Dynamic Estimation of OD Matrices for Freeways and Arterials

Transcription

1 Novembe 2007 Final Repo: ITS Dynamic Esimaion of OD Maices fo Feeways and Aeials Auhos: Juan Calos Heea, Sauabh Amin, Alexande Bayen, Same Madana, Michael Zhang, Yu Nie, Zhen Qian, Yingyan Lou, Yafeng Yin and Meng Li PREPARED FOR: Califonia Depamen of Tanspoaion Sacameno PREPARED BY: Insiue of Tanspoaion Sudies UC Bekeley Final Repo: ITS

2 TABLE OF CONTENTS LIST OF FIGURES... iv LIST OF TABLES... vii EXECUTIVE SUMMARY... viii 1. A DYNAMIC ORIGIN-DESTINATION MATRIX ESTIMATION ALGORITHM FOR FREEWAYS Inoducion Mehodology Noaion Sysem Repesenaion Sae augmenaion: Sae-Space Equaions Implemenaion of he Algoihm Implemenaions unde fee-flow condiions Implemenaion unde congesed condiions Compaison of he wo mehodologies Tansiion peiod: how o deec when affic condiions change How maix A affecs he model: sensiiviy analysis Tavel ime esimaion Consideaion of newok size Conclusions Bibliogaphy Appendix.1: Sae augmenaion ESTIMATING TIME-DEPENDENT FREEWAY O-D DEMANDS WITH VARIATIONAL INEQUALITIES Inoducion The dynamic O-D Esimaion Poblem The Poposed Esimaion Famewok and Soluion Algoihms The DoDE Fomulaion The Dynamic Newok Loading Pocess Algoihms fo Solving Vaiaional Inequaliy Poblems Numeical Resuls The Feeway Newok and Relaed Daa Tesing Scenaios and esuls Summay i

3 Refeences ESTIMATING TIME-DEPENDET O-D MATRICES FOR ARTERIALS Lieaue Review Backgond Exising Appoaches Closed-Newok-Oiened Appoach Open-Newok-Oiened Appoach Closed-Newok-Oiened Appoach Saisfaying equaliy and Inequaliy Consains Tavel Time Consideaions Incopoaing Muliple daa Souces Issues o be Fuhe Addessed Esimaion of Oigin-Desinaion flows fo Acuaion-Conolled Inesecions Inoducion Poblem Saemen and Noaions Poblem Saemen Noaions Convenional GLS Mehod Fomulaion Soluion Algoihm Impoved Two-Sep Mehod Fomulaion Soluion Algoihm Numeical Example Tacking Time-Vaying O-D Flows Esimaion of Oigin-Desinaion Flows fo Acuaion-Conolled Coidos Inoducion Model Fomulaion Model Pepaaion Decomposiion Scheme Sae-Space Repesenaion Numeical Expeimen Expeimen Seings Expeimen Resuls Real-Wold Applicaion Concluding Remaks Invesigaion of Dynamic sucue of O-D Demand Inoducion Hisoical Pespecives Empiical Invesigaion ii

4 Saisic Time Seies Analysis O-D Esimaos Daa Descipion Empiical Resuls Time Seies Model Specificaion Esimaion Resuls Conclusion Refeences DEVELOPMENT OF A PRACTICAL COMPUTER TOOL FOR DYNAMIC ORIGIN-DESTINATION MATRICES ESTIMATION iii

5 LIST OF FIGURES FIGURE 1.1 Implemenaion and validaion of he algoihm... 7 FIGURE 1.2 BHL secion of I-80W used fo he fis pa of his sudy... 8 FIGURE 1.3 Exac vs. Esimaed OD flow fo OD pai FIGURE 1.4 Exac vs. Esimaed OD flow fo OD pai FIGURE 1.5 Exac vs. Esimaed OD flow fo OD pai FIGURE 1.6 Exac vs. Esimaed OD flow fo OD pai FIGURE 1.7 I-90E Massachuses Tunpike FIGURE 1.8 Tue vs. Esimaed OD flow fo OD pai FIGURE 1.9 Tue vs. Esimaed OD flow fo OD pai FIGURE 1.10 Tue vs. Esimaed OD flow fo OD pai FIGURE 1.11 Compaison of wo diffeen levels of OD flows FIGURE 1.12 OD pai 1, unde congesed condiions FIGURE 1.13 OD pai 2, unde congesed condiions FIGURE 1.14 OD pai 3, unde congesed condiions FIGURE 1.15 OD pai 4, unde congesed condiions FIGURE 1.16 Visual compaison beween he mehodologies FIGURE 1.17 Visual ageemen beween ue and esimaed cuves fo fis OD pai BHL and fo diffeen avel imes FIGURE 1.18 Speed measued fom deeco and mode pediced by seing v h = 45 mph FIGURE 1.19 Imaginay newok FIGURE 1.20 Small newok wih 2 OD pais FIGURE 2.1 Simplifying highways FIGURE 2.2 Mege and divege nodes FIGURE 2.3 Cumulaive cuves consuced fom DNL FIGURE 2.4 A eal feeway newok FIGURE 2.5 The hee ypes of disibuions of demand ove ime FIGURE 2.6 ME, GEH and RMSE of couns, pah avel ime and O-D demand fo an oiginal scenaio FIGURE 2.7 ME, GEH and RMSE of couns, pah avel ime and O-D demand, hisoical iv

6 OD daa discaded since he 16 h ieaion FIGURE 2.8 Esimaed O-D demands fo fou OD pais fo apezoidal paen FIGURE 3.1 A ypical layou FIGURE 3.2 A ypical fou-way inesecion FIGURE 3.3 RMS eo beween acual and esimaed O-D paamees fo Expeimen 2 and Expeimen FIGURE 3.4 Convegence of O-D paamees b i1 fo Expeimen FIGURE 3.5 Convegence of O-D paamees b i2 fo Expeimen FIGURE 3.6 Convegence of O-D paamees b i3 fo Expeimen FIGURE 3.7 Convegence of O-D paamees b i4 fo Expeimen FIGURE 3.8 Compaison of flows o Leg FIGURE 3.9 Compaison of flows o Leg FIGURE 3.10 Compaison of flows o Leg FIGURE 3.11 Compaison of flows o Leg FIGURE 3.12 Compaison of RMS eos FIGURE 3.13 Numbeing convenion fo model fomulaion FIGURE 3.14 Numbeing convenion he hypoheical coido FIGURE 3.15 Acual vs. Esimaed O-D flows fo O-D pai FIGURE 3.16 Acual vs. Esimaed O-D flows fo O-D pai FIGURE 3.17 Acual vs. Esimaed O-D flows fo O-D pai FIGURE 3.18 Acual vs. Esimaed O-D flows fo O-D pai FIGURE 3.19 Acual vs. Esimaed O-D flows fo O-D pai FIGURE 3.20 Acual vs. Esimaed O-D flows fo O-D pai FIGURE 3.21 Acual vs. Esimaed O-D flows fo O-D pai FIGURE 3.22 Acual vs. Esimaed O-D flows fo O-D pai FIGURE 3.23 Illusaion of he Tesing Sie FIGURE 3.24 Modeling of he Coido FIGURE 3.25 Esimaed O-D flows fo O-D pai FIGURE 3.26 Esimaed O-D flows fo O-D pai FIGURE 3.27 Esimaed O-D flows fo O-D pai FIGURE h See-Univesiy Avenue inesecion FIGURE 3.29 Resuls of O-D flows desined o Leg 2 esimaed by O-D flow esimao.. 99 FIGURE 3.30 Esimaes of he hid-day O-D flows desined o Leg 2 wih O-D flow deviaion esimao v

7 FIGURE 4.1 Geneal famewok FIGURE 4.2 Simple use ineface FIGURE 4.3 Tesbed of I-80W a Bekeley FIGURE 4.4 Sofwae oupus vi

8 LIST OF TABLES TABLE 1.1 Compaison of boh mehodologies TABLE 1.2 Eo measues fo diffeen avel imes TABLE 2.1 Allocaion ules fo hee ypes of disibuions of demand TABLE 2.2 The esuls of all he 60 scenaios TABLE 3.1 Aveage RMS eos ove las 20 ieaions TABLE 3.2 Roo Mean Squae Eo nomalized RMSN TABLE 3.3 RMSN of esimaes of ime-vaying O-D flows TABLE 3.4 Model specificaions fo diffeenced O-D flows TABLE 3.5 Model specificaions fo diffeenced O-D flows deviaions TABLE 3.6 Model specificaions fo diffeenced O-D splis TABLE 3.7 Model specificaions fo diffeenced O-D splis deviaions TABLE 3.8 RMS of O-D esimao of splis and spli deviaions TABLE 3.9 RMS of O-D esimao of flows and flow deviaions vii

9 EXECUTIVE SUMMARY Oigin-Desinaion O-D maices povide infomaion on flows of vehicles aveling fom one specific geogaphical aea o anohe, and ae one of he ciical daa inpus o anspoaion planning, design and opeaions. Because i is vey ime consuming and labo inensive o obain hem hough household ineviews o oadside suveys, significan effos have been made o develop mahemaical models fo esimaing he maices fom link couns, which ae elaively easie o obain. So fa, up-o-dae commecial planning ools and simulaion sofwae have povided buil-in O-D esimaion modules. Howeve, mos of hese O-D esimaos ae only capable of esimaing saic O-D maices ahe han dynamic o ime-dependen O-D maices. The lae ae pe-equisies fo sho-em planning applicaions and affic opeaions sudies. The goal of his sudy was o bidge he gaps beween pacice and heoy in O-D esimaion. In paicula, his wok planned o develop he mehodologies fo deiving ime-dependen O- D maices fo linea newoks and implemen hem in a compue ool a linea newok is a sech of highway wih muliple enies and exis, whee hee would be no oue choices involved. The sudy was divided ino wo pas: O-D able esimaion on feeways and O-D able esimaion on aeials. To achieve he goal, he following asks wee pefomed on each pa: O-D esimaion on feeways: 1. A eview and compaison fo deemining appopiae echniques and mehods fom he lieaue was pefomed. A mehodology based on Kalman fileing echniques was chosen, whee he sae veco can be eihe he O-D flows fo each O-D pai o is deviaion fom a hisoical value. 2. A mehodology ha idenifies when affic condiions vay and makes use of exising models o esimae OD flows in a linea newok was implemened. The models wee esed using eal daa colleced fom wo diffeen newoks I-80W, CA, and I-90E, MA duing fee flow and congesed condiions. An algoihm o deec affic condiion changes, which makes use of he speed measuemens povided by loop deecos, was poposed. 3. The developmen of a compue O-D esimao ool wih a use-fiendly ineface has been saed. A peliminay vesion of his ool is povided. 4. A new O-D able esimaion appoach, based on vaiaional inequaliies, was poposed. This appoach akes ino accoun vaious levels of affic infomaion, such as link flow coun, hisoical O-D ables, saic planning O-D and obseved pah avel imes. A poion of feeway SR-41 in Fesno, CA 16.7 miles was used o es he appoach. Tue O-D ables ae no known, bu synheic imedependen O-D ables wee poduced. Diffeen demand paens wee esed and invesigaed. O-D esimaion on aeials: 1. A wo-sep appoach fo esimaing ime-vaying O-D flows fo acuaionconolled coidos wih incomplee infomaion abou eneing and exiing flows is poposed. A he fis sep, uning movemens fo each inesecion ae esimaed, and hen used a he second sep o consuc he measuemen viii

10 equaions o infe he coido O-D flows. The poposed appoach has been demonsaed and validaed on a es-case coido. The O-D esimao has subsequenly been applied o a segmen of El Camino Real, San Maeo, CA wih hee inesecions: 28h, 27h and 25h Avenue. 2. An empiical analysis of he O-D demand sucues and examinaion of hei impacs on he O-D esimaion a a single inesecion level have been conduced. Daa fom he 34h See-Univesiy Avenue inesecion in Gainesville, Floida in 2001, was used fo his pupose. Sofwae developmen: 1. The objecive of his pa of he pojec was o implemen he models and algoihms developed ino a compue ool o allow paciiones o apply he poposed models and algoihms. The wok ealized on vaiaional inequaliies was no implemened, since he coesponding esuls will need fuhe developmens befoe hey become applicable in he fom of a ool. All ohe esuls wee implemened. The sofwae cuenly compue he O-D flows fo 30 seconds inevals using he Bekeley Highway Lab daa used o es he feeway algoihm. 2. The cuen vesion of he sofwae is aached o his epo CD. The fundamenal conclusions fom his eseach ae he following: O-D esimaion on feeways: 1. Models using he deviaion of he O-D flow fom an hisoical value insead of he O-D flow diecly yield moe accuae O-D esimaes. The esimaions mach he acual daa well. 2. Accuae avel ime esimaes ae no essenial fo he accuacy of he poposed mehod. Peliminay evidence ha he accuacy of avel ime esimaes does no impac he esuls gealy, and a disincion beween fee flow and congesed condiions is enough fo he puposes of his wok was shown. 3. Fo he vaiaional inequaliy-based appoach, compuaional expeimens showed ha affic couns ae indispensable fo O-D esimaion, and ha he qualiy of he esuls inceases wih he numbe of measuemens. If he numbe of couning locaions is small, he locaion of he sensos migh be cucial fo he esimaion. Taffic couns alone ae no sufficien o obain accuae O-D demand esimaes. Hisoical ime-dependen O-D demands can dasically impove he qualiy of he esimaes. Saic O-D ables can also be used in he esimaion pocess. Finally, coune o ou expecaions, pah avel imes do no conibue significanly o impoving O-D demand esimaes. O-D esimaion on aeials: 1. Fo a single inesecion, he poposed wo-sep appoach oupefoms he convenional genealized leas squaes appoach. Moeove, he fome is moe efficien as well. As in he feeway case, boh appoaches would benefi fom accuae pio knowledge o paial O-D infomaion. 2. A he coido level, he esimao is able o ack he end of ime-vaying O-D flows and poduce esimaes elaively close o he acual values in an aveage sense. The esimao is less sensiive o he changes of he O-D flows and hus esimaes ae geneally no as flucuaing as he ue values. The eason is ha he poposed appoach a he coido level only makes use of he localized ix

11 infomaion. By doing so, he poblem can be decomposed o simplify he fomulaion and impove lagely he compuaional efficiency. 3. Since eal O-D obsevaions ae no available fom El Camino Real, he accuacy of he esimaes can no be veified. Howeve, he applicaion does demonsae ha he esimao is able o eadily wok wih acual field loop daa. 4. The compaison of esimaos wih diffeen sae vaiables suggesed ha he esimao wih sae vaiable of O-D flow oupefoms he ohes in he paicula case invesigaed. We fully ecognize ha O-D paens would be sie-dependen, and he esuls of his case sudy should no be genealized. 5. Demands o flows a diffeen O-D pais may possess diffeen sucues, which ae vey ofen no fis-ode auo egessive. Incopoaing all of hese ue sucues ino he Kalman fileing algoihm makes he model fomulaion vey complicaed. On he ohe hand, he simple fis-ode auo egessive assumpion poduces accepable esuls in ou empiical expeimens and pevious sudies. Theefoe, unless hee ae sufficien O-D daa ha sugges ohewise, one migh simply use he sae vaiable of O-D flows o splis and assume ha hey ae fisode auo egessive. Ou ecommendaions fo fuue wok ae as follows: 1. Alenaive souces of measuemens. Can measuemens fom pobe vehicles aleady available, fo example FasTak ansponde daa be incopoaed in ou esimaion algoihms? Paial O-D infomaion is now diecly available fom FasTak eades a a high peneaion ae in he Bay Aea and could incease he accuacy of O-D esimaions. 2. Senso placemen algoihms. Fo limied numbes of sensos o be deployed, whee should he sensos be deployed o povide maximal esimaion accuacy? 3. GPS-based measuemens. Invesigaion of he possibiliy of inegaing make diven infomaion such as cellula phone daa ino he esimaion algoihms. The applicabiliy of his new ype of GPS-based infomaion has implicaions which goes beyond O-D esimaion, in paicula fo: a. Tavel ime esimaion fo changeable message signs b. Congesion esimaion fo amp meeing c. High qualiy infomaion fo HOV lanes The use of GPS based cellula phone measuemens could: a. Impove he qualiy of esimaions fo he above quaniies, b. Povide Calans wih a iche daabase, addiional o PeMS and FasTak, c. Povide Calans wih measuemens whee infasucue is no available, d. Pogessively become an alenaive o cosly loop deeco deploymen o mainenance. This ecommendaion is in ou view he mos impoan, as i povides Calans wih a cos efficien long em alenaive o loop deecos. The pogessive peneaion of he cellula phone make by GPS equipped devices is diven by compeiion beween cellula phone companies. Majo cellula phone companies ae aleady developing hei own daa eieval infasucue, fom which Calans could poenially benefi eveywhee no only in uban o sububan aeas. x

12 CHAPTER 1 A DYNAMIC ORIGIN-DESTINATION MATRIX ESTIMATION ALGORITHM FOR FREEWAYS Pepaed by: Juan Calos Heea, Sauabh Amin, Alexande Bayen, and Same Madana Insiue of Tanspoaion Sudies Univesiy of Califonia, Bekeley 1

13 1.1 Inoducion Oigin-Desinaion OD maices 1 ae needed fo sho-em planning applicaions and fo opeaional sudies. In paicula, hey can be used o develop some conol saegies of he newok unde sudy in ode o impove is pefomance. Fo insance, a bee affic signal coodinaion may be achieved on an uban coido, an adapive amp meeing saegy can be developed on a feeway, o hey can be used o povide dives wih bee infomaion. The aim of his pojec is o implemen an algoihm o compue dynamic OD maices on a linea newok 2. In paicula, ou focus is he feeway case. The appoach used hee is based on he appoaches developed in [1] and [2]. These appoaches which ae quie simila esimae OD flows assuming he newok is in he same affic condiion. Tha is, hee is no shockwave in he newok. The appoaches descibe he sysem wih a se of linea sae-space equaions. Some paamees of hese equaions will depend on he affic condiions along he newok i.e. hey will change if he newok is unde fee-flow o congesed condiions. Fo his eason, we fis idenify he affic condiions, selec he coesponding sae-space equaions and solve hem unil a change in he affic condiions is deeced. If he affic condiions ae no he same along he whole newok, he model will no wok popely. The eason fo his is ha he sysem would no be linea, and hus, i would no be popely descibed by a se of linea sae space equaions. As a consequence, duing his ansiion peiod 3 which should no be vey long he model will pobably no yield good esimaes. The es of his documen is oganized as follows: appoaches adoped o epesen linea sysems wih sae-space equaions will be biefly descibed on Secion 1.2. Secion 1.3 descibes he implemenaions of he appoaches pesened in Secion 1.2 using eal daa and hei esuls. These implemenaions assume he same affic condiions all he ime i.e. hee is no need o idenify affic condiions in ode o selec he appopiae sae-space equaions. Secion 1.4 will addess he idenificaion of affic condiions. A bief discussion egading newok size is given in Secion 1.5. Finally, Secion 1.6 pesens he main conclusions of his wok. 1.2 Mehodology This secion pesens he noaion used and descibes how he sysem is epesened unde he wo appoaches. 1 Elemen ij of an OD maix is he numbe of vehicles going fom i o j duing some ime ineval. 2 A linea newok is a newok whee hee is no oue choice fo any OD pai. Fo insance, a highway wih on- and off-amps o an uban aeial can be modeled as a linea newok. 3 Peiod when a shockwave is in he newok. 2

14 1.2.1 Noaion N : Se of nodes in he newok, L : Se of links in he newok, n L : Numbe of links in L ha ae equipped wih couning saions each saion is denoed by l L, n OD : Numbe of OD pais, x h : Numbe of vehicles beween he -h OD pai ha lef he oigin in ime ineval h, x H h: hisoical esimae fo x h, x h : Deviaion of x h fom he coesponding hisoical esimae, x h : n OD 1 veco of all OD flows a ime ineval h, x H h : Associaed hisoical OD flow veco, x h : Veco of deviaions = x h x H h, y lh : Obseved affic couns a link saion l a ime ineval h, y h : n L 1 veco of such couns, y H h : associaed hisoical link coun veco, y h : Veco of deviaions of all couns = y h y H h, f qh : n OD n OD maix of coefficiens descibing he effecs of flows a ime ineval q on flows a ime ineval h, w h : n OD 1 veco of andom eos in he sae ansiion equaion defined in he following subsecion, Q: coesponding covaiance maix, p : maximum ode of he auoegessive model in he sae ansiion equaion, a qh : n L n OD assignmen maix descibing how OD flows x q conibue o link flows y h, v h : n L 1 veco of measuemen eos in he measuemen equaion defined in he following subsecion, R: coesponding covaiance maix, u: maximum numbe of ime inevals needed o avel beween any OD pai. 3

15 1.2.2 Sysem Repesenaion The sysem can be descibed by he following equaions [1]: x h+1 = y h = h k=h p+1 f kh x k + w h 1.1 h a kh x k + v h 1.2 k=h u The fis equaion elaes cuen OD flow wih he pevious ones, and i ies o capue he coelaion ove ime of he OD flows. This kind of coelaion aises fom unobseved phenomena such as changes in affic demand, and/o changes in he anspoaion newok. This equaion will hold ue as long as affic condiions change gadually smoohly. The second equaion elaes OD flows wih link couns hough he assignmen maices a kh. These maices conain he infomaion of how he affic evolves ove ime and space along he newok. The dimension of each assignmen maix is n L n OD, and hee ae as many maices as he maximum numbe of inevals needed o go fom any oigin o any desinaion poin in he newok. Each elemen l of he maix a qh epesens he conibuion of link coun l a peiod h o OD flow ha eneed he newok a h q. Ashok and Ben-Akiva [2] found ha Equaion 1.1 can only capue empoal inedependencies among OD flows, and does no epesen sucual infomaion abou ip paens. Insead of using x h as he sae vaiable, hey decided o use is deviaion fom a hisoical OD flow. By doing so, he esimaion and pedicion pocess would have aken ino accoun indiecly all he expeience gained ove pevious esimaions and i would be iche in is sucual conen. Also, deviaions can be eihe posiive o negaive, so hey can be appoximaed by a nomal disibuion, which is also a useful popey fo a ool such as Kalman File which will be used lae o solve he equaions. The coesponding equaions ae given by: x h+1 = y h = h k=h p+1 f kh x k + w h 1.3 h a kh x k + v h 1.4 k=h u Noe ha y H h = h k=h u a kh x H k Thee ae wo aspecs o noe abou he pevious equaions: 1. Assumpions: The following assumpions ae made abou he eo ems in pevious equaions: E[w h ] = 0, fo all h, 4

