometric ad traffic characteristics associated with the mergig situatio lead to a model framework that is differe from that of the freeway mailie lae selectio preseed i Chapter 4. The chapter is orgaized as follows: the backgroud of the research is preseed i Sectio 5.1. I Sectio 5.2, the structure of the late pla mergig model is detailed i three sub-sectios: the model compoes are preseed i 5.2.1, ad the descriptios of how these compoes lead to differe plas ad actios are preseed i ad respectively. The details of the model estimatio are preseed i Sectio 5.3. This sectio icludes descriptio of the data used to estimate the model parameters, the likelihood fuctio ad the estimatio results. The compariso of the goodess-of-fit of the late pla model agaist a reduced form model is also preseed i this sectio. The chapter cocludes with the validatio results withi the microscopic traffic simulator MITSIMLab ad a summary of the fidigs Backgroud Freeway mergig ivolves a complex decisio process. The target lae for freeway mergig i right had drive traffic is always the rightmost lae i the mailie. A o- 4 The model preseed i this chapter has bee developed as part of NGSIM program of the FHWA. The results preseed i this chapter have bee reported i Choudhury et al. (2006, 2007a, 2007b). The results of the simplified mergig models have bee developed by Lee (2006) ad Rao (2006). 105

106 ramp driver approachig the mailie seeks suitable gaps i the target lae for mergig. The merge is executed whe the gaps i the target lae are acceptable. However, i cogested situatios, whe acceptable gaps are ofte ot available, more complicated mergig pheomea may be observed. For example, i highly cogested situatios, due to restricted maeuverability i the logitudial directio, it may ot be possible for a driver to prepositio himself to a o-adjace gap ad he/she may decide to merge to the adjace gap through courtesy of the lag driver i the target lae or decide to force i ad compel the lag driver to slow dow. I such situatios, the chose mergig tactic dictates the pla/state of the driver, which i tur affects the driver s mergig behavior. The executio of the pla ivolves acceptace of available adjace gaps. The gap acceptace behavior models may differ depedig o the mergig tactic. For example, the acceptable gaps are smaller i case of courtesy mergig compared to ormal mergig sice there is less risk associated with it. However, the chose pla/state is uobserved ad oly the actio, that is the executio of the merge through gap acceptace, is observed. Further, the pla/state may evolve dyamically as the immediate executio of the chose mergig pla may ot be feasible. For example, a driver may begi with a pla of ormal mergig ad the chage to a pla of forced mergig as the mergig lae is comig to a ed. The probabilities of trasitios from oe pla to aother are affected by the perceptio of risk associated with the merge (aicipatio), the iertia to coiue the previously chose mergig pla (state-depedece) as well as the late characteristics of the driver like impatiece, urgecy ad aggressiveess. Existig microscopic traffic simulators, such as AIMSUN (TSS 2004), Paramics (Quadstoe 2004) ad VISSIM (PTV 2004), use basic or modified versios of their lae chagig models to model freeway mergig behavior. These models cosider gaps created by adjace vehicles, ad i some cases model reduced gap acceptace thresholds uder cogested coditios, but they do ot explicitly cosider all three mergig tactics i a sigle framework. Thus the existig models ofte fail to capture these pheomea i the mergig viciity ad represe cogestio icorrectly. The literature review (i Chapter 2) shows that several disjoi models have bee developed specifically to model the cooperative lae chagig ad forced mergig behaviors (Ahmed 1999, Hidas 2002, Wag et al. 2005), but oe of these models 106

107 iegrate the three mergig mechaisms io a sigle framework. Hidas (2005) developed a mergig model that icludes both cooperative ad forced merge compoes but the cooperative lae chage part oly cosists of modelig the decisio of the lag driver (whether or ot to provide courtesy to the mergig driver) ad ot the decisio of the mergig driver (whether or ot to iitiate or execute the courtesy lae chage based o the behavior of the lag driver). The uified decisio framework of the mergig driver is thus ot addressed i ay of these models. Therefore, these models fail to capture drivers' trasitio from ormal to cooperative or forced merge. The limitatios of the existig models ad the eed for improvig them is also reflected i the fidigs of the NGSIM study o Ideificatio ad Prioritizatio of Core Algorithm Categories, where developme of freeway mergig ad weavig model was raked fourth i importace by both model developers ad users (Alexiadis et al. 2004). 5.2 Model Structure The discussio i the previous sectio demostrates the eed to iroduce the choice of mergig tactic i the decisio framework of the driver. The mergig driver may merge through ormal gap acceptace, merge through courtesy of aother driver or decide to force i. I the case of a courtesy merge, the lag driver decelerates voluarily whereas i the case of a forced merge, the lag driver is forced to decelerate. The executio of the merge ivolves acceptace of available gaps. The pla ad the decisio process of the driver are late ad oly the ed actio of the driver (chage to the lae i the mailie) is observed. The framework of the proposed combied mergig model is summarized i Figure 5.1. Late decisios are show i ovals ad observed actios are show i rectagles. The choice of mergig tactic is hierarchical. The model hypothesizes four levels of decisio-makig: ormal gap acceptace, decisio to iitiate courtesy mergig, decisio to iitiate forced mergig ad gap acceptace for courtesy or forced mergig. The mergig driver first compares the available lead ad lag gaps i the mailie to the correspodig miimum acceptable gaps (critical gaps) for ormal gap acceptace. Critical gaps are fuctios of explaatory variables related to the subject driver ad 107

108 his/her eighborig coditios. If both the lead ad the lag gaps are greater tha the critical gaps, the merge ca be executed. Figure 5.1: Structure of the combied mergig model If the gaps are ot acceptable, the mergig driver evaluates the speed, acceleratio ad relative positio of the through vehicles ad tries to evaluate whether or ot the lag driver is providig courtesy. The courtesy or discourtesy of the lag driver is reflected i the aicipated gap. If the lag driver has decided to provide courtesy to a mergig vehicle ad has started to decelerate, the aicipated gap icreases. The aicipated gap of a particular driver also depeds o the legth of the time horizo over which it is estimated. Differeces i perceptio ad plaig abilities amog drivers are captured by the distributio of the legth of the time horizo. If the aicipated gap is acceptable, the mergig driver perceives that he/she is receivig courtesy from the lag driver ad iitiates a courtesy merge. The immediate completio of the iitiated courtesy merge however may ot be possible due to uacceptable adjace gaps. If the aicipated gap is uacceptable, the driver decides whether to force his/her way to the mailie compellig the lag driver to slow dow or ot. This decisio ca deped o the urgecy of the merge, driver characteristics (e.g. risk averseess) ad traffic coditios. Similar to courtesy merge, the immediate completio of the iitiated forced merge may ot be possible due to uacceptable adjace gaps. 108

109 If the driver does ot iitiate a courtesy or forced merge, the eire decisio process is repeated i the ext ista. However, if the driver has iitiated a courtesy or forced merge, ad is adjace to the same gap, the subseque decisios oly ivolve evaluatio of the adjace gaps for completio of the iitiated merge. After decidig to iitiate a courtesy or forced merge, the choice of mergig tactic is ot reevaluated uless there is a sigifica chage i eighborhood coditios e.g. the lead ad/or lag i the mailie chages ad the driver is adjace to a ew gap. The decisio tree of the driver thus differs depedig o the previously chose pla ad actio. The driver may be i oe of the followig three states at ay ista: Normal mergig (l t =M), Courtesy mergig (l t =C) ad Forced mergig (l t =F). If the driver has ot iitiated a courtesy or forced merge previously, the state is ormal. If the driver iitiates a courtesy merge but the adjace gaps are ot immediately acceptable for executig the merge, there is a trasitio to the courtesy mergig state. Similarly, if the driver iitiates a forced merge but the adjace gaps are ot acceptable for immediate executio of the merge, there is a trasitio to forced state. Figure 5.2: Decisio tree for ormal iitial state 109

110 If the driver is i the ormal state at a ista, the full decisio tree is i effect. That is, the decisio process starts from the top of the tree preseed i Figure 5.1 ad ormal, courtesy ad forced mergig plas are evaluated sequeially. The iermediate decisios are detailed i Figure 5.2. As show i the figure, while beig i the ormal state, the driver may perform ay of the followig: I. Chage laes ad complete the merge through ormal gap acceptace, II. Chage laes by iitiatig a courtesy merge ad immediately complete it, III/VII. Iitiate a courtesy merge but do ot complete it immediately, IV. Chage laes by iitiatig a forced merge ad immediately complete it, V/VII. Iitiate a forced merge but do ot complete it immediately, or VI. Do ot iitiate a courtesy or forced merge. A lae chage (I, II or IV) deotes the ed of the mergig process. If there are o lae chages, the decisio process coiues but there ca be a trasitio to courtesy (III) or forced (V) mergig state or the state ca remai the same (VI). Further, if a courtesy or forced merge has bee iitiated but ot completed from the ormal state ad the driver is adjace to a ew gap, the state of the driver is reset to ormal (VII). I the courtesy lae chagig state, if the driver is adjace to the same gap, the mergig pla is ot reevaluated ad the full decisio tree is ot active. Rather, the decisios oly ivolve evaluatio of the adjace gaps to complete the courtesy merge (Figure 5.3). Figure 5.3: Decisio tree for courtesy iitial state 110

