REAL-TIME ESTIMATION OF O-D MATRICES WITH PARTIAL TRAJECTORIES FROM ETC TAG DATA

Transcription

1 REAL-TIME ESTIMATION OF O-D MATRICES WITH PARTIAL TRAJECTORIES FROM ETC TAG DATA Jimyoung Kwon 1 Institut of Trnsporttion Stuis Univrsity of Cliforni, Brkly 6 Evns Hll, Brkly, CA Tl: (510)64-1; Fx: (510)64-9 kwon@stt.brkly.u Prvin Vriy Dprtmnt of Elctricl Enginring n Computr Scinc Univrsity of Cliforni, Brkly, Tl:(510)64-50; Fx:(510)64-15 vriy@cs.brkly.u Trnsporttion Rsrch Bor 4 th Annul Mting Jnury 005 Wshington, DC 400 wors (xcluing Figur cptions) Plus Figurs (150) totl 5950 wors 1 Corrsponing uthor

2 Kwon,Vriy 1 ABSTRACT Th origin-stintion (O-D) mtrix of trffic ntwork is usully stimt from link trffic counts combin with smpl survy. Prtilly obsrv vhicl trjctoris obtin with vhicl r-intifiction tchniqus such s lctronic tgs provi nw t sourc for rl-tim O-D mtrix stimtion. Howvr, u to incomplt smpling, ccurt stimtion of O-D mtrics from ths t is not trivil. W vlop sttisticlly soun, unbis stimtor for O-D stimtion, bs on th mtho of momnts. Th lgorithm prforms vry wll unr simultion, compr with simplr stimtors. Appli to t from vhicls with lctronic toll collction tgs in th Sn Frncisco By Ar, th lgorithm proucs rlistic tim sris of th hourly O-D mtrix. Kywors: rl-tim O-D mtrix stimtion; vhicl rintifiction; lctronic toll collction tgs; mtho of momnt stimtion

3 Kwon,Vriy 1 INTRODUCTION Knowlg of th origin-stintion (O-D) mtrix of trffic ntwork is usful for vrious plnning n oprtions tsks. Th mtrix givs th numbr of vhicls trvling btwn iffrnt zons in rgion. Tritionlly, th O-D mtrix is stimt from link trffic counts combin with smpl survy. Approchs to O-D mtrix stimtion using ths t inclu ntwork quilibrium, grvity, n istribution-ssignmnt mols (1). Vhicl loctions obsrv using vhicl r-intifiction tchniqus provi nw t sourc for rl-tim O-D mtrix stimtion. Tchniqus bs on vio imgry, inuctnc loop signturs, n lsrbs systms rport whn n whr crtin vhicl is obsrv (). Similr t my b obtin from vhicls with Elctronic Toll Collction (ETC) tgs n tg rrs instll t vrious loctions in th rgion. ETC tg systms in us inclu th E-Z pss in th Est Cost n FsTrk in th Sn Frncisco By Ar (). Bcus vhicl r-intifiction systms o not tct ll vhicls, vhicl trjctoris r only prtilly obsrv. Du to this incomplt smpling ccurt stimtion of O-D stimtion from such t is not sy, s will b m clr in Sction. This ppr proposs sttisticlly soun, unbis O-D mtrix stimtor, bs on th momnt stimtor from such t. Th rst of th ppr is orgniz s follows. Sction brifly scribs th ETC tg rr systm ploy in th By Ar n points out th smpling problms it crts. Sction scribs th sttisticl mtho in til. Sction 4 invstigts th prformnc of th propos lgorithm using simult t n rl t st from Sn Frncisco By Ar. Sction 5 conclus th ppr. BAY AREA ETC TAG READER SYSTEM Th FsTrk systm oprts ccoring to th Cltrns Titl 1 stnr (4). Th systm hs two lmnts, rr n trnsponr. Th rr is mount on pol, th trnsponr is mount on th vhicl winshil. Th rr trnsmits RF triggr puls to turn th trnsponr on. Aftr short tim ly, th rr trnsmits n nco, polling signl, which, upon tction n coing by th trnsponr, provis initil informtion to th trnsponr incluing th typ of trnsction th rr wishs to conuct. Th rr thn trnsmits n unmoult CW (continuous wv) RF signl for th trnsponr to moult with rply t mssg, incluing th trnsponr ID, whil bckscttring to th rr. Th rr thn trnsmits n nco cknowlg mssg n rqusts tht th trnsponr not rspon to th sm polling mssg gin for 10 scons (4). A rr systm srvs four trnsponr-ring loctions, ch with its own ntnn (5). In th By Ar ploymnt, ll four ntnns r mount on th sm sign structur, loct in th frwy ivir. Thr of th four ntnns point in on frwy irction th mjor irction, n on ntnn points in th opposit irction th minor irction. Th r covr in th minor irction, n th numbr of tct vhicls, is consquntly smllr. Th ntnns r mount t hight of btwn 16 n 4 ft, unlik t th toll booth whr th ntnn is clos to th vhicl. As rsult, only frction of tgg vhicls r tct. Th pssg of tgg vhicl gnrts rcor, comprising scrmbl vrsion of th tg (to protct privcy) n squnc of rr loctions whr th vhicl is tct n th corrsponing tim stmp. Th rsulting t st is incomplt for two rsons. First, th pntrtion rt, th frction of vhicls tht crry tgs, is btwn 15 n 40 prcnt, pning on th tim of y. Scon, th tction