16 E[w h w T m] = Q h δ hm whee δ hm is he Konecke s dela and Q h is n OD n OD covaiance maix a ime ineval h, E[v h ] = 0, fo all h, E[v h v T m] = R h δ hm whee R h is n L n L covaiance maix, E[w h v T m] = 0, fo all h, m, ha is, ansiion and measuemen eos ae uncoelaed. When used in pacical applicaions, elaionships beween OD flows o deviaions acoss diffeen OD pais ae ignoed. 2. Paamee esimaion: The maix f h can be esimaed fom hisoical daa by esimaing linea egession models fo each OD pai and Q can be appoximaed fom he esiduals of hese egessions. Fo Equaion 1.3 and 1.4, vecos x H h fo all h ae obained fom a daabase of he OD maix ceaed off line fom pevious esimaions. The maix R can be appoximaed fom hisoical daa. Howeve, compuaion of he assignmen maix a ph is a complicaed execise [3] as i is a nonlinea funcion of he oue choice assumpions, newok opology and avel ime. We do no deal wih he oue choice poblem because ou newok is linea i.e. hee is no oue choice. When he avel imes in he newok ae unobsevable, he assignmen maix is endogenous o he model. In ode o addess he endogeneiy of he assignmen maix, [2] and [1] use an ieaive OD esimaion and assignmen maix compuaion appoach. In his appoach, fo he cuen OD esimae, a affic simulaion model is used o compue he assignmen maix which is hen used o compue he new OD flow esimaes by he fileing algoihm. As poined in [3], such an appoach does no guaanee convegence and can poenially lead o biased esimaes. The auhos of [3] also develop a igoous appoach o esimae he assignmen maix based on sochasic mapping beween dynamic OD flows and link couns Sae augmenaion: Sae-Space Equaions A discee linea sae-space sysem is ofen descibed as follows: x h+1 = A x h + B k 1.5 y h = C x h + D k 1.6 Equaion 1.5 is efeed o as he sysem o ansiion equaion, while Equaion 1.6 is known as he oupu o measuemen equaion. x h is he sae veco a ime ineval h, y h is he oupu o measuemen veco a ime ineval h, h is he conol inpu a ime ineval h, and A, B, C, and D ae maices. Equaions , and Equaions sugges ha sae augmenaion is needed o fully uilize all available infomaion and o epesen ou sysem as a linea sae space model. If s = maxu, p 1, hen he augmened sae veco should be n OD s We denoe he n OD s augmened sae and he hisoical sae vecos by X h and Xh H, he n L n OD s + 1 augmened assignmen maix by A h, he n OD s+1 n OD s+1 augmened and appopiaely modified auoegessive paamee 5

17 maix by F h, he n OD s augmened eo veco by W h wih boom n OD s elemens as zeos and he n OD s+1 n OD s+1 covaiance maix by Q h. In addiion, we define following noaion: Y h := y h, X h := X h X H h, B h := A h X H h y H h. Following [2], he augmened sae ansiion and measuemen equaions can be wien as follows an example of how sae augmenaion is done can be found in Appendix A: X h+1 = F h X h + W h Y h = A h X h + B h + v h } 1.7 In Equaion 1.7 boh expessions ae in sae-space fom. In he implemenaions ha will be pesened in Secion 1.3 we assume ha maices F, W, A, and v ae ime invaian, and hen he subindex can be omied. Equaion 1.7 can now be diecly fed ino a Kalman fileing algoihm [4] o give minimum leas squaes esimaes of he sae vaiable X h+1. Depending on he sae vaiable chosen x h o x h, we would end up wih wo diffeen sae space epesenaions in he fom of Equaion 1.7. Soluion echniques such as Kalman fileing used in [1], [2], and [3] and ecusive leas-squae appoaches such as [5] ae based on one of hese sae-space epesenaions 4. Noe ha Equaion 1.7 imply ha each OD flow will be esimaed s + 1 imes duing each ime ineval. Howeve, when n OD is lage e.g., in a lage newok and/o when s is lage e.g., when he newok is in congesed egime, his pocedue migh become compuaionally inensive. To addess his poblem, [2] poposes an appoximaion scheme based on he assumpion ha much of he infomaion abou an OD flow is incopoaed he fis ime i is couned. Finally, one emak egading he obsevabiliy of he sysem should be made. Complee obsevabiliy efes o he abiliy o uniquely deemine he iniial sae veco fom a given se of measuemens. The facos affecing he obsevabiliy of he sysem ae: i he aio n L /n OD, ii degee of linkage beween OD flows and saion couns o he ank of he assignmen maix and iii he degee of linkage beween he OD flows ove ime o he ank of he ansiion maix. 1.3 Implemenaion of he Algoihm This secion descibes an acual implemenaion of he model descibed in Secion Equaion 1.7 and is main esuls. We pesen wo implemenaions pefomed duing fee-flow affic condiions BHL 5 and MT 6, and one duing congesed condiions BHL. Boh he mehodologies wih and wihou deviaions wee ied in wo of he implemenaions in ode o compae pefomance. Fo he implemenaion, he algoihm equies a hisoical OD maix, vehicles 4 In some pas of his documen we will efe o he model deived fom Equaion 1.1 and 1.2 as he mehodology wihou deviaions, and o he model deived fom Equaion 1.3 and 1.4 as he mehodology wih deviaions. 5 Bekeley Highway Laboaoy. 6 Massachuses Tunpike. 6

18 couns, and avel imes beween deeco saions as inpus. The oupu is he esimaed OD maix, which fo validaion puposes only will be compaed agains he eal OD maix o assess he pefomance of he algoihm. Some indexes of pefomance can be also compued in ode o compae alenaive appoaches. Figue 1.1 pesens an ouline of he mehod followed in his sudy. Real OD maix Hisoical OD maix Vehicle couns Algoihm Esimaed OD maix Evaluaion eo measues Pefomance of he algoihm Tavel ime beween deeco saions Implemenaion Validaion Figue 1.1: Implemenaion and validaion of he algoihm. The hisoical OD maix will be used o obain he ansiion maix by esimaing a linea egession models fo each OD pai, see Secion Depending on he mehodology used, his hisoical maix will be also used as pa of he sae vaiables he mehodology wih deviaions defines he sae vaiable as he deviaion of he esimaed OD flow fom he hisoical one. Tavel imes beween deeco saions ae needed o compue he assignmen maices and vehicle couns coesponding o he oupu veco y h. In all he cases, we ae dealing wih siuaions whee he avel ime is consan i.e. same affic condiions ove he simulaion peiod Implemenaions unde fee-flow condiions The sudy wih he fis newok BHL is implemened using he mehodology wihou deviaions, while boh mehodologies wih and wihou deviaions will be used on he second newok MT 7. Fo he second newok, howeve, we will pesen hee only he esuls fom he mehodology wih deviaions. Resuls fom he mehodology wihou deviaions wee only used o compae boh mehodologies Secion Inesae-80 Wesbound BHL Sie descipion and daa collecion. The newok chosen has wo oigins and wo desinaions fou OD pais and coesponds o highway I-80W beween Ashby and Powell. Figue 1.2 depics he geomey of he sie. The OD flows wee numbeed in he following way: 1-3 OD pai 1 7 This is due o he lack of hisoical daa fo BHL unlike he MT case. 7

19 Upseam 1 Downseam 3 HOV On-amp miles Off-amp 4 Figue 1.2: BHL secion of I-80W used fo he fis pa of his sudy. 1-4 OD pai OD pai OD pai 4 In he same way, link numbes coespond o he deeco saion numbes. The daa was colleced using BHL s cameas insalled on he oof of a building locaed nex o he highway. All he daa needed o un he algoihm wee exaced fom he video ecodings. This pocess is ime consuming. Daa wee ecoded duing one hou unde fee flow condiions 10:00 o 11:00am fo wo consecuive weekdays. The ime ineval chosen was 30 seconds i.e. a new OD maix esimaion is done evey 30 seconds. The fis day of daa was used o exac he hisoical OD maix, and he second one was used o implemen he algoihm. Tha is, fo he second day and in addiion o he exacion of he eal OD maix we had o coun vehicles evey 30 seconds a each eny and exi poin of he newok. A consan avel ime of 18 seconds fom upseam/on-amp o downseam/off-amp was used o compue he assignmen maices. Tansiion equaion. A fouh ode auoegessive model i.e. p = 4 fo each of he OD pais was used o compue he coefficiens of each maix f qh and hen F h. Ohe odes wee ied, bu p = 4 povides he bes fi i.e. lowes sum of squaed esiduals. Afe sae augmenaion Secion 1.2.3, he maix F h has dimensions u uns ou o be 1, hen s = maxu, p 1 = 3. Random eos wee compued as descibed in Secion Measuemen equaion. Assignmen maices wee compued as follows. Since he avel ime is 18 seconds 8, each vehicle eneing he newok duing he fis 12 seconds of ineval h will exi he newok duing he same ineval h. All he vehicles eneing he newok in he las 18 seconds of ineval h will exi he newok in he nex ineval h + 1. Assuming a unifom disibuion of he flow duing 30 seconds which is a faily easonable assumpion, 0.4 =12/30 of he vehicles eneing in h leave he newok 8 The avel ime was compued by inspecing he videos. Five andomly seleced vehicles, evey five minues wee used o compue he aveage avel ime. 8

20 duing h and he es of he vehicles eneing in h 0.6=18/30 leave he newok in h + 1. Fo his eason, hee ae wo assignmen maices u = 1. A h h = Ah 1 h = Based on he amoun of eo ha migh have occued while couning vehicles fom video daa, he value of he measuemen eos v h ae 8, 4, 8, and 4 fo y 1h, y 2h, y 3h, and y 4h, especively. Tha is, fo each obsevaion evey 30 seconds, we expec an eo of he ode of ±8 vehicles in he coun a a given saion including all lanes. Since he couns ae appoximaely 80 vehicles evey 30 seconds, his means an eo of abou 30%. The same easoning is applied o he amp couns, bu assuming a lage eo. Resuls. Even hough he daa is available fo one hou, o un he algoihm we used only he fis 35 minues 70 inevals of daa mainly because he daa-exacionpocess is cumbesome. Figues 3 o 6 show he esimaed doed line and he eal solid line OD flows fo he fis 35 minues of he second day fo evey OD pai please noe ha diffeen figues ae a diffeen scales # of vehicle pe ineval Exac Esimaed Time Ineval Figue 1.3: Exac vs. Esimaed OD flow fo OD pai 1. Figue 1.5 shows a vey good ageemen beween exac coun and esimae. The fis ae good fo he ohe OD flows, bu hey ae no as good as he case of he hid OD pai. I has o be noed ha he eal OD flow fo OD pai 4 akes only hee values 0, 1, and 2 see Fig.1.6. In fac, mos of he ime, he flow is zeo. Fo his eason, he 9

21 12 10 Exac Esimaed # of vehicle pe ineval Time Ineval Figue 1.4: Exac vs. Esimaed OD flow fo OD pai Exac Esimaed # of vehicle pe ineval Time Ineval Figue 1.5: Exac vs. Esimaed OD flow fo OD pai 3. 10

22 2 1.8 Exac Esimaed # of vehicle pe ineval Time Ineval Figue 1.6: Exac vs. Esimaed OD flow fo OD pai 4. esimaed values fo his OD pai ae always less han 1. This fac migh explain he vey good ageemen fo he hid OD pai. Inesae-90 Easbound Massachuses Tunpike Sie descipion and daa collecion. The lengh of he newok is 75.6 miles and i conains 10 oigin/desinaion poins see Figue 1.7. The fis one is only an oigin poin, and he las wo ae only desinaion poins. The ohe 7 poins in beween ae oigin and desinaion poins. Tha is, hee ae 8 eny poins and 9 exi poins, which yields 44 OD pais i.e. n L = 17 9 and n OD = mi 29.8 mi 5.3 mi 3.3 mi 2.1 mi 3.7 mi 7.8 mi 15.7 mi Figue 1.7: I-90E - Massachuses Tunpike The daa consiss of he OD flows evey 15 minues, duing 6 days, fo each OD pai. Tha is, he OD maix fo 6 days is known, bu he couns a he eny and exi poins ae no. The daa has o be manipulaed in ode o obain vehicle couns a hese locaions. The coun a any eny poin can be easily compued fom he oiginal daa, bu assignmen maices ae needed o esimae coun a exi poins. A consan speed of 55 mph was assumed o compue hese maices We ae assuming ha hee ae no deecos along he feeway. We have assumed ha hee ae deecos only a he eny and exi poins. 10 Since he secion unde sudy aely ge congesed, his is a easonable assumpion. 11

23 Fo his newok, boh mehodologies wih and wihou deviaions wee esed. In his secion, howeve, we will show only he esuls obained wih he mehodology wih deviaions he ohe implemenaion was pefomed fo compaison puposes. Tansiion Equaion. The fis day was used as he hisoical day, and he ansiion maix was compued using daa fom he fis wo days. Tha is, fo each OD pai an auoegessive model using he deviaion of OD flows of he second day fom hose of he fis one was pefomed in ode o compue he ansiion maix. The ode of he auoegessive model is no he same fo evey OD pai, and i depends on he value which gives he bes fi fom an saisical poin of view i.e. leas esiduals. In ou case, p maximum ode is 4. As we will see lae, s = maxu, p 1 = 6, and hen he augmened ansiion maix is an squae maix of dimensions n OD s + 1 n OD s + 1 see Secion The covaiance maix Q was compued using he esidual of he auoegessive models see Secion 1.2.2, and i has he same dimensions as he augmened ansiion maix. Measuemen Equaion. Assignmen maices wee compued assuming a consan speed of 55 mph in he whole secion 11. The avel ime fo he longes OD pai is abou 82.5 minues, which means ha 5.5 inevals 12 is he maximum ime ha any ip will ake in ou newok. Then, u = 6 and seven assignmen maices ae needed s = maxu, p 1 = 6. The dimensions of each one of hese seven assignmen maices ae n L n OD. Since s = 6 he augmened assignmen maix see Secion has dimensions These assignmen maices wee used o compue all he exi couns. Since some elemens of he assignmen maices ae no inege, we will obain facional couns. If we wok wih hese facional couns, hen he covaiance maix R would be zeo because he diffeence beween yh H and h k=h u a kh x H k would be zeo. Howeve, we have decided o ound he coun o he neaes inege 13. The diffeences wee hen used o compue he n L n L maix R. Noe ha deecos on he eny poins have no eo. Resuls. The hid day was used o es how he algoihm woks. The esul of he implemenaion of he algoihm is a maix ha conains he deviaions of he esimaed OD flow fom he hisoical one OD flow fo he fis day fo evey OD pai and fo evey ime ineval. Then, he esimaed OD flow is jus he sum of he deviaion and he hisoical OD flow. Figues 1.8 o 1.10 show he ageemen beween he exac and esimaed cuve fo hee OD pais ha have diffeen OD flow levels. Figue 1.8 coesponds o OD pai numbe 9, which is he lages one i.e. fom he fis eny o he las exi. Pai Disances beween oigin/desinaion poins wee also known. 12 We ae using 15 minues inevals. 13 Since loop deecos always give an inege numbe of vehicle coun, his seems o be a easonable hing o do. 12

24 Figue 1.9 is fom he second eny o he las exi, and pai 31 Figue 1.10 goes fom he fifh eny o fis exi afe i Tue Esimaed # of veh. pe ineval Time h Figue 1.8: Tue vs. Esimaed OD flow fo OD pai Tue Esimaed # of veh. pe ineval Time h Figue 1.9: Tue vs. Esimaed OD flow fo OD pai 17. I is impoan o noe ha he scale fo he hee gaphs is no he same. If fac, Figue 1.11 shows boh he ue and esimaed cuves fo OD pai 9 and 17 in he same gaph. 13

25 14 12 Tue Esimaed 10 # of veh. pe ineval Time h Figue 1.10: Tue vs. Esimaed OD flow fo OD pai Tue OD9 Esimaed OD9 Tue OD17 Esimaed OD # of veh. pe ineval Time h Figue 1.11: Compaison of wo diffeen levels of OD flows. Unlike he BHL case, his new newok has seveal OD pai, which makes he esimaion pocess a lile bi hade han befoe. The algoihm, howeve, sill pefoms vey well and he OD flows esimaed follow he end of he ue OD flows. 14

26 1.3.2 Implemenaion unde congesed condiions The BHL newok descibed in Secion was used o validae he algoihm unde congesed condiions. Thee days of video daa fom 3-6pm wee ecoded. Fo ou puposes, howeve, only 30 minues wee implemened, and ime inevals of 30 seconds wee used. As wih he MT case, boh mehodologies wih and wihou deviaions wee implemened on hese daa. We will pesen hee, howeve, only he esuls of he mehodology wih he deviaions esuls using he ohe mehodology wee used o compae he wo mehodologies, see Secion Again, he fis day was used o exac he hisoical OD maix, he second day was used o compue he ansiion maix F using he deviaion of his OD maix fom he hisoical one. The algoihm was hen implemened on he hid day. The assignmen maix A was compued in he same way as befoe see Secion 1.3.1, bu now he avel ime is lage because of congesion. Afe inspecion of he video daa, i is impoan o noe ha avel ime duing he peiod unde analysis and acoss lanes is no consan, and i has a significan vaiabiliy. As a fis appoach, howeve, we have decided o un he algoihm assuming a consan avel ime of 1 minue. This means ha a vehicle eneing in ime ineval h leaves he newok in ime ineval h + 2. Resuls. Figues 1.12 o 1.15 show boh he eal and he esimaed OD flow fo each OD pai noe ha he gaphs use diffeen scales Tue Esimaed 60 # of veh. pe ineval Time ineval Figue 1.12: OD pai 1, unde congesed condiions. Unfounaely OD pai 4 Figue 1.15, fom he on-amp o he off-amp has only one ineval wih one vehicle. As we saw in he fee flow case, his migh make he esimaion pocess easie. 15

27 9 8 Tue Esimaed 7 6 # of veh. pe ineval Time ineval Figue 1.13: OD pai 2, unde congesed condiions Tue Esimaed # of veh. pe ineval Time ineval Figue 1.14: OD pai 3, unde congesed condiions Compaison of he wo mehodologies Fee-flow daa fom MT and congesed daa fom BHL wee implemened wih boh mehodologies. Two eo measues wee compued in ode o compae pefomances beween he mehodologies fo each case [2]: - Roo mean squae eo RMS = i x i ˆx i 2 N 16

28 1.2 Tue Esimaed # of veh. pe ineval Time ineval Figue 1.15: OD pai 4, unde congesed condiions. - Roo mean squae eo nomalized RMSN= N i x i ˆx i 2 i x i whee, x and ˆx ae ue and esimaed OD flows especively and he summaion anges ove all OD pais and all inevals ove which analysis is pefomed. Table 1.1 shows he esuls. Table 1.1: Compaison of boh mehodologies MT fee flow BHL congesed W/O deviaions Wih deviaions W/O deviaions Wih deviaions RMS RMSN In boh cases he mehodology wih he deviaions povides bee esuls. Even hough he saisics fo he wo mehodologies ae close in he MT case, Figue 1.16 shows ha he mehodology wih deviaion povides bee esuls. The figue shows he ageemen beween he ue and he esimaed OD flows, fo OD pai 9 he same as in Figue 1.8, using he wo mehodologies. Clealy he mehodology wih deviaions agees much bee wih he eal daa. Because of he eason saed in Secion 1.2.2, his esul is no supising. 1.4 Tansiion peiod: how o deec when affic condiions change Tavel ime is a good indicao of he affic condiions on a secion of highway: he lage he avel ime is, he moe congesed is he secion. Assignmen maix A in Equaion 17

29 110 Wihou Deviaion 110 Wih Deviaion OD flow OD flow Tue Esimaed Tue Esimaed Time h Time h Figue 1.16: Visual compaison beween he mehodologies 1.7 depends on avel imes and hus, on affic condiions. If we knew avel imes beween saions, we would be able o diecly compue he assignmen maix A and un he algoihm. Two quesions aise a his poin: How accuae does he avel ime esimaion need o be? If he model is vey sensiive o he maix A, he avel ime esimaion should be vey accuae. On he ohe hand, if he maix A does no affec he model in a significan manne, we can ely in less accuae bu easie-o-implemen mehods o esimae avel ime. How can we obain avel imes beween saions in eal ime? This, of couse, is going o depend on he answe o he fis quesion How maix A affecs he model: sensiiviy analysis In ode o deemine how accuae he maix A and hen, avel imes needs o be, a sensiiviy analysis was pefomed. Tha is, implemenaions of he algoihm using diffeen avel imes wee pefomed, which yield diffeen assignmen maices A. The idea is o see how wose he esul would be by assuming a wong avel ime. Fo he BHL newok six diffeen avel imes wee ied in he congesed egimen: 18 fee flow, 30, 50, 70, 90 and 120 seconds 14. As was saed in Secion 1.3.2, he acual avel ime is 60 seconds. The 18 seconds avel ime is ied o see wha he esimaion esuls would be if fee flow condiions ae assumed when affic is acually congesed. The esuls in ems of he eo measues descibed in Secion RMS and RMSN ae shown in Table 1.2. Thee ae hee ineesing poins o menion hee. Fis, 90 seconds avel ime gives he bes eo measues. This migh be due o he fac ha avel ime was no consan 14 In a 0.3 miles sech of highway, hese imes imply he following speeds especively: 65 fee flow, 40, 24, 17, 13, and 10 mph. 18

30 Table 1.2: Eo measues fo diffeen avel imes 60 sec eal 18 sec ff 30 sec 50 sec 70 sec 90 sec 120 sec RMS RMSN duing he peiod unde sudy as menioned in Secion 1.3.2, and 90 seconds was a bee esimaion of he avel ime. The second poin o noe is ha eo measuemens fo avel imes ha assume ceain degee of congesion i.e. geae o equal o 30 seconds do no vay oo much fom one simulaion o anohe. In fac, he visual ageemen beween ue and esimaed cuves ae quie simila in all cases Figue 1.17 a and b show he cases when avel ime is 30, and 120 seconds fo he fis OD pai Tavel ime = 30 sec ; v = 40 mph Tavel ime = 120 sec ; v = 10 mph Tavel ime = 18 sec fee flow; v = 65 mph a 65 b 65 c # of veh. pe ineval Time ineval # of veh. pe ineval Tue Esimaed Time ineval # of veh. pe ineval Time ineval Figue 1.17: Visual ageemen beween ue and esimaed cuves fo fis OD pai BHL and fo diffeen avel imes. Lasly, if fee flow condiions ae assumed avel ime 18 seconds, he RMS and RMSN incease significanly. As can be seen fom Figue 1.17 c, in his case he visual ageemen beween ue and esimaed cuves is no good. Tha is he eason fo he lage RMS and RMSN obseved in Table The second and hid poins sugges ha we need o be able o disinguish beween fee flow and congesed egimes. Howeve, if hee is congesion, an accuae avel ime esimaion is no needed because diffeen maices A will yield quie simila esuls. The impoance of his esul will become clea in he nex secion Tavel ime esimaion Diffeen mehods o esimae avel ime can be found in he lieaue. The following ae hee diffeen appoaches: If we know he unifom speed along he whole secion, we can use = d is ime, v d is disance, and v is speed o esimae avel imes. The speed could be diecly measued fom he deecos using dual-loop deecos, fo insance. We can hen assume ha his speed is he same along he segmen. Howeve, assuming ha he speed a he deeco saion locaion poin speed coesponds o he speed 19

