Estimation of Time Dependent OD Matrices From Traffic Counts

Transcription

1 Estimtion of Time Dependent OD Mtices Fom Tffic Conts Hioyki ONEYAMA() Mso KUWAHARA(2) Tosio YOSHII(3) ()Pblic Woks Resec Institte, Ministy of Constction, Asi, Tkb, Ibgi305, Jpn Pone: , Fx: , Emil: (2)Institte of Indstil Science, Univesity of Tokyo, 7-22-, Roppongi, Minto-k, Tokyo, 4, Jpn Pone: , Fx: , (3)Institte of Indstil Science, Univesity of Tokyo, 7-22-, Roppongi, Minto-k, Tokyo, 4, Jpn Pone: , Fx: , ABSTRACT We popose te model fo estimting time-dependent OD mtices fom tffic conts in genel netok it ote coice ctivities Te model consists of to pts : () constction of te eltionsip beteen te timedependent OD volmes nd tffic conts t links nd (2) estimtion of niqe time-dependent OD mtices In te fist pt, e define tee-dimensionl netok to elte OD mtices to tffic flo on links We ten popose metod of estimting ote coice pobbilities In te second pt, e employ te Entopy Mximizing metod fo te sttic OD mtices estimtion nd extend it to time-dependent model As n extension, e popose simplified metod of estimting ote coice pobbility nd metod to tilize ggegted pio OD infomtion At te lst, e pply te model to test netok nd exmine its vlidity INTRODUCTION A model fo estimting time-dependent OD(Oigin-Destintion) mtices is eqied to elize te optimm tffic contol nd plnning Mny dynmic tffic simltion models ve been developed in ode to epodce tffic conditions nd evlte policies of tffic contol, signl contol, one-y tffic contol nd so on Sc dynmic model needs time-dependent OD, especilly one composed of smll OD zones Hoeve, it is d to estimte OD flos diectly fom OD svey In tis stdy, e ts popose dynmic estimto sing time-dependent tffic conts to obtin time-dependent OD volmes Tee s been some models estimting o pdting timedependent OD mtices fom tffic conts Tese models cn be divided into to types, te Intesection models nd te Netok models Te Netok model is moe complicted tn te fome becse of te considetion of ote coice bevio Ngyen et l ) poposed te model on genel tnsit netok Yng et l 2) estimted OD mtices so s to minimize te integted sqed eo beteen obseved nd pedicted link tffic flos it n efficient soltion metod developed bsed on Foie tnsfomtion Hoeve, tese models ve not inclded timedependent dive s ote coice bevio In te model developed by Cscett et l 3), ltog tey conside te dive s ote coice bevio, s enmetion of ll otes is eqied to estimte ote coice pobbilities, clcltion does not seem to be pplicble to lge netok Asok nd Ben-Akiv 4) ve lso poposed te on-line estimto sing te Klmn filteing Hoeve, tio of tffic flo on ec ote nd pio OD flos e eqied in tis estimto TIME-DEPENDENT OD MATRICES ESTIMATION MODEL Otline of te Model Tis stdy develops te model to estimte time-dependent OD volmes in genel netok, ic consists of links nd nodes In tis model, ec link s time dependent link tvel time ic is oeve flo-independent nd mst be inpt in dvnce bsed pon te field obsevtion In te fist pt of te model, te eltionsip beteen OD flo nd link flo is estblised by intodcing ote coice pobbility detemined fom te time dependent link tvel times In te second pt, time-dependent OD mtix is niqely estimted by pplying te entopy mximizing metod nde constints of te eltionsip beteen OD nd obseved link flos obtined in te fist pt