4 Kwon,Vriy rt, th frction of tgg vhicls tht r tct by rr is bout 0% in th mjor irction n 40% in th minor irction. Consquntly, if tgg vhicl psss unr svrl rrs, only som of ths my tct it. A propr sttisticl nlysis must tk into ccount th incompltnss of th t. METHOD Consir rgion comprising svrl zons, inx j = 1,, J. For two zons or nos j n k, writ j k if j is immitly upstrm of k. This rltion fins n g in th frwy ntwork viw s irct grph (6). An O-D pir is ny pir [j, k], somtims writtn [jk]. Not tht j = k is llow, in which cs w simply writ [j]. A pth is squnc of nos (j 1,, j A ) connct by gs. An O-D pir [j, k] is trvrsbl if pth xists strting from j n ning t k. It is uniquly trvrsbl if such pth is uniqu. A pth contins nothr pth if th lttr is inclu in th formr. An O-D pir contins nothr O-D pir if ny pth tht contins th lttr is contin by pth for th formr. W ssum th frwy ntwork stisfis th following rquirmnts: 1. Th grph is connct, i.., thr is pth conncting vry pir of nos;. An O-D pir is uniquly trvrsbl if it is trvrsbl;.th grph hs no cycl,i.., pth which strts n ns t th sm no. W iscuss how to rlx ths rquirmnts in Sction 4. Inx tim prios, sy hours of th y by t = 1,, T. Dnot by N(t) th totl numbr of vhicls tht mk trip uring t. Inx vhicls by i = 1,, N(t). Ths vhicls r group into isjoint sts of vhicls corrsponing to iffrnt O-D pirs. Th O-D volum btwn zons j n k uring prio tis not by N jk (t). At t th O-D mtrix is simply {N jk (t), j, k = 1,, J}. Lt ψ b th pntrtion probbility, i.., th probbility tht n iniviul vhicl is quipp with tg. Lt π j b th probbility tht cr with tg trvling th link j is tct by th tg rr. W ssum tht for vry vhicl, rgrlss of its O-D, th vnt tht it is quipp with tg n th vnt tht it is tct r inpnnt. Although our nrrtion is in trms of lctronic tgs, similr formultion pplis with othr vhicl rintifiction tchniqus. For n O-D pir [j, k], w fin M[jk] = Numbr of crs tct t j n k but not bfor j nor ftr k. (1) Not tht trms lik bfor (ftr) zon r wll fin thnks to rquirmnts bov on th frwy ntwork. W cn rwrit M[jk] s th sum M[jk] = 1(vhicl i hs O-D pth [l, m] () [l,m] i n vhicl i is quipp with tg n vhicl i is tct t j n k but not bfor j nor ftr k) ovr ll vhicls i ovr ll O-D pirs [lm] tht contin [jk]. Th nottion 1( ) is th inictor function. Although th ccounting intity () pprs complict, it llows th computtion of vrious momnts in xplicit form. Th xpcttion is givn by EM[j, k] = {N lm p l(jk)m : ovr ll O-D pirs [l, m] tht contin [j, k]}. ()