31 ove he whole secion does no seem o be a good appoximaion see fo insance Table 1 in [6]. Anohe opion is o consuc cumulaive cuves fom consecuive loop deeco saions N- cuves, see [7] o [8]. The hoizonal diffeence beween wo cumulaive cuves epesens he avel ime beween wo saions. This appoach assumes FIFO behavio which seems easonable and consevaion of vehicles beween saions no amps. The lae assumpion can be eaed in some way by adding amp couns o he mainline couns fo insance, bu i would inoduce some eo in he esimaion. The esimaion qualiy will also depend on he fequency of speed measuemens. A hid opion consiss in using a discee vesion of he q-k diagam. Insead of an infinie numbe of possible saes infinie numbe of k o speed, we ecognize hee o fou saes o modes. Each one of hese saes has a avel ime associaed wih i, and hus an assignmen maix A oo. Fo insance, if we ae unning he algoihm on he BHL newok, we would use maices compued in Secion when fee-flow condiions ae deeced, and maices fom Secion when congesion aises. This example assumes only wo modes: he whole secion feely flowing o he whole secion congesed a he same level. In [9], he auho assumes ha he mode canno be diecly obseved fom he daa. The mode jumps, howeve, follow a discee-ime Makov chain, wih ceain ansiion pobabiliy. Hee, he algoihm would joinly esimae he mode and he OD flows a each ime ineval. Given he esuls shown in Secion 1.4.1, i seems like we do no need an accuae avel ime esimaion. In ligh of he hid appoach discussed ealie, we could say ha speeds wihin a ceain ange and avel imes coesponding o hese speeds yield he same assignmen maix A. Tha is, he fundamenal diagam q k has been disceized ino a few modes. Then, he algoihm migh use loop deeco daa moving aveage of speed measuemen o deemine unde which mode he secion is woking, and use he coesponding maix A. We can idenify as many mode as we wan. Fo insance, one mode fo he feeflow condiions, and wo o hee modes fo diffeen levels of congesion on he whole secion 15 and hus, diffeen avel imes and diffeen A maices. The fis mode fee flow would coespond o speeds geae o equal o v 1, he second mode would include speeds beween v 1 and v 2, and he hid mode would conain speeds less han o equal o v 2. Then, we would se he following ule fo each obsevaion vk moving aveage of speed: If v 1 vk mode 1 pick A 1. If v 2 < vk < v 1 mode 2 pick A 2. If vk v 2 mode 1 pick A Fo easons saed in Secion 1.1, we will no conside modes whee a shock is aveling he secion. Fo insance, upseam congesed and downseam feely flowing is no a possible mode. Those peiods of ime should be vey sho especially if he newok is sho and can be ignoed. 20

32 Secion saed ha esimaions duing congesion do no change oo much fo diffeen avel imes as long as hey acually coespond o congesed condiions. This encouage us o hink of only wo modes, fee flow and congesed modes, and only one maix A associaed wih each mode. Two issues emain unsolved so fa: wha should be he speed heshold v h ha deemines he limi beween fee flow and congesed egimes; and wha speed should be used o compue he maix A in each mode. The wok of Vaaiya [10] can be used o addess boh quesions. In ha wok, he auho showed ha, on a ceain highways in Califonia I-10E, dives spen vey lile ime a ansiional speeds beween mph. On aveage, mos of he ime hey dive a 30 mph 35% of he ime and 60 mph 65%. If his is exapolaed o ohe highways in Califonia which seems easonable, he heshold speed v h could be se a 45 mph and maices A may be compued using a speeds of 60 and 30 mph fo fee flow and congesed condiions especively. In ode o es if v h = 45 mph is a good appoximaion o no, daa fom anohe deeco saion on BHL wee colleced. The daa consis on he 30-seconds aveage speeds acoss fou lanes no including he HOV lane colleced fom 4am o 8:30pm duing a weekday. The moving aveage ove 3 minues 16 was compued in ode o file some vaiabiliy of he daa. Then, each fileed obsevaion was compaed agains he heshold v h = 45 in ode o deemine he mode. Figue 1.18 shows boh he speed pofile ead by he deeco lowe cuve and he mode pediced using he algoihm jus descibed upe cuve. Using v h = 45 mph yields a mode sequence ha is in vey good ageemen wih wha happens in ealiy. 100 Speed mph Mode 1:fee flow; 2: congesion Time hous Figue 1.18: Speed measued fom deeco and mode pediced by seing v h = 45 mph. Fom he speed pofile, moning and evening ush peiods can be clealy idenified 6-10am and 2:30-5:30pm especively. The aveage speed is aound 40 mph in he moning and 30 mph in he evening. If we would have se he heshold a 40 mph, he moning ush peiod would have been jumping beween he wo modes. Secion 1.4.1, howeve, povided easons o believe ha fo ou puposes, moning and evening ush k 16 i=k 6 Tha is, he fileed obsevaion a ime k is given by: vk = vi 7, whee vi is he aveage speed a ineval i acoss all lanes. 21

33 peiods ae he same. Because of his, i would be pefeable o se he heshold no below 45 mph. In summay, given ha he esimaion esuls ae no oo sensiive o diffeen avel imes duing congesion, only wo maices A should be compued fo each newok: one fo he fee flow egime using a speed of 60 mph and he ohe one fo he congesed egime wih a speed of 30 mph. Using speed measuemens fom deecos, we have poposed a vey simple way o deemine which of hose maices should be used i.e. o deec when he affic mode changes. 1.5 Consideaion of newok size OD flows on a highway can be useful o implemen conol saegies such as amp meeing o o make opeaional sudies in ceain aea fo insance, on weaving secions. Fo his ype of use, we eally need o know how many vehicles ae going o make use of evey amp. Fo example, in Figue 1.19 we do no eally need o know how many vehicles ae going fom poin 1 o n 1 o n. We jus need o know how many vehicles ae going o ake one of he nex off-amps poins 4 and 6 and how many ae going hough he highway and he same is valid fo hose vehicles coming fom poin 2. N 1 N 2 N 3 N m n Deecos 6 n Figue 1.19: Imaginay newok Fo his eason, we hink ha i migh be easonable o divide he highway ino smalle newoks fo insance, N 1, N 2,,... N m in Figue 1.19, and hen un he algoihm on each one of hese newoks. Wha ae he pos and cons of doing ha? Pos: Tansiion peiod peiod when hee is shock aveling hough he newok, which ceaes wo modes in each newok: he shoe he newok, he shoe he ansiion peiod on ha newok is. In Secion 1.4, we saed ha hese peiods will be ignoed. If we conside he whole newok in Fig.1.19 as one newok using n = 8 and m = 3, a shock wih speed of 12 mhp would ake 15 minues 17 o cove he whole secion i.e. he algoihm would no wok duing 15 minues. Howeve, if we divided he newok ino N 1, N 2, and N 3, he algoihm would no wok only duing 5 minues fo each newok This assumes N 1, N 2 and N 3 ae 1 mile long 18 Each newok, in his case, is independen on he adjacen newok s esimaion. Tha is, we can un ou algoihm in N 1 and N 3, wihou infomaion fom N 2. 22

34 Clealy, fo lage newoks i.e. ignoed. lage n he ansiion peiod canno be Assignmen maix A esimaion: since he newok is sho, hee would be few OD pais and few link deeco saions. Then, he esimaion of he A maix would be easie. BHL newok is like any of he sho newoks in Fig.1.19, and we have aleady validaed he algoihm fo boh fee flow and congesed condiions in his newok. Cons: In Fig.1.19 we would no know he flows fom poin 1 o 6, n 1 o n, o fom poin 4 o n 1 o n, and so on. Howeve, as we saed befoe, his is infomaion ha we do no eally need fo conol puposes. 1.6 Conclusions Real-ime OD maix esimaion mehods epoed in he lieaue wok unde saionay condiions, i.e. hei pefomance when affic condiions change abuply is no good. We have implemened a mehodology ha idenifies when affic condiions change and hen makes use of exising models o esimae OD flows in a linea newok. The models wee esed using eal daa colleced fom wo diffeen newok and duing fee flow and congesed condiions. The esimaions mach he exac daa well. One paamee of he model used depends on affic condiions moe pecisely avel imes. We have shown peliminay evidence o suppo he idea ha an accuae esimaion of he avel ime is no needed, and a disincion beween fee flow and congesed condiions is enough fo ou puposes. A vey simple algoihm o deec hese condiions, which makes use of he speed measuemens povided by loop deecos, was poposed. We have no menioned he fac ha maix F in Equaion 1.7 also depends on affic condiions. Maix F assumes ha pevious OD flows affec he cuen one in a linea way auoegessive fom in Equaion 1.1 and 1.3. If we have evidence o believe ha hese influences o effecs ae diffeen fom fee flow o congesed egime, we will end up wih wo maices F insead of one. Esimaion of each one of hese maices would be as descibed in Secion A pacical implemenaion of he mehodology poposed should no be had. Suppose we ae ineesed in compuing OD flows fo a given newok fom 4am o 9pm on a weekday. The daa needed o implemen he mehodology would consis of: Two ses of hisoical couns a evey eny and exi poin of he newok: he fis se of couns will be used o compue a fis OD maix based on opimizaion models descibed in he lieaue such as he one descibed in [2]. If hee exis a hisoical OD maix fo he newok, hese couns would no be needed. This fis OD maix will be used o compue maix F in Equaion 1.7. The second se is o implemen he mehodology wihou deviaions in he newok using he maix F aleady esimaed. As a esul, we will have wo OD maices. 23

35 Couns and speed measuemens a evey eny and exi poin of he newok fo he peiod of inees: his is he infomaion needed o un he whole mehodology and o obain esimaions. Finally, hese daa migh be fed ino a compuaional ool ha will esimae he OD flows fo he peiod unde analysis. Based on he esuls epoed hee, hese esimaions should be accuae enough. 24

36 Bibliogaphy [1] J.Kogmeie S.R.Hu, S.Madana and S.Peea. Esimaion of dynamic assignmen maices and OD demands using adapive kalman fileing. ITS Jounal, 6: , [2] K.Ashok and M.Ben-Akiva. Alenaive appoaches fo eal-ime esimaion and pedicion of ime-dependen oigin-desinaion flows. Tanspoaion Science, 341:21 36, [3] K.Ashok and M.Ben-Akiva. Esimaion and pedicion of ime-dependen oigindesinaion flows wih a sochasic mapping o pah flows and link flows. Tanspoaion Science, 362: , [4] B.Andeson and J.Mooe. Opimal Fileing. Penice Hall Inc., [5] M.Bielaie and F.Ciin. An efficien algoihm fo eal-ime esimaion and pedicion of dynamic od flows. Opeaions Reseach, 521: , [6] B.Coifman. Esimaing avel imes and vehicles ajecoies on feeways using dual loop deecos. Tanspoaion Reseach A, 364: , [7] G.F.Newell. Applicaions of Queueing Theoy. Chapman & Hall, London, 2nd ediion, [8] C.F.Daganzo. Fundamenals of Tanspoaion and Taffic Opeaions. Elsevie Science Inc., New Yok, [9] X. Sun. Modeling, Esimaion, and Conol of Feeway Taffic. PhD hesis, Univesiy of Califonia, Bekeley, [10] P.Vaaiya. Reducing highway congesion: an empiical appoach. Euopean Jounal of Conol, 11: , Sae augmenaion This appendix aims o explain and show how he sae augmenaion pocess descibed in Secion woks. Fo his pupose we will do he whole pocess using a small newok. The newok in Figue 20 conains one oigin A and wo desinaions B, and C. Thee ae hee OD pais, which ae labeled as follows: 25

37 Fom A o B: OD pai 1. Fom A o C: OD pai 2. A B C Figue 20: Small newok wih 2 OD pais In his case, n OD = 2 and n L = 3. Fo simpliciy wih he noaion, le us assume ha p = 2 and u = 1 and hen s = maxu, p 1 = 1 and ha we ae woking wih he mehodology wihou he deviaions. Equaions 1.1 and 1.2 ae as follows: x h+1 = f hh x h + f h 1,h x h 1 + w h 9 y h = a hh x h + a h 1,h x h 1 + v h 10 Each em in pevious equaions ae descibed nex: Sae veco a ime ineval h: x h = x1h x 2h Two 2 2 maices descibing he effec of pevious OD flows on he cuen one: f h f hh = 11 0 f h 1 0 f22 h 11 0 f h 1,h = 0 f22 h 1 Noe ha hese foms assume ha OD pai 1 does no affec OD pai 2 and vicevesa. Random eo: w h = Veco of couns: y h = w11 w 21 y Ah y Bh y Ch Two 3 2 maices descibing how OD flows affec couns: a hh = a h 21 0 a h 1,h = a h 1 0 a h a h 1 32 Measuemen eos: v h = v 11 v 21 v 31 Sae augmenaion is done in ode o bing Equaions 9 and 10 ino he fom of Equaions 1.5 and 1.6, especively. The new epesenaion will be given by: Each em now is as follows: X h+1 = F h X h + W h 11 Y h = A h X h + v h 12 26

38 Sae veco a ime ineval h: X h = x 1h x 2h x 1h 1 x 2h 1 Squae maix F 4 4 descibing he effec of pevious OD flows on he cuen one: f11 h 0 f h fhh f F h = h 1,h 0 f22 = h 0 f h 1 22 I Random eo: W h = w h 0 0 = w 11 w Veco of couns same as befoe: Y h = y Ah y Bh y Ch Maix A 3 4 descibing how OD flows affec couns: A h = a hh a h 1,h = a h 21 0 a h a h 32 0 a32 h 1 v 11 Measuemen eos same as befoe: v h = v 21 v 31 Even hough we have used subindex h in maices F, W, A, and v, in pacice we assume ha hese maices ae ime invaian. Finally, Equaions 11 and 12 ae he ones ha ae finally fed ino he Kalman file o obain OD flows esimaes. 27

39 CHAPTER 2 ESTIMATING TIME-DEPENDENT FREEWAY O-D DEMANDS WITH VARIATIONAL INEQUALITIES Pepaed by: Michael Zhang, Yu Nie and Zhen Qian Depamen of Civil & Envionmenal Engineeing Univesiy of Califonia, Davis 28

40 2.1 Inoducion The Dynamic O-D Esimaion Poblem Tadiionally, an O-D able concens ips made ove a elaively long ime peiod e.g., moning peak ime wihin which he affic condiion is assumed o be homogeneous. Such O-D demands ae inended o be used wih saic avel foecasing models. Howeve, i is widely ecognized ha saic models ae inadequae o pedic he evoluion of affic paen ove ime of day. Taffic congesion is essenially a dynamic phenomenon. Fis of all, avel demands do flucuae ove ime of day. Duing moning commue, fo example, demand levels change subsanially as aveles adap o ime-vaying affic condiions by ouing and scheduling of depaue imes. Recuen affic congesion ofen seen in uban aeas is mainly a esul of he way such flucuaions ake place in space and ime. Namely, imbalanced disibuion of demand causes he shoage of oad supplies duing peak ime a vaious locaions bolenecks, whee queues develop and spead ove he newok. Theefoe, an O-D able ha easonably eflecs empoal disibuion is ofen indispensable fo dynamic avel foecasing models, which age a wide specum of applicaions anging fom sho-em planning o wihin-day affic conol/managemen. Howeve, geing eliable dynamic avel demands is nooiously difficul. In ypical avel diay daa, empoal infomaion e.g., saing ime and duaion is only available fo ips of ceain puposes mainly home-based wok and school. The dynamic disibuion facos used in pacice ae ofen aggegaed based on hese ips only, hus no necessaily epesenaive fo ips of ohe ypes, such as shopping and eceaion ips. To exac empoal infomaion fo hose ips call fo aciviy-based avel demand models, which emains a sae-of-he-a unil vey ecen, we noe ha he disibuion of demands in ime can naually aise fom foecasing models hemselves when individuals' depaue ime choices ae endogenized, as fis shown by William Vickey. Howeve, a demand paen esablished fom such a DTA model highly depends on individuals' pefeed aival ime windows and how hey pice unpuncual aivals. No supisedly, hese behavio paamees ae difficul o calibae, and he assumpions ha inend o simplify he poblem ae ofen oo song o be ealisic. Consequenly, alhough exising DTA models wih depaue ime choice may povide useful insighs o macoscopic policy analysis, hey hadly yield a dynamic O-D able moe han concepually meaningful. Moeove, commues' scheduling of depaue ime may no fully explain demand flucuaions duing he ush hou. In a nushell, deemining he up-o-dae ime-vaying avel demand paen in a highway newok emains a challenging and o some exen unesolved issue. On he one hand, avel demands obained fom lage-scale suveys no only come wih high pices in ems of moneay, ime and labo coss, bu also ae likely o be ou-of-dae. Moe impoanly, household suveys based on avel diaies do no ypically povide empoal ip infomaion wih a esoluion adequae fo dynamic avel foecasing. On he ohe hand, alhough avel foecasing models may be used o esablish he dynamic paen of avel demands, he oupus subsanially depend on individuals' 29

41 avel behavios and hus may no be eliable. This explains why subsanial eseach effos have been invesed in esimaing O-D demands fom vaious affic suveillance daa, which can be auomaically colleced a elaively low coss. Among diffeen affic daa, link flows e.g., affic couns fom loop deecos ae mos widely used. In he nex few secions, we will pesen a new Dynamic OD Esimaion DoDE poblem fo feeways based on vaiaional inequaliies. The feeway newoks we conside ae simila o hee one shown Figue 2.1, whee each O-D pai has a unique pah. Mainline Deeco Deeco Deeco Deeco On-amp Off-amp On-amp Off-amp Figue 2.1 Simplified highways On-amp Conside a feeway newok G N, A, whee N and A ae he ses of nodes and links especively. Nodes ae he locaions whee affic flow will mege ino he mainline o leave i. Links ae made up of amp and mainline links. Le R and S epesen he se of oigins he sa nodes of On-amps and desinaions he end nodes of Off-amps. The end of he feeway is consideed a special off-amp especively. The cadinaliies of he ses nodes, links and O-D pais ae denoed as N = m, A = n and R S = o especively. Le [ 0, T ] be an assignmen hoizon i.e., he analysis peiod. The newok is assumed o be empy a = 0, and only avel demands depaing wihin he assignmen hoizon ae consideed. Coesponding o he assignmen peiod, we define a loading hoizon [0,T ], whee T maks he ime when all affic cleas he newok. Le q s be he avel demand beween O-D pai s depaing a ime, and he oal demand fo he whole assignmen hoizon is T qs = qs d 0 Le c s denoes he ime-dependen avel ime fo OD pai s, f s he depaue flow ae fo OD pai s, all duing he assignmen ineval. Now, suppose ha loop deecos ae insalled a he enances of a se of seleced links, A o, so ha affic flow 30

42 eneing any link a A o a any ime [ 0, T ], x a, can be measued. Le n o = Ao be he numbe of he obseved links, φa be an assignmen ineval, a discee duaion duing which he depaue flow ae fo any O-D pai is assumed o be consan, ma, he numbe of assignmen inevals is given by T = m a φa, φm be he measuemen ineval, a discee duaion fo which he measued affic quaniies is aggegaed and ecoded a loading hoizon consiss of mm = T / φm measuemen φ inevals of unifom lengh, and l be he loading ineval, a discee duaion duing which newok condiions ae assumed o be saionay loading hoizon consiss of m loading inevals of unifom lengh, i.e., T l = mφ l l, hen by consevaion, we have he following elaionship beween O-D flow and measued link flow, assuming hee ae no measuemen eos: whee and and ia a s m p h f sa s = 1 = x h, a R, s S, a A, h = 1,2,... m h p sa = ia 1 δ i a d =1 if affic depaing a he dh loading ineval of he h assignmen ineval heading fo h 1 δ sa d = desinaion s aives link a duing he hh measuemen ineval 0 ohewise = φ a / φl. In veco fom, i becomes Pf = x 2.1 d h sa o m We emphasize ha he mapping P, which elaes ime-dependen O-D flow o ime-vaying link flow measuemens, is endogenously deemined. Tha is, P has o be updaed in accod wih he change of he pah flow paen f in he esimaion pocess. This in un equies ime-dependen link avel imes o be compued fom a given f, a poblem known as dynamic newok loading DNL. Undelying he DNL pocess is a newok affic model ha descibes he evoluion of affic flow. Dicaed by his affic model, he mapping beween P and f is ypically quie convolued and no in a closed, analyical fom. Geneally, 2-1 is undedeemined and hus has many soluions. To esolve his poblem, addiional infomaion should be supplied. Such infomaion can be oughly caegoized ino wo ypes: 1 A paial o complee base O-D able, which is ofen esablished fom exising suvey daa, called a hisoical O-D able. 31

43 2 Pah avel imes obained fom pobe vehicles. In he emainde of his epo, we ll pesen a new famewok fo DoDE poblem based on vaiaional inequaliies VI. In his famewok, he DoDE poblem is fis ansfomed ino a VI poblem, hen a DNL based on he LWR model is used o evaluae he mapping P and pah avel imes, finally wo soluion algoihms, he basic pojecion algoihm and he mehod of successive aveages MSA ae suggesed o solve he VI based DoDE poblem. 2.2 The Poposed Esimaion Famewok and Soluion Algoihms The DoDE Fomulaion Ou objecive is o obain a ime-dependen O-D demand paen in ems of ime-dependen pah flows ha, once loaded ono he newok, can epoduce obseved link affic couns and pah avel imes as closely as possible. This can be cas ino a psudo genealized leas squaes poblem of he following fom: min z f = 0.5w Pf x Mf q c f c 2 x wq wps p 2.2 subjec o: f 0 whee he maices P and M map ime-dependen pah flows hee also O-D flows ino ime-vaying link affic couns and hisoical O-D flow aes, especively, and he posiive scalas w w x q wp, and ae weighs placed on affic couns, hisoical O-D demands, and pah cos obsevaions, especively, and he posiive scala is added o popely scale wo diffeen ypes of quaniies, flow and cos in he objecive funcion. s p The opimaliy condiions of 2.1 ae: z f, f = 0 z f 0 f 0 2.2* Because he mapping P which depends on f and cf ae non-linea and possibly non-convex, he above poblem is exemely difficul o solve diecly. Wih he inoducion of coun, O-D and cos deviaions, howeve, he above opimaliy condiions can be case ino a vaiaional inequaliy VI, hence allowing VI soluion algoihms be employed o solve he DoDE poblem. Fo any given pah flow paen f 0 and a se of obseved link affic couns x > 0, h he coun deviaion dxa is defined fo each link a as 32

44 h ma h h xa s = 1 psa fs, h = 1, 2,... mm a Ao dxa = 0 ohewise 2.3 whee ia h 1 dh psa = δ sa ia d = 1 and d h 1 φm h 1 < ea φmh d δ sa = 0 ohewise 2.4 whee e d a is he ime when a vehicle would ene link a of he pah connecing an h O-D pai and depa a 1 φ a + d 0.5 φl. Howeve δ sa d is no a coninuous e d a funcion of he eny ime. The loss of coninuiy may cause non-exisence of soluions. To esolve his issue we eplace he dynamic pah-link incidence elaionship 2.4 wih he following one: δ dh sa φm = max 0, e d a φ m h 0.5 φm 2.5 Fo any given pah flow paen f 0 and a se of hisoical O-D demand paen q > 0, he O-D deviaion ds is defined fo each O-D pai s as qs fs, = 1, 2,..., ma If qs is given fo O - D pai s a dqs = 0 ohewise Fo any given pah flow paen f 0, and a se of obseved pah avel imes c = c, s he cos deviaion is defined as { } s, dc s =, s, cs f [ c f s s c s f ] Wih coun, O-D and cos deviaions, we can now define he pah deviaion follows: ia h h dx d a ds = wx δ sa + wqdqs + wpspdcs a d = 1 im and is maix fom is given by T T d f P w x Pf + M w q Mf + w s J c c = x q p p c whee J c is he Jacobian maix of cf wih espec o pah flow f. d s as I can be shown ha he opimaliy condiions fo 2.2 can be ansfomed ino he vaiaional inequaliy VI: ~ o m ~ a o m Find f R R a + such ha d ~ f, f f 0 fo all f F which lends he poblem o easy soluion hough vaious VI soluion algoihms. 33

