Traffic State Estimation from Aggregated Measurements with Signal Reconstruction Techniques

Transcription

1 raffc Sae Esmaon from Aggregaed Measuremens wh Sgnal Reconsrucon echnques Vladmr Corc, Nemanja Djurc, and Slobodan Vucec he esmaon of he sae of raffc provdes a dealed pcure of he condons of a raffc nework based on lmed raffc measuremens and, as such, plays a key role n nellgen ransporaon sysems. Mos of he esng sae esmaon algorhms are based on Kalman flerng and s varans, whch, sarng from he curren esmae, predc he fuure sae and hen correc on he bass of new measuremens. Mos ofen, raffc measuremens are aggregaed over mulple me seps, and hs procedure rases he queson of how o bes use hs nformaon for sae esmaon. A sandard approach ha performs he correcon only a he me sep when he aggregaed measuremen s receved s subopmal. Reconsrucng he hgh-resoluon measuremens from he aggregaed ones and usng hem o correc he sae esmaes a every me sep are proposed. Several reconsrucon echnques from sgnal processng, ncludng kernel regresson and a reconsrucon approach based on conve opmzaon, were consdered. he proposed approach was evaluaed on real-world NGSIM daa colleced a Inersae, locaed n Los Angeles, Calforna. Epermenal resuls show ha sgnal reconsrucon leads o more accurae raffc sae esmaon as compared wh he sandard approach for dealng wh aggregaed measuremens. I has been esmaed ha raffc congeson coss he world economy hundreds of bllons of dollars each year, ncreases polluon, and has a negave mpac on he overall qualy of lfe n meropolan areas. In order o solve hs emergng problem, ransporaon deparmens ncreasngly rely on sysems for real-me raffc conrol and managemen known as nellgen ransporaon sysems (IS). In addon o real-me operaons managemen, IS are valuable as a source of daa for ransporaon plannng and raveler nformaon sysems. he success of he IS grealy depends on he ably o measure raffc condons and esmae raffc saes a a fne spaal and emporal scale. Because of he mporance of raffc sae esmaon, he ransporaon research communy has developed and evaluaed many algorhms for esmaon of raffc varables such as raffc flow, speed, and densy. he spaoemporal evoluon of hese raffc quanes s descrbed by usng deermnsc raffc models and s esmaed Deparmen of Compuer and Informaon Scences, emple Unversy, 6 Wachman Hall, 85 Norh Broad Sree, Phladelpha, PA 922. Correspondng auhor: V. Corc, vladmr.corc@emple.edu. ransporaon Research Record: Journal of he ransporaon Research Board, No. 235, ransporaon Research Board of he Naonal Academes, Washngon, D.C., 2, pp. 2. DOI:.34/235-3 from measuremens wh he Kalman fler (KF) algorhm and s eensons (). For nsance, one of he mos commonly used raffc models for raffc sae esmaon s he cell ransmsson model (CM) (2, 3), he frs-order raffc model ha can be used for esmaon of raffc densy (4) or raffc speed (5). Snce raffc models are ypcally hghly nonlnear, he basc KF, desgned for lnear problems, canno be used. hs problem promped developmen of mehods for nonlnear sae esmaon ha are eensons of he basc Kalman flerng dea. For nsance, he eended KF (6) was successfully mplemened n RENAISSANCE, a real-me freeway raffc nework survellance ool (7 ), as well as n he second-order raffc flow model MEANE (8). When he raffc model s hghly nonlnear, alernave approaches such as unscened KFs (9), mure KFs (4), parcle flers (), and ensemble KFs (5) have been shown o perform que well. All KF-based approaches for raffc sae esmaon conss of predcon and correcon seps. In he predcon sep, sarng from he curren sae esmae, he raffc sae n he ne me sep s predced by he raffc model. In he correcon sep, f new measuremens are avalable, he predced raffc sae s correced accordng o he measuremens. However, he problem s ha he raffc measuremens provded by many ypes of sensors, ncludng he ubquous loop deecors, are mos ofen aggregaed across mulple me seps, makng dffcul o know how o use hem for sae esmaon. A sandard way o address hs ssue s o use he aggregaed measuremens for correcon only a he me seps when hey become avalable and o no correc he predced saes n he remanng me seps. However, hs approach s subopmal snce could lead o large uncerany and low accuracy of sae esmaon. o solve hs problem, Schreer e al. recenly proposed usng he aggregae measuremens for correcon a all me seps durng he aggregaon perod by posulang ha