TNK084 Traffic Thory sris Vol.4, numbr.1 May 008 Ral im simaion of raffic flow and ravl im Basd on im sris analysis Wi Bao Absrac In his papr, h auhor sudy h raffic parn and im sris. Afr ha, a im sris analysis and simaion ar mad. Kywords: Tim Sris, Esimaion I. Inroducion Ral-im and accura shor-rm raffic flow forcasing has bcom a criical problm in inllign ransporaion sysms. Basd on h hisorical and currn raffic flow daa, w can o sima h ravl im and raffic flows of shor-rm fuur, lik 5-30 minus is ofn rquird. In his projc, h raffic flows and ravl ims simaion and forcasing will b mad on h saisical aspc, mainly by h im sris analysis. Th aim of his projc is o idnify and analyz givn raffic daa by analyzing i as a im sris. Diffrn modls will b usd o dscrib h hisorical daa and currn daa, and hn mak som forcasing. By comparing h forcasing rsul o obsrvd valus, ry o figur ou h bs modl for a spcific road sgmn. II. Traffic parns A raffic sysm is a kind of muli-paricipaors, consanly changing, high-complxiy, non-linr sysm. On major characrisic of a raffic sysm is h high lvl of uncrainy wih rspc o raffic flows and ravl ims. Thr ar a lo of facors ha conribu o his uncrainy, for xampl, wahr and sasons, from an nvironmnal aspc; and raffic accidns and drivrs psychology, from a human aspc. Shor rm raffic forcasing is asily influncd by hs facors, vn du o h fac ha popl ar rying o avoid dlays and incidns. II.1. Traffic flows and ravl ims According our basic knowldg, ravl ims and spds ar a pair of rciprocal, and hr ar crain rlaionship bwn raffic flows and spd. Following figur dscrib h rlaionship vry wll: spd (km/ hour) spd/ f l ow 100 90 80 70 60 50 40 30 0 10 0 0 500 1000 1500 000 500 f l ow (vhi cl / hour) Figur 1, Spd/flow rlaionship of Snsor 1(3 lans), Tingsadsunnln E6, Göborg, 19h-nd, March, 001 Ths blu dos ar plod by h rcordd daa from h snsor, rd and yllow dos ar plod by h formula as (Equaion 1) k j q = ( u f u u ) u u f rd = 8 k j rd = 96 f u f yllow = 6 k j yllow = 115 II.. Sason, rnd and random componns Whn sudying h raffic daa in dail, i s asy o found ha h raffic daa has som spcialis lik sason, rnd and random componns. Sason: h way h daa changd priodically; Trnd: h way h daa ar likly o go; Random componns: daa changd wih ou sason or rnd aribus. Bcaus human lif is a 4 hours cycl aciv, h raffic daa also showd a 4 hours variy rpaing. Manwhil, h raffic daa showd som longr rm sason aribu, lik wk (Monday go o work wih high raffic flow in h ciy cnr, Sunday go o rlax caus high raffic flow in h rural ara) and monh(high raffic flow in h scnic spos on summr vacaion). Trnd is mor asily disinguishd from raffic daa, lik morning pak hour and afrnoon pak hour, raffic flows incras wih crain amoun, and 1
F. A. Auhor, S. B. Auhor hn dcrasd in h nigh. From a longr rm viw, a yar or dcad rnd will indica h sociy dvloping siuaion. Random componns in raffic daa ar mos common ons w can saw. As w discussd in h firs paragraph of chapr II, hr ar a lo of facors ha conribu o his uncrainy (or w calld random componns). Following figur illusra h sason/rnd/random clarly. h currn rsuls ar basd on h informaion known from h prvious im sp. Th rror rm is includd o h modl o capur nois in h daa. Th rror rm should hav h following propris, in ordr o mak a saionary modl: - h rror rm has a man of zro; - h rror rm has a consan varianc ovr im; - h rror rms corrsponding o diffrn poins in im ar no corrlad. In his projc, w will mak forcass of h fuur raffic flows daa basd on h hisorical and currn raffic flow daa. In his cas h ndognous modl is mor suiabl on doing his. If w also wan o includ mor informaion lik wahr or incoms, i would rquir an xognous variabl in h modl. Figur, on wk daa from Snsor 1(3 lans), Tingsadsunnln E6, Göborg, 19h-nd, March, 001 Basd on h figur, h sason and rnd could b disinguishd as following: Sason: 4 hours sason for vry whol day, from Monday (19 h, March 001) o Thursday ( nd, March 001). Trnd: Kp consanly in h middl nigh bfor 5:00 in h morning, hn incras sharply from 5:00 and rach h pak a 7:30, afr ha, h raffic