2 Definition of Time Axis nd Fomltion A veicle tjectoy on pt k is dn in te time-spce gp s in Fig, in ic time xis is divided into discete time-intevls of eql lengt t, nd time-intevl is defined s time intevl [ t, (+) t] T (), tvel time t link t time-intevl, is ssmed to be mltiple of n intege t Hence, t mst be sfficiently smll so tt cnge of link tvel time ove time cn be ell descibed It is lso ssmed tt T (), v (), link flo t link t time-intevl nd q (), OD flo depting fom oigin node of OD pi t time-intevl, do not vy ding ec time-intevl, ic mens tt tey sty constnt vles t te stt of time-intevl In Fig, veicle depting fom n oigin t time-intevl psses tog sevel links long te pt k nd entes link t time-intevl τ k( ), ic is ivl time t link en tffic on pt k of OD pi genetes fom oigin t te time Hee, since T () is pedetemined, ivl time t link, τ k( ), cn be clclted by smming link tvel times long te tjectoy time " ( +) t time peiod t t oigin Fige Time-Spce Gp (Descete Time Axis) "! =τ k ( ) pt k obsevtion point If veicles A nd B ctlly tvel s son in Fig, e nomlly expect tt tjectoies of veicles depting oigin ding time intevl e sped beteen tjectoies A nd B Hoeve, e ve to note tt, nde te ssmption bove, tose tjectoies sty beteen A nd B becse of te constnt tvel time in ny time peiod of t Using netok it te discete time xis, eltionsip beteen OD flo nd link flo cn be itten s: v t p q = τ τ δ t, () () () k k k ( ) ( k ()) ee p k () :pobbility tt tffic flo of OD pi depting fom its oigin ding time-intevl ses pt k, τ k- () :depte time-intevl fom n oigin en tffic flo on pt k of OD pi entes link ding time-intevl, δ k : ; if pt k of OD pi psses tog link, 0 ; oteise Tis eltionsip cn be lso itten s: v t = p, q t, (2) ( ) ( ) ( ) in ic p (, ) mens pobbility tt veicle depting fom oigin node of OD pi ding time-intevl entes link ding time-intevl Te p (, ) ence stisfies folloing: p(, ) = k Estimtion of Rote Coice Pobbility Let s ssme se's ote coice pobbility sc tt [ ] ( ) ( ) ( ) ( ) ( ) pk = Pob Ck + εk Cm + ε m, m,

3 ee C k ( ) : cost of pt k of veicle depting fom oigin of OD pi t time-intevl, ε k ( ) : n eo tem of C k ( ) If eo tem ε k ( ) s te Wible distibtion, e obtin te folloing ell knon Logit model : exp( θck( )) pk( ) =, (3) exp( θcm( )) m ee θ is te Logit pmete, nd it mst be given extenlly bsed on te ote coice ctivity If pt cost C k ( ) is ssmed line fnction of link costs long pt k, C k ( ) is itten s: Ck( ) = C ( τ k ( ) ) δ k (4), ee C () is te cost of link t time-intevl We ee conside only cse in ic te pt cost is line fnction of te link costs Since link tvel time T () is consideed one of te most epesenttive fcto, ic is ssmed given s mentioned elie, te link cost is lso ssmed to be given fo ll links t ll time peiods Te p k ( ) is obtined fom te optimiztion poblem (P) s son belo by extending te Fisk model on te sttic ssignment to te dynmic model PROBLEM P: min pk( ) log pk( ) C () p() θ +, k,,, st p() = pk( τ k () ) δ, k k, p p 0, pk( ) 0, k( ) =, ( ) k in ic p () is te smmtion of pobbilities enteing link t time-intevl fo ll OD pis We cn pove tt te soltion of poblem P is eqivlent to te Logit type pt coice pobbilities son in (3) (see Appendix) Time Axis Dmmy Node S Time Intevl t Dmmy Link R O Node D Node Fige 2 A Time-Spce Netok To solve P, let s intodce te tee-dimensionl time-spce netok, in ic te one-dimensionl time xis is