measuremens a all me seps are equal o he correspondng aggregaed measuremens (). Alhough hs approach can be very successful when raffc s n he free-flow regme, s no he bes choce durng he congesed regme or durng ransons beween he regmes. he use of sgnal reconsrucon echnques s eplored o re-creae he measuremen a every me sep from he avalable aggregaed measuremens. In hs lgh, he approach proposed by Schreer e al. can be nerpreed as sepwse reconsrucon of orgnal measuremens wh he aggregaed measuremens (). In addon o he sepwse reconsrucon, a few more sgnal reconsrucon algorhms wh ncreased compley, rangng from lnear nerpolaon o splne appromaon o reang sgnal reconsrucon as a conve opmzaon problem, are eplored. Fnally, all hese approaches are evaluaed on he hgh-qualy NGSIM raffc daa. 2

2 22 ransporaon Research Record 235 Problem Descrpon he followng nonlnear dynamcal sysem s consdered: z = M ( ) + = H ( ) + η ξ () where = vecor of sae varables, z = vecor of measuremens a me sep, mappng M = sae ranson model, H = measuremen model, η = sae process Gaussan nose wh zero mean and covarance Q, and ξ = observaon Gaussan nose wh zero mean and covarance R. In he raffc doman, he vecor represens a vecor of raffc varables such as speed, densy, or flow, and vecor z represens sensor readngs, n mos cases of speed or flow. Accordng o he CM approach (2, 3, 5), whch s used here, roads n he raffc nework are dvded no spaal cells of arbrary bu preferably equal or smlar lenghs. In he case of Euleran raffc sensors, such as fed loop deecors, cells are ypcally posoned n such a way ha he sensors are locaed a her downsream end and only a small subse of cells conans a sensor. he me s dscree, dvded no seps whose lengh s consraned by he Couran Fredrchs Lewy condons, sang ha a movng vehcle canno raverse more han one spaal cell durng one me sep (2). For eample, f he free-flow speed equals km/h and a srech of freeway s dscrezed no spaal cells each m long, he me sep mus no be longer han 6 s. he raffc sae esmaon problem s o esmae a sequence of rue saes, whch are no drecly observable, gven a sequence of measuremens z. If we choose funcons M() and H() o be lnear, he well-known KF closed-form soluon can be used for hs problem (); oherwse, here are many approaches ha have been proposed o deal wh he nonlneary of ranson or observaon funcons, or boh (5, 6, 9). All KF mehods work n a smlar way, by eravely performng predcon and correcon seps. In he predcon sep, he ne sae s predced gven knowledge abou he curren sae, and n he correcon sep, measuremens z of he curren sysem sae are used o reesmae (.e., correc) he predcon. For varous reasons, observaons mgh no be avalable a every me sep, n whch case only he curren sae can be predced whou he correcon sep. he dynamcal sysem (Equaon ) mples ha he measuremens z are obaned durng me sep. However, raffc measuremens are usually aggregaed over a perod of me o accoun for he nheren sgnal nose or o allow easer ransmsson and sorage of large amouns of measured daa. he aggregaed measuremens are repored every Δ me seps, where Δ s referred o as he aggregaon perod. As a resul, nsead of he observaons z, only he aggregaed measuremens y a me seps ha are mulples of Δ, {Δ, 2Δ,...} are avalable. Vecor y avalable a me s an aggregaon of z -values durng he prevous Δ me seps, Δ <. For nsance, n case of he volume measuremen, he aggregaed volume s he sum of all volume measuremens durng he aggregaon perod, and n he case of he speed measuremen, he aggregaed speed can be he average speed durng he aggregaon perod. o consder he second case n more deal, he aggregaed speed can be calculaed as follows: y = z = + Combnng Equaons and 2 resuls n y = H ( ) + = + = + ( 2) ξ () 3 where, by assumng ha he observaon nose ξ s sampled ndependen and dencally dsrbued, follows ha he nose varance drops by a facor of /Δ as compared wh he orgnal measuremens. Durng free flow, hs resul has a posve effec snce H( ) s sable and measuremen nose s reduced. However, durng he ranson or congeson perods, he gan n measuremen nose s offse by a loss of nformaon abou he fne-scale changes n he raffc sae. Fgure llusraes he measuremen aggregaon. Insead of he rue measuremen sgnal durng perod Δ, only s mean value y a he end of ha perod s gven. 35 y - rue measuremens sepwse reconsrucon aggregaed measuremen 25 y me sep FIGURE Problem descrpon.