111 Thus oce a trasitio is made from the ormal to courtesy mergig state, the state caot chage to forced mergig or ormal mergig uless the driver is adjace to a ew gap. If the driver is adjace to a ew gap, the state is however reset to ormal. Similar to the courtesy mergig state, i the forced mergig state, if the driver is adjace to the same gap, the eire decisio process is ot repeated. Rather, the decisios oly ivolve evaluatio of the adjace gaps to complete the forced merge (Figure 5.4). Thus, oce a trasitio is made from the ormal to forced merge state, the state caot go back to ormal ad it caot chage to the courtesy merge state uless the driver is adjace to a ew gap. The state of the driver is however reset to ormal if the driver is adjace to a ew gap. forced state Iitial State same adjace gap o chage ew adjace gap chage Forced Mergig Gap Acceptace t=t+1 forced state ormal state ed of merge Updated State Figure 5.4: Decisio tree for forced iitial state Thus whe the driver is adjace to the same gap i two subseque time istas, the followig state trasitios are possible: Normal to Normal ( l t+ 1 M lt = M Normal to Courtesy ( l t+ 1 C lt = M Normal to Forced ( l t+ 1 F lt = M Courtesy to Courtesy ( l t+ 1 C lt = C Forced to Forced ( l t+ 1 F lt = F Whe the driver is adjace to a ew gap, the followig trasitios are possible. 111

112 Normal to Normal ( l = t 1 M l = + t M ) Courtesy to Normal ( l = t 1 M l = + t C ) Forced to Normal ( l = t 1 M l = + t F ) The decisio compoes affectig the state trasitios ad subseque actios are discussed i Sectio These iclude ormal gap acceptace, decisio to iitiate a courtesy merge (aicipated gap acceptace), gap acceptace for completio of the courtesy merge, decisio to iitiate a forced merge ad gap acceptace for completio of the forced merge. Descriptios of how these compoes lead to differe plas ad actios are preseed i Sectios ad respectively Model Compoes Normal Gap Acceptace The ormal gap acceptace model idicates whether or ot a ormal merge is possible usig the existig gaps. I the model developed for this dataset, lead or lag vehicles are defied as the closest vehicles i the correspodig adjace laes withi the curre sectio of the subject vehicle (Figure 5.5). The lead gap is the clear spacig betwee the rear of the lead vehicle ad the fro of the subject vehicle. Similarly, the lag gap is the clear spacig betwee the rear of the subject vehicle ad the fro of the lag vehicle. Oe, or both, of these gaps may be egative if the vehicles overlap. Adjace gap Lag Vehicle Lead Lag gap Lead gap vehicle Legth vehicle lag lead lag lag G lead lead V, a Y G V, a Subject vehicle V, a Figure 5.5: Vehicle relatioships i a mergig situatio A available gap is acceptable if it is greater tha the critical gap. Similar to the critical gaps for gap acceptace for freeway lae chagig, the critical gaps for ormal 112

113 mergig are assumed to follow logormal distributios, the mea gap beig a fuctio of explaatory variables. This ca be expressed as follows: Mg Mg Mg Mg ( G ) G( X υ β α ) ε g { lead,lag} l =,,, + ( 5.1) Where, {, } Mg G = critical ormal gap, g lead lag X = vector of explaatory variables υ = idividual-specific radom effect : υ ~ N(0,1) Mg Mg β, α = parameters for ormal gap acceptace Mg Mg 2 ε = radom term for ormal gap acceptace: ε ~ N ( 0, σmg ) Gap acceptace ca be affected by the ieractio betwee the subject vehicle ad the lead ad lag vehicles i the adjace lae. It ca be also affected by the urgecy of the merge that ca be captured through the variable remaiig distace or time to the madatory lae chagig (MLC) poi. Cadidate variables affectig ormal gap acceptace iclude speed ad acceleratio of the subject, lead ad lag vehicles, distace remaiig to the MLC poi o the ramp, type of vehicles etc. The ormal gap acceptace model assumes that the driver must accept both the lead gap ad the lag gap to chage laes. Probability of driver i ormal state (M) makig a lae chage through ormal gap acceptace at time t ca be expressed as follows: ( =, υ ) ( =, υ ) P accept lead gap l M P accept lag gap l M t t lead M lead lag M lag ( =, υ ) ( =, υ ) lead M lead lag M lag l ( G )-G l ( G )-G =P G >G l M P G >G l M t t =Φ Φ σ σ Mlead Mlag ( 5.2) Decisio to Iitiate Courtesy Merge (Aicipated Gap Acceptace) If the adjace gaps are ot acceptable, the mergig driver evaluates the speed, acceleratio ad relative positio of the through vehicles ad aicipates the gap that will be available after τ secods. Because of the differece i perceptio amog idividuals, the aicipatio time τ may vary amog idividuals. The aicipated gap for idividual at time t is give by: 113

114 G lead lag lead lag 1 lead lag ( ) = G + G + Y + τ ( V V ) + τ 2 ( a a ) τ ( 5.3) 2 Where, as show i Figure 5.5 for idividual at time t, G Y lead lag G, G = available lead ad lag spacigs respectively lead lag V, V = lead ad lag speeds respectively lead lag a, a = lead ad lag acceleratios respectively = aicipated gap = legth of the subject vehicle The aicipated gap is thus calculated based o the assumptio that other drivers maiai their curre acceleratios. Therefore, if the lag driver of the mergig driver is deceleratig to provide courtesy, the aicipated gap is likely to icrease. If this aicipated gap is acceptable, the driver iitiates a courtesy merge. The aicipated gap is acceptable if it is larger tha the correspodig critical aicipated gap. Critical aicipated gaps are o-egative ad assumed to be log-ormally distributed 5. The mea of the distributio is a fuctio of explaatory variables. ( ) ( ) l G A = G X, υ, β A, α A + ε A ( 5.4) Where, A G = critical gap of idividual at time t for aicipated gap acceptace X A =explaatory variables ( ) υ =idividual-specific radom effect: υ ~N 0,1 ε =radom term for aicipated gap acceptace: ε ~N β A A, α =parameters for aicipated gap acceptace A 2 ( 0,σ A ) Cadidate variables affectig the decisio to iitiate a courtesy merge iclude: Status of the lag vehicle i the mailie: speed ad acceleratio of the lag vehicle, type of the lag vehicle (heavy vehicle or ot) etc. Traffic coditios: level of cogestio i the mailie etc. Probability of idividual iitiatig a courtesy merge at time t ca be expressed as follows: 5 Other o-egative distributios (trucated ormal, trucated logormal etc.) were also tested ad the log-ormal distributio had better fit tha other distributios. 114

115 P ( iitiate courtesy merge l = M, υ ) A ( ( τ ), υ ) A l ( G ( )) ( ) τ - G = P G > G l = M = Φ t σ A t ( 5.5) Decisio to Complete the Courtesy Merge After a driver iitiates a courtesy merge, the completio of the merge depeds o the acceptace of the immediate adjace gaps. A available gap is acceptable for courtesy merge if it is greater tha the correspodig critical gap (assumed to follow a logormal distributio). Cg Cg Cg Cg ( G ) G( X υ β α ) ε g { lead, lag} l =,,, + ( 5.6) Where, X Cg {, } G = critical courtesy gap, g lead lag υ = idividual-specific radom effect: υ ~ N(0,1) = vector of explaatory variables Cg Cg β, α = parameters for courtesy gap acceptace Cg Cg 2 ε = radom term for courtesy gap acceptace: ε ~ N ( 0, σcg ) Though the critical gaps for completio of courtesy merge have the same geeral fuctioal form as ormal merge, the variables ad the associated parameters ca be differe. Also, the critical gaps are assumed to be idepede of the iitial state that is the critical courtesy gap is assumed to be the same if the driver was i courtesy mergig state at the begiig of the decisio step (l t =C) or was at ormal mergig state i the begiig ad have just iitiated the courtesy merge (l t =M). I other words, it is assumed that the time elapsed after the driver has iitiated a courtesy merge does ot affect the critical gap for executio of the courtesy merge. The gap acceptace model assumes that the driver must accept both the lead gap ad the lag gap to chage laes. Probability of idividual executig a courtesy merge at time t give iitial state i ca be expressed as follows: 115

116 ( υ ) ( υ ) P accept lead gap l, P accept lag gap l, t=i t=i lead C lead lag C lag ( =, υ ) ( =, υ ) lead Clead lag Clag l ( G )- G l ( G )- G =P G >G l i P G >G l i t t =Φ Φ σclead σclag i M, C ( 5.7) Decisio to Iitiate a Forced Merge If the curre gaps are ot acceptable ad the driver perceives that a courtesy merge is also ifeasible (aicipated gap is ot acceptable), the driver evaluates whether or ot to iitiate a forced merge. By iitiatig a forced merge, the mergig driver imposes a deceleratio o the lag vehicle i the mailie. The utility of iitiatig a forced merge ca be expressed as follows: (,,,, ) F F F F U = U X υ β α ε ( 5.8) Where, U F F F β, α = parameters associated with iitiatig a forced merge F ε = radom term for iitiatig forced merge = is the utility of iitiatig a forced merge by idividual at time t Cadidate variables affectig the decisio to iitiate a forced merge iclude: Status of the mergig driver: distace to the MLC poi, delay (time elapsed sice the driver is i MLC coditio, as a proxy for impatiece), speed, type of vehicle (heavy vehicle or ot) etc. Status of the lag vehicle i the mailie: speed ad acceleratio of the lag vehicle, type of the lag vehicle (heavy vehicle or ot) etc. Traffic coditios: level of cogestio i the mailie, queue behid (mergig vehicles waitig behid the subject vehicle) etc. By assumig that the relatioship betwee the ifluecig variables are liear ad that the radom error terms ε F are idepedely ad ideically extreme value distributed, the probability of iitiatig a forced merge ca be modeled as a logit model ad ca be expressed as follows: 116

