Sensor Coverage and Location for Real-Time Traffic Prediction in Large-Scale Networks

Transcription

1 Sensor Coverge nd Loction for Rel-ime rffic Prediction in Lrge-Scle Networks Xing Fei, Hni S. Mhmssni, nd Stcy M. Eisenmn he bility to observe flow ptterns nd performnce chrcteristics of dynmic trnsporttion systems remins n importnt chllenge for trnsporttion gencies, notwithstnding continuing dvnces in surveillnce nd communiction technologies. As these technologies continue to become more relible nd cost-effective, demnd for trvel informtion is lso growing, s re the potentil nd the bility to use sensor nd probe informtion in sophisticted decision support systems for trffic systems mngement. his pper focuses on improving the efficiency of dt collection in trnsporttion networks by studying how sensor plcement ffects network observbility. he objective of this study is to identify set of sensor loctions tht optimize the coverge of origin destintion demnd flows of the rod network nd mximize the informtion gins through observtion dt over the network, while minimizing the uncertinties of the estimted origin destintion demnd mtrix. he proposed sensor models consider problems where the numbers of sensors re limited nd unlimited. he pper lso provides severl exmples to illustrte the reltive effectiveness of the proposed methodologies. he bility to observe flow ptterns nd performnce chrcteristics of dynmic trnsporttion systems remins n importnt chllenge for trnsporttion gencies, notwithstnding continuing dvnces in surveillnce nd communiction technologies. As these technologies continue to become more relible nd cost-effective, demnd for trvel informtion is lso growing, s re the potentil nd bility to use sensor nd probe informtion in sophisticted decision support systems for trffic systems mngement. Wheres probe dt bsed on cellulr-ssisted Globl Positioning System nd other cellulr phone technologies hold the promise of ner-ubiquitous informtion coverge in network, mesurements on system stte t given loctions using fixed sensors remin the bckbone of most trffic mngement centers for trffic mngement nd control purposes. Given the deployment nd mintennce costs of such instlltions, most gencies re clled on to determine the number nd loctions of such sensors cross given network. o improve the efficiency of dt collection in trnsporttion networks, it is criticl to understnd how sensor plcement ffects network observbility. Furthermore, new genertion of rel-time network trffic estimtion nd prediction systems is designed to Deprtment of Civil nd Environmentl Engineering, University of Mrylnd, Jeong H. Kim Engineering Building, College Prk, MD 74-. Corresponding uthor: H. S. Mhmssni, msmh@umd.edu. rnsporttion Reserch Record: Journl of the rnsporttion Reserch Bord, No. 9, rnsporttion Reserch Bord of the Ntionl Acdemies, Wshington, D.C., 7, pp.. DOI:.4/9- interct with rel-time sensor dt to support system mngement decisions nd through estimtion, prediction, nd control genertion cycles (). For exmple, rel-time dynmic trffic ssignment (DA) systems such s Dynsmrt-X nd DynMI use sensor mesurements on subset of the network links s bsis for estimting nd predicting trffic conditions on qusi-continuous bsis. In prticulr, the sensor mesurements re combined with current vlues nd historicl informtion to estimte previling origin destintion (O-D) ptterns nd predict their ner-term evolution, in ddition to predicting the network trffic ptterns ssocited with these O-D demnds. he objective of this study is to identify set of sensor loctions tht optimize the coverge of O-D demnd flows of the rod network nd mximize the informtion gins through observtion dt over the network, while minimizing the uncertinty in the estimted O-D demnd mtrix. his pper is composed of six sections. he second section provides review of erly reserch on the sensor loction problem. he third section presents frmework for pproching the sensor loction problem nd discusses models tht cn be used for the cses of unlimited nd limited numbers of sensors. he fourth section includes n nlysis tht illustrtes the informtion gins nd trde-offs ssocited with vrious sensor loction schemes. he fifth section exmines the results produced using the proposed models. he finl section concludes the pper nd delinetes some res of future work. BACKGROUND he growing need of gencies to obtin rel-time informtion on the trffic stte of key fcilities in the systems they mnge is driving interest in cost-effective deployment of sensor technologies cross the networks they mnge. his hs led to greter interest in the sensor loction problem. Understnding the trde-offs between sensor investments nd informtion gin is criticl to the gencies decision mking in this regrd. A number of reserchers hve ddressed limited versions of the sensor loction problem, primrily in the context of O-D mtrix estimtion using link counts from rod sensor sttions. he pst two decdes hve seen development nd ppliction of severl pproches for the O-D mtrix estimtion problem. In generl, these pproches fll into two ctegories: trffic-ssignment-bsed pproches nd sttisticl inference pproches. he first ctegory includes informtion minimiztion (entropy mximiztion) models. Vn Zuylen nd Willumsen () developed two models bsed on informtion minimiztion nd entropy mximiztion principles to estimte n O-D mtrix from trffic counts tht seeks to reproduce the observed link flows. Fisk () combined Vn Zuylen nd Willumsen s () mximum entropy model with user-equilibrium