dded to te to-dimensionl sptil netok, s son in Fig 2 On tis netok configtion, ctl link is decomposed into sevel links depending on enteing time peiods Since te veticl eigt of te link inceses by its link tvel time, slope of te link epesents te velocity And te FIFO cn be knon s condition tt links on te tee-dimensionl time-spce netok do not coss ec ote In te cse, veicle enteing link ding timeintevl cnnot be ovetken by veicle enteing link ding time-intevl + So, link tvel time T () needs to be given so s to stisfy:

4 ( + ) ( ) T T Hoeve, to djst te tee-dimensionl netok to te sl to-dimensionl netok, ne destintion node S mst be dded s in Fig2 To conside te poblem P on te tee-dimensionl netok, vibles in P ve to be conveted s follos Oigin node R on te tee-dimensionl netok coesponds to combintion of oigin node nd depte time peiod on te to-dimensionl netok, nd link A on te tee-dimensionl netok coesponds to combintion of link nd enteing time peiod on te to-dimensionl netok Ts, e cn convet te poblem P to te poblem on te tee-dimensionl netok by conveting vibles sc s p k ( ) to p KW, C () to C A, ee p KW is pobbility tt veicle ving OD pi W(oigin R, destintion S) ses pt K, nd C A is cost of link A on te teedimensionl netok Te poblem on te tee-dimensionl netok conveted fom te poblem P mentioned bove cn be solved in te sme y s P, nd te eslt is itten s: p KW = exp exp M ( θckw ) ( θc ) MW (5) Tis mens tt te flo-independent Dil s ssignment cn be pplied to solve te poblem on te teedimensionl netok Specificlly, e cn estimte p AW by pplying te Dil s ssignment in ic OD flo sets to on ny OD pi W on te tee-dimensionl netok Ten, p AW estimted bove is eql to p (, ) on te to-dimensionl netok Becse oigin R on te tee-dimensionl netok coesponds to oigin nd depte time on te todimensionl netok, destintion S to destintion s, nd link A to link nd enteing time peiod, espectively Estimtion of OD Mtices As discssed in te pevios section, eltionsip beteen OD flo nd link flo is fomlted s (3) sing te ote coice pobbility Sppose link flo is obseved s v ( ) ic consists of el link flo v () nd its eo tem ε (): v = v + ε = p, q + ε (6) () () () ( ) ( ) (), Unknons q ( ) mst be estimted so tt (6) is stisfied Te nmbe of conditions (6) is te nmbe of obseved link () x te nmbe of obseved time-intevls (), ic is nomlly less tn te nmbe of nknons, te nmbe of OD pis () x te nmbe of time-intevls ( ) Hence, e cn find mny sets of OD mtices ic stisfy (6) nd te poblem is o e sold coose niqe mtix mong te cndidte mtices stisfying (6) In tis stdy, e pply te Entopy Mximiztion metod 5) to tis time-dependent model in ode to coose niqe mtix Hee, pio OD flo depting fom te oigin of OD pi t time-intevl is denoted s ( ) Ten, OD flo q( ) nd link flo v ( ) q ( ) q ( ) X ( ) q cn be estimted s: p(, ) =, ( ) = ( ) ( ) v v X γ (8) in ic X () is pmete tt cn be obtined by solving p (, ) γ v ( ) X( ) = p(, ) q ( ) X( ),,,, (9) Te detiled deivtion of (7), (8) nd (9) is son in efeence 6) Hee, pmete γ mens eigt of link obseved eos Te lge γ is, te loe obseved eos e evlted, nd if γ =, it is te sme s noml entopy mximiztion metod in ic no eos e consideed (7)