117 P 1 ( iitiate forced merge l = M, υ t ) = ( 5.9) F F 1 + exp β X α υ ( ) It may be oted that the idividual-specific term υ is assumed to have a liear effect i the utility i this case. It ca have other o-liear forms as well (e.g. ieractio with other variables i the utility). Decisio to Complete a Forced Merge After a driver decides to iitiate a forced merge, the actual merge is executed oly whe the available gaps are acceptable i compariso with the critical gaps for the forced merge. Similar to ormal ad courtesy merge the critical gap for forced merge is assumed to be log-ormally distributed, the parameters beig differe from the other types of merge. Fg Fg Fg Fg ( G ) G ( X, υ, β, α ) ε { } g lead lag l = +, ( 5.10) Where, {, } Fg G = critical forced gap, g lead lag X = vector of explaatory variables υ = idividual-specific radom effect: υ ~ N(0,1) Fg Fg β, α = parameters for forced gap acceptace Fg Fg 2 ε = radom term for forced gap acceptace: ε ~ N ( 0, σfg ) Further, similar to courtesy mergig gap acceptace, the forced mergig critical gap is assumed to be idepede of the time the driver has bee i forced mergig state ad all else beig equal the probability of forced gap acceptace is the same if the iitial state at the begiig of the time period was forced (l t =F) or ormal (l t =M). Probability of idividual executig a forced merge at time t give iitial state i ca be expressed as follows: ( l =, υ ) ( =, υ ) P accept lead gap i P accept lag gap l i t t lead F lead lag F lag ( l, υ ) ( l, υ ) = P G > G = i P G > G = i t t ( ) ( ) lead F lead lag F lag l G - G l G - G =Φ Φ σflead σflag i M, F ( 5.11) 117

118 5.2.2 Choice of Pla: Selectig the Mergig Tactic The driver first evaluates whether or ot a lae chage is possible usig the existig adjace gaps without a courtesy or forced merge. So, the iitial state ad pla is always ormal. The driver the evaluates the courtesy merge ad forced merge plas sequeially. The trasitio probabilities from oe pla/state to aother are discussed ext. Courtesy Merge If the adjace gaps are ot acceptable uder ormal gap acceptace, the mergig driver evaluates the speed, acceleratio ad relative positio of the through vehicles ad decides whether or ot to iitiate a courtesy merge (Equatio 5.5). If the driver iitiates a courtesy merge but is uable to complete it immediately, there is a trasitio from the ormal mergig to courtesy mergig state. If the adjace gap is the same, the probability of a trasitio from ormal to courtesy merge is therefore the combied probabilities of ot acceptig the ormal gap, iitiatig a courtesy merge ad ot completig the courtesy merge ad ca be expressed as follows: ( + 1 = =, υ, τ ) P l C l M t t = 1 1 lead M lead lag M lag ( > l =, υ ) ( > l =, υ ) P G G M P G G M t t ( > =, υ, τ ) P G G l M A t lead Clead lag C lag ( > l =, υ ) ( > l =, υ ) P G G M P G G M t t ( 5.12) Where, the probabilities of the compoes ca be calculated usig Equatios 5.2, 5.5 ad 5.7. Oce the driver has iitiated a courtesy merge, as log as he/she is adjace to the same gap, the probability of beig i the courtesy merge state is 1. O the other had, if the driver has already iitiated a forced merge, ad is adjace to the same gap, the probability of iitiatig a courtesy merge is 0. However, if a courtesy merge has bee iitiated, but ot completed, ad the vehicle is adjace to a ew gap (i.e. the lead ad/or lag vehicle has chaged), the state of the driver is reset to ormal. The trasitio probabilities to courtesy merge state or pla are summarized i the followig equatios: 118

119 ( + 1 = =, υ, τ ) P l C l M t t lead M lead lag M lag ( > =, υ ) ( > =, υ ) = 1 P G G lt M P G G lt M P( G > GA lt = M, υ, τ) lead C lead lag C lag 1 P ( G > G lt = M, υ ) P ( G > G lt = M, υ ) δ P( lt+ 1 = C lt = C, υ, τ) lead C lead lag C lag = 1- P( G > G lt = C, υ ) P ( G > G l t = C, υ ) δ P( lt+ 1 = C lt = F, υ, τ) = 0 Where, δ = 1if the driver is adjace to the same gap at time t ad t+1, 0 otherwise. ( 5.13) Forced Merge If the curre gaps are ot acceptable ad the driver perceives that a courtesy merge is also ifeasible (aicipated gap is ot acceptable), the driver chooses whether or ot to iitiate a forced merge (Equatio 5.9). However, if the driver iitiates a forced merge, is uable to complete it immediately, ad is adjace to the same gap, the driver remais i the forced mergig state. I case of the same adjace gap, the probability of a trasitio from ormal to forced merge is therefore the combied probability of ot acceptig the ormal gap, ot iitiatig the courtesy merge, iitiatig a forced merge ad ot completig the forced merge ad ca be expressed as follows: ( + 1 = =, υ, τ ) P l F l M t t = 1 lead M lead lag M lag ( > l =, υ ) ( > l =, υ ) 1 1 P( G > GA lt = M, υ, τ) F F 1+ exp( β X α υ ) 1 P G G M P G G M t t lead F lead lag F lag ( > l =, υ ) ( > l =, υ ) P G G M P G G M t t ( 5.14) Where, the probabilities of the compoes ca be calculated usig Equatios 5.2, 5.5, 5.9 ad Similar to the courtesy merge, the probability that a driver is i the force merge state depeds o the previous state: the probability is 1 if the driver had already iitiated a forced merge to the same gap ad 0 if the driver had already iitiated a courtesy merge to 119

120 the same gap. However, if the driver caot complete a forced merge that has bee iitiated while he/she is adjace to the same gap, the state is reset to the ormal state. The probability of beig i forced mergig state ca therefore be expressed as follows: ( + 1 = =, υ, τ ) P l F l M t t lead M lead lag M lag ( > =, υ ) ( > =, υ ) = 1 l l P G G M P G G M t t 1 1 P( G > GA lt = M, υ, τ) F F 1+ exp( β X α υ ) lead F lead lag F lag ( > l =, υ ) ( > l =, υ ) 1 P ( lt+ 1 = F lt = C, υ, τ) = 0 P G G M P G G M δ t t ( + 1 = =, υ, τ ) P l F l F t t lead F lead lag F lag ( > l =, υ ) ( > l =, υ ) = 1- P G G t F P G G t F δ ( 5.15) Normal Merge If the driver does ot iitiate a courtesy or forced merge, the state remais ormal. The probability of a trasitio from the ormal to ormal state is therefore the combied probability of ot acceptig the ormal gap, ot iitiatig courtesy ad ot iitiatig a forced merge ad ca be expressed as follows: ( + 1 = =, υ, τ ) P l M l M t t = 1 lead M lead lag M lag ( > l =, υ ) ( > l =, υ ) P G G M P G G M t t 1 1 P( G > GA lt = M, υ, τ) 1 1 exp F F + ( β X α υ ) Where, the probabilities of the compoes ca be calculated usig Equatios 5.2, 5.5 ad 5.9. Further, wheever the driver is adjace to a ew gap, the state is reset to ormal. The probability of beig i ormal mergig state ca therefore be expressed as follows: ( 5.16) 120

121 ( + 1 = =, υ, τ ) P l M l M t t lead M lead lag M lag ( > =, υ ) ( > =, υ ) = 1 P G G l M P G G l M t t 1 1 P( G > GA lt = M, υ, τ) 1 δ + (1 δ) F F 1+ exp( β X α υ ) ( + 1 υ τ ) ( + 1 υ τ ) P l = M l = C,, = 1 δ t t P l = M l = F,, = 1 δ t t ( 5.17) Choice of Actio: Executio of the Merge The iitial state of the driver ca be ormal, courtesy or forced. The observed actio ivolves executio of the pla. The decisio tree of the driver ad the critical gaps vary with the iitial state. Normal State If the driver is i the ormal state at a ista, the driver ca execute a lae chage i three ways (Figure 5.2): 1. Chage laes through ormal gap acceptace, 2. Chage laes by iitiatig a courtesy merge ad immediately completig it, 3. Chage laes by iitiatig a forced merge ad immediately completig it. The probability of observig a lae chage coditioal o the iitial state beig ormal ca thus have the followig three compoes: Chage laes through ormal gap acceptace: A lae chage through ormal gap acceptace is possible if both lead ad lag gaps are acceptable for a ormal merge ad ca be expressed as follows: ( lead M lead l, υ ) ( lag M lag l, υ ) Compoe 1 P G G M P G G M = > = > = ( 5.18) t t These probabilities ca be calculated usig Equatios 5.1 ad 5.2. Chage laes by iitiatig a courtesy merge ad immediately completig it: These lae chages occur whe the adjace gap is ot acceptable for ormal merge but the driver perceives that the lag driver i the mailie is providig courtesy to him, iitiate a courtesy merge ad complete the courtesy merge i the same time step. The probability of such a lae chage is therefore the combied probability of ot acceptig 121