flow falls down a bi and incras slighly unil rach h afrnoon pak a 16:00. Thn i dcras consanly unil o h middl nigh. Boh 4 days ac almos h sam during h 4 hours. III. Tim Sris A im sris is a s of obsrvaion x, ach on bing rcordd a a spcific im. A discr-im sris is on in which h s T0 of im a which obsrvaion ar mad is a discr s, as is h cas, for xampl, whn obsrvaion ar mad a fixd im inrvals. Coninuous-im sris ar obaind whn obsrvaion ar rcordd coninuously ovr som im inrval, g, whn T0=[0,1]. III.1. Exognous modl and Endogn modl Exognous modls ar modls ha includ indpndn variabls. An xampl of a xognous modl is: Y = a + bx + whr X is h indpndn variabl, a is a consan and is a random variabl. An ndogn modl includs only ndognous variabls. An xampl of an ndognous modl wih im lag on Y = a + by 1 + is: This can b inrprd as III.. Smoohning, saionary and auocorrlaion Smoohning is mhod o produc daa s is o cra a funcion ha amps o capur imporan parns in h daa, whil laving ou nois. On of h mos common algorihms is h "moving avrag", ofn usd o ry o capur imporan rnds in rpad saisical survys. Saionary is a sa ha h join probabiliy disribuion dos no chang whn shifd in im. For xampl, h whi nois is saionary procss. Auocorrlaion is h srngh of a rlaionship bwn obsrvaions as a funcion of h im sparaion bwn hm. Mor prcisly, i is h cross-corrlaion of a signal wih islf. III.3. AR modl and MA modl[5] If procss of rror rm { } is auo corrlad, w can build a modl for capuring his informaion, xampl: Y = by 1 + = p 1 + v, -1<p<1, Th modl = p 1 + v is calld a firs ordr auo rgrssiv procss, AR(1). Th paramr p can b simad by: y by = 1 1 p = 1 Auo rgrssiv modl of highr ordr can b consrucd as = p1 1 + p + v AR() Anohr yp of modl for h rror rm is a moving avrag modl. Hr, h rror rm is assumd v o b dscribd by a wighd sum of arlir rms
F. A. Auhor, S. B. Auhor = v dv 1, d is h wigh paramr Anohr modl yp can b consrucd by combining h AR and h MA modls in an ARMA modl: p + v dv = 1 1 IV. Th Hol-Winr algorihm [5] According o Hol-Winr algorihm, h prdicd valu (4) is basd on h lvl (1), h rnd () and h sason (3). x 1 α)( x + ) + α( x s ),0 < = ( 1 1 p (1) Th nw lvl is simad by wighing h arlir simad lvl and rnd in addiion o h currn obsrvaion and h sason sima wih paramr α = (1 β) 1 + β( x x 1),0 < β < 1. () Th rnd is simad by h wo arlir lvl simas s = (1 γ) s + γ( x x ),0 < γ < 1 p (3) Th sason sima is basd on prvious sason sima and h diffrnc bwn h simad and h obsrvd lvl. xˆ x + s + δ = δ + s p+δ (4) This givs h prdicd valus δ im sps ahad in im. V. Mhodology Gnral approach o im sris modling:[1] VI. Plo h sris and xamin h faurs of h graph, chcking in paricular whhr hr is (a) a rnd, (b) a sasonal componns, (c) any apparn sharp changs in bhavior, (d) any oulying obsrvaion. VII. Rmov h rnd and sasonal componns o g saionary rsiduals. To achiv his goal i may somims b ncssary o apply a prliminary ransformaion o daa. VIII. Choos a modl o fi h rsiduals, making us of various sampl saisics including h sampl auocorrlaion funcion. IX. Forcasing will b achivd by forcasing h rsiduals and hn invring h ransformaion dscribd abov o arriv a forcass of h original sris. X. Ralizaion In h ralizaion par, h raffic daa rcordd by hn snsor 1 in Tingsadsunnln E6, Göborg, during 19h-nd, March, 001 will b usd. Th α< sofwar ITSM 000 v7.1 sudn vrsion, which also b usd in h rfrnc [1] will b usd for pars of h analysis. X.1. Daa Enring and chcking By looking h xcl fil providd by suprvisor, I found ha boh h raffic flow and raffic spd wr rcordd by 5 minus inrval, which mans 88 daass in on day and 115 daass during h whol wk priod (19 h - nd,march). Bu h sudn vrsion of ITSM000-v7.1 has a limiaion of no mor han 51 daass, so I jus combin four 5 minus inrvals ino on 0 minus inrvals and only us h firs 3 days daa, which maks 16 daass aloghr. (In his projc, only h raffic flow daa will b analysis as im sris) Afr combining and nring h original raffic flow daa ino h im sris sofwar, w can s h figur as following: 1 6 0 1 4 0 1 0 1 0 0 8 0 6 0 4 0 0 0 4 0 8 0 1 0 1 6 0 0 0 Figur 