5 SOME EXTENSIONS ON THE MODEL Aggegtion of Time Axis nd Simplified Estimtion of Rote Coice Pobbilities Te model poposed in pevios section needs to set time intevl t so sot s to descibe cnges of link cost (eg t ill be less tn bot 30 seconds) On te ote nd, e slly need to estimte OD mtices of longe time intevl, sc s 5-60 mintes Moeove, OD mtices t evey t time intevl ill not pefeble becse stocstic cnges e moe dominnt tn tend cnges Ts, it is pcticl to set mc longe time intevl tn t fo estimtion of OD mtices In tis section, e discss te simplifiction of estimting link se pobbility fo longe time intevl We efomlte eltionsip beteen n OD flo nd link flo fo longe time intevl ssming tt OD flo te does not cnge tog te long time intevl T = m] t, ee m is positive intege Fo long time intevl T, e define ggegted time-intevl H(i) s follos: H(i) = set of m sot time-intevls t inclded in te time section [(i-) T,i T], i =,2,,H/m If Q (i) is OD flo te fo OD pi t te i-t long time-intevl T, (2) becomes v t = Q i p t i H() i () () (, ) Ftemoe, let s ggegte link flo tog te j-t long time-intevl T So () () (,) v t = Q i p t H( j) i H( j) H( i) (0) If e define V (j) nd P (i, j) s son belo, te bove is simply eitten s: V j T = P i, j Q i T, () { } () ( ) () i, ee t V( j) = v( ) = v( ) ( ) T m, (2) H j H( j) P( i, j) = p( ) m, H() i H( j) (3) P (i, j) is pobbility tt veicle, depting fom O node of OD pi t te i-t time-intevl T, ill be obseved in te link t te j-t time-intevl T Since (2) nd () ve te exctly sme fom, OD mtices fo longe time intevl T cn be estimted by te sme metod s fo smll time intevl t Next e conside efficient estimtion of P (i, j) fo longe time intevl T Bsiclly, P (i, j) t te i-t nd j-t ggegted time-intevl T cn be estimted fom (3) if pobbility p (, ) e obtined t ll nd by te metod poposed in pevios section Hoeve, te estimtion of p (, ) fo evey nd sing smll time intevl t is qite tedios To decese clcltion times, e mke some ssmption tt ote coice ctivities o link se pobbility does not cnge ding longe time peiod of τ = c t Hee c is positive intege, ic is te nmbe of sot time-intevls t inclded in τ We ssme τ = c t m t = T nd m/c is intege Pcticlly speking, T nd τ migt be 60 mintes nd 30 mintes o so Te n-t time-intevl of τ mens [{(n- )c+} t,nc t] ic is denoted s N(n), nd te initil time of N(n) is itten s s(n)=(n-)c+ No e ssme tt ote coice pobbilities do not cnge t ec time-intevl of τ nd veicles depting fom te sme oigin ding te sme time-intevl τ ve te sme ote coice pobbilities Tt is, p (s(n) + l, ) = p (s(n), - l), l = 0,, 2,, c - Ts, p (, ) cn be itten s: p(, ) = p( s() n + l, ) = p( s() n, l), l = s() n, N() n,,, n, (4) Fom (2), p (, ) fo ll nd cn be estimted by clclting only pobbilities t te fist sot time-intevl t of ny te n-t time-intevl τ Ten, ggegted time link se pobbility P (i, j) cn be estimted fom (3)

6 Use of Sptilly o Tempolly Aggegted Pio OD Flo Te OD estimtion model poposed in te stdy needs pio OD mtices q ( ) fo te tget OD Hoeve, in mny cses, time-dependent pio OD mtices fo fine zone-to-zone level e not vilble, bt moe globl (ggegted) mtices sc s dily OD mtices fo lge zones In tis section, e discss o sptilly ggegted pio OD infomtion cn be tilized Fist of ll, sptilly nd/o tempolly ggegted OD cn be expessed s line smmtion of te minimm nit of OD flo, q ( ) Aggegted OD flo is denoted s g, U, in ic is nmbe of ggegted OD flo, nd U is set of ggegted OD flo Ten, ggegted OD flo cn be fomlted by sing minimm OD flo s g = (, ) q( ), (5), ee φ() : set of OD pi nd depte time (, ) inclded