122 the ormal gap, iitiatig courtesy ad acceptig the adjace gap for courtesy gap acceptace, all at the same time step. This ca be expressed as follows: lead Mlead lag M lag ( > =, υ ) ( > =, υ ) Compoe 2= 1 P G G lt M P G G lt M A lead Clead lag Clag ( > =, υ, τ ) P ( G > G = M, υ ) P ( G > G = M, υ ) P G G l M l l t t t These probabilities ca be calculated usig Equatios 5.2, 5.5 ad 5.7. ( 5.19) Chage laes by iitiatig a forced merge ad immediately completig it: This category of lae chage occurs whe the adjace gap is ot acceptable for ormal merge, ad the aicipated gap is ot acceptable for iitiatig courtesy but the driver decides to iitiate a forced merge ad the adjace gap is acceptable for immediate executio of the forced merge. The probability of this type of lae chage thus refers to the joi probability of ot acceptig the ormal gaps, ot acceptig the aicipated gap, decidig to iitiate a forced merge ad acceptig the lead ad lag gaps through forced gap acceptace. This ca be expressed as follows: Compoe 3 = lead Mlead lag lag ( > l =, υ ) ( > l =, υ ) 1 P G G M P G G M t M t 1 A 1 P( G > G lt = M, υ, τ) F F 1+ exp( β X α υ ) lead Flead lag Flag ( > l =, υ ) ( > l =, υ ) P G G M P G G M t t ( 5.20) This ca be calculated usig Equatios 5.2, 5.5, 5.9 ad Probability of makig a lae chage give the iitial state is ormal is the sum of the above meioed compoes. ( = 1 =, υ, τ ) P j l M t t = Compoe 1 + Compoe 2+ Compoe 3 ( 5.21) Where, the three compoes are give by Equatios 5.18, 5.19 ad 5.20 respectively. Probability of o lae chage coditioal that the iitial state is ormal merge ca be expressed as follows: ( = 0 l =, υ ) = 1 ( = 1 l =, υ ) ( 5.22) P j M P j M t t t t 122

123 Courtesy Mergig State I the courtesy merge state, if the driver is adjace to the same gap, the eire decisio process is ot repeated. Rather, the decisios oly ivolve evaluatio of the adjace gaps to complete the courtesy merge (Figure 5.3). Probability of a lae chage coditioal o the iitial state is courtesy merge ca therefore be expressed as follows: ( = 1 l =, υ ) P j C t t lead Clead lag Clag ( l, υ ) ( l, υ ) = P G > G = C P G > G = C t t ( 5.23) This ca be calculated usig Equatio 5.7. Probability of o lae chage coditioal o the iitial state is courtesy merge ca be expressed as follows: ( = 0 l =, υ ) = 1 ( = 1 l =, υ ) ( 5.24) P j C P j C t t t t Forced Mergig State Similar to the courtesy merge state, i the forced merge state, if the driver is adjace to the same gap, the eire decisio process is ot repeated. Rather, the decisios oly ivolve evaluatio of the adjace gaps to complete the forced merge (Figure 5.4). Probability of a lae chage coditioal o the iitial state is forced merge ca therefore be expressed as follows: ( = 1 l =, υ ) P j F t t lead Flead lag Flag ( l, υ ) ( l, υ ) = P G > G = F P G > G = F t t ( 5.25) This ca be calculated usig Equatio Probability of o lae chage coditioal o the iitial state is forced merge ca be expressed as follows: ( = 0 l =, υ ) = 1 ( = 1 l =, υ ) ( 5.26) P j F P j F t t t t Depedig o the chose pla ad decisio state, the lae actio ca thus have differe probabilities. 123

124 5.3 Model Estimatio Data Study Area The data used i the estimatio of the drivig behavior model represes travel o a meters orthboud sectio of Ierstate 80 (I-80) i Emeryville, Califoria (Figure 5.6). Figure 5.6: Estimatio data collectio site The data was collected ad processed as part of the FHWA s NGSIM program. The data was collected usig video cameras moued o a 30-story buildig adjace to I-80. The Uiversity of Califoria at Berkeley maiais traffic surveillace capabilities at the buildig ad the segme is kow as the Berkeley Highway Laboratory (BHL) site ft = m EB I ft = 3.6m ft = 3.6m shoulder 24ft = 7.3m 7 Powell St. O-Ramp Study Area of Trajectory Data 8 Ashby Off-Ramp Figure 5.7: Schematic of the estimatio data collectio site (ot i scale) 124

125 Complete vehicle trajectories were recorded at a resolutio of 10 frames per secod. 45 miutes of data were collected o April 13, 2005 at a resolutio of 0.1 secod durig the time iervals 4:00 to 4:15 p.m., 5:00 to 5:15 p.m., ad 5:15 to 5:30 p.m. The 4:00 to 4:15 p.m. period is represeative of a trasitioal traffic period i the build up to cogested coditios, ad the 5:00 to 5:30 p.m. period is represeative of cogested coditios. For data hadlig tractability, the combied dataset was sampled at the rate of 1 i 10 observatios, meaig the locatios of vehicles were kow at oe-secod time steps. The resultig dataset had 540 mergig vehicles with 17,352 observatios. Characteristics of the Estimatio Dataset As show i the schematic represeatio of the study area i Figure 5.7, there are o physical lae markigs separatig the o-ramp vehicles from the mailie vehicles. The absece of a physical lae demarcatio over a log stretch made it difficult to specify whe a lae chage has occurred, ad ecessitated the defiitio of a imagiary lae boudary. The madatory lae chagig (MLC) poi, as show i Figure 5.8, is defied as the poi where the width of the rightmost lae assumes the sigle lae width (3.6 meter). The defiitio of this poi is importa as it defies whether or ot a merge has occurred. A merge is classified as completed whe the ceer poi of the vehicle has crossed this imagiary lie/lae-mark (X i Figure 5.8). Figure 5.8: Defiitio of merge poi 125

126 The vehicle trajectory data coaiig the coordiates of the mergig ad mailie vehicles i the sectio were used to derive the required variables for estimatio of speed, acceleratio, average desity, etc. Speeds i the mergig sectio (the o-ramp ad part of lae 6 as defied i Figure 5.8) vary from 0 m/sec to a maximum of 20.7 m/sec with a mea of 4.2 m/sec. There are may stop-ad-go situatios prese i the dataset. Desities calculated 150 meters dowstream of the mergig vehicles i lae 6 rage from 0 veh/km/lae to veh/km/lae with a average of 61.9 veh/km/lae. 1.4 perce of the mergig vehicles i the dataset are heavy vehicles (trucks i this case). The distributios of speed, acceleratio, ad desity i lae 6 ad distace to the MLC poi i the eire dataset are show i Figure 5.9. Figure 5.9: Distributios of speed, acceleratio, desity, ad distace to MLC poi As defied earlier i this chapter, the lead gap is the distace betwee the fro of the subject vehicle to the rear of the lead vehicle i the target lae, ad the lag gap is the distace betwee the rear of the subject vehicle ad the fro of the lag vehicle i the 126

127 target lae (Figure 5.2). Negative gaps imply overlap betwee the subject ad the lead/lag vehicle. The statistics relatig to the subject vehicle are show i Table 5.1. Table 5.1: Statistics of variables related to the subject vehicle Variable Mea Std Dev Media Miimum Maximum Speed (m/sec) Average Desity d/s (veh/km/lae) Distace to MLC (km) Acceleratio (m/sec 2 ) Deceleratio (m/sec 2 ) Table 5.2 preses the descriptive statistics for the lead ad lag vehicle relative to the subject vehicle. Relative speeds are defied as the speed of the lead (lag) vehicle less the speed of the subject vehicle. The table summarizes statistics of the lead ad lag gaps (i.e. the gaps vehicle chaged laes io) both for the accepted gaps ad for the eire dataset (both accepted ad rejected gaps). Accepted lead gaps vary from 0.13 meters to meters, with a mea of 9.92 meters. Accepted lag gaps vary from 0.48 meters to meters. Statistics for the eire dataset are preseed i pareheses. Table 5.2: Statistics for the lead ad lag vehicles of mergig vehicles Variable Mea Std Dev Media Miimum Maximum Lead Relative Speed (m/sec) 0.24 (-0.29) 1.26 (2.15) 0.24 (0.01) (-16.80) 5.60 (8.13) Lead Gap (m) 9.92 (4.83) 9.01 (8.83) 7.57 (2.94) 0.13 (-19.43) (160.6) Lag Relative Speed (m/sec) (-0.41) 1.56 (2.15) (-0.15) (-14.25) 5.38 (18.09) Lag Gap (m) (5.25) (8.85) 8.43 (3.39) 0.48 (-19.9) (178.25) As expected, the mea accepted gaps are larger tha the mea gaps i the traffic stream for both the lead ad lag gaps. Similarly, mea lead relative speeds i the accepted gaps are higher tha those i the eire dataset ad mea lag relative speeds i the accepted gaps are lower tha those i the eire dataset. This implies that whe a gap is accepted, the subject vehicle is travelig slower tha the lead vehicle ad faster tha 127