5 Kwon,Vriy 4 Hr p l(jk)m = (1 π l ) (1 π j 1 )π j π k (1 π k+1 ) (1 π m ) (4) is th probbility tht vhicl with pth [l, m] is tct t j n k but not bfor j nor ftr k. Similrly, th lmnts of th vrinc-covrinc mtrix of th lmnts of M r Cov(M[j, k], M[j, k ]) = {N lm ψ [ 1([jk] = [j k ] ])p l(jk)m p l(jk)m p l(j k )m (5) : ovr ll O-D pirs [l, m] tht contin both [j, k] n [j, k ]} Dfin th vctors M = (M[jk] : for ll O-D pirs [jk]), (6) N = (N[jk] : for ll O-D pirs [jk]). () For lrg numbr of vhicls N jk, by th Cntrl Limit Thorm, M pproximtly hs multivrit norml istribution, M N(AN, Σ(N)), () in which th mtrix A n th function Σ( ) r spcifi by () n (5) rspctivly. Sinc hols, th mtho of momnt stimtor () of th unknown N is givn by E(M) = AN (9) η = A 1 M. (10) Th pproximt rror istribution of th stimtor η cn b obtin from ()-(10) s η N(N, A 1 Σ(N)(A 1 ) T ). Computing Σ(N) is complict, n so w us th bootstrp procur () to stimt th rror of th stimtor. Givn n stimt η, on simults bootstrp smpls M (b), b = 1,, B, for lrg B, ssuming η to b th tru prmtr. For ch of ths bootstrp smpls, on stimt η (b) gin n uss th bis n stnr rror of th bootstrp smpl s n stimt of th bis n stnr rror of th stimtor itslf. ANALYSIS W pply th lgorithm to simult t st s wll s th rl ETC tg t from Sn Frncisco By Ar. Simultion Consir rgion with thr zons j = 1,,. Th frwy ntwork is fin by gs 1 n. Th six O-D pirs r list in Tbl 1, togthr with th pths contining thm. Evn for this simpl ntwork, ll O-D pirs xcpt [1, ] r contin in multipl pths. Assum th tction probbility is 0.5 for ll thr rrs, i.. π 1 = π = π = 0.5 n N jk = 1, 000 for ll O-D [j, k]. Also ssum ll vhicls r quipp with tg, so th pntrtion rt ψ = 1. W simult M jk s th sum of bionmil rnom vribls with istribution bin(n lm, ψp lm ), summ ovr ll O-D pir [l, m] contining pth [j, k].

6 Kwon,Vriy 5 For comprison, w lso comput th niv stimtor η jk = Numbr of vhicls tht r obsrv t both j n k π j π k. (11) Th niv stimtor givs rough i bout th numbr of vhicls tht trvl btwn two loctions, but it is obviously incorrct for stimtion of N. Figur 1 n Tbl show th istribution of th stimts from th two mthos stimt with 00 simultion runs. Momnt stimtors r vry ccurt, xhibiting lss thn 1% bis for ll O-D pirs. Th rltiv rrors r lss thn 10%. As xpct, th niv stimtor prforms vry poorly by comprison. Excpt for O-D pir [1, ], it hs hug biss. On th othr hn, thir stnr rrors r of similr mgnitu s thos of th momnt stimtors. To summriz, niv pproch cnnot b us for nlysis of O-D pttrns, whil th momnt stimtor provis vry ccurt stimts for ll O-D pirs. By Ar ETC Dt Figur shows th topology of th frwy ntwork n ntnn loctions, which cn b viw s zon lbls. Th t wr collct ovr 4-hours of Thursy, 4 July, 00. Th currnt rr covrg is incomplt. It only inclus th northrn prt of Est By n smll prt of Sn Frncisco. Thus, for xmpl, zon inclus not only th r btwn zons n 9 but lso th hug r tht lis byon 50 Wst. Th rr covrg is bing xtn. Th grph rprsnttion shows ll th nos n gs. If w llow trips btwn nos (st n wst) n 5 (north n south), not by rrows in sh lins, thr r cycls (E 5N E n W W 5S), so w o not llow ths trips. For illustrtion, w consir only stboun/southboun trffic involving nos 1E, E, E, 9E, 5S, n E (gry nos). Th grph is thn compltly scrib by fiv gs: 1 5, 1,,, n 9. Thr r J = 6 nos n 15 trvrsbl O-Ds. For ch O-D (pth), th st of contining O-Ds (pths) is shown in Tbl. Not tht th pth-contining rltionship is much mor complict thn th ntwork consir for simultion. Th pntrtion rt ψ is tkn to b 15% n w us this vlu in th ppliction. Th tction rt prmtrs π j r stimt using th totl volum t obtin from th loop tctors instll t th loction clos to ch ntnns. W us th PMS systm () to obtin th loop t. From ths t, w fin π j = 0. for ll loctions. W first pply th lgorithm to stimt th O-D mtrix for th whol 4-hour prio. Tbl 4 shows th stimts n th bootstrp stimts of its bis n vrinc. Most biss r controll unr 1% of th stimts n th rltiv rrors unr 5%, xcpt for O-Ds [1, 1] n [, ]. Figur shows th bootstrp istribution of th stimts. W thn stimt hourly O-D mtrics by splitting smpls into hours ccoring to th vhicl s lst obsrv timstmp n pplying th lgorithm for ch hourly smpls. Figur 4 show th rsult. Th trn clrly rvls th chnging pttrn of vhicl trvl mn for ch O-D pir ovr th y. For xmpl, th O-D [1, ] (Sn Frncisco to Est By) [5] (Est By to Sn Jos) hs highr mn in th morning whil th O-D [] (Est By to Richmon) n [9] (Northst By to Th Crquins brig) hs highr mn in th vning. 5 DISCUSSION AND CONCLUSION W vlop mtho of momnts lgorithm for stimting tim-vrying OD mtrics from prtilly obsrv vhicl trjctoris t, togthr with bootstrp procur to stimt th stnr rror of th stimtor.

