Modeling Driver Behavior as a Sequential Risk-Taking Task

Transcription

1 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Modelig Driver Behvior s Sequeil Risk-Tkig Tsk Smer H. Hmdr Mri Treiber Hi S. Mhmssi Are Kesig TRB Aul Meeig Wshigo DC Jury 008

2 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Iroducio: Moivio/Objecive/Approch Cr-Followig Model Formulio: - Esimio of Collisio Probbiliy - Evluio Process: Vlue Fucio - Free-Flow Model Model Implemeio: - Iiil Plos - Asympoic Expsio for Efficie Implemeio Model Alysis d Assessme Coclusio d Fuure Reserch Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

3 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Moivio: 1- Exisig cr-followig models re ccide-free where sfey cosris re forced - There is eed for richer complee represeio of he cogiive processes uderlyig driver behvior i differe drivig codiios free-flow, cogesed d exreme codiios 3- Explici represeio of driver risk iudes is expeced o provide greer isigh io he role of risk-kig behviors i ccide-proe siuios Objecive: Explore d evlue cr-followig model h reflecs he psychologicl d cogiive specs d cpures risk kig behvior uder uceriy Approch: 1- Khem d Tversky s 1979 Prospec Theory provides heoreicl d operiol bsis o weigh driver s differe lerives. - Collisio probbiliy is esimed kig io ccou driver s risk iude. 3- Asympoic expsio llows efficie implemeio d ivesigio of model properies. 3- Numericl experimes re coduced Sesiiviy Alysis o sudy he poeil of he formuled model i erms of relism of he behvior geered. Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

4 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Cr-Followig Model Formulio Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

5 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Behviorl Frmework 1: Model Srucure I he Cr-Followig Process, hree behviors re possible: 1. Drivers ccelere. Drivers decelere 3. Drivers keep he sme speed Choosig bewee hese hree behviors is choosig bewee se of ccelerio vlues bewee d. mi mx Time is divided io differe ccelerio isces i = 1,.ec. A ech ccelerio isce i, driver sequeilly evlue he collisio risk ivolved i choosig he ccelerio d he correspodig gi or loss: 1. The risk is represeed by collisio probbiliy. The gi d losses re evlued usig prospec heory vlue fucio lloced o ech ccelerio vlue Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

6 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Behviorl Frmework 1: Prospec Theory Prospec heory llows choosig bewee differe se of lerives ccelerio vlues preseed for idividul id driver ime : C Ech lerive hs mesured objecive uiliy OC. A vlue fucio U PT [OC] is used o rsform he objecive uiliy io perceived subjecive vlue. The vlue fucio c represe: 1. Risk seekig versus versio iudes expressed by hums drivers.. Asymmery i weighig gis d losses. 3. The edecy o mesure gis d losses o i erms of bsolue vlues bu wih respec o referece poi. A probbiliy bili PC is ssocied o ech lerive. i A prospec fucio π[pc] rsforms he probbiliy erms o prospecs. Bsed o he prospec of ech lerive, i he lerive i wih he highes h π[pc] U PT [OC] Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

7 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Esimio of Collisio Probbiliy 1 σ v 1 v 1 pdf of v 1 v v 1 = f v = exp π σ v 1 σ v 1 es 1 Wh is he fuure speed of my leder? ves 1 : esimed subjecive fuure speed of led vehicle -1 s perceived by driver over iciped ime sp Follows orml disribuio wih sdrd deviio σ v 1, d me equl o cul velociy of he leder v 1. Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

8 Deprme of Civil d Eviromel Egieerig Smer Smer H. H. Hmdr Hmdr Jury 15, 008 Jury 15, 008 The crsh probbiliy is give by he probbiliy h Esimio of Collisio Probbiliy p + p y g y p y he gp ime is <= 0. Give cos ccelerio for he follower driver i quesio d cos velociy for he leder: = + L x x s quesio d cos velociy for he leder: + es s v v x x es + = + 1 v x x + + = + Wriig i erms of sdrdized orml disribuio, we < = + es v P p 1 v es 1 ge: Z v v v es = σ + Δ = Φ + Δ < = + s v s v Z P p Sdrdized Gussi Sochsic Vrible buled cumulive disribuio fucio Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio Φ < + v v Z P p σ σ 1 1 for he sdrdized Gussi

9 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Evluio Process 1 U The gi d losses re expressed here i erm of gis d losses i speed from he previous ccelerio isce i-1. If he gi d losses re expressed i erms of bsciss Δ& x = Δv =,he vlue fucio U PT is defied s follows: PT Normlizig Prmeer 1 m/s w + 1 w * h + 1 * = Posiive Prmeers o be Esimed 0 γ PT Vlue Fucio I collisio, he loss is ssumed o be reled o seriousess erm k v, Δv weighed by w c : whe he seriousess of he driver icreses, w c icreses. k v, Δv represes he sesiiviy o he loss cused by ccide. Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

10 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Evluio Process The vlue Fucio is: U = 1 p, i U PT p, iwck v, Δv + ε Where U PT = PT vlue fucio p,i = probbiliy of collidig wih rer-ed bumper of led vehicle give h o collisio ook plce i he i-1 h ccelerio isce ε = driver specific error erm ssumed o hve Weibull disribuio Usig coiuous logi model, he sochsic cr-followig ccelerio, cr followig + Δ i of vehicle is rerieved from he followig probbiliy desiy fucio: f exp mx = exp mi β U β U 0 ' d ' oherwise where β is free prmeer β>0 h reflecs he sesiiviy of choice o he uiliy. mi mx Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