3 Corc, Djurc, and Vucec 23 he man queson when one s dealng wh he aggregaed measuremens s how o use hem for sae esmaon. he problem wh mssng and aggregaed measuremens has also been recognzed n oher felds, such as hydrology and meeorology (3), where KF equaons are modfed o ncorporae aggregaed measuremens. In economcs, he same problem occurs when aggregaed unvarae (4) and mulvarae (5) economc me seres are analyzed, and s addressed by augmenng he sae space. However, sae space augmenaon would be oo cumbersome and oo compuaonally cosly for raffc sae esmaon, consderng he fac ha spaoemporal raffc models are nonlnear and hghly dmensonal. herefore, n hs work a much smpler, bu sll effecve, sgnal reconsrucon approach for dealng wh aggregaed raffc measuremens s eamned. he reconsruced sgnal a me sep s denoed ẑ, and he reconsrucon funcon A(), such ha zˆ A( y, ) ( 4) where y s he se of aggregaed measuremens unl me, y = {y,..., y }, such ha. he reconsruced measuremens ẑ are used n he KF correcon sep nsead of he unknown rue measuremens z defned n Equaon. Dependng on he form of he funcon A(), dfferen reconsrucons of he aggregaed measuremen can be obaned, and n he followng wo secons several dfferen choces for he reconsrucon funcon are descrbed. Esng Mehods A ypcal way of dealng wh he aggregaed measuremens s o apply he correcon sep only a he me seps n whch he aggregae measuremens become avalable and o use hem drecly as measuremens, ẑ = y, =. Consequenly, he aggregaed measuremen s used o correc he sysem sae a he end of he correspondng aggregaon perod, whereas durng he remanng Δ me seps only he predcon seps are used. hs mehod s referred o as he classc approach, snce s he mos commonly used approach n esng raffc sae esmaon algorhms. For hs approach, he reconsrucon funcon s defned as follows: y f = A( y, ) = ( 5) NaN oherwse here are several ssues wh hs mehod. Frs, z a me seps = Δ, 2Δ,..., KΔ does no necessarly equal y. Second, nformaon abou y s used a only one of he Δ seps, alhough y conans nformaon abou he aggregaed values a all Δ seps. As a resul, he sae esmaon wll be subopmal, characerzed by sudden jumps n he esmaes and by ncreasng uncerany a me seps beween he correcon seps. o address hs problem, Schreer e al. () recenly proposed a mehod o reconsruc all z vecors durng he aggregaon perod. Specfcally, nsead of usng he aggregaed measuremens y o correc only he predced sae a me sep, as n he classc approach, he mehod correcs all prevously predced saes. he reconsrucon funcon by Schreer e al. () s defned as follows: ( ) = < 6 A y, y ( ) and represens a sepwse reconsrucon of he aggregaed sgnal (see Fgure ). he auhors showed epermenally ha hs approach ouperforms he classc approach from Equaon 5. he sepwse reconsrucon from Equaon 6 s characerzed by poenally large dsconnues afer every aggregaon perod. hs feaure could lead o subopmal raffc sae esmaon, especally durng he ranson perods when he raffc undergoes regme changes or whn congesed perods. hs mehod, unlke he classc approach, ehbs esmaon delay of one aggregaon perod, snce was unl he new aggregaed measuremen y s acqured a me sep o reconsruc he measuremens z and esmae he raffc saes a me seps Δ <. hs mehod s referred o as he sepwse approach. he dea o reconsruc he orgnal measuremens and use hem n he correcon sep of a KF can be furher mproved by employng more sophscaed sgnal reconsrucon echnques. Several approaches for sgnal reconsrucon and how hey affec he accuracy of raffc sae esmaon from aggregaed measuremens are suded here. Sgnal Reconsrucon from Aggregaed Measuremens he sepwse approach from Equaon 6 s he smples measuremen