7 Kwon,Vriy 6 W rstrict our ttntion to frwys with rltivly simpl topology, in which th corrsponing grph is connct, ll trvrsbl O-Ds r uniquly trvrsbl, n thr is no cycl. Ths rquirmnts r too strict for mny rl worl frwy ntworks n cn b rlx with vrying ifficulty. For isconnct grph, on only ns to compos it into connct componnts n pply th lgorithm to ch componnt. For non-uniquly trvrsbl O-Ds, th ccounting qution () ns to b ppropritly moifi. If th rout choic probbilitis r known, such moifiction is rltivly strightforwr, but if thy r unknown, thy bcom xtr prmtrs in th mol n th xtnsion is not trivil nymor. Whn thr r cycls, on promising pproch woul b to rmov non-likly pths from consirtion to ruc th imnsion of M n N n simplify A. Ovrll, ths r ll intrsting chllngs n w r currntly working on xtning th currnt lgorithm to cop with such complx ntworks. Th propos lgorithm prform wll. Th simultion stuy shows tht th propos momnt stimtor n bootstrp stnr rrors r ccurt unr il sttings. Th ppliction to t from vhicls quipp with ETC trnsponrs n rrs instll t vrious loctions in Sn Frncisco By Ar, ls to cribl hourly OD mtrics. Combin with th wir ploymnt of tg rrs n grtr pntrtion of tgs in th vhicl popultion, th propos mthos woul nbl istrict-wi O-D rl-tim rporting in timly n ccurt mnnr. ACKNOWLEDGEMENT This stuy is prt of th PMS projct, which is support by grnts from Cltrns to th Cliforni PATH Progrm. W r vry grtful to John Wolf of Cltrns for his support. W thnk th Mtropolitn Trnsporttion Commision n Cho Chn for hlp in scuring th ETC t. Th contnts of this ppr rflct th viws of th uthors who r rsponsibl for th fcts n th ccurcy of th t prsnt hrin. Th contnts o not ncssrily rflct th officil viws of or policy of th Cliforni Dprtmnt of Trnsporttion. This ppr os not constitut stnr, spcifiction or rgultion. REFERENCES [1] T. Abrhmsson. Estimtion of Origin-Dstintion Mtrics using Trffic Counts A Litrtur Survy. Tchnicl Rport IR-901, Intrntionl Institut for Appli Systms Anlysis, ccss /19/004. [] C. Oh, S.G. Ritchi, n R. Jykrishnn. Rl-Tim Origin-Dstintion (OD) Estimtion vi Anonymous Vhicl Trcking. Tchnicl Rport Ppr UCI-ITS-TS-WP-0-0, Cntr for Trffic Simultion Stuis. Univrsity of Cliforni, Irvin., 00. [] Fstrk wbsit. ccss /19/004. [4] Cliforni Dprtmnt of Trnsporttion. lcsys/titl1/ocs/t1upt.htm, ccss /19/004. [5] Sirit Tchnologis. Automtic Vhicl Intifiction. Intity Flx Instlltion Mnul, n. [6] S. Evn. Grph Algorithms. Computr Scinc Prss, 199. [] P. Bickl n K.A. Doksum. Mthmticl Sttistics: Bsis Is n Slct Topics. Prntic-Hll, 000.