45 J c I should be noed ha does no have a closed fom and hus is evaluaion usually equies numeical appoximaion. Fo evey elemen in J c, we have: ia ia d cs f d cs f d = 1 d = 1 f s 2.9 = 2 f i s whee d is an loading ineval in he h assignmen ineval, and d is an loading ineval in he h assignmen ineval. Conside he maginal avel ime of adding one moe uni vehicle beween any wo loading inevals in he loading hoizon. Le be he depaue ime of a vehicle depaing a he h ineval beween O-D pai s, wihou he addional vehicle. Le denoe he ime he queue on ha link disappeas afe wihou he addiional vehicle if hee is no queue a, hen hee is no delay on his link. Le a denoe he aival ime of a vehicle depaing a he h ineval beween O-D pai s, wihou he addiional vehicle. Le b denoe he depaue ime of a vehicle depaing a he h ineval beween O-D pai s, wihou he addiional vehicle. Thee ae wo condiions o conside in evaluaing he maginal pah avel ime: 1 Fo O-D pai s, if is on-amp is ahead of he on-amp of O-D pai s o =, we shall fis seach he links along he pah of s o ge [, ], he effecive congesion ineval, fo evey link. Then, seach evey link along he pah of O-D pai s o obain is aival and depaue imes fo flows along ha pah, and if hese aival imes ae ealie han hei coesponding o lae han hei coesponding, congesion due o he flow fom s will no affec he flow of O-D pai s. When he flow of O-D pai s aives a any link duing ha link s effecive congesion ineval [, ], hen is conibuion o he maginal avel ime, i.e. he ecipocal of he capaciy of ha link, will be accumulaed along is pah. 2 Revesely, if is on-amp is afe he on-amp of O-D pai s, we again seach he links along he pah of s o ge hei effecive congesion inevals [, ]. Then, fom he lis of, find he one coesponding o he link along he pah of s ha meges wih he pah of O-D pai s. If his link is no congesed, hen congesion due o flows on pah s will no conibue o he maginal avel ime of O-D pai s. Ohewise, pefom he same adjusmen and compuaion as in 1. On he ohe hand, he use of he pah deviaion funcion povides a flexible famewok o fuse diffeen obsevaions ogehe. The opimaliy condiion, in he fom of a VI, is quie geneal and may be applied even when he esimaion poblem canno be cas as a mahemaical pogamming poblem. To solve he VI poblem 2-7, we need o evaluae he pah deviaion veco df, which equies dynamic newok loading, a pocess of loading ime-dependen O-D flows ono he newok accoding o a given model of affic flow dynamics. This pocess poduces he ime-dependen link flows and avel imes, and is descibed in he nex secion. a 34

46 2.2.2 The Dynamic Newok Loading Pocess As menioned ealie, he DNL poblem aims a obaining, on a congesed newok and ove a fixed ime peiod, he link cumulaive aival/depaue cuves hence ime-dependen link/pah avel imes coesponding o a given se of empoal O-D demands. DNL is an undelying componen of many dynamic newok poblems in which pahs coss depend on empoal pah flows in ways govened by affic flow dynamics. In he pas wo decades, DNL has aaced a gea deal of aenion fom anspoaion eseaches, simulaed by he need of boh simulaing uban affic and solving dynamic affic assignmen poblems. Accoding o how hey model affic flow dynamics, exising DNL pocesses may be classified ino hee goups: macoscopic, micoscopic and mesoscopic pocesses. A macoscopic DNL pocess employs macoscopic affic flow models o descibe affic dynamics, while a micoscopic DNL pocess uses micoscopic affic models, such as ca-following o paicle-hopping models, o descibe affic dynamics. A mesoscopic DNL pocess falls in-beween a macoscopic and a micoscopic DNL pocess in he sense ha i uses macoscopic models o descibe affic flow bu keeps ack of individual vehicula quana like a micoscopic DNL pocess does. A vehicula quana is an indivisible flow elemen which is acked in DNL like a vehicle in micoscopic simulaion. Howeve he size of he vehicula quana can be se abiaily small o eplicae analyical esuls as closely as desied. In his eseach, we make use of a polymophic, mesoscopic DNL pocess PDNL developed ove he yeas a UC Davis e.g., W. L. Jin 2003, X. J. Nie 2003, Yu Nie Since he feeway newok we conside in his eseach has a special sucue, he geneal PDNL pocess is consideably simplified o gain compuaional efficiency. In ou PDNL pocess, we use he hydodynamic affic flow model of Lighhill and Whiham 1955 and Richads 1956, known as he LWR model, wih he following speed-densiy s-k elaionship: s f k kc s = αs > f k k k k j c whee k is called ciical densiy, k he jam densiy and s fee-flow avel speed. c j α = k Hee c k j. This q k cuve was employed in Newell 1993 fo seamlining a gaphical LWR soluion and hen adoped by Daganzo 1994 in his cell ansmission model CTM. Ou PDNL pocess models he on-amp meges and off-amp diveges using he supply-demand mehod of Daganzo 1994,1995. The demand of a link, D, is he maximum possible exi flow ae ha wish o leave i D = min{ C, Q} And he supply of a link, S, is he maximum possible flow ae ha he link can accommodae S = min{ C, R} f 35

47 whee C is he flow capaciy depending on oad chaaceisics and/o conol saegies; Q is he ae of he flow ha is eady o exi; R is he maximum eny flow ae o he link pemied by he cuen affic condiion. Wih he inoducion of supply and demand funcions, one can deemine he feeway and amp flows on links 1-3 and 2-3 see see Figue 2.2,a ime, denoed as v 13 and v 23 beviy: whee D i S 3 fom he following maximizaion poblem he ime index is dopped fo max v = vi3, subjec o 0 vi3 Di, i = 1, 2, i i is he demand of link i, i = 1,2. is he supply of link 3, he downseam link. v i3 S D + D This pogam, howeve, does no have a unique soluion when 2 < Jin and Zhang 2003 poposed an alenaive disibuion scheme Di ai3 = D i i 3 1 S3 which yields he following simple soluion o he amp mege poblem: v i 3 = ai3 + S v, i, v = min{ D1 D2, 3}. Similaly, fo a amp divege wih feeway and off-amp links 1-2 and 1-3 see Figue 2.2, diveging flows v12 and v13 can be deemined by he following maximizaion poblem Daganzo 1995: max v = v + v, subjec o a v S, a v S v D whee a 1i , is called uning popoion. In he simples case e.g., fo evacuaion applicaions, uning popoions a 1 i a i can be deemined exogenously. Howeve, in he geneal conex of newok loading, 1 ae dependen on affic composiion, and vay wih ime and he demand paen. Thus, uning popoions have o be deived fom he desinaions of he vehicles eady o advance ino each diveging banch a any given ime. The soluion o he above mahemaical pogam is simply: S2 S3 v 12 = a12v, v13 = a13v, v = min{ D1,, } a a

48 Figue 2.2 Mege and divege nodes By pefoming DNL, we can ge a se of cumulaive aival and depaue flow cuves B fo evey link in he loading hoizon. Le us define a as he cumulaive flows o pass he enance of link a by ime, i.e., Ba = ua, [0, T ] 0 u whee a is he ae of flow eneing link a a ime. Similaly, E a denoes he cumulaive flows o pass he exi of link a by ime, i.e., Ea = va, [0, T ] 0 whee v a is he ae of flow leaving link a a ime. Figue 2.3 gives an example of cumulaive aival and depaue cuves. These cumulaive cuves ae vey useful because mos quaniies of inees o he descipion of affic flow can be deived fom hem. The avesal ime ha a vehicle would expeience if i enes he link a ime is he hoizonal sepaaion beween he wo cuves noe ha he hoizonal line cosses B a a ime. Mahemaically, τ a can be compued as τ a = ag min{ Ba Ea + τ } τ

49 Figue 2.3 Cumulaive cuves consuced fom DNL We emphasize ha 2.11 holds only when vehicles do no pass each ohe when avesing he link, known as he Fis-In-Fis-Ou FIFO ule. To see his, noe ha he fomula 2.11 is valid only if any vehicle will no exi he link unil all vehicles pesen on he link a is eny ime have lef. Pu i in anohe way, no vehicle can leave he link ealie han any vehicle which enes he link befoe i Algoihms fo Solving Vaiaional Inequaliy Poblems The DoDE poblem cas as he VI poblem 2-8 can be solved by a numbe of algoihms. Hee we pesen wo such algoihms, he basic pojec algoihm and he heuisic soluion algoihm known as he mehod of successive aveages MSA. Befoe we poceed o descibe hese algoihms, we simplify he noaion by using oma f f, W R+, c f d f and accodingly denoe he VI poblem by VI c,w. Le us define he mei o gap funcion of VI c,w as follows: ρ f : = min c f, g f = max c f, f g f g W is a soluion o VI c,w if and only if f solves he maximizaion pogam 1 Basic Pojecion Algoihm Le be a consan posiive scala, he main ieaion of his algoihm cenes aound compuing he following pojecion: g W 38

50 k + 1 k 1 k f = ΠW f c f This algoihm has been applied fo solving he DTA poblem in Wu e al 1998, whee 1 i was shown ha he pojecion mapping Π W f c f ensue he convegence of he above algoihm. has o be a conacion o 2 The Successive Aveages Algoihm In he mehod of successive aveages, he main ieaion of he algoihm concens he following flow updae: k + 1 k f = 1 λ f + λg f k whee 1 λ = k, and κc < 0 i f ci f gi f =, κ 1 0 ci f 0 In he nex secion, we will employ hese algoihms o solve an example DoDE poblem fo a synheic feeway newok wih eigh O-D pais, and compae he esimaion esuls unde diffeen levels of affic infomaion. 2.3 Numeical Resuls The Feeway Newok and Relaed Daa The feeway newok used in his sudy, which conains a nohbound poion of Feeway SR-41 in Fesno, CA fom N. Fesno S. o S. Golden Sae Blvd, 16.7 miles, is shown in Figue 2.4. The newok used in ou expeimen is a immed vesion which includes only he on/off amps and mainline links of he feeway newok as illusaed in Figue 1-1. The immed newok consiss of eigh inechanges, 12 off-amps, 17 on-amps, 31 feeway mainline links and 116 O-D pais. Thee ae hee main bolenecks in his newok due o lane dops. The assignmen hoizon is a wo-hou peak peiod, and he assignmen, measuemen and loading inevals ae se o 15 minues, 15 minues and 6 seconds, especively. Figue 2.4 A eal feeway newok The synheic ime-dependen O-D ables and affic measuemens in ou expeimens 39

51 ae poduced using he following pocedue. Fis, eal affic couns povided by Calans Disic 6 ae used o esimae an iniial ime-dependen O-D able. Second, oal avel demands fo he whole assignmen hoizon ae obained fo each O-D pai by summing all ime-dependen enies of he able. Thid, vaious ules ae applied o allocae he oal demands o diffeen assignmen inevals following a fla, apezoidal o a wo-peak paen see Figue 2.5 and Table 2.1, and he esuls ae used as he gound uh fo ime-dependen O-D demands he Tue O-D able. Fouh, a dynamic newok loading DNL based on he kinemaic wave heoy is pefomed o obain affic measuemens including affic couns and pah avel imes noe ha affic assignmen equals newok loading in he feeway case since no oue choice is involved. Finally, he synheic O-D able is unifomly peubed by 20% i.e., each eny of he synheic OD able imes 1.2 o geneae a synheic ime-dependen hisoical O-D ip able. The oal demands in he assignmen hoizon ae used as planning saic demands. Since we have synheic O-D demands which ae assumed o be he ue O-D flow in his case, we may obain synheic affic couns fo each link and pah avel ime fo each OD pai by pefoming PDNL wih he synheic OD, as descibed above. Regading he hisoical OD maix, we unifomly peubed he synheic O-D by 20% i.e synheic OD imes 1.2 unifomly. Moeove, we se he consan oal O-D demands in wo hous as he planning saic O-D. Thee ae all kinds of infomaion we can have o conduc he O-D esimaion, bu someime only some of hem ae available o us, o fo example, we may only obain he affic couns infomaion fo vey limied links main links o amps. Theefoe, we also would like o see he accuacy of demand esimaion given by paial infomaion based on combinaions of diffeen infomaion. Now we ae able o ceae diffeen scenaios o check which kind of infomaion is moe efficien in pedicing O-D demands. 40

52 apezoidal O-D demand fo one OD pai 30 O-D demandveh/ineval Assignmen ineval fla O-D demand fo one OD pai 25 O-D demandveh/ineval assignmen ineval wo-peak O-D demand fo one OD pai 35 O-D demandveh/ineval assignmen ineval Figue 2.5 The hee ypes of disibuions of demand ove ime 41

53 Table 2.1 Allocaion ules fo hee ypes of disibuions of demand Two-peak paen Ineval Popoion.1/16 1/8 3/16 1/8 1/16 1/8 3/16 1/8 Fla paen Ineval Popoion.1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 Tapezoidal one-peak paen Ineval Popoion.1/23 5/46 4/23 4/23 4/23 4/23 5/46 1/23 The following hee indices ae used o measue he qualiy of he O-D esimaion: 1 GEH: a meic poposed by Biish enginees o measue he qualiy of he esimaes Zhang e.al N 2 V pi Vi i= V pi + Vi / 2 GEH = 1 N whee Vpi = he i-h value pediced by he model and Vi = he i-h field measuemens. N is he oal numbe of obsevaions. A pefec mach will esul in a zeo GEH value Zhang e.al Mean Eo ME: ME = N i= 1 V pi V N V i i 100% 3 RMSE: he oo mean squae eo. RSME i= 1 = N Vpi Vi Sixy scenaios ae ceaed o compae he esimaion qualiy unde vaious daa coveage schemes. Fo each demand paen apezoidal o one-peak, fla and wo-peak, we conside eigh diffeen daa coveage schemes: 1 Taffic couns 2 Taffic couns and paial pah avel imes 3 Taffic couns and pah avel imes 4 Taffic couns and hisoical O-D 5 Taffic couns, hisoical O-D and paial pah avel imes 1 6 Taffic couns, hisoical O-D and pah avel imes 7 Taffic couns and planning saic O-D 8 Taffic couns, planning saic O-D and pah avel imes As fo affic couns, o see how many links of couns ae appopiae fo DoDE in N 2 1 Paial means ha pah avel imes ae available only on some assignmen inevals and/o O-D pais. 42

54 ems of accuacy, we also ied o give hee ypes of affic couns 1 All affic couns infomaion including all couns on mainlines and amps 2 Mainline affic couns infomaion 3 Random affic couns infomaion whee 30 ou of 60 mainline links and amps ae andomly picked fom a unifom disibuion. In all he expeimens, BPA is eminaed when he numbe of main ieaions exceeds 50 o he sandad deviaion of esimaed O-D ables in wo consecuive ieaions is less han 0.05, whicheve comes fis Tesing Scenaios and esuls Among he hee demand paens, he consan fla demand paen poduces no congesion in he newok, so is flow chaaceisics ae moe pedicable. Boh he apezoidal and he wo-peak paens cause affic congesion a hee disjoin bolenecks. The congesion lass half an hou in he wo-peak paen and one hou in he one-peak paen, and neve speads o ohe secions beyond he boleneck secions. Table 2-2 epos he ME, GEH and RMSE saisics obained fom he DoDE esimaion esuls in all 60 scenaios. Since he hee saisical measues show consisen esuls, i.e., when one measue is lowe in one case han in anohe case, he ohe measues also shae he same end, when will use only he ME saisics in ou subsequen discussions. The ole of affic couns Among he vaious foms of affic infomaion, affic couns ae he mos commonly available and hence povide he basic inpus o O-D esimaion. The quesions ae: ae hey sufficien o obain good O-D esimaes, and how many couning locaions ae needed? Ou expeimens indicae ha even unde a full se of affic couns, he mean eo obained sill anges fom 22% o 53%, alhough he affic couns hemselves ae closely epoduced by he model fo all he hee demand paens. Reducing he amoun of couning locaions, on he ohe hand, would lead o even pooe esuls. Fo example, he ME fo he one-peak demand paen inceases fom 49% o 64% when abou half of he links ae couned. In eihe case, jus affic coun alone seems inadequae o povide a eliable esimae of a ime-dependen O-D ip able, because he mapping beween ime-dependen O-D demand and he obseved link affic couns is no one-o-one. In ou expeimens, he numbe of couning locaions is educed by half, and in one case, he 30 couning locaions ae placed on all feeway mainline links, and in anhe case, hey ae andomly placed on eihe feeway o amp links. I seems ha when hee is sufficien numbe of couning locaions, whee o place hem is no a ciical issue. Bu his may change when he numbe of couning locaions is much fewe, and is woh fuhe invesigaion. 43

55 All couns Mainline couns. Random couns Table 2.2 The esuls of all he 60 scenaios Scenaios Tapezoidal demand Fla demand Two-peak demand ME GEH RMSE ME GEH RMSE ME GEH RMSE His SP His His SP No PT PPT PT No PT PPT PT No PT PPT PT His His His His His His His Sands fo hisoical O-D, PT sands fo pah avel ime, PPT sands fo paial pah avel ime and SP sands fo saic planning O-D The ole of hisoical O-D infomaion In all cases whee hisoical O-D ip ables ae used, he mean eos ae educed damaically. Noe ha in ou expeimens he hisoical O-D demands always eain he shape o pofile of he undelying avel demand paen because hey ae geneaed fom a unifom peubaion o he synheic O-D ha we ae ying o esimae. The emakable impovemen we obained wih he addiion of hisoical O-D should be lagely cedied o availabiliy of his sucual infomaion. Including saic O-D demands also impoved maginally he esimaes fo boh he one- and wo-peak demand paens, bu he impovemen is moe sizable o he fla demand paen. This is somewha expeced because a saic O-D ip able conains he shape of he fla-demand paen, bu no he one- o wo-peak paens. These esuls highligh he impoance of he knowledge of he pofile of he demand, no he volume of demand in O-D esimaion. Alhough hisoical OD infomaion ha conains he pofile of he O-D demands o be esimaed is vey useful in guiding he esimaion pocess o find he igh demand pofile, hisoical O-D demands hemselves may no be elied on a he lae sages of he esimaion because hey can be fa away fom he acual O-D demands. Theefoe focing he demand deviaion smalle acually could lead o lage esimaion eos. 44

56 Figue 2.6 shows he change of ME, GEH and RMSE of couns, pah avel ime and O-D demand ove all he ieaions in he one-peak paen whee all hee kinds of daa ae povided. Obviously, he esimaion sas o move in he wong diecion afe 15 ieaions when he deviaion of hisoical OD is compaaively lage. We can eliminae his poblem by using hisoical infomaion in he beginning pa of he esimaion pocess o shape he demand pofile, hen discad i and poceed wih ohe infomaion such as affic couns and/o pah avel imes. Figue 2.7 shows an example whee he above pocedue is caied ou, and one can see ha he values of ME, GEH and RMSE ae nealy monoonically deceasing wihou he obvious upun shown in Figue 2.6. The ole of pah avel imes As expeced, he use of avel imes did no impove he qualiy of O-D esimaes in he fla demand paen, since hee is no congesion in he newok in ha scenaio. Supisingly, ou expeimens showed ha he use of avel imes in he ohe wo cases whee hee is congesion in he newok also did no impove he O-D esimaes. This is somewha unexpeced because unlike affic couns, avel imes can eveal moe abou affic condiions on he newok. Upon a moe caeful examinaion, howeve, an explanaion can be found. Inuiively, if wo O-D pais go hough he same boleneck, i makes no diffeence o he queuing ime hence he pah avel ime whehe he addiional vehicle ha joins he queue is fom one O-D pai o he ohe. Tha is, he mapping beween O-D pah flow and pah avel ime is also no one-o-one. Thus he use of pah avel imes will no eliminae he unde-deemined condiion in he O-D esimaion poblem. Wha complicaes he poblem even moe is ha unde in he dynamic conex, affic couns and avel imes ae inicaely elaed hough he evoluion of ime, so he wo pieces of infomaion ofen ovelap each ohe. In fac, if hee ae seveal bolenecks in he newok as in ou case, he incease of one moe uni of avel ime on one pah will affec all he avel imes of O-D flows ha go hough he boleneck ha caused he one uni avel ime incease. Theefoe, i is difficul o idenify he O-D pai wih ha addiional vehicle ha caused he avel ime incease ha affeced he ohe O-D pais. I should also be noed ha binging in pah avel ime poses difficulies o he soluion algoihm as well, since he mapping is highly nonlinea and may no saisfy he monoone popeies equied by he basic pojecion algoihm. Moeove, he soluion o he DoDE poblem wih he pah avel ime deviaions consideed may no have a unique soluion. 45

57 Figue 2.6 ME, GEH and RMSE of couns, pah avel ime and O-D demand fo an oiginal scenaio Figue 2.7 ME, GEH and RMSE of couns, pah avel ime and O-D demand, hisoical OD daa discaded since he 16 h ieaion 46

58 Ohe findings Ou numeical expeimens also evealed some ineesing algoihmic convegence paens. When only affic couns ae used, he ME, GEH and RMSE saisics fo avel demand all convege monoonically, wih a shap dop in he 10~15 ieaions and a long, fla ail afe ha. Fo all ohe cases, hei convegence paens ae no monoonically deceasing: smalles ME, GEH and RMSE of O-D demands ae always obained aound 12~15 h ieaion, bu hen he ME, GEH and RMSE sas o ise befoe he paen ges fla again and he algoihm sops when he maximum numbe of ieaion is eached. Take he case of he one-peak demand paen wih affic couns, hisoical O-D and pah avel imes used in he esimaion, he lowes ME , GEH 0.33 and RMSE 2.05 values ae eached a he 15 h ieaion, bu ise o ME, GEH and RMSE when he pojecion algoihm sops a he 47 h ieaion. The possible easons fo hese diffeen convegence paens ae heefold: 1 when only affic couns ae used, he DoDE poblem is a quadaic opimizaion poblem, hence a unique soluion can be found hough he basic pojecion algoihm, hus we have monoone convegence; 2 when boh affic couns and hisoical O-D demands ae used, he DoDE is sill a quadaic poblem, bu he hisoical O-D can be quie inaccuae and focing he esimaed O-D demands o appoach he hisoical O-D demands by he pojecion algoihm would lead he soluion away fom he undelying O-D demand paen in lae ieaions, when he O-D deviaion is compaaively lage han he coun deviaion; and 3 when avel imes ae used, he DoDE poblem becomes highly nonlinea, and he mapping may no be monoone, heefoe he basic pojecion algoihm may no convege o may convege o a local soluion. We also found ha he qualiy of O-D esimaes vay consideably acoss O-D pais. Fo some O-D pais, good esimaes can be obained egadless of he ypes of daa used. Fo ohes, howeve, accuae esimaes can only be obained when muliple souces of daa ae used. Clealy, diffeen O-D pais have diffeen dependence on daa, bu hei empoal pofile can be capued in mos cases. As an evidence of his, Figue 2.8 shows he esimaed O-D demands fo fou OD pais andomly picked in all he 116 OD pais fo he one-peak demand paen. As we can see fom his figue, even when he magniudes of demands ae no esimaed well, he empoal pofiles of he O-D demands ae closely followed by he esimaes. 47

59 Figue 2.8 Esimaed O-D demands fo fou OD pais fo apezoidal paen 2.4 Summay We have poposed a vaiaional inequaliy appoach o esimae ime-dependen O-D demands fo a feeway newok wih eny and exi amps. This appoach akes ino accoun vaious levels of affic infomaion, such as link flow coun, hisoical O-D ables, saic planning O-D and obseved pah avel imes. The DoDE poblem pesened in his epo exclusively ages off-line applicaions, i.e., esimaing he empoal demand paen ha is elaively sable fom day o day. Wihin day flucuaions can be handled hough a Kalman file pocess o a olling hoizon saegy as descibed in Kang 1999 and Zhou Fom ou compuaional expeimens, some conclusions fo daa coveage can be dawn. Fis, affic couns ae indispensable o O-D esimaion, and he moe he bee. Bu when he numbe of coun locaions is sufficienly lage moe han half of he oal numbe of links in he newok in ou case, i appeas ha couning locaions does no mae. When he numbe of couning locaions is small, howeve, whee o place hose counes may be vial and is woh of fuhe invesigaion. Second, affic couns alone ae usually no sufficien o obain accuae O-D demand esimaes. Hisoical 48