reconsrucon scheme from he aggregaed daa. Now several more sophscaed sgnal reconsrucon approaches wll be consdered, ncludng hree sandard pecewse nerpolaon echnques (lnear, cubc splne, and cubc Herme splne nerpolaon), as well as kernel regresson and a conve opmzaon approach. Durng he nerpolaon process for all approaches oher han he conve opmzaon, he aggregaed measuremen y wll be assumed o be n he mddle of he aggregaon perod (Fgure 2a). More formally, when a me sep, s assumed ha y was obaned a me sep Δ/2. Wh hs assumpon, y can be reaed lke a sample from he measuremen me seres {z } flered wh a cenered wndow of lengh Δ. he followng sgnal reconsrucon echnques were suded. Pecewse Lnear Inerpolaon Lnear appromaon by a sragh lne beween he aggregaed measuremens (Fgure 2) s ( ) = + ( ) A y, y y y for 3 2, ( 7) I should be observed ha when y +Δ becomes avalable, he sgnal s reconsruced whn he nerval ( Δ/2, + Δ]. As a resul, he old sgnal reconsrucon n he nerval ( Δ/2, ] wll be modfed (compare hs nerval n Fgure 2, a and b), whch ypcally resuls n a more accurae reconsrucon. hs observaon wll be mporan when he delay allowed n he raffc sae esmaon s dscussed n he followng secon. Pecewse Cubc Splne Inerpolaon As an alernave o nerpolaon by a sragh lne, cubc splne nerpolaon reconsrucs he sgnal as pecewse cubc splnes. Splne nerpolaon uses low-degree polynomals (of degree 3 n hese

4 24 ransporaon Research Record y - rue sae lnear reconsrucon sepwse reconsrucon y + y me sep (a) 35 y - rue sae sepwse reconsrucon lnear reconsrucon y + y me sep (b) FIGURE 2 Pecewse lnear nerpolaon approach: (a) afer recevng measuremen a me and (b) afer recevng measuremen a me D.

5 Corc, Djurc, and Vucec 25 epermens) n each of he aggregaon nervals and chooses hem such ha hey f smoohly ogeher. Gven aggregaed measuremens, he splne funcon fs hese pons wh he splne curve such ha s made ou of cubc polynomals of he followng form: ( ) = = 2 3 f a b c d 2,,..., ( 8) where a, b, c, and d are coeffcens of he h polynomal found durng ranng. he reconsrucon funcon s gven by A y, f f, () ( ) = ( ) + where s defned as n Equaon 8. he sepwse mehod s a specal case of hs approach, obaned by pecewse -degree polynomals. Also, upon recevng new aggregaed measuremen y +Δ, he whole sgnal from me = unl me = + Δ needs o be reconsruced. Cubc Herme Splne Inerpolaon Cubc Herme splne nerpolaon s smlar o he prevously nroduced cubc splne nerpolaon, he dfference beng ha he polynomals (Equaon 8) are n he Herme form (6). Kernel Regresson Approach Kernel regresson s a common smoohng echnque (7 ) ha reconsrucs a sgnal as a weghed average of he neghborng aggregaed measuremens: ( ) = A y, y k, ( ) ( ) k, { }, 2,..., ( ) where k(,) s a kernel funcon. he common choce for he kernel funcon s he Gaussan kernel (used here), defned as follows: ( ) k (, ) = ep 2 ( ) 2 σ where σ s a kernel wdh parameer. For smaller values of σ only he closes neghbors are consdered, whereas for larger σ he reconsruced sgnal becomes very smooh. Conve Opmzaon Approach Fnally, he nerpolaon problem s defned as a conve opmzaon problem. Conve opmzaon relaes o a class of nonlnear opmzaon problems n whch boh he objecve o be mnmzed and he consrans are conve. Conve opmzaon problems are aracve because a large class of hese problems can now be effcenly solved (8). Gven he se of aggregaed measuremens y, s proposed o fnd smooh esmaes ẑ by solvng he followng conve problem: mn zˆ zˆ ẑ 2 ( + ) = subjec o zˆ = =,..., ( 2 ) y j j ( j ) + = j he objecve funcon ensures ha he appromaon s as smooh as possble, and he consran ensures ha he mean value of esmaed measuremens whn every aggregaon