3, Traffic flow daa figur By doing his, i s vry asy o poin ou h sasons and rnds aribus: a 4 hours sason and 4 major rnds in on day. X.. S r i s Daa pr-procssing In ordr o achiv h saionary sa, hos sasons and rnds aribus mus b rmovd from h daas. Th Classical Transform funcion of ITSM000 can b usd o achiv h saionary sa. Obviously h sason inrvals ar 7(41/4), so nr h sason priod:7 and choos Linr Trnd in Polynomial Fi, w can g h figur of saionary sa as: 0 1 0-1 0-0 0 4 0 8 0 1 0 1 6 0 0 0 S r i s Figur 4, Saionary sa X.3. Modl fiing Thr ways o find h suiabl for h saionary daas: by auo fi funcion or by looking ACF/PACF 3
F. A. Auhor, S. B. Auhor figur. An ARMA modl of paramr (p=4,q=4) is suggsd by h auo fi funcion. 1. 0 0 S a m p l A C F 1. 0 0 S a m p l P A C F From valu in abl 1, w bliv h forcas valu is qui clos o h masurmn daa, which mans a good modl has bn found.. 8 0. 8 0. 6 0. 4 0. 6 0. 4 0 X.5. Ral im. 0. 0 0 -. 0 -. 4 0 -. 6 0 -. 8 0-1. 0 0 0 5 1 0 1 5 0 5 3 0 3 5 4 0. 0. 0 0 -. 0 -. 4 0 -. 6 0 -. 8 0-1. 0 0 0 5 1 0 1 5 0 5 3 0 3 5 4 0 Figur 5, Sampl ACF and Sampl PACF I also can b s ha h Sampl ACF and Sampl PACF valus falls down afr 4 or 11. W choos an ARMA modl wih (p=4, q=4) o fi h original daa in his cas X.4. ARMA Forcasing[4] Onc h modl has bn fid, w can us h modl o forcas h fuur valu of daa. Figur 6 shows h forcasd valu of nx 4 hours by ARMA forcasing funcion: 1 6 0 1 4 0 1 0 1 0 0 8 0 6 0 4 0 So far, all h sudis ar sill basd on h hisorical daa, in a sld priod, bu h basic hory and approach ar almos h sam as in h ral im saus. All w nd o do in h ral im saus is o nring h currn raffic daa coninuously ino h original daas and run h forcas sp again, hn h nw forcas daa will com ou. In h raliy work, his could b don auomaically by a conrol compur conncing all h snsors. XI. Travl ims simaion To mos of h road usrs, ravl ims ar mor maningful o hm ohr han raffic flow or raffic spd. Onc h raffic flow is forcasd by our modl, w can us h Equaion 1 in Chapr II o calcula h raffic spd V. If w alrady h lngh of his road sgmn L, divid L by spd V, w can g a vry rough local ravl im T. If accura glob ravl ims simaion is ndd, hn w mus hav ravl flow forcass all ovr h nwork, which mans w nd larg moun currn raffic daa and build suiabl modl for vry road. This is a rally hard job, a br way o do his is build up a sysmic simaions modl. 0 0 5 0 1 0 0 1 5 0 0 0 5 0 Figur 6, 4 hours Forcas Ploing h forcas daa and masurd daa in nd, March in a sam figur, w can found h diffrnc bwn hm: 1800 1600 1400 100 1000 800 600 400 00 0 1 11 1 31 41 51 61 71 Figur 7, Diffrnc bwn masurmns and forcas Tabl 1 illusras h diffrnc vry wll, Masurmn Forcas Masurmn Forcas Diffrnc XII. Conclusion Alhough my rsul provd a nic simaion in spcify road, i sill has a long way o go o h pracical siuaion. To obain a good modl could dscrib h raffic parns, abundan original daa ar ncssary. Modl fiing and choosing ar h kys o rsul an accura simaion or a bad on, which mans popl mus vry familiar wih h ACF/PACF figur and hav nough xprinc in modl choosing. On inviabl shorcoming of im sris analysis and forcas in raffic aspc is ha i can nvr modl/forcas any raffic accidns. Acknowldgmns This work was suppord by Clas Rydrgrn Rfrncs Avrag Valu 65.97 687.805556 94.94% Sandard 464.81575 478.977401 97.04% Dviaion Corrlaion 0.9661 Tabl 1, Diffrnc valu 1. Pr J. Brockwll, Richard A. Davis, <<Inroducion o Tim Sris and Forcasing>>, scond diion. Charls W. Osrom,JR, <<Tim Sris Analysis>> 4
F. A. Auhor, S. B. Auhor 3. Sun Xianghai, Liu Tanqiu, <<A sudy on urban shor-rm raffic flow forcasing basd on a nonlinar im sris modl>> 4. HAN Chao, SONG Su, WANG Chng-hong, <<A Ral-im Shor-rm Traffic Flow Adapiv Forcasing Mhod Basd on ARIMA Modl>> 5. Clas Rydrgrn, cous slid of TNK084, Traffic Thory Tim consum Subjcs covrd and im plan: Saisics fundamnal: 10 hours Tim srious analysis sudying: 0 hours ITSM000 sofwar sudying: 5 hours Traffic daa analysis: 5 hours Modl buil and calibraion: 15 hours Forcasing and comparing: 10 hours Documn: 5 hours Toal work hours: 70 hours. 5