60 ime-dependen O-D demands, paiculaly hose ha eveal he empoal demand pofiles of he undelying demand paens, can dasically impove he qualiy of he demand esimaes. When a planning saic demand maix is povided, one can apply a empoal pofile o disibue he oal demand ino seveal ime inevals, and use i as he hisoical O-D maix, and finally, coune o ou expecaions, pah avel imes do no conibue much o impoving O-D demand esimaes. Moeove, due o he complex elaion beween pah flow and pah avel ime, he use of pah avel imes also desoys he nice quadaic sucue of he DoDE poblem and bings convegence difficulies o he basic pojecion algoihm. Theefoe, hei use in dynamic O-D esimaion is no ecommended. We suspec ha in he dynamic seing, affic couns and hei empoal disibuions capue much moe of he newok condiions han in he saic seing, endeing avel imes less needed in he dynamic O-D esimaion poblem. Ou fuhe wok will be dieced a how o place he affic counes when hei numbe is small compaed wih he oal numbe of links in a newok, and how o impove he compuaional efficiency of he DoDE soluion pocedue employed in his sudy. Refeences Daganzo, C.F.1994, The cell ansmission model: a dynamic epesenaion of highway affic consisen wih he hydodynamic heoy, Tanspoaion Reseach 28B, Daganzo, C.F. 1995, The cell ansmission model, pa II: Newok affic, Tanspoaion eseach 29B, Jin, W.L. & Zhang, H.M. 2003, On he disibuion schemes in he discee kinemaic wave model of meges, Tanspoaion Reseach 37B, Jin, W. L Kinemaic Wave Models of Newok Vehicula Taffic. PhD hesis, Univesiy of Califonia, Davis. Kang, Y. 1999, Esimaion and Pedicion of Dynamic Oigin-Desinaion O-D Demand and Sysem Consisency Conol fo Real-Time Dynamic Taffic Assignmen Opeaion, PhD hesis, The Univesiy of Texas a Ausin. Lighhill, M. J. & Whiham, J. B. 1955, On kinemaic waves. II. a heoy of affic flow on long cowded oads, Poceedings of he Roy Sociey A 229, Newell, G.F. 1993, A simplified heoy of kinemaic waves in highway affic, Tanspoaion Reseach 27B, Nie, Xiaojian The Sudy of Dynamic Use-Equilibium Taffic Assignmen. PhD hesis, Univesiy of Califonia, Davis. Nie, Yu A Vaiaional Inequaliy Appoach Fo Infeing Dynamic Oigin-Desinaion Tavel Demands. PhD hesis, Univesiy of Califonia, Davis. 49

61 Richads, P. I. 1956, `Shockwaves on he highway', Opeaion Reseach 4, Wu, J.H., Floian, M., Y. W. & Rubio, M. 1998, The coninuous dynamic newok loading poblem: a mahemaical fomulaion and soluion mehod, Tanspoaion Reseach 32B, Zhang, H.M Zhang, Ma, Jingao, and Dong, Hu 2006, Calibaion of Depaue Time and Roue Choice Paamees in Mico Simulaion wih Maco Measuemens and Geneic Algoihm, Tanspoaion Reseach Boad Annual Meeing 2006 Pape # Zhou, X. 2004, Dynamic Oigin-Desinaion Demand Esimaion and Pedicion fo Off-Line and On-Line Dynamic Taffic Assignmen Opeaion, PhD hesis, Univesiy of Mayland, College Pak. 50

62 CHAPTER 3 ESTIMATING TIME-DEPENDENT O-D MATRICES FOR ARTERIALS Pepaed by: Yingyan Lou and Yafeng Yin Univesiy of Floida 51

63 3.1 Lieaue Review Backgound Oigin-Desinaion O-D maices povide infomaion on flows of vehicles aveling fom one specific geogaphical aea o anohe, and ae one of he ciical daa inpus o anspoaion planning, design and opeaions. Because i is vey ime consuming and labo inensive o obain O-D maices hough household ineviews o oadside suveys, significan effos have been made fo decades o develop mahemaical models fo esimaing he maices fom link couns, which ae elaively easie o obain. So fa, up-o-dae commecial planning ools e.g. EMME/2 and simulaion sofwae e.g., Paamics have povided buil-in O-D esimaion modules. Howeve, mos of hese O-D esimaos ae only capable of esimaing saic O-D maices ahe han dynamic o ime-dependen O-D maices. The lae ae pe-equisies fo sho-em planning applicaions, affic impac analyses, and opeaions sudies. Fo example, as a new geneaion of planning ools, DynaMIT-P and DYNASMART-P ovecome he limiaions of saic models by capuing he dynamics of congesion fomaion and dissipaion associaed wih affic peak peiods. This enables he evaluaion of a wide aay of congesion elief measues, which could include boh supply-side and demand-oiened measues FHWA, To apply hese ools, ime-dependen O-D maices should be supplied as inpus. As anohe example, in feeway coido managemen, ime-of-day amp meeing algoihms equie O-D flow facions, and adapive amp conol saegies ofen need o know dynamic O-D flow in ode o disibue expeced flow educions fom a boleneck o vaious meeed amps upseam. Though he esimaion of saic O-D maices is well eseached, dynamic O-D esimaion is a much moe ecen opic. Fo he saic esimaion, hee ae exensive lieaue concening vaious fomulaions and soluion mehods, including infomaion minimizaion, enopy maximizaion, maximum likelihood, Bayesian infeence, and genealized leas squaes fo newoks wihou congesion, and bi-level pogamming fo newoks wih congesion see he epo by Chen e al., 2004 fo a ecen suvey of his opic. The saic appoach equies all ips saed in he modeled peiod o be compleed in he same peiod. This assumpion may be appopiae fo a long ime hoizon, bu ceainly no ealisic a all fo a vey sho ime a a geneal newok, say five o en minues wih a long sech of a highway coido. In view of ha eal-ime infomaion of O-D flows o he O-D maix fo each sho ineval is an essenial inpu fo sho-em planning applicaions and eal-ime affic opeaions and managemen, especially in he conex of inelligen anspoaion sysems ITS, a vaiey of models have been poposed o use ime-seies of link flow daa o deive dynamic O-D maices ove he las wo decades. This secion eviews he exising models fo dynamic O-D esimaion and hen aemps o idenify he issues ha may need fuhe eseach. The wo pevailing appoaches fo open newoks and closed newoks especively ae summaized in Secion Since he majo concen of he eseach pojec is on linea newoks a linea newok is a sech of highway wih muliple enies and exis, whee hee would be no oue choices involved whee eny and exi couns ae moe likely o be known heefoe closed newoks, Secion fuhe dwells on he closed-newok-oiened appoach, 52

64 elaboaing he issues of dealing wih consains in he famewok of ecusive esimaion, consideing avel ime and flow popagaion, and incopoaing muliple daa souces. Secion summaizes he issues o be fuhe addessed Exising Appoaches Geneally speaking, dynamic O-D esimaion is o use ime-seies of link flow daa o deive ime-dependen O-D maices. Based on he kenel measuemen elaionship used, he exising appoaches may be caegoized ino wo classes: closed-newok-oiened appoach and open-newok-oiened appoach Closed-Newok-Oiened Appoach Closed newoks ae newoks whee all he eny and exi couns ae known duing all measuemen inevals. Fo a geneal newok, his acually implies ha ime-vaying depaue aes ip poducion aes of all oigins and aival aes ip aacion aes of all desinaions ae known. Consequenly, he key feaue of his appoach is he diec esimaion of O-D splis fom ime-seies measuemens of newok eny, exi couns and someimes link flows. The fundamenal idea is ha he affic flowing hough a anspo faciliy is eaed as a dynamic auo-egessive pocess in which he sequences of eal-ime exiing couns depend, by causal elaionships, upon he sequences of eal-ime eneing couns. In his manne, addiional infomaion can be obained, which can be used besides he consevaions beween he exi and eny flows, o idenify he sucue and size of he flows inside he faciliy wihou using fuhe a pioi infomaion Cème and Kelle, Cème and Kelle 1981 shall be cedied fo hei fis applicaion of he above idea in idenifying uning flows a isolaed inesecions. The flow consevaion equaion is he basic sysem equaion in his appoach, epesening he elaionship beween he eal-ime exiing couns of a ceain desinaion j and eal-ime eneing couns of all he elaed oigins i: y j = bij qi + e j i Hee y j and q i ae he exiing and he eneing affic couns, b ij is he O-D spli, he popoion of affic flows eneing a oigin i and exiing a desinaion j, and e j is a andom eo. Since hee ae always much moe unknowns, namely he O-D splis, han he elaionships esablished, vaious sysem idenificaion mehods should be applied o esimae he unknowns. Cème and Kelle 1987 poposed fou mehods: 2 Thee exis ohe ems fo classifying hese models, such as non-assignmen-based vesus assignmen-based appoaches by Chang and Tao 1999, and inesecion-oiened vesus newok-oiened by Chen e al Hee we follow he eminology by Ashok and Ben-Akiva Noe ha hee is acually no clea dividing line beween hese wo appoaches, and he classificaion is moe o faciliae he pesenaion of he ideas. 53

65 coss-coelaion maices, consained opimizaion, ecusive esimaion and Kalman fileing fo solving his poblem. Concuenly, Nihan and Davis 1987 developed a ecusive pedicions eo RPE esimao of acking dynamic O-D paamees. These mehods can all be inepeed as ecusive o non-ecusive leas-squaes mehods, because hey shae he same assumpions and insighs ha can be aced back o Gauss s leas-squaes esimaion heoy Soenson, Also noe ha alhough he ecusive algoihms menioned above equie an iniial O-D maix o sa wih, he dependency on his iniial inpu in hei esimaes deceases as ime passes. Howeve, when majo shifs in demand paens occu such as fom peak o non-peak, he pefomance of hese algoihms degades. These models ignoe avel imes in he faciliies, which is jusifiable fo inesecions and vey small newoks. Bell 1991 exended he models by pemiing he disibuion of avel imes o span a numbe of diffeen inevals. His fis mehod employs a concep of plaoon dispesion in epesening he dynamic ineacions beween eny and exi flows while his second mehod assumes feely-disibued avel ime o addess avel ime vaiabiliy. Chang and Wu 1994 used nonlinea macoscopic speed-densiy-volume elaions o esimae avel imes and inoduced link-use popoions o esablish a new se of flow popagaion consains. Wih he assumpion ha he O-D paens ae auo-egessive, hese closed-newok-oiened models do no need a age O-D able, alhough hey equie he eny and exi couns of he newok a all ime poins. Li and Moo 2002 aemped o addess he issue of incomplee obsevaions by esimaing he O-D flows ahe han he O-D splis using a genealized leas squaes GLS appoach, which howeve involves a pioi age O-D maix. Appaenly limied by he equiemen of all eny/exi couns, his appoach is no vey pacical fo lage-scale geneal newoks Open-Newok-Oiened Appoach This appoach is inended fo being used o esimae dynamic O-D maices fo geneal newoks. The lieaue on his appoach is ahe limied, and he mos noewohy wok known o us ae hose of Willumsen 1984, Okuani 1987, Cascea e al. 1993, Ashok and Ben-Akiva 1993, 2000, 2002, Madana e al. 1996, Bell e al. 1996, Sheali and Pak 2001 and Hu e al This appoach consides he esimaion of ime-vaying O-D maices as he invese poblem of dynamic affic assignmen DTA poblem. Insead of using he simple flow consevaion elaionship, he assignmen maix fom DTA model seves as he kenel measuemen elaionship. Define y as he veco of he measued link couns, f as he veco of O-D flows o be esimaed, e as he andom eo veco and A as he assignmen maix fom DTA model, he sysem equaion is as follows: v y = A k, f k + e k= p The elemen of he assignmen maix A is he link-use popoion, which is defined as 54

66 he popoion of a paicula O-D flow depaing is oigin duing ineval k, pio o he cuen ineval by a mos p inevals, conibues o he flow on link l duing ineval. Since he esulan sysem of equaions is highly unde-deemined, pevious sudies have aken wo diffeen pahs o esolve he poblem. Noe ha no mae which pah is employed, he main difficuly is he deeminaion of he assignmen maix. The fis pah is o fomulae opimizaion poblems, nomally consained GLS poblems, wih using an a pioi O-D maix and hen o selec among he infinie numbe of poenial candidaes he one ha is closes o he a pioi O-D maix. Willumsen 1984 and Cascea e al fomulaed vey simila minimizaion poblems wih sligh diffeence in hei objecive funcions: Willumsen 1984 used he enopy funcion of O-D flows while Cascea e al used a combinaion of enopy and leas squaes of link flows. Thei majo diffeence, howeve, lies in he way of compuing link use aios. In Willumsen 1984, link use popoions ae no diecly compued. Rahe, a affic simulaion model CONTRAM is used o obain an accumulaion faco based on aios beween simulaed and obseved link couns, and uses his faco o updae O-D flows and foce convegence. On he ohe hand, Cascea e al esimaed he link-use popoions hough dynamic newok loading, which equies he knowledge of oue avel imes. Rahe han obaining avel imes based on esimaed O-D demands, Cascea e al pesumed ha hisoical avel imes ae available and uses hem o pefom dynamic newok loading. Moe ecenly, Sheali and Pak 2001 geneaed he ime-dependen link-use popoions bu did no explicily specify how, and fomulaed a consained GLS model whose objecive funcion has an addiional oal-cos-diven componen in ode o avoid using an a pioi O-D maix. A column geneaion appoach was developed o solve hei model. The second pah is o assume ha affic dynamics is auo-egessive. Boh Okuani 1987 and Ashok and Ben-Akiva 1993, 2000 use Kalman File o updae ime-vaying O-D daa wih he assumpion ha he assignmen maix A is known fom diecly-measued avel imes. Disincive fom all ohe wok, Ashok and Ben-Akiva 1993 assumed ha ahe han he O-D daa hemselves, he deviaions of cuen O-D daa fom hisoical O-D daa ae auo-egessive. Madana e al added a conol equaion o he model of Ashok and Ben-Akiva ha accouns fo he ime-vaying effecs of affic infomaion on avel demands via a binay choice on oue swiching. This binay oue swiching decision is eihe o exi hough he oiginal desinaion, o o exi hough an off-amp befoe eaching he oiginal desinaion. Hu e al poposed an adapive Kalman fileing ha uses ime-vaying assignmen maices geneaed by DYNASMART. Recenly, Ashok and Ben-Akiva 2002 fuhe evised hei model by using sochasic link-use popoions o addess he unceainy associaed wih he popoions. They also suggesed an ieaive O-D esimaion as an alenaive when he diecly-measued avel imes ae no available. Bu his pocess has wo majo defecs ha he convegence is no guaaneed and ha he esulan O-D flows could be biased. Mos ecenly, Bielaie and Ciin 2004 followed he fomulaion of Ashok and Ben-Akiva 2000 bu suggesed using he LSQR algoihm fis pesened by Paige and Saundes, 1982 insead of he Kalman fileing o solve his spase linea sysem. The ime-dependen pah flow esimao poposed in Bell e al is quie diffeen 55

67 fom he models eviewed above in he sense ha i is only quasi-dynamic and seady sae condiions ae assumed wihin each peiod. Moe specifically, ips saed in one peiod will always be compleed wihin he same peiod unless inadequae oad capaciies peven hem fom doing so. The queued vehicles, if any, will be caied fom one peiod o he subsequen. Excep his, he popagaion of affic flow and spaial and empoal evoluion of congesion ae simply ignoed. Regadless of such limiaions, Bell s ime-dependen pah flow esimao is compaably efficien and applicable o lage-scale geneal newoks Closed-Newok-Oiened Appoach This pojec is concened wih deiving ime-dependen O-D maices fo linea newoks. In linea newoks, eny and exi couns ae ofen known. Theefoe, he closed-newok-oiened appoach will be eadily applicable. This secion fuhe elaboaes he developmen of his appoach, which has followed hee majo pahs addessing issues of saisfying consains of he O-D splis, aking accoun of avel ime and flow popagaion in a newok and incopoaing muliple daa souces o incease he sysem obsevabiliy Saisfying Equaliy and Inequaliy Consains As afoemenioned, his appoach is feaued wih diec esimaion of O-D splis fom ime-seies eny and exi couns. The O-D spli b ij is he popoion of he affic eneing a eny i a ime ineval ha leaves a exi j. Theefoe, he paamees b ij ae obviously bounded beween zeo and one. Moeove, he sum of all he O-D splis fom a specific eny i a a specific ineval should be equal o one. These inequaliy and equaliy consains cause some difficulies when applying he ecusive leas-squae esimaion and Kalman fileing mehods because he basic sucues of hese mehods do no allow fo consains. Consequenly, exa cauion should be execised o guaanee he saisfacion of hese consains. Fo he inequaliy consains, noe ha he equiemen of he O-D splis less han one is edundan if he non-negaiviy consains and he equaliy consains ae saisfied. Nihan and Davis 1987 fis poposed a uncaion mehod fo he RPE esimao o guaanee he non-negaiviy of he O-D splis. In he ecusive esimao, he soluion of he O-D spli a he ime ineval, b ij, is equal o b ij -1 plus a coecion iem m ij. The uncaion mehod essenially ensues ha he absolue value of he coecion iem is always less han b ij -1 by muliplying he coecion iem wih a weighing faco. I can be seen ha he uncaion mehod acually leads o a loss of opimaliy in he ecusive esimaion. Bell 1991 suggesed a moe poweful consained ecusive leas squaes algoihm o handle he inequaliy consains. His basic idea is o deive he Kaush-Kuhn-Tucke KKT opimaliy condiion fo he leas squae pogam a each ime ineval, and apply an ieaive pocess o deemine he Lagange muliplies. A he ieaion if he non-negaiviy of he O-D paamee is violaed, an adjusmen o he associaed Lagange muliplie will be made unil all he consains ae me. 56

68 The equaliy consains ae moe difficul o deal wih because hey ae applied o each ow of he O-D maix while he closed-newok-oiened appoach esimaes he maix column by column, in view of he fac ha he affic flow exiing fom jh exi is only elaed o he jh column of he O-D maix. A nomalizaion mehod and a pojecion mehod afe he unconsained esimaion pocess have been poposed by Nihan and Davis The nomalizaion is o simply updae each ow elemen by dividing he ow sum, while he pojecion mehod is o pojec he esul of he unconsained esimaion ono he hypeplane defined by he equaliy consains. Again, hese wo appoaches ae heuisic and seem povide no guaanee fo an unbiased esimaion 3. Moe ecenly, Li and Moo 1999 poposed a ecusive appoach based on equaliy-consained opimizaion o addess he equaliy consains. Insead of esimaing he O-D maix column by column, hei fomulaion employs all he O-D paamees and hus he dimension of he poblem is inceased fom he numbe of he oigins o he poduc of he numbe of oigins and ha of he desinaions. Bu he advanage is ha he equaliy consains can be handled explicily while solving he leas squae poblem. Fo he sake of saving compuaion ime, he poposed appoach only pefoms one sep of ieaion in Bell s algoihm o coec fo inequaliy consains. Recenly, Simon and Chia 2002 developed a consained Kalman fileing mehod ha can be applied o deal wih he equaliy consains in he Kalman-fileing-based algoihm. Thei mehod is essenially a pojecion mehod ha can be viewed as he genealizaion of he pojec mehod used in Nihan and Davis They povided a igoous poof ha he pojecion is an unbiased sae esimao fo any known symmeic posiive definie weighing maix, and hen fuhe pesened a weighing maix ha has he smalles esimaion eo covaiance. Noe ha in ode o handle he equaliy consains i is ineviable o expand he dimension of he poblem so ha all he O-D paamees can be esimaed a he same ime Tavel Time Consideaion The empoal and spaial dispesion of affic is of gea impoance o deemine eihe he casual elaionship beween he eneing and exiing couns o he ime-dependan assignmen maix. Indeed epesenaion of flow popagaion and esimaion of avel ime ae ighly elaed. Incoec epesenaion of flow popagaion will lead o wong elaionships beween O-D paamees and link flows, esuling biased esimaes of hese paamees. The fis geneaion of he closed-newok-oiened models focuses on uning movemen idenificaion fo inesecions whee avel ime can be safely assumed negligible. When exending hese models o newoks, epesenaion of flow popagaion is he fis issue o esolve. Bell 1991 poposed wo mehods o allow fo disibuions of avel imes hough inesecion o newok ha span moe han one 3 Based on he wok by Simon and Chia 2002, he pojecion mehod uns ou o be an unbiased sae esimao, no he bes one hough. 57

69 ineval. The fis mehod assumes a geomeically disibued avel ime fo each exi. In his mehod, he paamees o be esimaed ae he plaoon dispesion faco ogehe wih he O-D splis. The second mehod consides feely-disibued avel imes, bu assumes vehicles fom any enance should each o a specified exi wihin hee ime inevals. Thus, he vaiables o be esimaed a each ime ineval ae exacly he O-D paamees fo he hee inevals befoe he cuen ineval. Chang and Wu 1994 used a se of non-linea macoscopic affic elaions o esimae avel imes, bu assumed vehicles ha each one exi duing an ineval come fom only wo consecuive ime inevals fo each enance. Moeove, he poposed mehod epesens flow popagaion by inoducing link-use popoions o be esimaed simulaneously wih he O-D paamees. Consequenly, no only he dimension of he poblem inceases, bu also he poblem iself becomes nonlinea. Exended Kalman fileing was adoped by Chang and Wu o idenify he nonlinea sysem Incopoaing Muliple Daa Souces Mos exising dynamic O-D esimaion models make use of ime-seies affic couns a enies and exis in he newok and some even equie a age O-D maix o guaanee he sysem obsevabiliy. The affic couns ae supposed o be colleced fom sensos such as he inducive loop deecos, and he age O-D maix is assumed o be available fom a hisoical O-D daa o a simple suvey. Wu and Chang 1996 and Chang and Tao 1996 included consains esablished fom dynamic sceenline and codonline flows o incease he obsevabiliy of he dynamic ineacions beween O-D paens and he esuling link flow disibuions. In ode o obain a moe eliable esimae, Chang and Tao 1999 pesened an inegaed model ha employs he inesecion uning flow daa o poduce an addiional se of consains in idenifying pah flows fom a DTA model. The adven of auomaic vehicle idenificaion AVI echnologies would benefi dynamic O-D esimaion wih poviding sampled complee o incomplee vehicle ajecoies. Fo example, if hee ae video deecion sysems insalled a seleced inesecions, explici uning movemens, including igh-un, hough and lef-un will be available. Alhough ou lieaue seach has no found a diecly-elevan eseach ha incopoaes such a daa souce ino he afoemenioned modeling famewok, many eseaches have invesigaed he possibiliy of aking advanage of hese AVI daa in vaious O-D esimaions. Fo example, eleconic oll collecion ag can povide paial ip ajecoies of vehicles equipped wih a ag. Due o he fac ha only a facion of agged vehicles can be sampled, Kwon and Vaaiya 2005 developed a saisical model o deive an unbiased esimao of he O-D maix essenially ag eade o ag eade inechange flows based on he mehod of momens. As anohe example, aea-wide AVI sysems, such as in-vehicle global posiioning sysem and cell phone acking, can povide paial, bu complee ajecoies of he vehicles. I has been poposed by seveal eseaches ha ogehe wih he make shae of such AVI equipmens, an off-line O-D can be obained fom hese AVI daa Asakua e al. 2000; Anoniou e al. 2004; Dixon and Rile 2005; Eisenman and Lis The applicaion of his idea o he Han-Shin expessway newok in Japan See Asakua e al povided a pacical case which indicaes he effeciveness of his mehod. 58