128 the lag vehicle. The distributios of the speeds ad spacig with respect to the lead ad lag vehicles for the eire dataset are show i Figures 5.10 ad 5.11, respectively, ad those for the accepted gaps are show i Figures 5.12 ad 5.13, respectively. Figure 5.10: Distributios of lead relative speed ad spacig i the full dataset Figure 5.11: Distributios of lag relative speed ad spacig i the full dataset Figure 5.12: Distributios of lead relative speed ad spacig for the accepted gaps 128

129 Figure 5.13: Distributios of lag relative speed ad spacig for the accepted gaps From the dataset, it was observed that more tha 80 perce of the merges occur whe the distace to the madatory lae chagig poi, as defied by the imagiary lae boudary, is less tha 100 m. Figure 5.14 shows the distributio of the umber of merges with distace to the madatory lae chagig poi i the sectio. Number of Merges Remaiig Distace (km) Figure 5.14: Distributio of umber of merges with distace to MLC poi 129

130 5.3.2 Likelihood of the Trajectory All model parameters were estimated joily usig a maximum likelihood techique. The likelihood fuctio that was maximized is preseed i this sectio. At ay time t, a idividual ca be i oe of the followig states: Courtesy mergig ( l t Forced mergig ( l t = F = C), ), or Normal lae chagig ( l t = M ). The lae chagig decisios of the driver depeds o the state. The state of the driver at ay ista depeds o his/her previous state(s). Accordig to the first-order Markov assumptio: The state at a give time period t depeds oly o the state at time (t-1) ad actio of all previous time periods (1: t-1). The lae actio at a give time period t depeds oly o the state at time period t. Further, i the mergig data, the observatio of a driver eds whe he makes a lae chage. That is, there is always a sequece of o chages followed by a lae chage i the last time step. Therefore, the fact that the driver is i state l t at time t coditioal that the previous state was l t-1 idicates the followig: The lae actio i the previous state l t-1 was o chage ( j t-1 = 0 ) ad There has bee a trasitio to state l t from state l t-1 at (t-1) th time step, where, l, l M, C, F. t t-1 The probability of beig i state l t is therefore the product of the probability of beig i state l t-1 at time (t-1), ad the joi probability of o chage at the previous time period ( j t-1 = 0 ) ad probability of a trasitio from state l t-1 to state l t at time (t-1). The lae actios at time t ( j t ) are coditioal o the state at time t ( l t ). As discussed i Sectio 5.2.3, may decisio state sequeces ca lead to the same state at time t. At time t for driver, the probability of observig a particular lae actio j is the sum of 130

131 probabilities that he/she is observed to execute lae actio j give that the selected mergig pla is l, over all sequece of plas that could have led to pla l t. P ( j j, υ, τ ) = P ( j l, υ ) P( l, j l, j, υ, τ ) = t 1: t-1 t t t t 1 t 1 1: t-2 ( l1,, lt ) ( l1,, lt ) P ( j l, υ ) P ( l l, j, υ, τ ) P( j l, υ ) t t t t 1 1: t-1 t 1 t 1 ( 5.27) The probability of observig the eire trajectory of driver ca be calculated recursively ad is give by the followig equatio: ( 1, K, T υτ, ) P j j = = = ( l1, K, l ) T ( T T ) ( ) ( ) ( ) P j l, υ P l, j l, j, υ, τ LP l, j l, j, υ, τ P l, j l, υ, τ T T 1 T 1 1: T P (, jt l ) (, 1 1, 1: 2,, ) ( 3, 2 2, 1,, ) ( 2, 1 1,, T υ P lt j ) T lt j T υ τ L P l j l j υ τ P l j l υ τ lt l T 1 l2 lt ( T, T υ ) P j l P l2 lt 1 ( l l 1, 1: 1,, ) ( 1 1, ) j υ τ P j l υ ( 3 2, 2, υ, τ ) ( 2 2, υ) ( 2 1, 1, υ, τ ) ( 1 1, υ) l = M; l M, C, F ; j = 1; j = 0 1 1:t T 1:T 1 T T T T T L P l l j P j l P l l j P j l ( 5.28) Where, the state trasitio probabilities are give by Equatios 5.13, 5.15 ad 5.17 ad the lae actio probabilities are give by Equatios 5.21 through The ucoditioal idividual likelihood is give by: L υ τ υ τ υ τ ( 5.29) = P ( j,, j 1 T, ) f( ) f( ) d d υ τ Where, f ( υ) = stadard ormal probability desity fuctio f ( τ ) = probability desity fuctio of a doubly trucated ormal distributio with mea µ ad variace σ τ 2 τ Assumig that the observatios from differe drivers are idepede, the loglikelihood fuctio for all N idividuals observed is give by: N L = l( L ) ( 5.30) = 1 The maximum likelihood estimates of the model parameters are foud by maximizig this fuctio. 131

132 5.3.3 Estimatio Results All model parameters: the parameters of the gap acceptace models, the pla/state trasitio models ad the age effect are estimated simultaeously with detailed vehicle trajectory data usig maximum likelihood estimatio techique as described i the previous sectio. However, i order to simplify the preseatio, estimatio results for the various compoes of the model are preseed ad discussed separately. The preseatio order follows the hierarchy of the hypothesized decisio-makig process: the ormal gap acceptace model is preseed first, followed by the iitiatio ad executio of courtesy merge models, ad iitiatio ad executio of forced merge models. The summary of estimatio results is preseed i Table 5.3. Table 5.4 preses the parameter estimates of the ormal merge model, Tables 5.5 ad 5.6 prese the results of the courtesy merge model ad Tables 5.7 ad 5.8 prese the results of the forced merge model. Table 5.3: Estimatio results of the mergig model Fial log-likelihood Iitial log-likelihood Number of cases 540 Number of observatios Number of parameters 42 Adjusted rho-bar square 0.88 The state-depede mergig model is compared with a reduced form model with o late mechaism (Lee 2006). The istaaeous model aims at capturig the ormal, forced ad courtesy behavior of drivers through a sigle gap acceptace level by icludig variables releva to all three types of merges i a sigle critical gap fuctio. The model structure is show i Figure The model is estimated with the same trajectory data. The late pla model is a extesio of the sigle level model. The summary statistics of the estimatio results for the two models, preseed i Table 5.4, show a improveme i the fit of the model, eve whe accouig for the larger umber of parameters i the late model. 132

133 Figure 5.15: Framework of the sigle level mergig model (Lee 2006) Statistic Table 5.4: Model compariso Sigle Level (R) Combied Mergig (U) Likelihood value Number of parameters (k) Akaike iformatio criteria (AIC) Adjusted rho-bar square ( ρ ) The model with explicit target lae choice has larger values i terms of both AIC ad ρ 2 (detailed i Chapter 4). This idicates that the iclusio of the late plas i the decisio framework results i a improved goodess-of-fit eve after discouig for the icrease i the umber of parameters. The detailed estimatio results of the model compoes are preseed below: Executio of Normal Merge I the hypothesized decisio makig process, the driver first evaluates the adjace lead ad lag gaps to decide whether or ot to merge through ormal gap acceptace. I order for the gap to be acceptable both the lead ad lag gaps, must be acceptable. The critical lead ad lag gaps are fuctios of the relative speeds ad acceleratios of the adjace vehicles ad the remaiig distace to the madatory lae chagig poi. The estimated coefficies are preseed i Table

134 Table 5.5: Estimatio results of the ormal gap mergig model Normal Lead Gap Variable Parameter t-stat Normal lead costa * Relative average speed (positive) (m/sec) * Relative lead speed (egative) (m/sec) * Remaiig distace fuctio Distace to MLC poi (10 m) Costa Heterogeeity coefficie, RemDistLead α Stadard deviatio for ormal lead gap, σ M lead M lead Heterogeeity coefficie for ormal lead gap, α Normal Lag Gap Normal lag costa * Relative lag speed (positive) (m/sec) * Relative lag speed (egative) (m/sec) * Remaiig distace fuctio Distace to MLC poi (10 m) Costa Heterogeeity coefficie, RemDistLag α * Lag acceleratio (positive) (m/sec 2 ) Stadard deviatio for ormal lag gap, σ M lag M lag Heterogeeity coefficie for ormal lag gap, α * same coefficies i ormal, courtesy ad forced gap acceptace levels The lead critical gap is a fuctio of the average speed i the mailie relative to the subject vehicle s speed, the relative speed of the lead with respect to the subject ad the remaiig distace to the madatory lae chagig poi ad ca be expressed as follows: G Mlead lead ( ) ' V 0.505Mi 0, V + d = exp 1 + exp( υ) Mlead 0.819υ+ ε (5.30) 134