2 rnsporttion Reserch Record 9 model into single mthemticl problem in bilevel progrmming formultion. Recognizing tht the number of O-D pirs (unknown vribles) in the O-D estimtion problem is normlly greter thn the link trffic sttions, it hs become common prctice to integrte the priori O-D mtrix with the link counts to identify unique estimted O-D mtrix. he second ctegory includes mximum likelihood (ML) pproches, generlized lest-squres (GLS) pproches, nd Byesin inference pproch. Spiess (4) ssumed the O-D demnd to be reliztion from independent Poisson distributions with unknown mens. A ML model ws formulted to estimte these mens to reproduce the estimted link flows consistently with the observed flows. Cscett () proposed GLS estimtor combining with trffic counts vi n ssignment model. Bell () presented n lgorithm for the constrined GLS problem nd estblished its convergence. Mher (7) ssumed tht the prior O-D mtrix nd the observed link counts follow multivrite norml distributions nd proposed Byesin sttisticl pproch to updte the prior O-D mtrix. More recently, new sources of informtion produced by emerging technologies, such s utomtic vehicle identifiction, hve been used to estimte the O-D mtrix with point sensor dt (8 ). Awre of the inherent connection between the O-D estimtion problem nd link count observtions, severl reserchers hve pproched the sensor loction problem s n O-D covering problem. Lm nd Lo () proposed trffic flow volume nd O-D coverge criteri to determine priorities for locting sensors. By using concept of mximum possible reltive error (MPRE) to bound the ctul reltive error, Yng et l. () formulted simple qudrtic progrmming problem nd showed tht, if n O-D pir is not covered by sensor, the MPRE is infinite. On the bsis of the MPRE, Yng nd Zhou () proposed four bsic rules for the sensor loction problem: Rule. O-D covering rule: A certin portion of trips between ny O-D pir should be observed. Rule. Mximl flow frction rule: For prticulr O-D pir, link with the mximl frction of tht O-D flow should be selected. Rule. Mximl flow-intercepting rule: Under certin number of sensors constrint, the mximl number of O-D pirs should be observed. Rule 4. Link-independence rule: he resultnt trffic counts on the selected links should not be linerly dependent. Yim nd Lm (4) evluted the mximl net O-D cpture rule nd the mximl totl O-D cpture rule on lrge-scle network. Binco et l. () proposed n itertive two-stge procedure tht focuses on mximizing coverge in terms of geogrphic connectivity nd size of the O-D demnd popultion. Chootinn et l. () formulted biobjective model for locting trffic counting sttions for the purpose of O-D mtrix estimtion. hey considered Yng nd Zhou s () mximl covering rule while minimizing the number of sensors s two conflicting criteri nd proposed multiobjective method to obtin good compromise solution. Ehlert et l. (7) extended Yng nd Zhou s () work, tking the existing sensors into ccount nd thereby seeking second-best solution. Yng et l. (8) extended their work to the screenline-bsed sensor loction problem nd formulted n integer liner progrmming model. Prvinvongvuth et l. (9) proposed methodology for selecting the most preferred pln from the set of Preto optiml solutions obtined from solving multiobjective utomtic vehicle identifiction reder loction problem constrined by the resource limittion s well s the O-D flow coverge. he previously mentioned pproches were ll proposed or implemented under the ssumption of error-free mesurements, where the objec- tive is to mximize O-D coverge. None of these studies considered reducing the uncertinty in the O-D mtrix estimtes. Eisenmn et l. () proposed conceptul Klmn-filtering-bsed frmework to mximize the informtion gin nd minimize the error of the estimted O-D demnd mtrix to find sensor loctions. hey used simultion-bsed pproch to evlute the vlue of vrious sensor loction schemes for rel-time network trffic estimtion nd prediction pplictions in lrge-scle network. Zhou nd List () focused on locting limited number of trffic-counting sttions nd utomtic vehicle identifiction reders in network to mximize expected informtion gin for the subsequent O-D demnd estimtion problem solution. he pproch presented here seeks to identify set of sensor loctions tht optimize the coverge of O-D demnd flows of the rod network nd mximize the informtion gins through observtion dt over the network, while minimizing the uncertinty in the estimted O-D demnd mtrix. It stnds prt from most pproches in the literture in tht it explicitly considers time-vrying O-D demnd. FRAMEWORK his section presents methodologic pproches to two vrints of so-clled sensor loction problem. he first methodology is focused on solving the sensor loction problem with n unlimited number of sensors (unconstrined). he second methodology is focused on solving the sensor loction problem with limited number of sensors (constrined). Unlimited Network Sensors Yng nd Zhou () formulted binry integer progrm to determine the minimum number of sensor loctions required to stisfy n O-D covering rule for rod network with given prior O-D mtrix nd pth selection. minimize subject to δwz w W Α z A z =, A where z = if sensor is locted on link nd is otherwise, nd δ w = if some trips between O-D pir w pss over link A nd otherwise. It cn be shown tht the resultnt sensor loction solution stisfies the O-D covering rule nd tht selected links will be independent. A lrge network contining mny O-D zones nd significnt number of links my be difficult to solve with this formultion. A heuristic used to solve the proposed formultion might find only set of fesible or suboptiml solutions insted of the optiml set. his is due to the trde-off between computtion time nd solution qulity. In ddition, Yng nd Zhou s model is bsed on sttic trffic ssignment nd considers n O-D pir covered once sensor is locted on single link of the pths between tht prticulr O-D pir. In relity, the pth set between O-D pirs evolves with time of dy. hus, Yng nd Zhou s O-D covering model does not gurntee result in which ll O-D pirs re covered t ll times throughout the dy.