70 3.1.5 Issues o Be Fuhe Addessed Fom he above eview of elaive lieaues, we idenify he following issues ha may need fuhe eseach: 1 Impovemen of he usage of AVI daa. Mos sudies ea he AVI daa as an off-line esouce fo he age O-D maix. Howeve, he eal-ime AVI daa may seve as anohe infomaion souce which will povide a leas he oigins of he vehicles and an appoximaion of he link volume. Also, ohe affic infomaion such as uning movemens on he seleced inesecion fom video deecion sysems may be employed o impove he sysem obsevabiliy. I would be necessay o invesigae how o fuse hese diffeen souces of AVI daa o maximize he accuacy of he esimaion. 2 Link use popoions. Because hee is no oue choice in linea newoks, i is elaively easie o compue moe accuaely link-use popoions. Wih an exogenous eliable souce of eal-ime avel ime infomaion, efficien ways should be invesigaed o compue link-use popoions. Wihou such an exogenous souce, an ieaive pocedue beween O-D esimaion and flow popagaion acing should be conduced o povide endogenous esimaes of avel ime and link use popoions. Alhough such an ieaive echnique is expeced o be quie difficul, i migh be sill feasible fo linea newoks. 3 Incomplee infomaion. Fo linea newoks, cuen models equie all he eny and exi couns. Howeve, he obsevaion infomaion is ofen incomplee. Fo example, i is unlikely o obain exi couns fom he ypical seings of loop deecos fo acuaed signal conol sysems. I is necessay o invesigae how o ensue he sysem obsevabiliy in such a conex, in addiion o inoducing a age O-D. 4 Measuemen eos. Ou expeience wih he affic loop deecos suggess ha he loop deecos geneally have only 70%-80% accuacy. Such sysemaic measuemen eos can no be epesened by he eo em in he Kalman Fileing. Theefoe, eseach effos need be made o miigae he impacs of he inaccuacy of affic daa. One possible soluion is o seek a obus counepa of he O-D esimaion opimizaion poblem ha will oleae changes in he affic daa, up o a given bound known a pioi. 59

71 3.2 Esimaion of Oigin-Desinaion Flows fo Acuaion-Conolled Inesecions Inoducion This secion addesses he eal-ime esimaion of O-D flows splis o uning popoions fo isolaed inesecions. The pupose is o use ime-seies daa of affic couns o deive ime-independen o ime-vaying O-D flows. Esimaing O-D flows fo isolaed inesecions is a saing poin fo he newok O-D esimaion. A single inesecion can be viewed as he smalles newok sysem wih muliple oigins and desinaions. Moeove, individual inesecion is he key elemen of a lage-scale newok. If uning movemens of all he inesecions in he newok ae known, he affic siuaion wihin his sysem can be eplicaed o simulaed. Theefoe, inesecion O-D esimaion poblem seves as he basis fo sysem idenificaion, monioing and conol. A vaiey of esimaos have been developed fo esimaion of dynamic inesecion O-D, equiing all he eneing and exiing couns a all ime poins. Howeve, even as small as an isolaed inesecion, he obsevaion infomaion is ofen incomplee. Fo example, i is unlikely o obain exiing couns fom he ypical seings of loop deecos fo acuaed signal conol sysems. Li and De Moo 2002 poposed a consained genealized leas squaes GLS mehod o addess he issue of incomplee obsevaions. Thei mehod equies a pioi age O-D maix, and he esuls will heavily depend on his iniial value, which migh lead o a biased esimae. This chape pesens a new wo-sep opimizaion pocedue fo poblems wih complee eneing couns bu incomplee exiing couns, a common infomaion paen fom acuaion-conolled inesecions. The fomulaion is sill based on he noion of GLS, bu makes full use of he available infomaion, heeby simplifying he compuaion and paially eliminaing he dependence on he pio infomaion. To faciliae he pesenaion of he idea on dealing wih infomaion incompleeness, we fis focus on he siuaion whee he O-D maix o be esimaed is consan and hen exend he famewok o ack ime-vaying O-D maices by modeling he O-D splis as a andom walk pocess Poblem Saemen and Noaions Poblem Saemen A big segmen of inesecions in he U.S. ae acuaion-conolled. Acuaed signal conolles eceive calls o acuaions ha eques sevice fo a paicula movemen, ypically fom inducive loop deecos cu ino he pavemen suface. Fo he pupose of signal opeaions, he conolle does no need o deemine if he call is due o a single 60

72 vehicle o a lage plaoon of vehicles. Howeve, he advancemen and deploymen of elecommunicaion and ITS echnologies have made affic couns and occupancies moe eadily available fom acuaed conol signal sysems. Fo example, in Califonia, second-by-second euns of signal saus and loop deeco daa can be obained fo all phases. Fig. 3.1 shows a ypical loop layou fo he majo appoach o an acuaion-conolled inesecion. Thee ae wo se of loops, advance loops and pesence loops, insalled fo he hough-movemen, and he lef-un bay is equipped wih loop deecos as well. Noe ha fo he mino appoach nomally only pesence loops ae insalled. Fig A ypical loop layou Wih such a ypical seing, all he eneing couns can be obained fom eihe he advance o pesence loops. Howeve, exiing couns ae geneally no available. Fo he majo appoaches, we may esimae he exiing flows fom advance loops fo he downseam inesecion wih he assumpion ha hee ae no diveways in beween. Fo he mino appoach, i is less likely o even esimae he couns. In summay, depending on he specific loop seings, hee ae muliple infomaion paens available fom acuaed conol sysems. In his chape, we focus on a pevalen case in which all of he eneing flows and exiing flows on majo appoaches ae known while he exiing couns on mino appoaches ae missing Noaions Consisen wih he noaions by Nihan and Davis 1987 and Li and De Moo 2002, we le: J i denoe he se of exis j, which is pemissible fo vehicles eneing a enance i; I j denoe he se of enances i, which pemis vehicles o ake exi j; OJ is defined as he index se of exis whee he affic couns ae available; y j k and q i k denoe he exiing couns of exi j and aiving couns of enance i especively, duing ime ineval k; 61

73 v T y = [ y... y...], j OJ 1 j v q = [ q q ] T vt vt vt vt Q = diag q, q, q, q b ij denoes he O-D spli, i.e. he popoion of affic couns eneing via enance i and leaving via exi j. By definiion, b ij mus saisfy: b = 1, i 3.1a j J i ij b ij 0, i, j 3.1b b v j is defined by he jh column of he O-D maix; v v v T b = [ b ] T b T is a column veco; D = [ I I I I ] is a maix wih 4 ows and 16 columns faciliaing he consains 1a o be ewien in he veco fom Convenional GLS Mehod Fomulaion Fo he exis whee exiing couns ae available, he measuemen equaion can be expessed as: y j k = bijqi k + e j k, j OJ 3.2 whee e j k is he andom measuemen eo wih zeo mean. i I j The famewok of he esimaion is o find O-D splis such ha he mean squaes of he eo em ae minimized. Howeve, since some of he exiing couns ae no obsevable, he coesponding b ij become inesimable unde he leas squae famewok. To deal wih he above issue, a pioi infomaion of he O-D splis o be esimaed has o be inoduced ino he objecive funcion of he GLS fomulaion. Essenially, he esimaes of O-D splis ae deemined by he pevious esimae O-D and hen ae coeced by he cuen obsevaions. A each ime ineval k, given he esimae of b ij in he pevious ineval, bˆ ij k 1, and he weighing faco ζ ij, he GLS fomulaion can be wien as: 2 [ bij bˆ ij k 1 ] + y j k min ζ ij bijqi k j i I j j OJ i I j 3.3 subjec o 3.1a and 3.1b Noe ha he above model is poposed by Li and De Moo 2002 in a diffeen bu equivalen fom. 2 62

74 Soluion Algoihm To handle he non-negaiviy consains 3.1b in he leas squaes fomulaion, Bell 1991 suggesed a poweful ieaive algoihm. The basic idea is o deive he Kaush-Kuhn-Tucke opimaliy condiion fo he opimizaion poblem a each ime ineval, and hen apply an ieaive pocess o deemine he Lagange muliplies. Duing he ieaion if he non-negaiviy of he O-D paamee is violaed, an adjusmen o he associaed Lagange muliplie will be made unil all he consains ae me. Li and De Moo 1999 fuhe exended his mehod o deal wih equaliy consains 3.1a. In his chape, insead of following hei pocedue, we adop Kalman fileing as he soluion algoihm. In view of wo facs ha he weighing faco w ij should be deemined accoding o he covaiance sucue of boh ems in he objecive funcion in ode o obain he bes linea unbiased esimao Cascea, 1984, and ha he Kalman fileing epesens essenially a ecusive soluion o he oiginal leas squaes poblem Soenson, 1970, we ansfom he oiginal opimizaion poblem 3.3 ino he following equivalen consained Kalman fileing poblem. Since bˆ ij k 1 can seve as an available obsevaion a each ime ineval k, we wie he sae-space equaions as follows: v v b k = b k 1 v b ˆ k 1 I v v w k 3.4 v = b k + v y k Q k e k whee w v k and e v k ae he coesponding andom eo ems, subjec o consains 3.1a and 3.1b: Db v k = 1 3.5a b v k 0 3.5b To apply he Kalman fileing algoihm, we can assume he covaiance maix of e v k is known, denoed as Rk his assumpion is ealisic since he measuemen eo of he loop deecos can be possibly esimaed pio o he opeaion of he file and ha w v k and e v k ae independen. Noing ha w v k is acually he deviaion of he esimae value fom he ue value a ime ineval k-1, he covaiance maix of w v k v vˆ v v T ˆ should be equal o E[ b b k 1 b b k 1], which is denoed by P k 1 and can be esimaed duing he Kalman fileing pocedue. vˆ b k 1 I Le Zk denoe v and H k denoe and apply Kalman fileing wih y k Qk equaliy consains using he maximum pobabiliy mehod Simon and Chia, 2002 o his sysem, he soluion can be expessed as: 63

75 whee, T [ H k P k H k + V k ] T 1 K k = P k H k ~ v vˆ vˆ b k = b k 1 + K k Z k H k b k 1 v ~ ~ ˆ v 1 v T T b k = b k P k D [ DP k D ] Db k 1 P k + 1 = [ I K k H k ] P k v w k V k = E v e k P k 1 0 v v [ w k e k ] = 0 R k I should be poined ou ha we discad 3.5b hee fo he compuaion simpliciy. The inequaliy consain 3.5b is vey likely o be me since he O-D splis ae consan and ha hei iniial values can be well chosen o be all posiive. If necessay, he uncaion mehod poposed by Nihan and Davis 1987 can be applied o guaanee he non-negaiviy Impoved Two-Sep Mehod Fomulaion Fo he exis whee he affic couns ae available, he coesponding column of he O-D maix can be diecly esimaed fom model 3.2, which is an unbiased esimao and is dependency on he iniial inpu diminishes as ime passes. Howeve, model 3.4 is v v unbiased only if he mean value of he eo em [ w k e k ] T is zeo. We can assume safely he measuemen eo e v k has a zeo mean, bu whehe w v k mees his equiemen depends on he choice of he iniial values of he O-D splis. A poo iniial value will always lead o a biased esul fo all b ij even hose could be unbiasedly esimaed by using model 3.2. Anohe limiaion of convenional GLS model 3.4 is he compuaional difficuly. To obain he ecusive soluion 3.6, an invesion of maix should be conduced a each sep. The dimension of he maix depends on he size of b v and y v. I may be accepable fo he isolaed-inesecion poblem, bu is no applicable fo a lage-scale newok. To impove he model in hese wo aspecs, we popose a new wo-sep pocedue. The pocedue esimaes he jh column of O-D maix associaed wih he exi j OJ using model 3.2, and hen obains he emaining unspecified O-D paamees using a consained leas squaes model. Consequenly, he fis sep esimao is unbiased and does no depend on he iniial inpus. This wo-sep model is essenially applying he noion of GLS, bu aemps o deemine as many O-D splis as possible fom he obsevaions, diffeen fom he convenional way of using he obsevaion as a coecion o he pio O-D infomaion. Such a decomposiion scheme no only guaanees he accuacy of he fis-sep esimao, bu also makes i possible o conve he oiginal veco idenificaion poblem ino seveal independen scala poblems in 64

76 boh seps, which ceainly impoves he compuaion efficiency. Given he esimae ineval k is as follows: Sep 1: Sep 2: ˆ k 1 in he peceding ineval, he fomulaion a each ime b ij min y j bq ij i, j OJ i I j subjec o b > 0, i I ij 2 ij ij ij min b b k 1, i, j OJ subjec o b = 1; b > 0, j OJ j J i 2 ij j Soluion Algoihm Based on he same pocedue of conveing model 3.3 o model 3.4, he fis-sep model 3.8 can be epesened by he following sysem equaions: bj k = bj k 1 v, j OJ 3.10 T yj k = q k bj k + ej k whee he covaiance of he andom eo ej k is assumed o be known, denoed as j k. Since he esul of his esimao is expeced o be vey close o he ue value a convegence, we ignoe he inequaliy consains and apply he unconsained Kalman fileing fo each j OJ sepaaely o idenify his sysem: v v T v 1 K j k = Pj k q k [ q k Pj k q k + j k ] ˆ = ˆ v T b 1 + [ ˆ j k bj k K j k y j k q k bj k 1 ] 3.11 v T Pj k + 1 = [ I K j k q k ] Pj k I is appaen fom 3.11 ha afe he decomposiion, no maix invesion is needed when compuing he Kalman gain K j k since he iem in he backe is only a scala. Thus, his fomulaion impoves he compuaional efficiency o a gea exen. The second-sep model is an equaliy-consained leas-squaes poblem fo each ow i of he O-D maix, afe omiing he non-negaiviy consains. I is essenially a pojecion of he pevious O-D paamees o a plane in he b ij space govened by he esul fom sep 1. Since he exiing couns fo he wo majo appoaches of a ypical fou-way inesecion ae known, wo columns of he O-D maix will be esimaed fom sep 1. Theefoe, he second-sep fomulaion is a mos a wo-dimensional minimizaion poblem fo each ow i, which can be easily solved. 65

77 If he lef/igh-un splis on he majo appoach can be diecly measued o addiional infomaion of he O-D paen is given, such as U-un is known o be pohibied, he numbe of unknown vaiables can be fuhe educed by sepaaely esimaing hose O-D paamees wih available obsevaions. As a consequence, he esul is expeced o be moe accuae Numeical Example In his secion, a simulaion example is povided o illusae he wo mehods of esimaing O-D splis fo an acuaion-conolled inesecion wih incomplee exiing couns. Conside a ypical fou-way inesecion pohibiing U-un, as shown in Fig All eneing couns ae available, bu he exiing couns ae only available a majo appoaches, leg 1 and leg 3. The duaion of he simulaion was se as 100 ime inevals. Fig A ypical fou-way inesecion To mimic he opeaion of he esimao, we adoped he hypoheical ue O-D maix used by Nihan and Davis 1987 shown as follows: The acual eneing couns wee geneaed andomly fom 0 o 100 fo each ime ineval, and obseved exiing couns fo exis 1 and 3 wee geneaed based on 3.2. The measuemen eos fo boh legs 1 and 3 wee assumed o be independen nomal andom vaiables wih zeo mean; and he vaiance j k is se equal o 15 pecen of he acual exiing affic. 66

78 The iniial value of he O-D paamees was given as: And he P maices wee always iniialized as ideniy maices wih zeo elemens on he diagonal coesponding o he O-D paamees b ii = 0. Five simulaion expeimens wee conduced fo boh he convenional GLS mehod and he wo-sep mehod. The oo mean squaes RMS of he diffeence beween he ue and he esimaed O-D paamees wee compued as a measuemen of esimao pefomance a each ime ineval. Table 3.1 shows he aveage RMS ove he las 20 ieaions fo boh esimaos, leading o he conclusion ha he wo-sep mehod oupefoms he convenional GLS mehod. Fig. 3.3 displays he RMS eos acoss ieaions fo expeimen 2 and 3. And he convegence of he O-D splis fo expeimen 5 is ploed in Fig. 3.4 o Fig Table 3.1 Aveage RMS Eo ove Las 20 Ieaions Expeimen Convenional GLS Mehod Two-Sep Mehod Expeimen RMS Diffeence Ieaion 0.08 Expeimen RMS Diffeence Ieaion Convenional GLS Mehod Two-Sep Mehod Fig RMS eo beween acual and esimaed O-D paamees fo Expeimen 2 and Expeimen 3 Fig. 3.4 and Fig. 3.6 show ha fo hose exis wih available affic couns, he wo-sep 67

79 mehod povides moe accuae esimaes of hei coesponding columns of he O-D maix, which do no depend on he iniial value. Noe ha b 22 and b 44 ae known o be zeo, leaving only one decision vaiable in he second sep fo boh ow 2 and ow 4. Theefoe, he accuacy in he fis and he hid column esimaion of he O-D maix will esul in an unbiased esimaion of b 24 and b 42 See Fig. 3.5 and Fig Moe specifically, in he wo-sep mehod only fou O-D paamees ae dependen on he iniial value while in he convenional GLS mehod, all he welve splis o be esimaed ely on he iniial value b21 value Ieaion b31 value Ieaion b41 value Ieaion Convenional GLS Mehod Two-Sep Mehod Acual Value Fig Convegence of O-D paamees b i1 fo Expeimen b12 value Ieaion 0.5 b32 value Ieaion 0.9 b42 value Ieaion Convenional GLS Mehod Two-Sep Mehod Acual Value Fig Convegence of O-D paamees b i2 fo Expeimen 5 68

80 b13 value Ieaion 0.9 b23 value Ieaion b43 value Ieaion Convenional GLS Mehod Two-Sep Mehod Acual Value Fig Convegence of O-D paamees b i3 fo Expeimen b14 value Ieaion b24 value Ieaion 0.7 b34 value Ieaion Convenional GLS Mehod Two-Sep Mehod Acual Value Fig Convegence of O-D paamees b i4 fo Expeimen Tacking Time-Vaying O-D Flows Above we pesened a wo-sep fomulaion fo eal-ime esimaion of O-D maix of isolaed acuaion-conolled inesecions wih complee eneing couns and incomplee exiing couns. Though ou discussion focuses on he consan O-D maix esimaion poblem, he poposed famewok can be easily exended o ack ime-vaying O-D maices by modeling he O-D splis as a andom walk pocess. Accodingly, he covaiance maix of he andom deviaion should be inoduced o he Kalman fileing in ode o coec he P maices a each sep. 69

81 We applied he exended wo-sep appoach o he same inesecion used in Secion 3.2.5, and Figs compae he esimaes of he new appoach and convenional GLS wih he ime-dependen ue values, and Fig epos he RMS eos of he wo appoaches. I can be obseved ha he wo-sep appoach sill oupefoms he convenional GLS appoach. Moeove, he fome is moe efficien as well. I should also poined ou ha boh appoaches would benefi fom accuae pio knowledge o paial O-D infomaion X21 value Ieaion X31 value Ieaion X41 value Ieaion Convenional GLS Two-Sep Acual Fig Compaison of flows o Leg X12 value Ieaion X32 value Ieaion X42 value Ieaion Convenional GLS Two-Sep Acual Fig Compaison of flows o Leg 2 70

82 80 60 X13 value Ieaion 100 X23 value Ieaion X43 value Ieaion Convenial GLS Two-Sep Acual Fig Compaison of flows o Leg 3 X14 value Ieaion 50 X24 value Ieaion X34 value Ieaion Conveional GLS Two-Sep Acual Fig Compaison of flows o Leg 4 71

83 7 Convenional GLS Mehod Two-sep Mehod 6 5 RMS Diffeence Ieaion Fig Compaison of RMS eos 72

84 3.3 Esimaion of Oigin-Desinaion Flows fo Acuaion-Conolled Coidos Inoducion The model pesened in he pevious secion focuses on uning movemen idenificaion fo isolaed inesecions whee avel ime can be safely assumed negligible. Exending he model o a coido moe pecisely, linea newok will ineviably involve deeminaion of vehicle avel imes and descipions of flow popagaion fo consucing he casual elaionship beween coun measuemens and O-D flows o be esimaed. Exising closed-newok models ofen make assumpions on he plaoon dispesion o flow popagaion and hen endogenously esimae boh O-D flows and he popagaion paamees. In he lieaue, Bell 1991 poposed wo mehods o allow fo disibuions of avel imes hough inesecion o newok ha span moe han one ineval. The fis mehod assumes a geomeically disibued avel ime fo each exi. In his mehod, he paamees o be esimaed ae he plaoon dispesion faco ogehe wih he O-D splis. The second mehod consides feely-disibued avel imes, bu assumes vehicles fom any enance should each o a specified exi wihin hee ime inevals. Thus, he vaiables o be esimaed a each ime ineval ae exacly he O-D paamees fo he hee inevals befoe he cuen one. Chang and Wu 1994 used macoscopic affic models o esimae avel imes, and hen assumed vehicles ha each one exi duing an ineval come fom only wo consecuive ime inevals fo each enance. Moeove, he poposed mehod epesens flow popagaion by inoducing link-use popoions, which ae esimaed simulaneously wih he O-D paamees. Consequenly, no only he dimension of he poblem inceases, bu also he poblem iself becomes nonlinea. Exended Kalman fileing was adoped o idenify he nonlinea sysem. In he open-newok appoach, he sysem equaion used is as follows: v y = A k, f k + e k= p whee y is he veco of he measued link couns; f is he veco of O-D flows o be esimaed; e is he andom eo veco and A is he assignmen maix, which encapsulaes oue choice and flow popagaion. The elemen of he assignmen maix A is he link-use popoion, which is defined as he popoion of a paicula O-D flow depaing is oigin duing ineval k, pio o he cuen ineval by a mos p inevals, conibues o he flow on link l duing ineval. The complexiy of descibing oue choice and flow popagaion is ofen avoided in many pevious sudies by simply assuming ha he maix is known and offeing some geneal discussions ha he maix can be compued using simulaion, o DTA models o he analyical equaions, if he avel imes ae known e.g., Okuani, 1987; Ashok and Ben-Akiva, 1993, In fac, even wih he esicive assumpions ha avel imes ae known, and uses ae homogenous and hee is no oue choice, he assignmen maix canno be exacly deemined fo a newok wih acive bolenecks and muliple O-D pais because he 73

85 bolenecks may peven vehicles depaing ogehe aiving a he same ime ineval. Addiional assumpion has o be made on he plaoon dispesion. Fo example, Cascea e al assumed ha vehicles wihin a goup depaing a ineval k using pah p ae unifomly compised wihin he depaue duaion T and say wihin he ineval as hey move acoss he newok. Ashok and Ben-Akiva 2002 fuhe elaxed he assumpion o pemi he effecs of seching and squeezing of packes as hey avese he newok. In his chape, we esimae ime-dependen O-D flows fo acuaion-conolled coidos whee all of he eneing flows and exiing flows on majo sees ae known while he exiing couns on mino sees ae missing. We assume ha avel imes ae known fom exogenous souce, e.g., a affic suveillance sysem o ae esimaed using affic simulaion. Fo he lae, we may fis decompose he newok ino a sysem of isolaed inesecions, and hen apply he model poposed in he pevious pape o esimae he uning movemens fo each inesecion. The esimaed movemens can be fuhe fed ino a affic simulaion package, such as VISSIM, o esimae avel imes. Based on he esimaed avel imes, flow popagaion can be descibed fo he linea newok wih an assumpion of plaoon dispesion, and he O-D flows can hen be infeed accodingly. This appoach may eliminae he poenial inconsisency beween he esimaed O-D maix and he avel imes, which is one of he majo poblems in many pevious dynamic newok O-D esimaion models Model Fomulaion Model Pepaaion Conside a coido consising of n inesecions, hen he dimension of he O-D maix is 2 n n + 1, and he node numbeing convenion is pesened in Fig