135 Where, G Mlead ' lead = critical lead gap for the ormal gap acceptace level (m) V =relative average speed factor (m/sec) V =relative speed of the lead vehicle with respect to the subject (m/sec) d =remaiig distace to the madatory lae chagig poi (10 m) υ = idividual-specific radom effect Mlead ε Mlead = radom error term associated with ormal lead gap : ε ~ N 0, ( ) The lag critical gap is a fuctio of the subject vehicle speed relative to the lag vehicle, the remaiig distace to the madatory lae chagig poi ad the acceleratio of the lag vehicle. This ca be expressed as follows: G lag lag ( ) ( ) d Max( a ) Max 0, V Mi 0, V = exp , υ + ε 1+ exp( υ) Mlag lag M lag Where, Mlag G = critical lag gap for the ormal gap acceptace level (m) lag V =relative speed of the lag vehicle with respect to the subject (m/sec) d =remaiig distace to the madatory lae chagig poi (10 m) lag 2 a =acceleratio of the lag vehicle (m/sec ) υ = idividual-specific radom effect Mlag ε Mlag = radom error term associated with ormal lead gap: ε ~ N 0, ( ) (5.31) The lead critical gap icreases with the average speed of the mailie. As the mailie average speed icreases, the driver eeds larger critical gaps to adjust the speed to the speed of the maistream. However, critical gap does ot icrease liearly with icreasig average speeds i the mailie (Figure 5.16), rather it icreases as a dimiishig fuctio β avg ' V, where, V =, avg 1+ exp( Max( 0, V )) ' 1 relative speed betwee the average mailie ad the subject vehicle (m/sec). avg V beig the 135

136 2 Media Lead Critical Gap (m) Relative Average Speed (m/sec) Figure 5.16: Lead critical gap as a fuctio of relative average speed i the mailie The lead critical gap is larger whe the lead vehicle is movig slower tha the subject sice the driver perceives a icreased risk whe the lead is slowig dow ad he/she is gettig closer to the lead vehicle (Figure 5.17). 10 Media Lead Critical Gap (m) Relative Lead Speed (m/sec) Figure 5.17: Lead critical gap as a fuctio of relative lead speed The lag critical gap icreases with the relative lag speed: the faster the lag vehicle is relative to the subject, the larger the critical gap (Figure 5.18). The lag critical gap icreases as the acceleratio of the lag vehicle icreases (Figure 5.19), due to the higher perceived risk of mergig io the maistream whe the lag vehicle is acceleratig. 136

137 10 Media Lag Critical Gap (m) Relative Lag Speed (m/sec) Figure 5.18: Lag critical gap as a fuctio of relative lag speed Media Lag Critical Gap (m) Acceleratio of Lag Vehicle (m/sec 2 ) Figure 5.19: Lag critical gap as a fuctio of lag vehicle acceleratio Both the lead ad lag critical gaps decrease as the distace remaiig to the madatory lae chagig poi decreases. This is because as the driver approaches the poi where the ramp eds, the urgecy to make the merge icreases ad he/she is willig to accept lower gaps to merge. To capture drivers heterogeeity, a idividual-specific radom term has bee iroduced i the coefficie of the remaiig distace. Aggressive ad timid drivers ca thus have differe critical gaps, the remaiig distace beig equal. The aggressiveess/timidity of the driver captures the heterogeeity amog the driver populatio ad is assumed to have a coiuous distributio (trucated ormal i this case) rather tha discrete havig a discrete class membership. All other variables havig o effect, the lead ad lag critical gaps as a fuctio of remaiig distace for aggressive drivers are much smaller tha the gaps for timid drivers. Thus, aggressive drivers ca fid lead ad lag gaps to be acceptable eve whe they are far from the MLC 137

138 poi. O the other had, timid drivers have large critical gaps till they reach the ed of the ramp. The sesitivity of the lead ad lag critical gaps as a fuctio of the remaiig distace accordig to the idividual characteristics of the driver is show i Figure 5.20 ad Figure 5.21 respectively. As see i Figure 5.20, the timid drivers have a uusually large critical lead gap till they are closer to the MLC poi, implyig that they do ot cosider lae chages at the begiig of the o-ramp. It may be oted that the sig of the uobserved driver characteristics is cosiste for both gaps as well as other choice dimesios. The t-statistics for the liear part of the coefficie of remaiig distace is foud to be very sigifica both for lead ad lag gaps. Media Lead Critical Gap (m) 30 Aggressive Driver Timid Driver Remaiig Distace to MLC (m) Figure 5.20: Lead critical gap as a fuctio of remaiig distace to MLC poi Media Lag Critical Gap (m) 12 Aggressive Driver Timid Driver Remaiig Distace to MLC (m) Figure 5.21: Lag critical gap as a fuctio of remaiig distace to MLC poi 138

139 Estimated coefficies of the uobserved driver characteristics ( υ ) are egative for both the lead ad lag critical gaps. This implies that a aggressive driver requires smaller gaps for lae chagig compared with a timid driver. Iitiatio ad Executio of Courtesy Merge If the available lead ad lag gaps are ot acceptable for ormal merge, the mergig driver evaluates the speed, acceleratio ad relative positio of the through vehicles ad tries to evaluate whether or ot the lag driver is providig courtesy to him/her. The courtesy or discourtesy of the lag driver is reflected i the aicipated gap which is defied as the total gap after time τ (aicipatio time): G lead lag lead lag 1 2 lead lag ( τ ) = G + G + Y + τ ( V V ) + τ ( a a ) (5.32) 2 Where, for idividual at time t, G = aicipated gap, m Y = legth of the subject vehicle, m lead lag G, G = available lead ad lag spacig respectively, m lead lag V, V = lead ad lag speeds respectively, m/sec lead lag 2 a, a = lead ad lag acceleratios respectively, m/sec The aicipated gap is compared agaist the critical aicipated gap ad if deemed acceptable, the mergig driver perceives that he/she is receivig courtesy from the lag driver ad iitiates a courtesy merge. The aicipated gap is acceptable if it is larger tha the correspodig critical aicipated gap. Critical gaps are assumed to be log-ormally distributed (a better fit tha other o-egative distributios). The mea of the distributio is a fuctio of explaatory variables: the relative lag speed, remaiig distace, ad desity of the traffic stream. The estimated parameters are preseed i Table

140 Table 5.6: Estimatio results of the iitiate courtesy model Iitiate Courtesy Merge Variable Parameter t-stat Aicipated gap costa Relative average speed (positive) (m/sec) Relative lead speed (m/sec) Remaiig distace fuctio Distace to MLC poi (10 m) Costa Heterogeeity coefficie, RemDistA α Stadard deviatio for aicipated gap ( σ ) A A Heterogeeity coefficie for aicipated gap ( α ) Mea of aicipatio time ( µ ) τ Stadard deviatio of aicipatio time ( σ ) τ The estimated fuctioal form of the critical aicipated gap is give by: G A lag Max( 0, V ) 0.153ρ + d = exp 1+ exp( υ) A 0.231υ + ε Where, A G = critical aicipated gap for the iitiatig courtesy merge (m); lag V =relative speed of the lag vehicle with respect to the subject (m/sec); d ρ υ A = remaiig distace to the madatory lae chagig poi (10 m); = desity i the rightmost lae of the mailie (veh/10 m); ad =uobserved driver characteristics. A 2 ε ~ N 0, ε = radom error terms ( ) (5.33) Similar to ormal critical gaps, the critical aicipated gap is higher at higher lag speeds. It decreases as the remaiig distace decreases ad it is smaller for aggressive drivers tha timid drivers. Courtesy yieldig/mergig more commoly occurs i dese traffic coditios ad hece the probability of mergig through courtesy icreases with the desity of mailie traffic. The critical aicipated gap therefore decreases with 140

141 desity of traffic i the rightmost mailie lae. Media critical aicipated gap as a fuctio of desity is preseed i Figure Media Critical Aicipated Gap (m) Desity (veh/km ) Figure 5.22: Media critical aicipated gap as a fuctio of desity i target lae O iitiatig a courtesy merge, the driver decides whether to complete the merge by acceptig or ot the available gap based o the respective lead ad lag critical gaps. For ideificatio purposes, except for the costa ad the uobserved driver characteristics, the coefficies of variables i these levels are restricted to be the same as for the ormal gap acceptace level (Table 5.7). The estimated fuctioal form of the lead ad lag critical gaps for courtesy ca be expressed by the followig equatios: G ε Clead Clead G ε 2 ( ) lead ( ) + + = exp Clead 0.054υ + ε ~ N 0, ' V 0.505Mi 0, V d 1+ exp( υ) 2 ( ) lag lag ( ) ( ) Max 0, V Mi 0, V = exp d Max( 0, a ) 0.554υ + ε 1+ exp( υ ) Clag lag C lag Clag Where, ~ N 0,0.554 Clead C lag G, G =lead ad lag critical gaps for the courtesy gap acceptace level respectively ε Clead, ε Clag = radom error terms (5.34) (5.35) 141

142 Table 5.7: Estimatio results of the courtesy gap acceptace model Courtesy Lead Gap Variable Parameter t-stat Courtesy lead costa *Relative average speed (positive) (m/sec) *Relative lead speed (egative) (m/sec) *Remaiig distace fuctio Distace to MLC poi (10 m) Costa Heterogeeity coefficie, RemDistLead α Stadard deviatio for courtesy lead gap, σ Clead Heterogeeity term for courtesy lead gap, Clead α Courtesy Lag Gap Courtesy lag costa *Relative lag speed (positive) (m/sec) *Relative lag speed (egative) (m/sec) *Remaiig distace fuctio Distace to MLC poi (10 m) Costa Heterogeeity coefficie, RemDistLag α *Lag acceleratio (positive) (m/sec 2 ) Stadard deviatio for courtesy lag gap, σ Heterogeeity term for courtesy lag gap, Clag Clag α * same coefficies i ormal, courtesy ad forced gap acceptace levels The estimatio results show that all other thigs held costa, a driver is more willig to accept smaller lead ad lag gaps whe he/she is i the courtesy mergig state tha i ormal or forced mergig states. This is iuitive sice i case of courtesy mergig, the lag vehicle is slowig dow ad therefore, a smaller buffer space is sufficie. 142