nerval s equal o he correspondng aggregaed measuremen. In hs case, here s no analycal form for he reconsrucon funcon A(); raher, he soluon of he opmzaon problem s used nsead: ( ) = 3 A y, zˆ ( ) where, wh a slgh abuse of noaon, ẑ s he h elemen of he soluon of he conve problem (Equaon 2). Algorhm o dscuss he echncal deals of he raffc sae esmaon algorhm based on Kalman flerng, whch employs he proposed sgnal reconsrucon echnques, wll be assumed ha he newes aggregaed measuremen was receved a me and ha he measuremen vecors z have been reconsruced up o me pon. I wll also be assumed ha all he esmaed saes up o me pon are saved. Once he aggregae a me + Δ becomes avalable, he reconsrucon of he enre measuremen sequence s repeaed up o me pon + Δ. In he case of he sepwse reconsrucon, he reconsruced ẑ a me seps < wll reman unchanged. herefore, Kalman flerng s performed from me sep o + Δ by usng he reconsruced sgnal ẑ. Wh all oher reconsrucon echnques proposed n he prevous secon, he reconsruced ẑ a me seps < wll be modfed. Snce he saes durng me perod [ Δ, ] were esmaed by usng he old reconsruced sgnal, one can back up n me and re esmae he saes for perod [ Δ, ] by usng he new reconsruced measuremen. hs reesmaon of he saes s done o beer prepare he KF for he curren esmaon perod [, + Δ]. Once he sae re esmaon for perod [ Δ, ] s compleed, he sae esmaon proceeds by usng he reconsruced measuremens for perod [, + Δ]. he algorhm for he KF wh he proposed sgnal reconsrucon echnques s summarzed n he followng (hs algorhm requres an esmaon delay of one aggregaon nerval, Δ). me sep s assumed and furher ha he sae of he KF a me sep Δ s sored n memory. Repea:. Wa unl aggregaed measuremen y +Δ becomes avalable. 2. Reconsruc he measuremen up o me sep + Δ usng all avalable aggregaed measuremens. 3. Load from memory he KF sae a me sep Δ. 4. Reesmae he saes by usng he KF from me sep Δ + o. 5. Sore o memory he KF sae a me sep. 6. Esmae sysem saes from me seps + o + Δ by usng he KF (and repor hs o he user). 7. Se + Δ. Dependng on he allowed delay n makng he raffc sae esmaon, several versons of he algorhm can be obaned. In addon

6 26 ransporaon Research Record 235 o he foregong one-nerval delay algorhm, s neresng o consder he offlne verson of he algorhm, whch allows an arbrary delay n he sae esmaon. hs mode of operaon s called he analyss mode. In hs mode, s assumed ha a sequence of hsorcal aggregaed measuremens s gven and ha he goal s o esmae he sequence of hsorcal raffc saes. hs mode s useful when he hsorcal daa are analyzed and s also used as a benchmark for he onenerval delay mode, snce beer resuls wh he use of he analyss mode are epeced. Algorhmcally, he dfference beween hs mode and he foregong algorhm s ha only Seps 2, 6, and 7 are eraed. Velocy Cell ransmsson Model In order o es he proposed sgnal reconsrucon approaches for dealng wh aggregaed measuremens, he velocy cell ransmsson (CM-v) model (5) was employed because of s smplcy. I s based on he Lghhll Whham Rchards model (9, ) and s defned as follows: g g = + ( ) (, (, ) ) = 2,,..., L ( 4) where = speed a h cell a me sep, L = oal number of cells, and g = numercal flow funcon defned as follows: ( ) = g, 2 ( ) R 2 f 2 R( c) f c R( ) f c ma ( R( 2), R( ) ) f 2 c 2 2 ( 5) where c = ma /2, R() = 2 ma, and ma s he mamal speed allowed by he CM-v model. Boundary condons before he frs cell and afer he las cell modeled by Equaon 4 are modeled as random walk (7 ), such ha wo ghos cells are added a he begnnng and a he end of he road segmen: = + ξ M+ L+ L+ = + ξ ( 6) he CVM-v model represens he model ranson funcon M() from Equaon by combnng