86 2 4 2i 2n 1 2n i+1 2n+1 Inesecion # i a. Coido b. Inesecion Fig Numbeing convenion fo model fomulaion The following noaion is used fo he model fomulaion: f ij : numbe of vehicles eneing he coido fom oigin i o desinaion j, i, j {1,2,...,2 n + 2}, duing ime ineval ; k θ ijl : facion of f ij ha aives a inesecion l duing ime ineval k, i, j {1,2,..., 2 n + 2}, l { 1,2,..., n} 4 ; η ij,l : uning movemens fom leg i o leg j duing ime ineval a inesecion l, i, j {1,2, 3,4}, l { 1, 2,..., n} ; T : maximum avel ime fom any inesecion i i < j o inesecion j; 1 j T : maximum avel ime fom any inesecion i i > j o inesecion j; 2 j Decomposiion Scheme We popose a wo-sep decomposiion scheme o esimae ime-vaying O-D maices fo coidos wih incomplee infomaion. The essenial idea is o ackle he poblem in wo seps o levels: inesecion and coido levels. A he inesecion level, he decomposiion scheme pesened in he pevious chape is applied o infe uning movemens fo each individual inesecion; a he coido level, he oiginal poblem is fuhe decomposed ino a seies of sub-poblems wih espec o each desinaion o oigin, using he uning movemens esimaed a he inesecion level. As saed in he 4 In he poposed mehod, we assume his faco is known fom plaoon dispesion models so ha he poblem emains linea. Ohewise, exended Kalman fileing may be adoped o infe boh O-D flows f and he dispesion facos θ. 75

87 pevious chape, he decomposiion scheme makes full use of available infomaion and hence educes he dependency on he qualiy of he pio O-D infomaion a he inesecion level. A he coido level, i does no even equie any pio O-D infomaion. The wo-sep appoach impoves significanly he compuaional efficiency by decomposing he oiginal high-dimensional poblem ino much smalle poblems. Specifically, a he inesecion level, fo each ime ineval k, he uning movemen k η ij,l, i, j {1,2,3,4}, l {1,2,..., n} can be esimaed a all inesecions. As a consequence, 2n+8 O-D flows can be explicily esimaed, including mino-o-mino k k k k k k O-D flows, fo each inesecion l,,,, fo inesecion 1 f2 l,2l+ 1 f2 l+1, 2l f2 n+ 2,2n+ 1 f2 n+2, 2n f2 n,2n+ 2 f2 n, 2n+ 1 k k k k and,,, fom inesecion n. Among hose esimaes, k k k k only fou ae biased, namely f 12, f 13, f2 n+ 2,2n+ 1 and f2 n+ 2,2n, as shown in he pevious chape. A he coido level, o impove he compuaional efficiency, we decompose he oiginal high-dimensional poblem ino a seies of sub-poblems wih espec o each desinaion o oigin. In ohe wods, we esimae he O-D maix column by column wih espec o each desinaion o ow by ow fo each oigin. These wo esimaes can be fuhe combined hough a weighed aveage o impove he qualiy of he final esimae. The weighing facos may be chosen based on he vaiances of he esimaion f 12 f 13 f 21 f 31 eos, which ae updaed ieaively duing he Kalman fileing pocess. Le fˆo and fˆd be wo esimaes obained using he ow by oigin and column by desinaion decomposiion especively and êo and êd he coesponding esimaion eos especively. The final esimae may ake he foma as ˆ ˆ ˆ Vaˆ ed f o + Vaˆ eo f d f = Vaˆ e + Vaˆ e o Assume fˆo and fˆd ae independen esimaes, i can be shown ha he vaiance of Vaˆ ed Vaˆ eo he final esimaion eo is, less han ha of eihe oiginal Vaˆ eo + Vaˆ ed esimaes. Fo ow esimaion, eihe he oal inflow a each oigin o he uning movemens obained fom he inesecion level poblem can seve as obsevaions. Howeve, o avoid he accumulaion of esimaion eos, i is moe favoable o esimae he O-D flows based on he fis-hand oal inflow infomaion. Theefoe, he causal elaionships ae: 2n + 2 d = 1 k q o = f k whee is he inflow fom oigin o a ime ineval k. q 0 k o, d d 76

88 The column esimaion is moe complicaed since hee is no diec obsevaion of he oal depaues. Fo each desinaion 2j, j { 1,2,..., n}, wo esimaes of uning movemens esuled fom he inesecion level poblem, η12, j and η 32, j, may be seleced and he causal elaionships beween uning movemens and O-D flows ae: k 2 j k,2 j, j 1 = k T j o= 1 k 2 n+ 1 k, k 32, j = θo,2 j, j 2 = k T o= 2 j+ 1 k, η 12, j = θo η Fo individual desinaion 2j+1, j { 1,2,..., n}, η14, j and η 34, j may be seleced o infe he O-D flows desined o he node: η η j k 2 j k, k 14, j = θo,2 j+ 1, j 1 = k T j o= 1 k 2 n+ 1 k, k 34, j = θo,2 j+ 1, j 2 = k T j o= 2 j+ 1 f o,2 j f f o,2 j o,2 j+ 1 f o,2 j+ 1 In addiion, η31, 1 O-D maix: and η 13, n may be used o esimae fo he fis and las column of he η η k 2 n+ 1 k, k 31,1 = θ o,1,1 2 = k T o= 2 k 2n+ 1 k, k 13, n = θo,2n+ 2, n 1 = k Tn o= 1 1 f f o,1 o,2n+ 2 N oe ha η 12,1 = f12, η 14,1 = f13, η 32, n = f2n+ 2, 2n and η34, n = f2n+ 2,2n+ 1. Theefoe, he oiginal poblem is decomposed ino 4n = 4n 2 sub sysem idenificaion poblems in oal Sae-Space Repesenaion Fo he fomulaion, we assume ha U-un is pohibied a each inesecion and f ii = 0. Moeove, he O-D flows and uning movemens ae assumed o be fis-ode auo-egess pocesses, alhough moe geneal sucues can be accommodaed if hisoical O-D daa ae available o deemine he sucue. Due o he flow popagaion, uning movemens a paicula ineval can be aibued o O-D flows a pevious muliple inevals. Fo example, he uning movemen k, j k 2 j 1 + η 12 = θ = k T o= j 1 1 1, k o,2 j, j f o,2 j, which suggess ha each O-D flow be esimaed muliple imes. As in Ashok and Ben-Akiva 1993, 2000, we use sae augmenaion o achieve his, and he augmened space veco fo his sub-poblem would be: 77

89 f k T j 1 1,2 j 1 k T j 2 j 1,2 j 1 k T j + 1 1,2 j f f f f 1 k T j j 1,2 j f k 1,2 j k 2 j 1,2 j M } } M M M M... } 1 k T j k T k j As in he pevious secion, he Kalman Fileing is adoped o infe he ime-vaying O-D maices. Noe ha he sae veco is diffeen fo each ime ineval. Afe k-1 ime inevals, we have he k-1 h esimae fo ime dependen O-D flows f, { 1,2,..., k 1}, denoed as f k 1. A ime ineval k, since ij k f ij 1 1 T j ij k f ij is now included in he sae veco meanwhile is excluded, an iniial value of his ime k O-D flow is needed fo he k h k k 1 esimaion. We le f ij k 1 = fij k 1 be he iniial value k of f ij. Consequenly, f ij k, { 1,2,..., k}, is equal o f ij k 1 plus he coecion iem Numeical Expeimen Expeimen Seings We demonsae and veify he poposed wo-sep appoach on a hypoheical coido wih fou inesecions and 90 O-D pais in oal, as shown in Fig Two scenaios wee esed, one fo consan O-D flows and he ohe fo ime-vaying O-D flows, geneaed based on he fis-ode auo-egessive assumpion. The maximum avel 1 imes fom any inesecion o inesecion j wee se as T j = j 1+ 2 = j + 1 and 2 T j = n j + 2 = 4 j + 2 = 6 k j. Assuming no plaoon dispesion i.e, θ is eihe zeo o one, indicaing whehe O-D flow f ij passes inesecion l a ime k o no, we used a affic loading pocedue o deemine he eneing, exiing flows and uning movemens a each inesecion. Finally, measuemen eos wee andomly added o hose flows, which wee hen used by he poposed appoach o infe he O-D flows. ijl 78

90 a. Coido b. Inesecion Fig Numbeing convenion fo he hypoheical coido Expeimen Resuls We fis applied he poposed appoach o infe consan O-D flows. The iniial values of he O-D flows wee andomly geneaed. Figues compae he acual and esimaed O-D flows fo seleced O-D pais, including end o end 10-1, long-disance mino o mino 9-2, medium-disance mino o mino 2-6 and uning movemen in he same inesecion I can be found ha he esimaes convege quickly wihin 10 inevals and hee is vey good ageemen beween he acual and esimaed values fo all cases wihin 5% deviaion. 79

91 O-D flows Acual Esimae Time ineval Fig Acual vs Esimaed O-D flow fo O-D pai Acual Esimae 50 O-D flows Time ineval Fig Acual vs Esimaed O-D flow fo O-D pai

92 Acual Esimae O-D flows Time ineval Fig Acual vs Esimaed O-D flow fo O-D pai Acual Esimae 65 O-D flows Time ineval Fig Acual vs Esimaed O-D flow fo O-D pai 8-10 Table 3.2 pesens he oo mean squae eo nomalized RMSN fo each O-D pai. I shows ha mos of he O-D pais expeience ahe small eos while ceain O-D pais, such as 4-1, 8-1 and 3-5, suffe highe eos, likely due o he poo qualiy of he pio O-D infomaion. 81

93 Table 3.2 Roo Mean Squae Eo Nomalized RMSN We hen infeed ime-vaying O-D flows. The iniial values of he O-D flows wee andomly geneaed as well, and he compaisons beween esimaes and ue values fo seleced O-D pais ae pesened in Figs The RMSN of he O-D flows esimaed by desinaion is 0.223; he values ae and fo he oigin-based and he weighed-aveage esimaes. Table 3.3 fuhe pesens RMSN of he weighed-aveage esimae fo each O-D pai. We have he following wo obsevaions: Boh he column and ow decomposiions ae able o ack he end of ime-vaying O-D flows and poduce esimaes pey close o he acual values in an aveage sense. The ow decomposiion is moe compuaionally efficien bu he column decomposiion povides bee esimaes in ems of he RMSN. The eason is ha he ow decomposiion only makes use of he aival infomaion and infes each ime-dependen O-D flow once i leaves while he column decomposiion essenially uses boh he aival and depaue infomaion. Howeve, he poposed column decomposiion appoach a he coido level only focuses on he localized infomaion i.e., he uning movemens a he inesecion conaining desinaion node and discads ohe infomaion available upseam as he vehicles shall be obseved a all hese inemediae inesecions. The weighed aveage of he wo esimaes eplicaes acual O-D flows bee. The RMSN dops by 6.5% and 3.4%, compaed wih he ow and column decomposiion especively. Figues 7-10 also sugges ha he weighed-aveage esimae follows he acual O-D paen well. Consequenly, he RMSN fo each O-D pai pesened in Table 3.3 is compaable wih is counepa in Table 3.2 fo he consan O-D case. 82

94 O-D flows Acual Esimae by desinaion 35 Esimae by oigin Esimae weighed aveage Time ineval Fig Acual vs Esimaed O-D flow fo O-D pai O-D flows Acual 35 Esimae by desinaion Esimae by oigin Esimae weighed aveage Time ineval Fig Acual vs Esimaed O-D flow fo O-D pai

95 O-D flows Acual 30 Esimae by desinaion Esimae by oigin Esimae weighed aveage Time ineval Fig Acual vs Esimaed O-D flow fo O-D pai O-D flows Acual 30 Esimae by desinaion Esimae by oigin Esimae weighed aveage Time ineval Fig Acual vs Esimaed O-D flow fo O-D pai

96 Table 3.3 RMSN of Esimaes of Time-Vaying O-D Flows Real-Wold Applicaion We applied he poposed O-D esimao o a segmen of El Camino Real, San Maeo, CA using he daa colleced on Febuay 1 s, Since we do no have eal O-D obsevaions fom he coido, he accuacy of he esimaes canno be veified. Theefoe, he pupose of he applicaion is o demonsae ha he esimao is able o eadily wok wih acual field loop daa. The esing sie consiss of eigh signalized inesecions as shown in Figue The disances beween inesecions in mees ae pesened in he illusaion. Thee ae fou on/off amps beween he 17 h Ave. and he 20 h Ave. accessing J. Ahu Younge Feeway. Howeve, since he vehicle couns ono/off he feeway ae no available, i is assumed in his applicaion ha he amp flows ae zeo. N 12h Baneson 17h 20h 25h 27h 28h 9h Fig Illusaion of he Tesing Sie In ode o avoid possible paen shifs beween peak/off-peak hous, only off-peak loop couns fom 9:49-14:04 wee used. The aw daa wee aggegaed ino a five-minue esoluion. Among hese eigh inesecions, hee is no loop infomaion 85

97 available fo 17 h Ave., and he couns fo he eas and he noh bounds of 28 h Ave. inesecion ae missing as well. Theefoe, wih he assumpion ha hee ae no uning movemens a he 17 h inesecion, he coido is modeled as a fou-inesecion sysem as shown in Figue Noe ha 9 h Ave. and 27 h Ave. inesecions ae excluded fom he sysem as hei main line depaue couns obained fom adjacen inesecion advanced loop couns ae no available. 12h Baneson 20h 25h Fig Modeling of he Coido The iniial values of he O-D flows wee geneaed manually based on he couns of he fis ime ineval. Since he avel ime esimaed based on he speed limi of 35mph beween any of he wo inesecions is less han he duaion of a single ime ineval, we assume hee he flow popagaion is neglecable and all he vehicles aive a hei desinaions wihin he same ime ineval as hey depa. Seleced esuling esimaes ae displayed in Figues O-D flows Esimae by desinaion Esimae by oigin Esimae weighed aveage Time ineval Fig Esimaed O-D flow fo O-D pai

98 O-D flows Esimae by desinaion Esimae by oigin Esimae weighed aveage Time ineval Fig Esimaed O-D flow fo O-D pai O-D flows Esimae by desinaion Esimae by oigin Esimae weighed aveage Time ineval Fig Esimaed O-D flow fo O-D pai Concluding Remaks We have pesened a wo-sep appoach fo esimaing ime-vaying O-D flows fo acuaion-conolled coidos wih incomplee infomaion abou eneing and exiing flows. A he fis sep, inesecion uning movemens fo each inesecion ae 87

99 esimaed, and hen used a he second sep o consuc he measuemen equaions o infe he coido O-D flows. The poposed appoach has been demonsaed and veified on a hypoheic coido. The esuls sugges ha he esimao is able o ack he paens of he O-D flows and povide esimaes close o he acual values. Moeove, we have applied he O-D esimao o a segmen of El Camino Real, San Maeo, CA. The applicaion demonsaes ha he esimao is able o eadily wok wih acual field loop daa. The appoach can be fuhe enhanced by addessing he following issues: Incopoaing plaoon dispesion in deemining he assignmen maix o simulaneously esimaing he popagaion paamees ogehe wih O-D flows using he nonlinea Kalman fileing. Developing an efficien mehod o incopoae mainline hough movemen infomaion available a upseam inesecions. 3.4 Invesigaion of Dynamic Sucue of O-D Demand Inoducion Pevious wo secions have pesened appoaches fo eal-ime esimaion of dynamic O-D flows fo acuaion-conolled inesecions and coidos. The echnique adoped is Kalman fileing and he fundamenal idea is o fomulae sae-space equaions whee he sae vecos o be esimaed ae assumed o be a dynamic auo-egessive pocess. In his manne, infomaion of he sucue of he O-D demand is obained in addiion o he measuemen equaions o esimae he demand wihou using fuhe a pioi infomaion Ceme and Kelle, This secion addesses a fundamenal issue in applying he above echnique. In Kalman fileing, he sae vecos can be and have been specified as O-D splis, defined as he pecenage of ips geneaed fom an oigin o specific desinaions e.g., Ceme and Kelle, 1987; Nihan and Davis, 1987; Bell, 1991 and Chang and Wu, 1994, O-D flows e.g., Okuani, 1987, he deviaions of O-D flows fom hisoical daa e.g., Ashok and Ben-Akiva, 1993 and Hu e al., 2001 o he deviaion of O-D splis e.g., Ashok and Ben-Akiva, The selecion of sae veco is ciical because i esuls in diffeen model sucue ha equies diffeen amoun of compuaion effo. Moe impoanly, i deemines he pefomance of he esimao. If he undelying auo-egessive assumpion does no eplicae ealiy well, he esimao may no be able o povide easonable esimaes. Ineesingly, many of hese pevious sudies make he choice pimaily o faciliae he model fomulaion wihou enough jusificaion. A he same ime, few sudies have been done o offe insighs how o make he choice. This secion is an empiical invesigaion on he selecion of sae vecos in he fileing pocess, using he affic daa colleced fom a single inesecion in 88

100 Gainesville, Floida. The pupose of his invesigaion is no o deemine which sae vaiable should be seleced which seems impossible, bu o offe some obsevaions of he O-D demand sucues and he esimao pefomances, and hopefully shed lighs on how o choose he mos appopiae sae veco and he coesponding esimaion model. To his aim, we conduc saisic ime seies analysis o examine he dynamic popey of he O-D demand, and compae he esimaion esuls fom using diffeen sae vecos Hisoical Pespecives As peviously saed, exising dynamic O-D esimaos wih fileing echniques can be casually caegoized ino wo classes: he closed-newok and he open-newok oiened appoaches, based on he kenel sysem equaions adoped. Fo closed newoks, all he eny and exi couns ae known duing all measuemen inevals. Naually, he O-D splis may be consideed as sae vaiables, esimaed diecly using he flow consevaions epesening he elaionship beween he eal-ime exiing couns of a ceain desinaion j and eal-ime eneing couns of all he elaed oigins i: y j = bij qi + e j i Hee y j and q i ae he exiing and he eneing affic couns, b ij is he O-D spli, he popoion of affic flows eneing a oigin i and exiing a desinaion j, and e j is he andom measuemen eo. The O-D splis have been assumed o be auo egessive in he pioneeing sudies, e.g., Ceme and Kelle 1987 and Nihan and Davis Boh sudies model he ime-dependen O-D splis as a fis-ode auo-egessive AR1 ARq denoes he qh-ode auo-egessive model pocess: b = b 1 w whee ij ij + w ij is a seies of whie noise ems wih zeo means and known covaiance. ij Unde addiional assumpion ha he O-D splis ae independen among all O-D pais, i.e. hee is no coelaion beween all he eo ems w ij, he O-D esimaion poblem can be decomposed ino smalle idenificaion poblems, each concening only he splis elaed o one specific exi j. Such decomposiion inceases he compuaion efficiency by avoiding maix invesions in updaing he Kalman gain. Aenion should be paid o saisfying consains associaed wih O-D splis. The spli b ij is he popoion of he affic eneing a eny i a ime ineval ha leaves a exi j. Theefoe, he paamees b ij ae obviously bounded beween zeo and one. Moeove, all he O-D splis fom a specific eny i in a specific ime ineval should add up o one. These inequaliy and equaliy consains cause some difficulies when applying Kalman fileing because he basic sucue of his algoihm does no allow fo 89

101 consains. How o deal wih he consains is one of he bigges issues fo he closed-newok appoach. Alhough pevious sudies have poposed appoaches o saisfy hose equaliy and inequaliy consains, e.g., Nihan and Davis 1987, Bell 1991 and Li and De Moo 1999, hee is no effecive algoihm o addess boh ses of consains simulaneously. One may hink of adjusing he peliminay esuls accoding o boh ses of consains ieaively, bu his heuisic mehod will be compuaionally demanding. Moeove, he exisence of he convegence and whehe he esuls ae unbiased emain unpoved. Fo open newoks, no all eny and exi couns ae known, and he infomaion available is nomally link flows ecoded by deecos. Unde his infomaion paen, he esimaion of O-D flows is egaded as a evese poblem of he dynamic affic assignmen DTA poblems. The sysem equaion is as follows: x = A f + e = m whee x is he measued link volume veco; m is he maximum numbe of ime inevals needed o avel hough he sysem; f is he veco of O-D flows; coesponding measued eo veco and A e is he is he assignmen maix models mapping he O-D flows of ime ineval o he cuen link volume. The assignmen maices incopoae he flow popagaion and oue choice infomaion bu ae difficul o obain and have been assumed o be known fom exogenous esouces such as diec obsevaions o DTA models. Consequenly, he O-D flows ahe han he O-D splis ae he sae vecos o be esimaed. Define G as he ansiion maix descibing he effec of pevious O-D flow f on he cuen O-D flow a, some eseaches assumed he O-D flows follow he moe geneal auo-egessive sucue: 1 f = G f + w = p whee p indicaes he maximum numbe of lags and w is a seies of whie noise vecos wih zeo mean e.g., Okuani, This assumpion is moe ealisic han he AR1 model fo he O-D splis in he sense ha i allows O-D flows fom moe han one pevious peiod o have coelaions wih he cuen O-D flow. And he esimaion is only consained by he non-negaiviy condiions which, because he magniude of O-D flows is much geae han he O-D splis, ae aely acivaed. Howeve, hisoical daa ae equied o esimae he ansiion maices. Disincive fom he above assumpion, Ashok and Ben-Akiva 1993 assumed ha ahe han O-D flows hemselves, he deviaions of cuen O-D flows fom hisoical O-D daa ae auo-egessive. They agued ha he adiional assumpion only capues 90

102 he empoal inedependencies while he moe complicaed sucue of O-D paens has been ignoed. By incopoaing hisoical daa ino he esimaion, he spaial and ohe popeies of he specific demand paens may be epesened. One moe advanage is ha he deviaions may be bee appoximaed by he auo-egessive model since hey can ake boh negaive and posiive values. Accodingly, he non-negaiviy inequaliy consains ae no necessay. The only siuaion whee he deviaions need o be adjused is when he absolue value of he negaive deviaion is geae han he hisoical flow, bu his happens vey occasionally in he expeimens. Theefoe, no equaliy and inequaliy consains ae imposed on he flow deviaion vaiables. Le he supescipion H denoe he hisoical counepas, he sae-space equaions should be efomulaed as follows: 1 H H f f = G f f + w = p H H x x = + A f f e = m Decomposiion of his sysem is impossible unless he newok sucue is simple and some ceain links ae only elaed o limied O-D flows. Theefoe, such appoach is moe compuaionally demanding. Besides, he qualiy of hisoical daa may affec he efficiency of he esimao. Poo hisoical basis is expeced o degade he esimao s pefomance. Any O-D flow can be expessed by he poduc of ip poducion a he oigin and he O-D spli. Ashok and Ben-Akiva 2000 fuhe obseved ha hese wo componens exhibied diffeen vaiabiliy wih ime. The ip poducion may be highly vaiable while he O-D splis ae elaively sable. Allowing fo his diffeenial vaiabiliy in he esimaion pocess could aguably incease he pefomance of he esimao. Consequenly, wo ses of ansiion equaions can be specified as follows: 1 H H d d = Φ d d + w1 = p 1 H H b b = Ψ + b b w2 = m whee d is he ip poducion veco; Φ is he ansiion maix descibing he effec of pevious ip poducion on he cuen poducion; b is he spli veco and Ψ is he coesponding ansiion maix; w and w ae he eo vecos and p and m ae he ode of he auo-egessive pocesses. In summay, he sae vecos can be specified as O-D splis, O-D flows, O-D flow deviaions and O-D spli deviaions. Alhough he choice of sae vecos is vey ciical o he sucue, compuaion complexiy and pefomance of he esimao, hee is no pevious sudy o eveal ininsic sucues of he O-D demands and offe insighs on