8 Kwon,Vriy [] PMS wbsit.

9 Kwon,Vriy LIST OF TABLES 1 O-D pth n pths contining th O-D pth for th simult linr frwy Bis n stnr rror of th Momnt n Niv Estimtors from simultion with 00 runs whn tru N = 1, O-D pth n pths tht contining th O-D pth for th By Ar ntwork Estimt of 4-hour O-D mtrix n stnr rrors from By Ar ETC t for 4 July,

10 Kwon,Vriy 9 LIST OF FIGURES 1 Simultion rsults for mtho of momnt stimtor (top) n niv stimtor (bottom). For ch O-D pir, th istribution of th stimtor is shown s boxplot. Th vrticl lin is th tru vlu N jk = 1, Mp of th Stuy Sit in Sn Frncisco By Ar with pproximt loction of th tg rrs (lft) n grph rprsnttion of th frwy Momnt stimtor n bootstrp istribution of th stimtor for By Ar t Estimt of hourly O-D volums for By Ar t. Two blck soli lins corrspon to O-D pirs (1,) n (5) tht hv highr morning trffic n blck ott lins corrspon to () n (9) tht hv highr vning trffic

11 Kwon,Vriy 10 TABLE 1 O-D pth n pths contining th O-D pth for th simult linr frwy O-D pth Pths contining th O-D pth (1) (1), (1,), (1,,) (1,) (1,), (1,,) (1,,) (1,,) () (1,), (1,,), (), (,) (,) (1,,), (,) () (1,,), (,), ()

12 Kwon,Vriy 11 TABLE Bis n stnr rror of th Momnt n Niv Estimtors from simultion with 00 runs whn tru N = 1, 000 Bis SE O-D Momnt Niv Momnt Niv pir Estimtor Estimtor 1, , , , , ,

13 Kwon,Vriy 1 TABLE O-D pth n pths tht contining th O-D pth for th By Ar ntwork O-D pth Pths contining th O-D pth (1) (1), (1,), (1,,), (1,,,9), (1,,), (1,5) (1,) (1,), (1,,), (1,,,9), (1,,) (1,,) (1,,), (1,,,9) (1,,,9) (1,,,9) (1,,) (1,,) (1,5) (1,5) () (1,,), (1,,,9), (), (,9), (,), (,,9) (,9) (1,,,9), (,9), (,,9) () (1,), (1,,), (1,,,9), (1,,), (), (,), (,,9), (,) (,) (1,,), (1,,,9), (,), (,,9) (,,9) (1,,,9), (,,9) (,) (1,,), (,) (5) (1,5), (5) () (1,,), (,), () (9) (1,,,9), (,9), (,,9), (9)

14 Kwon,Vriy 1 TABLE 4 Estimt of 4-hour O-D mtrix n stnr rrors from By Ar ETC t for 4 July, 00 O-D Momnt Bootstrp Bootstrp Bootstrp Bootstrp pth Estimtor Bis Rltiv S.E Rltiv (vhicls/y) Bis Error (1) (1,) (1,,) (1,,,9) (1,,) (1,5) () (,9) () (,) (,,9) (,) (5) () (9)

15 Kwon,Vriy 14 Momnt Estimtor,, O D pir, 1, 1, 1, Niv Estimtor,, O D pir, 1, 1, 1, FIGURE 1 Simultion rsults for mtho of momnt stimtor (top) n niv stimtor (bottom). For ch O-D pir, th istribution of th stimtor is shown s boxplot. Th vrticl lin is th tru vlu N jk = 1, 000.

16 Kwon,Vriy 15 Rt 4 (9) 0 9W 9E 50 I 50 split () 4/90 split () 4 50 W E W E 50 E&W split () split (5) W E 5N 5S By Brig 1W 1E 5th St (1) FIGURE Mp of th Stuy Sit in Sn Frncisco By Ar with pproximt loction of th tg rrs (lft) n grph rprsnttion of th frwy.

17 Kwon,Vriy ,,,9, O D,9 1,5 1,, 1,,,9 1,, 1, Volum FIGURE Momnt stimtor n bootstrp istribution of th stimtor for By Ar t.

18 Kwon,Vriy 1 Estimt O D Counts (vh/hour) , 1,, 4 1,,,9 5 1,, 6 1,5,9 9 0,,,9 b, c 5 9 c c c 9 c c c c c c c c c c c c b c b 6 6 b b c c c b 9 4 b b b c b b b b b b b b b c c c b b b b b b Tim of Dy (Hour) FIGURE 4 Estimt of hourly O-D volums for By Ar t. Two blck soli lins corrspon to O-D pirs (1,) n (5) tht hv highr morning trffic n blck ott lins corrspon to () n (9) tht hv highr vning trffic.