143 Iitiatio ad Executio of Forced Merge If the driver perceives that a ormal lae chage is ot possible ad there is o courtesy yieldig of the lag driver (aicipated gap is ot acceptable), the driver chooses whether or ot to iitiate a forced merge. As described i Sectio 5.2.1, this is modeled as a biary logit model. Table 5.8: Estimatio results of the iitiate forced merge model Iitiate Forced Merge Variable Parameter t-stat Iitiate force costa Heavy lag vehicle dummy F Heterogeeity term for iitiated forced merge( α ) The decisio to iitiate a forced merge was foud to be depede o the aggressiveess of the driver ad whether the lag vehicle i the mailie is a heavy vehicle or ot. I particular, the coefficie of aggressiveess has a sigifica impact o the decisio to iitiate a forced merge. If the lag is a heavy vehicle, the probability of iitiatig a forced merge decreases, as the driver perceives a higher risk i udertakig such a maeuver. The variable remaiig distace (urgecy of the merge) ad delay (impatiece) of the driver were assumed to impact forced merge, but the estimated coefficies of these two variables did ot have the expected sigs. This may be due to the fact that i the estimatio dataset, may of the forced merges actually occurred i the begiig of the sectio as opposed to the ed. The probability of iitiatig a forced merge is give by the followig equatio: P F Where, 1 = (5.36) 1 + exp hv ( δ υ ) hv δ = heavy lag vehicle dummy, 1 if the lag vehicle is a heavy vehicle, 0 otherwise Similar to courtesy mergig, o iitiatig a forced merge, the driver decides whether to complete the merge by acceptig the available gap or ot based o the respective lead ad lag critical gaps. For ideificatio purposes, except for the costa ad the 143

144 uobserved driver characteristics, the coefficies of variables i these levels are restricted to be the same as for the ormal gap acceptace level (Table 5.9). Table 5.9: Estimatio results of the forced mergig model Forced Lead Gap Variable Parameter t-stat Forced lead costa *Relative average speed (positive) (m/sec) *Relative lead speed (m/sec) *Remaiig distace fuctio Distace to MLC poi (10 m) Costa Heterogeeity coefficie, RemDistLead α Stadard deviatio for forced lead gap, σ Flead Flead Heterogeeity term for forced lead gap, α Forced Lag Gap Forced lag costa *Relative lag speed (positive) (m/sec) *Relative lag speed (egative) (m/sec) *Remaiig distace fuctio Distace to MLC poi (10 m) Costa Heterogeeity coefficie, RemDistLag α *Lag acceleratio (positive) (m/sec 2 ) Stadard deviatio for forced lag gap, σ Flag Heterogeeity term for forced lag gap, Flag α * same coefficies i ormal, courtesy ad forced gap acceptace levels The estimated fuctioal form of the lead ad lag critical gaps for courtesy ca be expressed by the followig equatios: G ε Flead Flead ' lead V 0.505Mi( 0, V ) + d exp 1 exp( υ ) = + + Flead υ ε + 2 ( ) ~ N 0,7.95 (5.37) 144

145 G ε 2 ( ) lag lag ( ) ( ) Max 0, V Mi 0, V = exp d Max( 0, a ) υ + ε 1+ exp( υ) Flag lag F lag Flag Where, ~ N 0,0.465 (5.38) Flead Flag G, G = lead ad lag critical gaps for the forced gap acceptace level respectively ε Flead ad ε Flag = radom error terms The costa term for the lag critical gap for forced mergig is smaller tha for the ormal ad courtesy merges. However, the lead critical gap for the forced mergig case is foud to be larger tha the case of the ormal merge. This reflects the fact that oce the driver has iitiated a forced merge (pushed the fro bumper establishig the right of way), the lead gap plays a domia role i the completio of the merge. Oce iitiated, the forced merge is completed oly whe the lead gap is sufficiely large sice the maeuver ivolves sigificaly higher risk tha for ormal gap acceptace. Distributio of Aicipatio Time The aicipatio time is assumed to follow a doubly trucated ormal distributio. Estimatio results idicated that it is ormally distributed withi 0 to 4 sec. 6 The estimated distributio of aicipatio time is 1 τ 1.87 φ if 0 τ 4 f ( τ ) = otherwise (5.39) 6 Differe values betwee 0 to 6 sec were tested as the upper limit of aicipatio time ad the selected value (4 sec) provided the best goodess-of-fit. 145

146 5.4 Model Validatio Figure 5.23: Distributio of aicipatio time Both the late pla model ad the sigle level model were implemeed i the microscopic traffic simulator MITSIMLab (Yag ad Koutsopoulos 1996) for aggregate validatio. I the validatio process, part of the aggregate data was first used to calibrate the overall behavioral parameters of MITSIMLab. The calibrated MITSIMLab outputs were the compared with the remaiig data Data U.S. 101 dataset was collected o a 2100 feet (640 meter) southboud sectio of U.S. Highway 101, i Los Ageles (Califoria) with five mailie ad oe auxiliary lae coectig to the Veura o-ramp ad the Cahuega off-ramp (Figure 5.24) ft (640 m) Veura O -Ramp 698 ft (213 m) Study Area of Trajectory Data Lakershim Off -ramp Figure 5.24: Validatio data collectio site This site has a auxiliary lae after the oramp, which was ot the case for the I-80 site. 45 miutes of data (7:50 a.m.-8:35 a.m.) were available. Based o the trajectory 146

147 data, syhetic sesor data was created i three locatios (Figure 5.24). This sesor data replicated cous ad speeds (aggregated over every five miutes) that would have bee recorded by sesors located i these locatios Aggregate Calibratio The aggregate calibratio problem ca be formulated as a optimizatio problem which seeks to miimize a fuctio of the deviatio of the simulated traffic measuremes from the observed measuremes (Toledo et al. 2004). The umber of behavioral parameters i the simulatio model is very large ad it is ot feasible to calibrate all of them. Based o previous experiece ad sesitivity test results, the followig parameters of the combied model were selected for calibratio: Acceleratio ad deceleratio costas Desired speed mea ad sigma Iercepts (costas) ad stadard deviatios (sigma s) of ormal critical gap Aicipatio time mea ad sigma Probability of yieldig of the mailie vehicle Costa for forced mergig Idividual-specific radom errors i the remaiig distace terms The fit of the combied model to the calibratio data are preseed i Table Table 5.10: Calibratio results of the combied model Lae-specific Cous Before After Improveme Calibratio Calibratio RMSE (vehicles/15 mis) % RMSPE (%) % Lae-specific Speeds Before After Improveme Calibratio Calibratio RMSE (m/s) % RMSPE (%) % 147

148 5.4.3 Aggregate Validatio The validatio process ivolved data that was ot used for calibratio. A compariso of the followig simulated ad observed statistics was coducted: Lae-specific poi speeds for the remaiig 15 mis (8:20-8:35 am) Lae-specific flows for the remaiig 15 mis (8:20-8:35 am) Distributio of locatio of merges (summarized from aggregate trajectory data) The measures of performace of the combied model were compared agaist the performace of the reduced form sigle level model. The OD flows used for this step were calculated directly from the trajectory data. Lae-specific Sesor Speeds A separate set of lae-specific speed measuremes from sesors (ot used for calibratio) was used for validatio purpose. The comparisos of the goodess-of-fit measures are preseed i Table As is evide from the RMSE ad RMSPE, the performace of the models improved with complexity of the model: the late pla mergig model performed better tha the sigle level model. Table 5.11: Compariso of lae-specific speeds Sigle Level Model Combied Mergig Model Improveme RMSE (m/s) % RMSPE (%) % Lae-specific Sesor Cous The simulated lae-specific sesor cous of the late pla mergig model were compared agaist the actual observatios ad the simulated cous of the sigle level model. As observed i Table 5.13, the combied mergig model had a sigificaly better match with the actual observatios. 148

149 Table 5.12: Compariso of lae-specific cous Sigle Level Model Combied Mergig Model Improveme RMSE (vehicles/5 mis) % RMSPE (%) % Locatio of Merge The simulated locatios of merges were compared agaist the observed locatios. The late pla model had a sigificaly better predictio of the locatio of merges tha the sigle level model (see Figure 5.17). I particular, the simpler model teds to over predict merges occurrig toward the ed of the auxiliary lae sice courtesy ad forced mergig plas are ot cosidered explicitly i this model % of 40 Merges % of Merges Observed Late Combied Full Pla Reduced Sigle Normal Level Form Remaiig distace to ed of mergig lae (m) 5.5 Summary Figure 5.25: Compariso of merge locatios The detailed structure, estimatio results ad validatio results of a late pla based combied mergig model has bee preseed i this chapter. The model iegrates ormal, cooperative ad forced mergig types io a sigle framework. Parameters of the models are estimated with detailed vehicle trajectory data collected from I-80, i Califoria. The effect of uobserved driver/vehicle characteristics o the lae chagig process was captured by driver-specific radom terms icluded i differe model compoes. 149