103 how o deemine sae vecos Empiical Invesigaion In he following we examine he affic daa colleced fom an inesecion of 34 h See and Univesiy Avenue in Gainesville, Floida See Fig fo he inesecion layou and he numbeing of he fou appoaches o eveal he ue sucue of he O-D demand a he inesecion. To povide moe pagmaic compaison, we esimae he O-D demand using each sae vaiable unde he simple fis-ode auo-egessive assumpion and hen compae he esimae wih he ue value. Leg 2 Mino Leg 1 Majo Leg 3 Majo Leg 4 Mino Fig h See-Univesiy Avenue inesecion Majo = Univesiy Avenue, Mino = 34 h See Saisic Time Seies Analysis Each uning movemen is consideed as an independen sochasic pocess. Since U-un is pohibied, hee ae 12 ime seies available fo each sae vaiable, and hey ae modeled sepaaely. We use he ARIMA module in SPSS vesion 13.0 o do he ime seies analysis O-D Esimaos We esimae he O-D demand using each sae vaiable unde he simple fis-ode auo-egessive assumpion. Fo he sae vaiable of O-D splis, he esimao can be epesened as: b = b 1 + w y = Q b + e 92

104 whee,,, q q q q diag Q T T T T v v v v = and ohe elemens denoe he same vaiable in veco fom. This sysem consiss of fou obsevaion equaions and 12 unknowns. Denoe V as he covaiance maix of eo veco w and R he covaiance maix of e, he unconsained Kalman file soluion is: [ ] [ ] [ ] + = + + = + = V P Q K I P b Q y K b b R Q k P Q Q P K T T v v v In acual applicaion, V can be egaded as a paamee of he esimao if unknown. Taking accoun of he equaliy consains in veco fom, = b D v, he oiginal soluion should be coeced by: [ ] [ ] 1 ˆ 1 = k Db D DP D P b b T T v v v Wih he O-D spli deviaions as he sae vaiable, he esimao can be fomulaed as: + = + = 1 1 e b b Q y y w b b b b H H H H The soluion can be saighfowadly deived by eplacing b and y in he O-D splis soluion by b b H and y y H. To esimae O-D flows diecly, he fou enies and exis of he inesecion ae consideed as eigh viual links. Consequenly, he eneing and exiing couns ae eaed as he link volumes. Theefoe, he numbe of obsevaion equaions inceases fom fou o eigh in his appoach. The flow popagaion can be ignoed since a cycle is consideed one ime ineval. Aanging hese 12 O-D flows in he same ode as he O-D splis, he sae-space equaion becomes: + = + = 1 e f D H q y w f f whee [ ] = diag H Simila fomulaion fo he deviaions of O-D flows is as follows: + = + = 1 1 e f f D H q y q y w f f f f H H H H H Daa Descipion Acual O-D flows uning movemens of he 34 h See - Univesiy Avenue inesecion in Gainesville, Floida wee colleced cycle by cycle fom he videos 93

105 ecoded in Since oally diffeen auo-egessive paens may exis fo diffeen imes of day, only he off-peak uning movemens wee couned in ode o moe likely obain flows wih homogenous O-D paens. Wihou loss of genealiy, one cycle is egaded as a ime ineval, in which he eneing and exiing couns of each appoach wee aggegaed fom he uning movemens. The O-D spli b ij was calculaed as he coesponding O-D flow f ij divided by he eneing flow of appoach i. Limied by he videos available, 46 daa poins wee obained fom 9:00AM o 11:00AM fo he fis day July 17, 2001, 27 daa poins fom 9:00AM o 10:00AM and 28 daa poins fom 10:00AM o 11:00AM wee colleced fo he second day July 18, 2001 and hid day July 20, 2001 especively. Seveal assumpions wee made in he following saisical analysis. To examine he O-D flows and splis, i was assumed he auo-egessive paen emains consan ove hese hee peiods so ha one can assemble hee sho ime-seies daa ino one longe seies o have enough obsevaions. To evaluae he deviaions of O-D flows o splis, he fis-day daa wee chosen as he basis while he second- and hid-day daa wee combined ogehe. I was also assumed ha all he lane goups wee independen. In he esimaion pa, expeimens wihou he independence assumpion wee also conduced o show he impac of he covaiance maix of noise veco, V, on he esimaes Empiical Resuls Time Seies Model Specificaion Alhough mos of he saisic sofwae has buil-in ime seies analysis modules, he peliminay es of he daa and he ansfomaion o saionaiy canno be done auomaically. Sequence gaphs and hisogams wee ploed as a efeence o check whehe he oiginal daa ae saionay. The sequence gaphs indicaed no song end and peiodiciy in he seies of O-D flows, splis, flow deviaions and spli deviaions. Bu he hisogams fo hese vaiables wee no symmeically disibued and wee fa away fom he ypical Gaussian maginal disibuion. To ake accoun of his effec, fis-ode diffeencing was applied o eliminae possible unevealed complicaed ends. The new seies wee ceaed as: New diffeenced O-D daa = Oiginal O-D daa Oiginal O-D daa -1, =2, 3 94

106 Table 3.4 Model Specificaions fo Diffeenced O-D Flows Lane Goups AR1 AR2 AR3 AR4 Appoach 1 Lef Tun Appoach 1 Righ Tun Appoach 1 Thu Appoach 2 Lef Tun Appoach 2 Righ Tun Appoach 2 Thu Appoach 3 Lef Tun Appoach 3 Righ Tun Appoach 3 Thu Appoach 4 Lef Tun Appoach 4 Righ Tun Appoach 4 Thu Fo O-D splis and flows, he new diffeenced seies consis of 100 daa poins while he deviaion seies include only 45 daa poins. Sequence gaphs and hisogams suggesed hese diffeenced vaiables wee moe likely o be saionay. Theefoe, he saisical analyses wee based on hese new seies. Noe ha he fac ha he diffeence seies is kh-ode auo-egessive model does no necessaily mean he oiginal daa is k+1h-ode model. The ineupion iem plays a vey impoan ole in saisic model esimaions. The coefficiens of he bes fi models fo all he 12 O-D movemens fo hese fou sae vaiables ae lised in Tables 3.4 o 3.7. Inconsisency beween diffeen lane goups is obseved. Moeove, he esuls indicae ha fo his paicula daa se, he simple fis-ode auo-egessive model canno fi he oiginal daa well. Moe sophisicaed auo-egessive models up o he fouh ode ae needed o descibe he O-D sucues. Though complicaed, i is sill feasible o incopoae hese ansiion elaionships ino he Kalman fileing esimao. 95

107 Table 3.5 Model Specificaions fo Diffeenced O-D Flow Deviaions Lane Goups AR1 AR2 AR3 Appoach 1 Lef Tun Appoach 1 Righ Tun Appoach 1 Thu Appoach 2 Lef Tun Appoach 2 Righ Tun Appoach 2 Thu Appoach 3 Lef Tun Appoach 3 Righ Tun Appoach 3 Thu Appoach 4 Lef Tun Appoach 4 Righ Tun Appoach 4 Thu Esimaion Resuls The fou esimaos descibed in Secion wee applied o esimae he O-D demand, and he oo mean squae RMS of he eo beween he esimaes and acual O-D flows was calculaed as a pefomance measuemen fo each esimao. Fo esimaos wih sae vaiable of O-D spli o deviaions, he O-D flows wee compued as he poducion of esimaed O-D splis and he eneing couns o summaion of he esimaed deviaion and he hisoical daa. 96

108 Table 3.6 Model Specificaions fo Diffeenced O-D Splis Lane Goups AR1 AR2 AR3 AR4 Appoach 1 Lef Tun Appoach 1 Righ Tun Appoach 1 Thu Appoach 2 Lef Tun Appoach 2 Righ Tun Appoach 2 Thu Appoach 3 Lef Tun Appoach 3 Righ Tun Appoach 3 Thu Appoach 4 Lef Tun Appoach 4 Righ Tun Appoach 4 Thu Diffeen combinaions of iniial values and he esimao paamee V epesening he covaiance of he sae ansiion equaion wee esed. Fo he O-D splis, he iniial value was se as The ue O-D splis a he fis cycle and he aveage values ove all he ime inevals wee seleced as wo ohe iniial scenaios. They ae denoed especively as Basic, Fis Cycle and Mean iniial condiions. Basic iniial condiion fo O-D flows abiaily se all he componens equal o one. V maix was se as eihe he esimaed covaiance of he sample O-D daa Sample Covaiance o he ideniy maix Ideniy. Noe ha unde he Sample Covaiance condiion, he O-D daa of diffeen lane goups canno be egaded as independen since he esimaed sample covaiance would aely be a diagonal maix. 97

109 Table 3.7 Model Specificaions fo Diffeenced O-D Spli Deviaions Lane Goups AR1 AR2 Appoach 1 Lef Tun Appoach 1 Righ Tun Appoach 1 Thu Appoach 2 Lef Tun Appoach 2 Righ Tun Appoach 2 Thu Appoach 3 Lef Tun Appoach 3 Righ Tun Appoach 3 Thu Appoach 4 Lef Tun Appoach 4 Righ Tun Appoach 4 Thu The esimaion esuls ae pesened in Tables 3.8 and 3.9. Noe ha he second- and hid-day O-D demands wee esimaed sepaaely if deviaions ae sae vaiables. Repoed RMS of he coesponding wo esimaos he O-D spli deviaion esimao and he O-D flow deviaion esimao is he aveage RMS of hese wo ses of esimaion. I can be obseved fom hese ables ha in his paicula case sudy, assuming O-D flows o be fis-ode auo-egessive leads o he bes esimaion unde he same iniial condiions and esimao paamees. The esimaos focusing on he O-D daa hemselves oupefomed hei counepas of deviaions. See Figs 3.29 and 3.30 fo a visualized compaison of O-D flow esimao and flow deviaion esimao. The lae coesponds o he las 27 inevals in he fome figue. 98

110 6 f12 value Time Ineval 30 f32 value Time Ineval f42 value Time Ineval Fig Resuls of O-D flows desined o Leg 2 esimaed by O-D flow esimao 4 3 f12 value Time Ineval 20 f32 value Time Ineval f42 value Time Ineval Fig Esimaes of he hid-day O-D flows desined o Leg 2 wih O-D flow deviaion esimao Impac of he ansiion funcion was fuhe exploed by esimaing he O-D flows using he esimaed AR1 of he diffeenced seies. The ansiion maix G shall be 99

111 inoduced o he Kalman fileing. The sae-space equaions ae: + = + = e f f D H q y q y w f f G f f Soluion o his model is: + = + + = + = V G P D H K I G P f f G D H q y q y K f f G f f R D H k P D H D H P K T T T The RMS unde Mean iniial and Sample Covaiance condiion is 1.35, which is 211% moe han ha of he oiginal O-D flow esimao unde he same condiions. This esul seems no consisen wih ha of he ime seies analysis. A plausible explanaion is ha he ime seies analysis was conduced afe he subjecive ansfomaion pocess o saionaiy and does no necessaily eveal he ue sucue. On he ohe hand, i may sugges ha easonable appoximaion of he O-D paens is able o lead o accepable esuls. The expeimens also veified ha he iniial value does no have a pimay impac on he pefomance of he esimao and he effec of covaiance maix V seems even moe negligible. On he ohe hand, pefomance of he O-D spli and spli deviaion esimaos may impove when hey wee consained Conclusion We have conduced an empiical analysis of he O-D demand sucues and examined hei impacs on he O-D esimaion using he daa fom he 34 h See-Univesiy Avenue inesecion in Gainesville, Floida in Auo-egessive models of he diffeenced O-D flows, flow deviaions, spli and spli deviaions wee esimaed, and he bes fi models ange fom AR1 o AR4. The compaison of esimaos wih diffeen sae vaiables suggesed ha he esimao wih sae vaiable of O-D flow oupefoms he ohes in his paicula case. We fully ecognize ha O-D paens would be sie-dependen, and he esuls of his case sudy should no be genealized. Howeve, ou empiical invesigaion does offe he following obsevaions, which may be of use fo fuue sudies and pacices: 100

112 The idea of deviaions, moe specifically, he deviaions of O-D flows o splis do no necessaily bee epesen he O-D demand paens. If hisoical daa ae available, saisical analysis may be conduced o eveal he ininsic sucue of he O-D demand; Consequenly, he esimaos wih sae vaiables of deviaions do no necessaily poduce bee O-D esimaes; Demands o flows a diffeen O-D pais may possess diffeen sucues, which ae vey ofen no fis-ode auo egessive; Incopoaing all of hese ue sucues ino he Kalman fileing makes he model fomulaion vey complicaed. On he ohe hand, he simple fis-ode auo egessive assumpion poduces accepable esuls in ou empiical expeimens and pevious sudies. Theefoe, unless hee ae sufficien O-D daa ha sugges ohewise, i may be sensible o simply use he sae vaiable of O-D flows o splis and assume ha hey ae fis-ode auo egessive. 101

113 Esimao O-D Splis Deviaion of O-D Splis Table 3.8 RMS of O-D Esimao of Splis and Spli Deviaions Iniial Values Paamee V Basic Fis Sample Consains RMS Mean Ideniy Cycle Covaiance X X No 2.43 X X Equaliy 2.17 X X Inequaliy 2.12 X X No 2.27 X X Equaliy 2.01 X X No 2.85 X X Equaliy 2.55 X X No 2.77 X X Equaliy 2.32 X X No 2.74 X X Equaliy 2.41 X X No 2.72 X X Equaliy 2.31 X X No 2.73 X X Equaliy 2.38 X X No 2.74 X X Equaliy

114 Table 3.9 RMS of O-D Esimao of Splis and Spli Deviaions Iniial Values Paamee V Esimao Fis Sample RMS Basic Mean Ideniy Cycle Covaiance X X 2.30 X X 1.80 X X 1.13 O-D Flows X X 0.58 X X 0.78 X X 0.64 X X 2.40 X X 1.92 Deviaion X X 1.60 of O-D X X 1.05 Flows X X 1.42 X X

115 Refeences Anoniou, C., Ben-Akiva, M.E. and Kousopoulos, H.N Incopoaing auomaed vehicle idenificaion daa ino oigin-desinaion esimaion. Tanspoaion Reseach Recod No. 1882, Asakua, Y., Hao, E. and Kashiwadani, M Oigin-desinaion maices esimaion model using auomaic vehicle idenificaion daa and is applicaion o he Han-Shin expessway newok. Tanspoaion, 27, Ashok, K. and Ben-Akiva, M.E Dynamic oigin-desinaion maix esimaion and pedicion fo eal-ime affic managemen sysem. In Poceedings of he 12h Inenaional Symposium on Tanspoaion and Taffic Theoy, ed. C. F. Daganzo, , Bekeley. Elsevie, NY. Ashok, K. and Ben-Akiva, M.E Alenaive appoaches fo eal-ime esimaion and pedicion of ime-dependen oigin-desinaion flows. Tanspoaion Science Vol. 34, Ashok, K; Ben-Akiva, ME 2002 Esimaion and pedicion of ime-dependen oigin-desinaion flows wih a sochasic mapping o pah flows and link flows. Tanspoaion Science, 362, Bell, M.G.H The eal ime esimaion of oigin-desinaion flows in he pesence of plaoon dispesion. Tanspoaion Reseach, 25B, Bell, M.G.H., Lam, W.H.K and Iida, Y A ime-dependen muli-class pah flow esimao. In Poceedings of he 13h Inenaional Symposium on Tanspoaion and Taffic Theoy, , Elsevie Science. Bielaie, M. and Ciin, F An efficien algoihm fo eal-ime esimaion and pedicion of dynamic OD ables. Opeaions Reseach 52, Cascea, E., Inaudi, D., and Maquis, G Dynamic esimaos of oigin-desinaion maices using affic accouns. Tanspoaion Science, 27, Chang, G.L. and Tao, X Esimaion of dynamic newok O-D disibuions fo uban newoks. In Poceedings of he 13h Inenaional Symposium on Tanspoaion and Taffic Theoy, 1-20, Elsevie Science. Chang, G.L. and Tao, X An inegaed model fo esimaing ime-vaying newok oigin-desinaion disibuions. Tanspoaion Reseach, 33A, Chang, G.L. and Wu, J Recusive esimaion of ime-vaying O-D flows fom affic couns in feeway coidos. Tanspoaion Reseach, 28B, Chen, A., Chooinan, P., Recke, W. and Zhang, H.M Developmen of a pah flow esimao fo deiving seady-sae and ime-dependen oigin-desinaion ip ables. Califonia PATH eseach epo, UCB-ITS-PRR , Sepembe

116 Cème, M. and Kelle, H Dynamic idenificaion of flows fom affic couns a complex inesecion. Poceedings of he 8h Inenaional Symposium on Tanspoaion and Taffic Theoy, V.F. Hudle e al., eds., Univesiy of Toono Pess, Toono. Cème, M. and Kelle, H A new class of dynamic mehods fo he idenificaion of oigin-desinaion flows. Tanspoaion Reseach, 21B, Dixon, M. P. and Rile, L. R Populaion oigin-desinaion esimaion using auomaic vehicle idenificaion and volume daa. Jounal of Tanspoaion Engineeing, 1312, Eisenman, S. M. and Lis, G. F Using Pobe Daa o Esimae OD Maices. IEEE Inelligen Tanspoaion Sysems Confeence, Washingon, D.C., USA, FHWA 2004 hp:// Fedeal Highway Adminisaion Dynamic Taffic Assignmen Reseach Pogam. Hu, S.R., Madana, S., Kogmeie, J.V. and Peea, S Esimaion of dynamic assignmen maices and OD demands using adapive Kalman fileing. ITS Jounal, Vol.6, Kwon, J. and Vaaiya, P Real-ime esimaion of O-D maices wih paial ajecoies fom eleconic oll collecion ag daa. 84 h Annual Meeing of Tanspoaion Reseach Boad, Washingon, D.C., USA, 2005 Li, B., Moo, B.D Recusive esimaion based on he equaliy-consained opimizaion fo inesecion oigin-desinaion maices. Tanspoaion Reseach 33B, Li, B., Moo, B.D Dynamic idenificaion of oigin-desinaion maices in he pesence of incomplee obsevaions. Tanspoaion Reseach, 36B, Madana, S.M., Hu, S.R., and Kogmeie, J 1996 Dynamic esimaion and pedicion of feeway O-D maices wih oue swiching consideaions and ime-dependen model paamees. Tanspoaion Reseach Recod 1537, Nihan, N.L. and Davis, G.A Recusive esimaion of ogin-desinaion maices fom inpu/oupu couns. Tanspoaion Reseach, 21B, Okuani, I The Kalman fileing appoaches in some anspoaion and affic poblems. Poceedings of he 10h Inenaional Symposium on Tanspoaion and Taffic Theoy, N.H. Gane and N.H.M. Wilson eds., Elsevie, New Yok, Paige, C.C and Saundes, M. A LSQR: An algoihm fo spase linea equaions and spase leas squaes, ACM TOMS 81, Sheali, H.D. and Pak, T Esimaion of dynamic oigin-desinaion ip ables fo a geneal newok. Tanspoaion Reseach, 35B, Simon, D. and Chia, T.L Kalman Fileing wih Sae Equaliy Consains. IEEE Tansacions on Aeospace and Eleconic Sysems, Vol. 38,

117 Soenson, H.W Leas-squaes esimaion: fom Gauss o Kalman. IEEE Specum, Vol. 7, Willumsen, L.G Esimaing ime-dependen ip maices fom affic couns. Poceedings of he 9h Inenaional Symposium on Tanspoaion and Taffic Theoy, he Nehelands, Delf Univesiy. Van De Zijpp, N Dynamic oigin-desinaion maix esimaion on mooway newoks. Ph.D. Disseaion. Delf Univesiy of Technology, Delf, Nehelands. Wu, J. and Chang, G.L Esimaion of dynamic newok O-D wih sceenline flows. Tanspoaion Reseach, 30B, Zhou, X Dynamic oigin-desinaion demand esimaion and pedicion fo off-line and on-line dynamic affic assignmen applicaion. Ph.D. Disseaion. Univesiy of Mayland, College Pak, MD. 106

118 CHAPTER 4 DEVELOPMENT OF A PRACTICAL COMPUTER TOOL FOR DYNAMIC ORIGIN-DESTINATION MATRICES ESTIMATION Pepaed by: Meng Li Califonia PATH Univesiy of Califonia, Bekeley 107

119 The objecive of his sub-ask is o implemen he models and algoihms developed in he pevious ask and develop a ool wih a use-fiendly ineface o allow paciiones o apply hese developed models and algoihms. The developed ool is designed o esimae he dynamic OD maices fo a peiod of inees when affic flow is in a seady sae and fo a linea newok whee loop deeco coun infomaion is available. In his sudy, he loop deeco coun infomaion is assumed o come fom he Feeway Pefomance Measuemen Sysem PeMS. The geneal famewok is shown in Figue 4.1. I is composed of fou subsysems Ineface, Sofwae, Newok ID, and PeMS. Figue 4.1 Geneal Famewok The simple use ineface is shown in Figue 4.2. The fis sep afe loading he sofwae is o define he newok. Paciiones can choose fom Inesae-80 Wesbound a Bekeley which is he esbed seleced by his sudy as shown in Figue 4.3, o Cusomized newok using PeMS sysem. Fo he cusomized newok, wo hisoical OD maices o hisoical couns fo a leas wo days ae equied as he inpu. Figue 4.2 Simple Use Ineface 108

120 Figue 4.3 Tesbed of I-80W a Bekeley The nex sep fo uses is o selec he dae and peiod of ineess fo he esimaion. Wih he given selecions, he sofwae will ead he daa files, which conains 30-second loop deeco coun infomaion fom PeMS. Afe obaining all he equied inpus wih deviaions i.e. A, V, F, W, and y k, he sofwae will un he developed mehodology o esimae he OD maix. I is impoan o noe ha he sofwae has o soe he esimaed OD maix, because i will be used as he second hisoical OD maix o compue he maix F fo he nex ime. Tha is, once he peiod ends, he sofwae will use he diffeence o deviaion beween he jus-esimaed OD maix and he OD maix esimaed he day befoe o compue a new maix F. In he fuue, he sofwae could be coninuously unning so he hisoical OD maix could be self-leaned and accodingly updaed. The oupus of he sofwae ae pesened by he boom pa of Figue 4.4. Wih he given peiod of ineess, he sofwae calculaes a new OD maix fo each 30-second ineval. Uses can choose 1:Display esimaed OD o 2:Display model pefomance. If 1 is seleced, uses can fuhe selec an ineval fo display. The esimaed OD flow fo each OD pai will be displayed. If 2 is seleced, he esimaed OD will be compaed wih he given eal OD maix. Then he oo mean squae eo RMS and he oo mean squae eo nomalized RMSN will be calculaed and displayed. I is noe ha he funcion o display he model pefomance is only available fo he given peiod of ime a he I-80W esbed, because he special daa has only been colleced using Bekeley Highway Lab BHL s cameas insalled on he oof of a building locaed nex o he esbed. The eal OD maix was hen exaced fom he video ecodings. This pocess is ime consuming. 109

121 Figue 4.4 Sofwae Oupus The cuen use ineface is sill vey simple and he funcions ae vey limied. Theefoe, he use ineface, some of he daa collecion and pocessing pocedues, and moe convenien funcions will be fuhe developed and debugged. 110