3 Fei, Mhmssni, nd Eisenmn O-D Covering Problem with ime-vrying Network Flows o ccount for sensor loction problems on lrge-scle networks with time-vrying flows (e.g., determined with DA methodology), method is proposed tht considers time-vrying pth determinnt. his model will result in set of sensor loctions on the links long the pths covering subset of O-D pirs tht experience O-D demnd flows in excess of minimum number of trips, ζ, where ζ is threshold termed s the degree tht defines the relevnt O-D pirs in ny time intervl. he following binry integer progrm formultion of the sensor loction problem (SLP) is presented to provide coverge of the O-D pirs with flow beyond preset relevnt degree ζ. SLP- where z = if sensor is locted on link during deprture time nd otherwise, nd δ w = if some trips with deprture time between O-D pir w trverse link A nd otherwise. is the plnning horizon for sensor dt collection. Algorithm Step. Run DA simultion softwre [Dynsmrt-X () in this study] given prior O-D demnd mtrix to get δ w, A, w W,, =. ζ = ζ minimize subject to δwz w W, with dw ζ, Α z =,, A δ w z A = ssignment d from DA, A, w W, w Step. If <, filter out the O-D pirs with flow less thn ζ. Run brnch-nd-bound (BnB) method to solve the binry integer model to obtin the solution pth set z of SLP-. Otherwise, if, Z = {z }, stop. Step. Set =+, ζ to stisfy the O-D coverge percentge in time intervl ; go to Step. o illustrte the proposed model, Figure shows the sensor loctions for two networks: Fort Worth, exs, with 47 sensors tht cover O-D pirs [ trffic nlysis zones (AZs)], including 8 nodes nd 44 links, nd Irvine, Cliforni, with 8 sensors tht cover, O-D pirs ( AZs), including nodes nd links. he priori relevnt degree ζ = under the DA. he time period of interest is the morning pek from : to 8:.m. Figure presents the solution results for the sttic model proposed by Yng et l. (8). he sme networks using sttic informtion result in hving sensors nd 44 sensors, respectively. he results of the dynmic model show tht, to cover ech O-D pir in the network cross time, more sensors re needed thn those obtined by solving the sensor loction problem bsed on sttic trffic ssignment. Figure shows the minimum number of required sensor loctions for ech deprture time intervl over the nlysis horizon. Figure shows the sensor loctions on two rel trffic networks covering network O-D pirs in different deprture time intervls from : to 8:.m. Although the sensor loctions found by the lgorithm for SLP- cn cover ll the O-D pirs cross time, there might exist more thn one sensor covering the sme O-D pir becuse the proposed problem hs two dimensions: temporl nd sptil. Insted of considering the sensor loctions for ech time period seprtely, the SLP- model integrtes the constrint conditions during different deprture intervls into one constrint set nd solves the binry integer model using the BnB lgorithm once the simultion ssignment is completed nd the resulting routing policies become vilble in ll the deprture time intervls. Becuse it is ssumed in this section tht the number of sensors is unlimited, only solutions to model SLP- re nlyzed. () FIGURE Sensor loctions by DA in () Fort Worth network nd Irvine network.

4 4 rnsporttion Reserch Record 9 () FIGURE network. Sensor loctions by sttic model in () Fort Worth network nd Irvine Model SLP- is formulted s follows. SLP- subject to minimize δ wz w W, with dw ζ, Α z =,, A δ w z A = ssignment d from DA, A, w W, w where z = if sensor is locted on link during deprture time nd is otherwise, nd δ w = if some trips with deprture time between O-D pir w pss over link A, nd otherwise. is the plnning horizon for sensor dt collection. Sensitivity Anlysis on the Number of Sensors nd Percentge O-D Coverge A sensitivity nlysis ws conducted to explore the reltionship between the number of sensors nd level of O-D coverge in network. he purpose of this nlysis is to explore the mrginl vlue, in terms of percentge coverge, of dding sensors to the network. he nlysis lso provides pltform to investigte the effect of sensor loction on the O-D demnd coverge rte. By setting n pproprite ζ in ech deprture time intervl nd solving the corresponding SLP- model, Figure 4 shows different sensor numbers required to provide different levels of O-D cover- 4 Sensor Number Sensors on Fort Worth Network Sensors on Irvine Network Deprture ime FIGURE Number of sensors for ech time period.