Equaons 5 and 6. In he CM-v model, s assumed ha sensors are measurng speed (e.g., hey are double loop deecors) a a subse of cells. As a resul, he observaon mar H from Equaon s a lnear funcon of sysem varables, snce he sae of he sysem s drecly measured. he measuremen nose varance R s assumed o be consan n me and s denoed smply R. Ensemble Kalman Fler he ensemble KF (EnKF) was frs nroduced by Evensen as an alernave o he eended KF, whch performs poorly when he sae ranson funcon s hghly nonlnear (2). he EnKF belongs o a group of subopmal esmaors ha use Mone Carlo or ensemble negraon. he EnKF uses a collecon of sae vecors (called he ensemble members of sysem saes) o propagae he sae forward n me and o compue he mean and covarance needed for he correcon sep. he covarance esmaed n hs way s used o compue he Kalman gan, and he correcon sep equaon says he same as n he radonal KF. he EnKF algorhm employng he CM-v model s summarzed n he followng seps. In Sep, samples ha are generaed represen he pror knowledge abou he nal sae and hs sep represens nalzaon of he sysem. Seps 2 hrough 4 represen he predcon phase, whereas Seps 5 and 6 represen he correcon phase.. Generae N ensemble members of sysem saes s n, by drawng N samples from a Gaussan dsrbuon, where n =, 2,..., N, and nde denoes he nal me sep. 2. Make a predcon usng he CM-v model: ŝ n M s = ( ) + n η n 3. Compue he mean of he ensemble: = N N n= sˆ n 4. Use he mean of he ensemble o compue he covarance of he predced sae: P = E N ( E ) where mar E s defned as E = [ŝ,..., ŝ N ]. 5. Calculae he Kalman gan as follows: K = P( H) H P ( H) + R 6. Use he measuremen o oban a new ensemble: n n n n s = sˆ + K z Hsˆ + ξ 7. Go o Sep 2. Daa Se he EnKF wh he descrbed sgnal reconsrucon approaches was esed on he NGSIM daa se (22) colleced a Inersae ( Hollywood Freeway), locaed n Los Angeles, Calforna. hs srech of hghway s 6 m long and consss of fve lanes and one on- and one off-ramp. rajecores of all vehcles were colleced on June 5, 5, beween 7: a.m. and 8:5 a.m. o collec such hgh-qualy daa, egh vdeo cameras were used o monor hs secon of hghway, and from he recorded vdeo daa, coordnaes for each vehcle were eraced every. s. Compared wh he ypcal double loop deecor daa, where lengh of hghway s usually several klomeers and dsance beween deecors s hundreds of meers, he poron of he hghway n he NGSIM daa se s very shor n boh me and space. o make hs daa se more suable for raffc sae esmaon, he lefmos lane was dvded

7 Corc, Djurc, and Vucec 27 no 32 cells m long wh vrual deecors placed every m. As a resul, sarng wh he Cell Number 2, every ffh cell conans a vrual sensor ha provdes speed measuremens. he whole 5-mn me nerval was dvded no me seps of.6 s, whch s conssen wh he Couran Fredrchs Lewy condons (2). he frs me seps were dscarded because of mssng daa. As a resul, he fnal dscrezaon of me and space amouns o 32 spaal cells and, me seps. o smulae he aggregaon, was assumed ha sensors repor aggregaed speed every Δ me seps. he repored y aggregaed values were obaned as he average speed durng he Δ me seps plus a random Gaussan nose wh mean zero and varance. Epermenal Seup he sudy objecve was o esmae he rue raffc sae avalable n he orgnal NGSIM daa gven he aggregaed measuremens and o preprocess hem by usng varous sgnal reconsrucon approaches dealed earler. he repored performance measure s mean absolue error (MAE), defned as follows: esm MAE = L K, rue,, ( 7) where esm, and rue, are esmaed and rue speed, respecvely, for he h spaal cell a me sep, where =, 2,..., K, and =, 2,..., L. Varables K and L represen he oal number of me seps and spaal