in ggegted OD flo g, U, (, ) q ( ) : pmete tt expesses contibtion of q( ) to g (, ) is if (, ) φ() nd 0 oteise No e se g fo pio ggegted OD flo, nd η fo devition of g fom g: g = g + η = (, ) q( ) + η (6), Te (6) is vey simil to (6) : U coesponds to (, ) W, g coesponds to (, ) nd (, ) coesponds to p(, ) Teefoe, by sbstitting (, ) fo p(, ) nd g fo (, ) in (7),(8),(9), te folloing eqtion cn be obtined Fom tis eqtion, X cn be solved, nd ( ) ( ) ( ) ( ) (, ) γ g X = (, ) q ( ) X,, q cn be obtined s son belo by sbstitting X () fo X in (7): p (, ) q q X g X (, ) =, NUMERICAL EXAMPLES We pply te model discssed bove to test netok s son in Fig3 Te time intevl sed in te model is t=0 sec, τ=30 min, nd T= o nd e estimte OD mtices fo evey o Te expeiment is cied ot in te folloing y Fist, link tvel time e given fo ll links t evey timeintevl t Link se pobbilities P (i, j) fo evey T= o e estimted by te Dil s ssignment on te tee-dimensionl netok fom (5) nd (4) Hee, te Dil s pmete is ssmed θ=000[/sec] Next, el OD demnd fo evey o is ssigned to ec link bsed on link se pobbilities P (i, j) Tese link se pobbilities nd link flos e sed in te model s inpt dt All link flos ssmed to be obseved nd oly OD flo is sed fo te pio mtices of te model Using tese dt, OD mtices ee estimted by poposed OD estimtion metod sing pmete γ = 0 We exmine te model bevio by comping OD estimtes it el OD mtices ssmed We exmine sevel cses on te diffeent conditions of obseved link flo nd pio OD flo Tee pttens e consideed on obseved link flo: no eo, mximm 0% eo, nd 20% eo In simil y, fo pttens e consideed on te pio OD flo: no devition, mximm 0% devition, mximm 20% devition, ppoximted pio OD onded off to te nded Fige 3 Test Netok Tble Reslt of Simltion pio obsevedcoeltion RMSE PRMSE OD eo coefficient (Ve/) (%) Cse ¾ ¾ Cse 2 ¾ Cse Cse 4 À Cse 5 À ¾ Cse 6 Á ¾ Cse 7 Â ¾ Cse 8 Â Cse 9 Â À pio OD A:no pio OD, B:no devition,c:mx 0%, D:mx 20%, E:ond off ndeds obseved eoa:no eo, B:mx 0%, C:mx 20%

7 OD volmes (Ve/) % 0% ssmed No devition Time (o) OD volmes (Ve/) % 0% ssmed No devition Time (o) ()No Link Obseved Eo Cses (b) No Pio OD Devition Cses Fige 4 Time Vition of Estimted OD flo digit We exmine nine cses in combintion it tese bove cses Te eslts of tese expeiments e son in Tble, ic descibes coeltion coefficients beteen estimted OD mtices nd ssmed el OD, Root Men Sqe Eos (RMSE ) nd Pecent Root Men Sqe Eos(PRMSE) Fist, let s see te coeltion coefficient beteen OD estimtes nd el OD mtices In te cse tt no pio OD flo is given (Cse ), te coeltion coefficient is vey lo ; oeve, e cn get te good eslt in te cses it pio OD flos (Cse2-9) ic gees it te sme eslt s pevios stdies on sttic OD estimtion In te Entopy Mximizing Metod, to get moe pecise OD mtices, it is mjo fcto to give moe pecise pio OD infomtion close to el OD mtices Secondly, let s exmine effect of eos obseved on link flos Te coeltion coefficient get loe nd RMSE(o PRMSE) inceses en obseved eo gets lge(cse 2,3,4 nd Cse 7,8,9) Hoeve, in te cse tt obseved eo is mximm 0%(Cse 3,8) te coeltion coefficient, RMSE nd PRMSE is lmost sme in compison it te cse tt obseved eo does not exist(cse 2,7) Altog e cnnot sy so definitely, it sos tt effect of obseved eos of link flos on te cccy of te model is not so lge if obseved eo is sfficiently smll Tidly, conside effect of devition of