5 Fei, Mhmssni, nd Eisenmn 8 Sensor Number on Fort Worth Network Sensor Number on Irvine Network 4 Sensor Number 8 4 FIGURE 4 % % % 4% % % 7% 8% 9% O-D Coverge Percentge Number of sensors needed to cover given percentge of O-D demnd. ge in the Fort Worth nd Irvine networks under the dynmic model. As expected, to cover more O-D pirs, more sensors hve to be instlled in the network. hese results lso indicte tht obtining greter thn % O-D coverge for either network requires significnt increse in the number of sensors. he results lso indicte tht firly low number of judiciously plced sensors cn provide substntil coverge. Figure shows sensors covering % of the O-D demnd flow on the Fort Worth testbed network nd sensors covering % of the O-D demnd flow on the Irvine testbed network. he sensors re mostly distributed long freewys, where the links hve higher flows thn on rteril streets. he results revel tht, if resources re constrined, deploying sensors long the freewy would mke sense in terms of mximizing the O-D demnd coverge. Limited Network Sensors (Constrined) his section exmines the sensor loction O-D coverge problem when finite number of sensors re deployed. he solutions for the unlimited (unconstrined) sensor cse tht re presented in Figures nd show lrge number of sensors locted on rterils nd much smller number on the freewys. he sensitivity nlysis in the preceding subsection showed tht few wellplced sensors on freewys could provide high percentge of the O-D coverge. Becuse freewys tend to hve higher link flows thn rterils, it mkes sense tht lrger percentge of O-D pirs could be covered by smller number of links. If the number of sensors tht cn be plced in the network is limited, the gol becomes one of both covering the O-D pirs nd intercepting s mny O-D flows in the network s possible. () FIGURE Prtil O-D demnd coverge on () Fort Worth nd Irvine networks.

6 rnsporttion Reserch Record 9 Nottions nd Problem Definition N = set of zones, consisting of n zones; I = set of origin zones, consisting of n zones; J = set of destintion zones, consisting of n zones; A = set of links, consisting of links; W = set of O-D pirs; N od = number of O-D pirs, N od = I J ; L = set of links with mesurements; = subscript for link in network, A; w = subscript for O-D pir in network, w W; i = subscript for origin zone in network, i I; j = subscript for destintion zone in network, j J; C = vector of mesurements (L ); H = mpping mtrix (L N od ), mpping demnd flow to link counts; D = demnd vector, consisting of N od entries d(i, j) D; ( ) = priori estimted demnd vector, consisting of N od entries dˆ (i,j) ( ) ( ); (+) = posteriori estimted demnd vector, dˆ (i,j) (+) (+); D (+) = posteriori estimted demnd error mtrix; D ( ) = priori estimted demnd error mtrix; P ( ) = priori vrince covrince mtrix of demnd mtrix; P (+) = posteriori vrince covrince mtrix of the demnd mtrix; nd = vector of rndom noise quntities N(, R) corrupting the mesurements. Generlized Lest-Squres O-D Demnd Estimtor Assume the reltionship between the unknown O-D demnd flow nd mesurements cn be expressed s liner combintion with rndom, dditive mesurement error. he mesurement process is modeled s follows: C = HD+ () he objective is to minimize the devition between observed link flows nd estimted link flows, ccording to the GLS estimtion, J = C H R C H With J ( ) ( ) = the resultnt GLS estimtor is = ˆD H R H H R C Assuming the mesurement errors re uncorrelted (e.g., R = I), it is esy to prove tht = 4 ˆ D H H H C For ny mtrix H, the rnk(h) = rnk(h H) = rnk(hh ), so tht if mtrix H is of full rnk, then the lest-squres solution ( ) is unique nd minimizes the sum-of-squred residuls. In other words, () () the link counts on ech observed link need to be linerly independent of ech other. According to Aitken s theorem (), the GLS estimtor ( ) is the minimum vrince liner unbised estimtor in the generlized regression model. With time-vrying weighting mtrices K nd K, the recursive form cn be expressed s = ( ) + () + K KC becuse ( + )= D+ D ( + ) ( )= D+ D ( ) () Substituting Equtions nd into Eqution gives D ( + )= K ( D+ D ( ) )+ K( HD+ ) D = ( K + KH I) D+ K D ( )+ K (7) ( ) or (+) is unbised. ht is E( ( + ))= E( D+ D ( + ))= D+ E D + E( ( ) )= E( D+ D( ) )= D+ E D By definition, E( ) =, Equtions 7 nd 8 give K = ( I KH) (9) Substituting Eqution 9 into Eqution 7 D ( + )=( I KH) D ( )+ K () By definition, the posterior error vrince covrince mtrix P E D E D D E D ( + )= ( ( + )) ( ( + ))( ( + )) ( ( + )) = E D D ( ( + ) ( + ) ) () Substituting Eqution into Eqution, = (( ) ( ) + )(( ) ( ) + ) P I KH D K I KH D K + o minimize P (+), the first-order optimiztion condition needs to be stisfied, P + K = I KH Pˆ I KH KRK () hus, the optiml weight mtrix, which is referred to s the Klmn gin mtrix is P H HP H R ( ) D ˆ ( ) + = = I KH P H KR () ( ( ) + ) + D ( ) ( ) E( D ( + ))= E( D ( ) )= (8) (4)