cells, respecvely. he embedded MALAB mplemenaons were used for lnear, cubc splne, and cubc Herme splne nerpolaons (nerp funcon), kernel regresson was mplemened n MALAB, and he conve opmzaon approach was solved by usng CVX, he MALAB oolbo for dscplned conve programmng (23). he epermens were repeaed fve mes, and he average MAE and he sandard devaon were repored. he nalzaon of EnKF was done such ha he ensemble members are sampled from a Gaussan dsrbuon wh mean km/h and varance km/h. he mean and he varance were chosen on he bass of he documenaon provded wh he daa se. he mamum speed on hs srech of hghway was deermned o be 5 km/h and he ensemble sze was se o. he remanng parameers of he EnKF model were se o he followng values: measuremen nose varance, km/h; sae nose varance, km/h for ghos and 5 km/h for all oher cells. hese parameer values were obaned by model calbraon. Resuls Frs, he performance of wo esng sgnal reconsrucon mehods (classc and sepwse) was compared wh he performance of he fve proposed reconsrucon echnques. Separae epermens were run for dfferen lenghs of he aggregaon nerval Δ, where Δ {, 2, 4,,, }. When Δ =, he aggregaon s no performed and he orgnal, vrual loop deecor measuremens are used n he correcon sep. Alhough hs s no a realsc seng, s very useful as a benchmark. Resuls for he one-nerval delay and he analyss mode are presened n able. Confrmng he resuls from Schreer e al. (), he sepwse approach conssenly acheves beer performance han he classc approach, demonsrang he benefs of he measuremen reconsrucon. However, alhough he sepwse and classc approaches acheve reasonable resuls for shorer aggregaon nervals n boh modes of operaon, for longer nervals he proposed sgnal reconsrucon mehods work beer. For eample, when he aggregaon nerval Δ s se o, he reconsrucon approach based on conve opmzaon acheves % mprovemen over he sepwse reconsrucon. When he analyss and he one-nerval modes are compared, he repored MAE s nearly he same for all nerpolaon mehods, ecep for cubc splne nerpolaon. he reason s poor nerpolaon of he cubc splne mehod for nerval ( Δ/2, ], semmng from he use of hgher-degree polynomals. However, n he analyss mode only nerpolaon s performed, whch leads o mproved performance. he performance of he conve opmzaon approach and he lnear nerpolaon approach s very smlar n all sengs. For he remanng epermens, he aggregaon nerval was fed o Δ =. o ge beer nsgh no he sae esmaon performance, Fgure 3 shows he reconsruced measuremen sgnal for he cell wh a sensor obaned wh dfferen reconsrucon mehods. he dfference beween he proposed approaches and he sepwse approach s clearly vsble; proposed approaches follow he rue (unobserved) measuremen much closer and whou sudden jumps. For furher nsgh, Fgure 4 demonsraes he dfference n speed esmaon beween he sepwse and cubc Herme nerpolaon approaches on wo represenave ypes of cells, one wh and one whou a vrual sensor. When he approaches are appled o he cell wh a sensor, he sep funcon n he sepwse approach does no model raffc speed well durng he ranson perods, whereas he cubc approach handles hese suaons well. Boh approaches fal ABLE MAEs for Dfferen Aggregaon Inervals Mode Δ Classc Sepwse Lnear Inerpolaon Splne Inerpolaon Herme Inerpolaon Opmzaon Kernel Regresson One-nerval 4.73 ± ± ± ± ± ± ±.7 delay ± ± ± ± ± ± ± ± ± ± ± ± ± ±.2 6. ± ± ± ± ± ± ± ±.3 5. ± ± ± ± ± ± ± ± ± ± ± ± ±.2 Analyss 4.73 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.4 6. ± ± ± ± ± ± ± ±.3 5. ± ± ± ± ± ± ± ± ± ± ± ± ±.2