11 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Cr-Followig versus Free-Flow The free-flow ccelerio is defied simply by:, free flow + Δ = V, desired ed Δ v The fil ccelerio is: + Δ = mi, free flow + Δ,, cr followig + Δ Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

12 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Model Implemeio Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

13 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig All N+1 drivers = 0, 1, N re ssumed o hve ideicl prmeers where s is he correspodig gp d v is he relive speed whe pprochig. The esimio uceriy σ v of he velociy of l = α v l he leder is proporiol o he velociy iself, i.e., he relive error vriio coefficie α is cos. The icipio ime horizo is ssumed o be he miimum bewee he ime-o-collisio TTC = s / Δv d some mximum vlue mx : = s, Δv = s Δv mx s Δv oherwise mx Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

14 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Asympoic Expsio sympoic expsio of he ccelerio probbiliy disribuio of his model will give: v& = ~ N *, σ he disribuio of ccelerios is pproximely give by Gussi disribuio whose momes re: * = rgmx U, σ 1 = βuβ U '' * U ' d U '' c be clculed lyiclly give by derivives of Φz h is desiy of Gussi. The vlue iself eeds o be clculed umericlly. * i is o gureed h * is uique olieriies i U PT. However, ll ivesigios preseed i his pper show h i is uique for he prmeers chose while ssessig he model. Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

15 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Efficie i Implemeio i We ssume: γ = w =1 lier vlue fucio Accordigly: U ; s, v, Δv = wcφ z where 0 1 s Δv + z = αv As ecessry codiio for mximizio i d miimizio i i problems, U ' eeds o be zero: U ' = 1 0 w c f N z z ' z / Where: f z = e d N 1 π z' = αv s * = Δv + αvz * ; z* = rgmx U z = l mx mx Seig he Newo s mehod is used o fid he opimum; 0 = * is +1h ierio is defied by Newo s Mehod ierios. wc ' π Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio 0 z

16 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Model Alysis d Assessme Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

17 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Iiil Plos 1 Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

18 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Iiil Plos Severl Relioships Vlue Fucio versus Spce Hedwy, Vlue Fucio versus Speed, Vlue Fucio versus Relive Speed were esed. Remrkbly, i sochsic equilibrium, pproxime ime hedwys of 1.5 secods re kep cos i he cr-followig regime mily iflueced by α. mx Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

19 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Iiil Plos 3 50 mile/hr Decrese i s v icreses: reurig o sfey gp 15 mile/hr Sedy Se Velociies Sedy Se Spce Hedwys 40 m 18 m Icrese i Vrice wy from = 0 : PT vlue fucio Hrsh decrese i empig voidig collisio higher h v /s if v>10 If res d s = 30 m, emp ccelere slowly o rech soppig hedwy 1- m/s Coour plos of he ccelerio probbiliy desiy equio s fucio of v, s d v op lef, op righ d boom righ. Coour for siuio wih sdig vehicle or red rffic ligh, v = v for s = 30 m boom righ Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

20 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Movig Trffic wih s = 0 m : ll vehicles use he mximl decelerio vlues whe 18 m / s v 40 m / s d 7 m / s Δv 0 m / s. The lowes vrices re observed mosly for smll v d whe v decreses below zero: less disurbces is cused by ccelerig mosly reled o he vehicle properies he decelerig mosly reled o drivers persoliy. Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

21 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Coclusio d Fuure Reserch Needs The model implemeed showed promisig resuls i erms of sochsic equilibrium. The sympoic exesio of he cr-followig equios is possible lyiclly d llows more efficie implemeios d fser execuio. Decisio mkig heories such s prospec heory llow more solid psychologicl bckgroud for he preseed model, relig i o rich lierure o ye exploied i he rffic modelig domi: sochsiciy, risk kig d ccides re well icorpored i he modeled behvior of he drivers. Fuure Reserch Needs: Clibrio d vlidio by compriso wih rel-life rjecory d Sudyig he resulig flow-desiy relioships s well s oher mcroscopic performce mesures verge rvel imes, verge dely ec The free-flowflow d he le chgig behviors re o fully developed i his sochsic frmework. Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

22 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Thk you for your eio! Quesios? Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

23 Deprme of Civil d Eviromel Egieerig Loss versio see i he seeper slope wih losses h wih gis: Asymmery i weighig gis versus losses Dimiishig sesiiviy o icresig gis d losses Used i his Model Evluio of oucomes relive o referece poi, ke s he speed i previous ccelerio isce Oulie Khem d Tversky, 1979 Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

24 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig s 0 = Δ v + α vz * mx mx U s he spce hedwy s chges Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

25 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig s 0 = Δ v + α vz * mx mx U s he speed v chges Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

26 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig s 0 = Δ v + α vz * mx mx U s he relive speed v chges Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

27 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio

28 Smer H. Hmdr Jury 15, 008 Deprme of Civil d Eviromel Egieerig Efficie Implemeio: Newo s Numericl Approximio We se: 0 = * The Newo s mehod is used o fid he opimum; is +1h ierio is defied by: = F / F' + 1 Where F = U ' = U w f z z' F ' = U '' PT = U '' c N z z ' z '' PT w c f N z + Oulie Iroducio Model Formulio Model Implemeio Model Alysis d Assessme Coclusio