7 Fei, Mhmssni, nd Eisenmn 7 Substituting Eqution 4 into Eqution, the miniml updted vrince covrince mtrix is Pˆ( + )= I KH Pˆ D A simple form of Klmn gin mtrix cn be expressed s P + H R Eqution cn be lso expressed s P + P H R H = ( ) + More detiled derivtions nd nlysis of the optiml estimtion nd filtering reltionship cn be found elsewhere (4). If it is ssumed tht the mesurement error is independent, then R is digonl mtrix. So, Eqution cn be written s follows: K P H + = R he mtrix H is mpping mtrix, mpping the O-D demnd flow to the link counts; if it is ssumed to be n identity mtrix, one gets K P + = R ( ) () D he Klmn gin mtrix in the sensor loction problem cn be interpreted s the summtion of informtion gin contributed by ech O-D pir pssing over tht observtion link. It is proportionl to the uncertinty in the estimte nd inversely proportionl to the mesurement noise (4). he reltionship of Eqution declres tht, given the priori vrince covrince estimted demnd error, the lrger the gin link hs, the more informtion it cn collect to correct the estimted error. Compred with the posteriori vrince covrince mtrix, the gin mtrix is much more sensitive to the mesurement errors tht influence the estimted results. On the bsis of the preceding nlysis, the objective in the constrined sensor coverge model is to find the set of links tht cn mximize the totl informtion gins, constrined by the link independence nd resource constrints. o mintin the unbisedness of the O-D flow estimtor, link independence should be stisfied. o find the mpping mtrix, H or the so-clled time-dependent link flow proportion mtrix, the priori estimted trffic demnd should be ssigned to the network ccording to some ssignment rules (). In this study, the drivers in the network were ssumed to tke the pths consistent with those generted under the dynmic user equilibrium ssignment. Dynmic user equilibrium nd system optimiztion procedures re importnt components of Dynsmrt-P (), which is used to solve the dynmic user equilibrium ssignment problem nd find corresponding simulted time-dependent link flows nd mpping mtrix H in this study. Sensor Loction Model nd Algorithm () (7) (8) (9) If L Lˆ, where Lˆ is n optiml solution to SLP-, the sensor loctions could cover the relevnt subset of O-D pirs. he problem cn be formulted s SLP-. A SLP- mximize Kz SLP- subject to z L w W SLP-b A If L < Lˆ, only prtil relevnt O-D pirs cn be covered; thus, the problem is formulted s SLP-. SLP- mximize Kz subject to A Considering the mgnitude of the O-D demnd flow reltive to the vrinces, the model cn be formulted s follows: If L Lˆ, SLP- mximize E D + trce P ˆ ( + ) subject to z L w W A δw A If L < Lˆ ( ) SLP- mximize E D + trce P ˆ ( + ) subject to A z L w W H H P + P = + R ( + )= D+ D ( + ) z =,, A z w W, H H P + = P ( )+ R ( + )= D+ D ( + ) z =,, A δ w z L w W A P H K =, A w W HP H r + z =,, A w A P H K =, A SLP-ssignment d w W HP H r ( ) + δ w = w from DA, SLP-e A, w W, z =,, A δ z w W, SLP- ( ) = ssignment dw from DA, A, w W, ( SLP- f )

8 8 rnsporttion Reserch Record 9 he proposed models re computtionlly intensive. he mjor difficulty is ssocited with clculting the Klmn gin mtrix, becuse mtrix inversion occurs t ech time intervl. he computtionl intensity is especilly noticeble in lrge-scle network. he sequentil lgorithm by Chui nd Chen () ws designed to void direct computtion of the inversion of the mtrix, HP ( )H + R by ssuming independence of the link mesurement errors. Figure illustrtes the sequentil lgorithm for the sensor loction problem [similr to the sequentil lgorithm of Chui nd Chen ()]. BnB is common strtegy used to solve integer progrms. BnB lgorithms hve been developed in vriety of res. Becuse of its dptbility, BnB hs been used in vriety of serch lgorithms, such s best-first serch nd depth-first serch, s well s others. o ccommodte solving the sensor loction problem in lrgescle network, Algorithm integrtes the BnB lgorithm with the sequentil lgorithm. hrough the use of n efficient serch lgorithm, Algorithm cn be used to solve the SLP in lrge-scle network. However, s the size of the network grows, the efficiency of nlyzing different sensor loction strtegies will be reduced. Algorithm (Sequentil Algorithm + BnB Algorithm) Step (initiliztion). Running DA simultion softwre () given prior O-D demnd mtrix to get link flow proportion H under user equilibrium ssignment, A. Given P ( ) = Pˆ, which cn be from historicl O-D dt sttistics or trffic-plnning gents. Step (gin clcultion). Using the sequentil lgorithm clculting the link gins cross the network. Step (BnB lgorithm). With the clculted link gins cross the network, solving the proposed model s binry integer model using the BnB lgorithm. An intuitive notion to solve the proposed model is selecting L links every time from the network G(V, A), clculting the totl link gins ech time nd selecting the loctions hving the lrgest link P D ( ) P D ( )H K = H P D ( )H + r K D(+) = D( ) + K(C HD( )) D(+) P D (+) P D (+) = (I KH )P D ( ) FIGURE Sensor loction problem sequentil lgorithm flowchrt. gins. However, the number of combintions of L links from totl of A links is A A ( L) =! L! A L! nd the complexity of the computtion will become nonpolynomil. Bsed on this notion, Heuristic is developed to find the best fesible solution. Heuristic Step (initiliztion). Run DA simultion softwre () given priori O-D demnd mtrix to obtin link flow proportions H under user equilibrium ssignment, A. Clculte nd sort the informtion gin on ech link of the network using the previously mentioned sequentil lgorithm. Given P ( ) = Pˆ, which could be obtined from historicl O-D dt sttistics or trffic-plnning gencies. Set LS =φ, LS = A LS, S od =φ, S od = W, where S od is set of O-D pirs covered by sensors in set LS. N sensor =, where N sensor is the totl number of sensors in set LS. Step (stopping criterion). If number of links (N sensor ) in set LS, N sensor = L, where L is given sensor number or S od =φor the preset computtion time is reched, stop. Output the best fesible solution in set LS, Otherwise, go to Step. Step (mximl gin selection). If LS =φ, select link l i, so tht gin i gin j, j, i j, i, j LS. Insert the selected link l i into link set LS nd let N sensor = N sensor +, S od = {N li }, S od = W S od, where N li is the set of O-D pirs newly covered by the selected link l i, delete the selected link l i from set LS, go to Step. Otherwise, if LS φ, go to Step. Step (link selection). For ech link in link set LS, the O-D pirs covered by the link nd in set S od re mrked by + nd the O-D pirs covered by l i nd in set S od re mrked by. n (i) is the number of + vlues nd n (i) is the number of vlues. Select link l i so tht n (i) n ( j), j, i, j LS. If there exist links l i, l j so tht n (i) >, n ( j) >, nd n (i) > n ( j) =, i, j LS, select link l j. Else if n (i) >, n (j) > nd n (i) = n ( j) =, i, j LS. If link gin on link i > link gin on link j, select link l i. Else if link gin on link i link gin on link j, select link l j. Else if there exist links l i, l j so tht n (i) > n ( j) >, i, j LS, select link l i. Else if there exist links l i, l j so tht n l (i) = n l ( j) >, i, j LS. If n (i) > n ( j), select link l j. Else if n (i) < n ( j), select link l i. Else if n (i) = n ( j), select the link with less mesurement error. Else if n (i) = n ( j) =, select the link with lrger link informtion gin. End if Moved the O-D pirs covered by the selected link from set S od to set S od. N sensor = N sensor +. Go to Step. As mentioned in lst section, the link counts on ech observed link need to be linerly independent of ech other. he bsic ide in the proposed heuristic is to select links with the lrgest informtion gin while keeping the rnk of link proportion mtrix H full. hus, the complexity of the proposed heuristic is determined by the complexity of finding mximl O-D coverge given the simulted link flow proportions, H, A from Dynsmrt-P nd