150 Importa explaatory variables were foud to be the urgecy of the driver (e.g. distace remaiig to the ed of the mergig sectio), the relatio of the mergig vehicle with eighborig vehicles (e.g. lead ad lag speed ad positio etc.), the traffic coditios (e.g. average speed ad desity) ad driver heterogeeity. Statistical comparisos of estimatio results idicate that the estimated combied model has sigificaly better goodess-of-fit compared to a reduced form simpler model that does ot explicitly cosider courtesy ad forced mergig. This was supported by a validatio case study where the performace of the two models was compared i a differe etwork settig. The combied model performed sigificaly better tha the simpler models across all measures. I the curre model, the late plas were assumed to iclude oly lateral decisios ivolved with the mergig decisio. The exte of the improvemes obtaied with the ehacemes i the mergig models preseed i this applicatio idicates that further advaces i mergig models may lead to improvemes i their ability to replicate observed vehicle trajectories. I particular, icludig target gap choice ad acceleratio i the model is a possible future directio of research. As observed i the validatio results, the combied model was better at replicatig cous tha speeds. Iclusio of target gap choice ad speed adjustme to reach a targeted gap i the decisio framework of the mergig driver may improve the match of speed as well as esurig a better predictio of merge locatios. 150

151 Chapter 6 Lae Selectio o Urba Arterials The late plas ivolvig the lae selectios of drivers o urba arterials are ivestigated i this chapter. The specific models discussed here are the lae choice model for urba iersectios ad the lae chagig model for arterial mailie sectios. The geeral structure of both models is the same as preseed i the previous chapters: late pla followed by observed actios. However, because of differeces i geometric ad operatioal characteristics, the late plas ivolvig lae selectio of drivers o urba mailie ad side streets are likely to be quite differe tha those of freeway mailie (discussed i Chapter 4) ad o-ramps (discussed i Chapter 5). The chapter is orgaized as follows: the backgroud of the research is preseed i Sectio 6.1. Descriptio of the estimatio data ad the details of the model estimatios are preseed i Sectio 6.2: the iersectio lae choice model i Sectio ad the mailie lae chagig model i Sectio Each sectio icludes the model structures, the likelihood formulatios ad the estimatio results of each of the two models. The chapter cocludes with the aggregate calibratio ad validatio results withi the microscopic traffic simulator MITSIMLab ad a summary of the fidigs Backgroud Travelers o arterial etworks face special challeges regardig lae positioig strategies. Arterial corridors have a set of varied drivig activities that differ by lae ad locatio. These activities ecompass trip destiatio activities (parkig, eerig trasit 7 The model preseed i this chapter has bee developed as part of the NGSIM program of FHWA. The results preseed i this chapter have bee reported i Choudhury et al. (2007). A simplified versio of the mailie lae chagig model has bee developed by Ramaujam (2007). 151

152 stops, right turs, left turs etc.), trip origiatio activities (exitig a parkig spot, exitig trasit stops etc.), ad complex routig behaviors (permissive left turs, pedestriaimpeded right turs etc.). Drivers familiar with the etwork may be aware of these activities ad be midful about how they vary by lae ad locatio. These drivers ofte make appropriate tactical lae positioig decisios to miimize their travel times ad drivig efforts o these complex facilities. The familiarity ad plaig ability of the drivers that is: how far they look-ahead or pla-ahead affect their tactical plas ad thus impact their drivig decisios. Due to situatioal costrais, immediate executio of the tactical lae selectio pla may ot be possible. For example, at a particular ista, coflicts with other vehicles ca delay moveme to the target lae. Further, chages i circumstaces may lead to chages i the tactical pla: a log queue build-up i the chose target lae for example ca lead to amedme to the origial target. The chose target laes are thus uobserved ad oly the immediate choice of laes is observed. The lae positioig decisios geerally maifest themselves iside lae chagig models i existig simulatio systems. Lae chagig models are ofte geeralized betwee freeway ad arterial facilities. O arterial etworks, existig models rely o stadard lae chagig logic to determie vehicle positioig behavior. Some models address the pre-positioig of drivers for path-pla cosideratios usig rule based lae chagig models (Ji et al. 1999, Wei et al. 2000). But the complexity of the tactical plas behid the immediate decisios of the drivers, as well as the heterogeeity i their plaig behaviors, is igored i the existig models. Moreover, oe of the arterial lae selectio models ivolve rigorous statistical estimatio usig detailed traffic data. These weakesses of the existig models ofte lead to urealistic spillbacks ad ueve queue distributios across laes. This has bee also reflected i the fidigs of the NGSIM study o Ideificatio ad Prioritizatio of Core Algorithm Categories, where developme of arterial lae selectio model has bee ideified as the most importa research area by both model developers ad users (Alexiadis et al. 2004). Lae selectios i urba arterials iclude lae chages withi the arterial mailie sectios as well as lae choices at iersectios. The lae chages i the arterial mailie sectios ivolve repeated decisios of drivers while iersectio lae choices are 152

153 iermitte decisios that are evaluated oly whe the driver is turig at iersectios (sice chagig laes withi the iersectio are ot permissible). Further, the ature of coflict with other vehicles while turig at a iersectio is distictly differe from that of lae chagig i a mailie sectio. These lead to differeces i the detailed frameworks of the iersectio lae choice ad mailie lae chagig models i this study. A driver turig at a sigalized iersectio is likely to choose the lae that he/she perceives to be the best ad plas to move to that lae. However, because of coflicts with other vehicles havig the right of way, it may ot be possible to execute the pla immediately. The immediate lae choice of the driver, that is the lae where he/she is observed just after turig, thus may ot be the same as the origially targeted lae. The pla of the driver is thus uobserved ad oly the immediate lae selectio is observed. The iersectio lae selectio is therefore a two level decisio: Choice of target lae (pla) Choice of immediate lae (actio) I a usigalized iersectio, the choice may ivolve additioal levels like gap acceptace, target gap selectio ad/or decisio whether or ot to move towards the target lae usig alterate gap acceptace tactics (e.g. by courtesy of aother driver or by forcig i). Oce the driver gets o the mai arterial, he/she is likely to have a late pla based lae chagig decisio structure similar as i the freeway mailie: target lae selectio followed by gap acceptace to reach the target lae. However, i urba arterials where average speeds ad headways are sigificaly lower tha the freeway, duratio of the lae chagig maeuver may be loger tha that of a freeway. Drivers tryig to reach their target lae therefore are ot istaaeously observed to complete the lae chage i the directio of the target lae eve if a acceptable adjace gap is available. Rather, a lae chage is observed i the directio of the chose target lae i presece of a acceptable adjace gap whe the executio of the lae chage has bee completed (ceer poi of the vehicle has passed the lae boudary). 153

154 The simplest lae chagig maeuver of drivers i the arterial mailie ca thus be defied as a three stage decisio: Choice of target lae (pla) Decisio to accept available gaps (pla) Executio of the lae chage ( actio) The compoes of the chose pla (target lae selectio ad gap acceptace) are late or uobserved ad oly the completio of the executio of the lae chage is observed. It may be oted that i cogested urba arterials, the pla may iclude additioal levels like target gap selectio ad choice of lae chagig tactics i the decisio framework. As meioed, i a urba arterial with closely spaced turs, the pla-ahead distace of the driver geerally has a strog ifluece o the lae selectio. The tactical plas of the driver are affected by ifluecig factors withi the pla-ahead distace of the driver. For a driver familiar with the etwork ad turig i a subseque sectio, this implies that he/she does ot cosider the path-pla i lae selectio uil the desired tur is withi the pla-ahead distace. It may also imply that the driver does ot look beyod the pla-ahead distace while cosiderig lae-specific variables such as average speeds, desity ad queue legths. The pla-ahead distace of the driver is expected to vary amog the driver populatio ad ca deped o differe factors such as persoal traits, etwork familiarity, cogestio level etc. While the coiuous pla-ahead distace is more appropriate for lae chagig scearios where the decisios are evaluated coiuously, a discrete approach (where pla-ahead distaces are multiples of sectio legths) is more releva i case of iersectio lae choice sice the decisios are take iermittely at distict pois i the etwork. 6.2 Model Estimatio Estimatio Data Study Area The two arterial models i discussio: the iersectio lae choice ad the withi sectio lae chage model are estimated from data collected from Lakershim Boulevard 154

155 i Los Ageles, Califoria. Vehicle trajectory data was collected i 2005 as part of the FHWA s NGSIM project o a segme of the arterial located ear the iersectio with US highway 101 (Hollywood Freeway) (Figure 6.1). Figure 6.1: Lakershim Boulevard arterial sectio The study site is approximately 1600 feet (488 m) i legth. It cosists of four sigalized iersectios, ad three to four through laes i each directio i each sectio. Five video cameras were used to collect the trajectory data for a 22 miute period (8:28 am to 9:00 am). These cameras were moued o top of a 36-story buildig, 10 Uiversity Plaza, located adjace to the US 101 ad Lakershim Boulevard ierchage. Figure 6.2: A schematic represeatio of the arterial stretch (ot i scale) Figure 6.2 shows a schematic of the arterial segme costitutig the study area. It also provides details regardig the referece idices used for demarcatig the 155