8 28 ransporaon Research Record 235 rue sae reconsruced sgnal rue sae reconsruced sgnal 9 me sep (a) 9 me sep (b) rue sae reconsruced sgnal rue sae reconsruced sgnal 9 me sep (c) 9 me sep (d) FIGURE 3 rue sae and reconsruced measuremens for dfferen nerpolaon approaches for cell wh sensor: (a) sepwse approach, (b) lnear nerpolaon approach, (c) cubc Herme nerpolaon approach, and (d) opmzaon approach. rue sae esmaed sae rue sae esmaed sae 9 me sep (a) 9 me sep (b) FIGURE 4 rue and esmaed speed for sepwse and cubc nerpolaon approaches for (a, b) cells wh sensor. (connued)

9 Corc, Djurc, and Vucec 29 rue sae esmaed sae rue sae esmaed sae 9 FIGURE 4 (connued) me sep (c) 9 me sep rue and esmaed speed for sepwse and cubc nerpolaon approaches for (c, d) cells whou sensor. (d) o model shor, sudden jumps n speed a me sep. because of he aggregaon of measuremens. However, when he approaches are appled o he cell whou a sensor, hey are boh less accurae snce measuremens are no avalable for hs cell, and he correcon sep s never performed. able 2 repors he MAEs for dfferen ypes of spaal cells. More specfcally, Cell represens he error for all cells ha have vrual deecors, Cell 2 for all cells ha are locaed mmedaely downsream of he cells wh deecors, and Cell 5 s for cells locaed mmedaely upsream of he cells wh sensors. From able 2, can be concluded ha mos approaches acheve he lowes error for cells wh deecors and he hghes error for he cells locaed he furhes from he deecors. Cells ha are locaed downsream of sensors have lower error han cells locaed upsream of sensors. hs fndng can be eplaned by he fac ha here are several backward shocks n he daa se, whch he CM-v model reproduces well. he classc approach has a very hgh error for all cells, even cells n whch he deecors are locaed. I s mporan o menon ha he ghos cells modelng boundary condons are no aken no consderaon for hs analyss. Concluson In he raffc doman, measuremens a every me sep are ypcally no avalable; nsead, only aggregaed measuremens over predefned aggregaon nervals are gven. hs procedure can pose a problem when raffc saes are esmaed, snce he popular KF mehods acheve sae-of-he-ar performance only when he measuremens are avalable whou gaps. o solve hs ssue, was proposed o reconsruc he hgh-resoluon measuremen by usng several approaches. I was furher demonsraed how he reconsruced sgnal s used wh he EnKF employng he CM-v n wo modes of operaon: onlne one-nerval delay mode and offlne analyss mode. he resuls show he benefs of he proposed approach, whch ouperformed he esng mehods and mproved he accuracy of he underlyng raffc model. Acknowledgmen hs work was funded n par by a Naonal Scence Foundaon gran. ABLE 2 Repored MAEs for ypes of Cells References Varable Cell Cell 2 Cell 3 Cell 4 Cell 5 Classc Sepwse Lnear nerpolaon Splne nerpolaon Herme nerpolaon Opmzaon Kernel regresson Kalman, R. A New Approach o Lnear Flerng and Predcon Problems. Journal of Basc Engneerng, Vol. 82, 9, pp Daganzo, C. he Cell ransmsson Model: A Dynamc Represenaon of Hghway raffc Conssen wh he Hydrodynamc heory. ransporaon Research Par B, Vol. 28, No. 4, 994, pp Daganzo, C. he Cell ransmsson Model, Par II: Nework raffc. ransporaon Research Par B, Vol. 29, No. 2, 995, pp Sun, X., L. Muñoz, and R. Horowz. Hghway raffc Sae Esmaon Usng Improved Mure Kalman Flers for Effecve Ramp Meerng Conrol. Proc., 42nd IEEE Conference on Decson and Conrol, Mau, Hawa, IEEE, New York, 3, pp Work, D., O. ossavanen, S. Blandn, A. Bayen,. Iwuchukwu, and K. racon. An Ensemble Kalman Flerng Approach o Hghway raffc Esmaon Usng GPS Enabled Moble Devces. Proc., 47h IEEE Conference on Decson and Conrol, Cancun, Meco, Dec. 9, IEEE, New York, 8, pp

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