9 Fei, Mhmssni, nd Eisenmn 9 sorted link informtion gins. In generl, this problem cn be clssified s mximum rnk mtrix completion problem, which ssigns vlues from some set to mximize the rnk of the mtrix. It cn be shown tht the computtionl complexity of the proposed lgorithm is O(n ), where n is the number of links in the network. NUMERICAL EXAMPLES A series of exmples bsed on six-node network is used to demonstrte the proposed methodology. o fcilitte the bility to compre the results of this reserch with the recent results of Zhou nd List (), the sme exmple network ws used. he first exmple is single-point sensor loction, ccording to the setup in Figure 7. O-D Pir is from Node to Node nd O-D Pir is from Node to Node ; O-D Pir hs two routes; 7% of the flow trvels long pth { 4 } nd the remining % of the flow trvels long pth { 4 }. Both O-D pirs hve flow volume of units. Assume P ( )= 4 mening tht O-D Pir hs lrger priori vrince thn O-D Pir. he stndrd devition of the mesurement error for sensor is ssumed to be % of the corresponding true flow volume. he sensor in Figure 7b covers O-D Pir with lrger vrince producing lrger gin thn tht in Figure 7c. Becuse the sensor in Figure 7d covers both O-D pirs nd intercepts the lrgest O-D flows in these scenrios, it collects the lrgest gin through the observtion counts even though the sensor in Figure 7d brings lrger error thn tht in Figure 7b nd c. If the error in Figure 7d is reduced to, similr to tht in Figure 7b nd c, it hs R =, [.7.7], nd gin = 8.7, producing lrger informtion gin. Figure 8 presents exmples of single sensor loctions with route choice. Figure 8 presents n error-free link proportion estimte nd the mesurement error proportionl to the link flow scenrio. he gin in tht scenrio is.49, which is greter thn ll the scenrios in Figure. his indictes tht the mesurement error could result in reduction of the link informtion gin. Figure 8b considers the link proportion estimtion error, which reduces the informtion gins. Although the sensor in Figure 8c covers both O-D pirs, it still cnnot produce the lrgest informtion gin becuse of the lrgest mesurement error in the three scenrios. Even when the mesurement error is reduced to, the gin mtrix is [.8.47] nd gin =.9. Figure 9 presents exmples of two sensor loctions. Figure 9 covers O-D Pir, Figure 9b covers O-D Pir, Figures 9c nd d cover both O-D pirs, nd Figure 9e covers O-D Pir but the two sensors hve mesurement error correltion between them. As expected, Figure 9c collected lrger gins thn the other scenrios becuse it covers both O-D pirs. Although Figure 9d covers both O-D pirs s well, the informtion gin is smller thn in Figure 9c P D = () 4 R = I H = [ ].8 Gin = = K =.8 4 (c) R = I H = [ ]. Gin = = K =. 4 (d) R = Gin = H = [ = ].7.7 K =.84 Zone Loop detector FIGURE 7 Single-point sensor loctions: () bse cse, one sensor for O-D pir ( ), (c) one sensor for O-D pir ( ), nd (d ) one sensor for both O-D pirs.