pio OD mtices Fo cses(cse 2,5,6,7) e tested in cses itot obseved link flo eo Fom tese eslts, te lge devition of pio OD mtices is, te ose cccy of te model is It sos tt devition in pio OD mtices ve n effect on te cccy of te model Fig4 indictes oly OD volmes fom O node 6 to D node Tis is te cse tt oly OD volmes is te lgest of ll OD pis in te netok nd its pio OD volmes ve positive devition fom ssmed ones Fig4() is te cse ee obseved link flo eo does not exist (Cse 2,5,6), nd (b) is te cse ee te devition of pio OD mtices does not exist (Cse 2,3,4) Tese figes indicte good fitness of time vition beteen estimted OD flos nd ssmed one Note tt te estimted OD flo vies nifomly in ccodnce it te devition of pio OD mtices in te cse of (), bt not nifomly in te cse of (b) CONCLUSION In te stdy, e popose te model estimting time-dependent OD mtices fom tffic conts in te genel netok Te model consists of to pts : () constction of te eltionsip beteen te time-dependent OD volmes nd tffic conts t links nd (2) estimtion of te time-dependent OD Fo te pcticl se, e popose te simplified metod of estimting ote coice pobbility nd te metod to se ggegted pio OD infomtion We pply te model to test netok Te min eslts tt ve been mde in tis exmples e s follos: ()We cn get te pefeble OD estimto in te cse tt pio OD mtices e given (2) It is ssmed tt effect of obseved eos of link flos on te cccy of te model is not so lge if obseved eo is sfficiently smll, bt devition of pio OD mtices ve n effect on te cccy of te model (3) We cn see good fitness of time vition beteen estimted OD flos nd ssmed one sing te model We cnnot mention te teoeticl stdy on estimtion eos o csed by link obseved eo, link se pobbilities eo nd so on So fte stdy on sc e mst be mde bsed on elibility nlysis of te model

8 REFERNCES Ngyen, S, Moello, E nd Pllottino, S Discete Time Dynmic Estimtion Model fo Pssenge Oigin/destintion Mtices on Tnsit Netoks Tnsp Res Vol22B No4, pp25-260, Yng, H Iid, Y nd Sski, T Dynmic Estimtion of Rel Time OD Tip Mtix fom Tffic Conts Poc of Infstcte Plnning No3, pp ,990(in Jpnese) 3 Cscett, E, Indi, D nd Mqis G Dynmic Estimtos of Oigin-Destintion Mtices Using Tffic Conts Tnsp Sci Vol27 No4, pp , Asok, K, Ben-Akiv, ME Dynmic Oigin-Destintion Mtix Estimtion nd Pediction fo Rel-Time Tffic Mngement Systems Tnsp nd Tffic Teoy, pp , Vn Zylen, HJ, Willmsen LG Te Most Likely Tip Mtix Estimted fom Tffic Conts Tnsp Res Vol4B, pp28-293, Oneym, H, K, M, Estimtion of Time Dependent OD Mtices fom Tffic Conts to be pblised in Tffic Engineeing (in Jpnese) APPENDIX Te, Lgngen of poblem (P), L, is: L = pk( ) log pk ( ) + C () p () + λ ( ) pk( ) θ k,,,, ( ) + µ () p() pk τ k () δ k, k, p k, e = + + λ µ τ δ Ptil deivtives of L it espect to bot nknon vibles, k( ) nd p ( ) L log pk ( ) k p ( ) θ k { } ( ) ( ( )) L p () () µ () = C + Ts, by Kn-Tcke s condition, { log p ( ) } λ ( ) µ ( τ ( k + + k ) ) δ k = 0 θ cn be deived if p k ( ) is positive Ts, ote coice pobbility of pt k is itten s: pk ( ) = exp θ C ( τ k ( ) ) δ k { ( ) } exp θ λ Using pk( ) =, te bove eqtion cn be eitten s: k exp{ θ C( τ ( )) } k δ k pk( ) = exp{ θ C( m( )) } m τ δ m exp{ θck( )} = exp θ C m { m( )} Tis eqtion coesponds to te ote coice pobbility defined in(4) Ts, it is poved tt te soltion of poblem (P) gives te ote coice pobbility defined in te fome section k k