10 rnsporttion Reserch Record 9 4 () [.49 ] R =.7 Gin = w= H = [.7 ] R = (.7 +.) [.949 ] Gin = K w= H = [.7 ] = (c) R = (.) H = [. ] [.94.79] Gin = w= K =.7 Zone Loop detector FIGURE 8 Single-point sensor loctions with route choice: () ssignment error-free with link proportion.7, stndrd of link proportion estimtion errors (from trffic ssignment) =., nd (c) ssignment error-free with link proportion of.. becuse of the liner dependence of the two observtions. Compring Figure 9 nd e, the correltion of mesurement errors provided some reduction in the informtion gin. Figure presents exmples of three sensor loctions. Figure e collected the lest informtion gin becuse the three sensors cover only one O-D pir, wheres the other scenrios cover both O-D pirs. Figure produces the best gin becuse of the link independent of the sensor dt. More sensors do not lwys result in more informtion gin. Figure 9c with two sensors (gin =.) hs lrger gins thn most scenrios in Figure. Even if the two cses hve the sme mesurement errors, in Figure e one O-D pir hs , gin =.9, which is less thn tht in Figure 9c. his exmple is lso used to demonstrte the procedure of Algorithm for finding the best fesible solution of sensor loctions. Step (initiliztion). Link. K Link. K R = H =[ ] = [. 7. 7] gin = K =. 84 w= R = 7. H =[ 7. ] = [. 49 ] gin = K =. 49 w= R = (. ) H = [. ] Link. K Link 4. K = [ ] gin = K = w= R = H =[ ] gin = 8. w= R = H =[ ] Link. K gin = K =. = [. ] P D = ˆ w= = [ 8. ] 7. LS φ, LS 4,,,,, Sod φ, S od,, Nsensor =,H = NULL = ={ } = ={ } 4. One sensor in the network. First, select Link, which hs mximl gin over ll five links, H = [.7 ], rnk(h) = + =, so LS = {}, LS = {,,4,}, S od = {}, N sensor =, then go to Step, the number of selected links equls given sensor number, stop. Compring the sensor on Link tht covers both O-D pirs, the sensor on Link covers only O-D Pir. However, the root-men-squred error (RMSE) of the sensor on Link is.4, which is less thn.78, the RMSE of the sensor on Link. hus, the sensor on Link cn provide more relible O-D mtrix thn the estimted y sensor dt on Link.

11 Fei, Mhmssni, nd Eisenmn 4 () R = I H = Gin = = K = R = I H =.. Gin = = K =. 4 (c) R = I H =.8. Gin = = K =. 4 (d) R = I H = Gin = = K =.99 R =.. H = (e) Gin = = K =.848 Zone Loop detector FIGURE 9 wo-point sensor loctions: () two uncorrelted sensors for O-D pir ( ), two uncorrelted sensors for O-D pir ( ), (c) two uncorrelted sensors for both O-D pirs, (d ) two uncorrelted sensors for both O-D pirs, nd (e) two prtilly correlted sensors for O-D pir ( ).

12 rnsporttion Reserch Record 9 4 () R =.I Gin = = H = K = R =.I H = Gin = = K =.7 4 (c) R =.I Gin =.7.9 = H =. -.4 K = (d) R =.I Gin = = H = K = R =.I.9 H =.9.9 (e) Gin = = K =.8889 Zone Loop detector FIGURE hree-point sensor loctions: () three uncorrelted sensors for both O-D pirs, three uncorrelted sensors for both O-D pirs, (c) three uncorrelted sensors for both O-D pirs, (d ) three uncorrelted sensors for both O-D pirs, nd (e) three uncorrelted sensors for O-D pir ( ).

13 Fei, Mhmssni, nd Eisenmn. wo sensors in the network. After finding the first sensor on Link, go to Step. n () = n () = n () =, n () = n () =, n () =. Link is selected, LS = {,}, LS = {,,4}, S od = {,}, N sensor =. Go to Step, stop. he totl gin of sensors on Links nd is.49, wheres the totl gin of sensors on Links nd is.9, nd the totl gin of sensors on Links nd is.. he Irvine network (Figure ) ws used to demonstrte the proposed model. he simultion experiments were implemented on n Intel Xeon centrl processing unit.-ghz 4-bit mchine with 8G memory. he historicl -h time-dependent O-D volumes were integrted into one demnd tble s n priori men estimte becuse of the limited sensor number constrint. It is ssumed tht the stndrd devition of the priori demnd vrince is % of the corresponding demnd volume in the historicl demnd tble. he stndrd devition of the mesurement error is ssumed constnt. Dynsmrt-P () ws gin used to ssign the O-D volumes onto the network nd represent trffic flow evolution. Figure shows the network covered by sensors. he two O-D pirs crrying the lrgest O-D volume re (, 4) nd (47, ), ccounting for bout % of the totl historic O-D demnd. Sensor nd Sensor covered these two O-D pirs. In fct, s illustrted in the previous smll network, the independent sensors usully gve high link informtion gins. It is not sufficient to minimize the vrince of the estimted O-D demnd for one prticulr O-D pir while leving other O-D pirs uncovered. he proposed method tries to find the sensor loctions tht grner the lrgest possible link informtion gins in conjunction with mximizing network coverge. he sensors shown in Figure covered bout % of the totl O-D trips nd the sensors shown in Figure b covered bout 4% of the totl O-D trips. As discussed in the first section, only if the link mpping mtrix H hs full rnk is the O-D demnd estimtor ( ) the best liner unbised estimtor. he gin mtrix ws derived bsed on the best liner unbised estimtor, which explined why the independent sensor dt lwys produced the lrgest gins. he following observtions re mde from those exmple results to mximize informtion gins:. he sensors need to be locted on the links tht cn intercept the most O-D flows.. he sensor observtion dt should be linerly independent.. More sensors do not necessrily men lrger informtion gins. 4. he lower the mesurement error, the greter the gin the system my ttin. ANALYSIS MEASURES o ssess the impct of different sensor loction strtegies in conjunction with the O-D demnd estimtor error reduction, the RMSE of the O-D demnd will be clculted to check the qulity of the estimted O-D mtrix. he RMSE is simply the squre root of the men squred error (MSE). Proposition he proposed models lwys produce the miniml MSE cross ll other O-D estimtors. Proof In sttistics, the MSE is defined (7) s follows: = + ( )( ) MSE θθˆ vr θ ˆ θ ˆ θ θ ˆ E θ As mentioned previously, the GLS O-D demnd estimtor is unbised; thus, its MSE mtrix is its covrince mtrix. Recll Eqution 7 the MSE of the O-D estimtor is P (+) = P ( ) + H R H. Becuse P ( ) is the priori vrince covrince mtrix of the demnd mtrix nd the objectives of both of the SLP- nd SLP- Sensor Sensor () FIGURE Sensor loctions by link gin selection in Irvine network: () sensors nd sensors.

14 4 rnsporttion Reserch Record 9 models re indirectly minimizing P (+), the MSE bsed on the proposed models is thus the miniml sttistics inference cross ll other estimtors. his completes the proof. Under the time-dependent condition, the time dimension needs lso to be considered; the clcultion is s follows: RMSE = ( d dˆ w w) w W where dw = ground truth O-D trips of O-D pir w t deprture time, dˆ w = estimted O-D trip of O-D pir w t deprture time, W = set of O-D pirs, L = totl number of O-D pirs in the set, nd = number of deprture time intervls. In given sensor loction strtegy, the RMSE is clculted cross ll the O-D pirs nd cross ll the time intervls. ble shows the RMSE of the previously mentioned numericl exmples. CONCLUDING REMARKS his pper presents models tht use the Klmn filtering method to explore time-dependent mximl informtion gins cross ll the links in the network. he reserch proposes two types of sensor loction models to solve n O-D coverge problem nd mximl informtion gin driven problem. he focus is on solving the sensor loction problem s n O-D coverge problem under DA. A sensitivity nlysis is conducted to explore the reltionship between the number of sensors nd the level of O-D coverge in network. he gol is to produce qulity estimted O-D mtrix tht integrtes link observtion dt tht minimize the vrince of the O-D flow estimtor. his reserch constructed n unbised generlized lestsqures estimtor, using liner reltionship nd link flow proportions obtined from dynmic simultion-ssignment procedure (Dynsmrt-P). he models were developed to identify link sensor loctions tht produce mximl informtion gins nd mximl O-D pir relibility. In ddition, sequentil lgorithm ws developed to ABLE RMSE for Numericl Exmples Scenrio () (c) One sensor coverge without route choice RMSE One sensor coverge with route choice RMSE Scenrio () (c) (d) (e) wo sensor coverge RMSE hree sensor coverge RMSE solve the proposed models. Severl smll numericl exmples were used to demonstrte the proposed methodology. Finlly, the detector configurtion ws evluted on the bsis of RMSE to prove the efficiency of the proposed methodology. Recognizing the importnce of sensor loction nd its reltionship to the qulity of n estimted O-D mtrix, this pper estblished connection between the two criticl issues of trffic sensor loction nd estimtion error bsed on Klmn filtering. Ongoing reserch is exploring efficient lgorithms tht cn be used to solve the sensor loction problem for O-D estimtion in rel-time setting, using rel-time observtion dt. In ddition, n efficient frmework is being designed to embed the proposed methodology into simultion-bsed rel-time network trffic estimtion nd prediction system (Dynsmrt-X) tht relies on DA methodology to tke full dvntge of the dynmic sensor loctions to estimte nd predict O-D mtrices in lrge-scle networks. REFERENCES. Mhmssni, H. S., nd X. Zhou. rnsporttion System Intelligence: Performnce Mesurement nd Rel-ime rffic Estimtion nd Prediction in Dy-to-Dy Lerning Frmework. In Advnces in Control, Communiction Networks, nd rnsporttion Systems, in Honor of Prvin Vriy (E. Abed, ed.), Birkhäuser, Bsel, Switzerlnd,, chpt... Vn Zuylen, H. J., nd L. G. Willumsen. he Most Likely rip Mtrix Estimted from rffic Counts. rnsporttion Reserch Prt B, Vol. 4, 98, pp Fisk, C. S. On Combining Mximum Entropy rip Mtrix Estimtion with User Optiml Assignment. rnsporttion Reserch B, Vol., 988, pp Spiess, H. 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