ESTIMATING TRAVEL DEMAND BY OBSERVED FLOWS ON REAL SIZED NETWORK: METHODS AND STATISTICAL PERFORMANCES

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1 ESTIMATING TRAVEL DEMAND BY OBSERVED FLOWS ON REAL SIZED NETWORK: METHODS AND STATISTICAL PERFORMANCES Massmo D Gang Dept. of Archtecture, Plannng and Transportaton Infrastructures - Unversty of Baslcata C.da Maccha Romana POTENZA (EU Italy) E-Mal: dgang@unbas.t Mchele Ottomanell Dept. of Hghways and Transportaton Polytechnc Unversty of Bar Va Orabona, BARI (EU-Italy) E-Mal: m.ottomanell@polba.t 1 INTRODUCTION Travel demand estmaton represents one of the man topcs n transportaton systems plannng. In order to obtan accurate and correct estmates of current O-D travel demand and to forecast ts varaton, n lterature a great deal of methods have been proposed based on dfferent theoretcal frameworks. These methodologes can be dvded n two man groups: drect estmaton methods and model estmaton (ndrect estmaton methods). Both groups lead to travel demand estmates that are usually represented by means of the so-called Orgn-Destnaton matrces (or tables). Methodologes belongng to the frst group allow to estmate the current travel demand and are based on statstcal technques (usually regresson methods) to extend to unverse the data collected by samplng surveys. Model estmaton approach allows to estmate both current and future demand by usng a set of mathematcal models whose functonal structures have to be specfed and calbrated. The most used travel demand models are based on the random utlty theory. Both approaches need expensve and tme consumng ad hoc dsaggregate samplng surveys. Thus, n partcular traffc studes the avalable budget for surveys and and/or the requested computatonal effort could be a strong constran. To overtake these problems, great attenton has been pad to novel O-D matrces estmaton methods that use as nput nformaton cheap and easy to collect aggregate data. The most attractve of them are the traffc counts (TC) based methods that use the traffc flows observed on the transportaton network lnks to reduce and/or elmnate the dsaggregate data n the estmaton procedures. Informaton provded wth traffc counts can be used n drect or ndrect way. Drect methods employ traffc counts to estmate, update and/or adjust O-D travel matrces (target matrces). Recently ndrect methods have been proposed. They employ a set of TC to calbrate the

2 parameters of travel demand models to be used to estmate the O-D matrces (Cascetta and Russo, 1997; Cascetta, 2001; Ottomanell, 2001). In lterature the two approaches are always presented separately and sometmes wth respect to small-test networks. In ths paper ther statstcal performances and robustness are analysed and compared wth reference to a mddle szed urban area by usng real data. An heurstc drect method for calbratng O-D matrces s presented and mplemented. The results obtaned by the dfferent methods are compared and dscussed. 2 METHODOLOGY Let the road and transt networks be defned by means of a graph consttuted by a set of n nodes N and a set of l lnks L. We defne as centrods the nodes that are orgn and destnaton of trps, so that d s the average amount of trps between the par =(o,d) where o s the orgn of trp and d s the destnaton. We ndcate wth d the vector of the O-D travel demand (O-D matrx), made up of the elements d. Let f l be the traffc flow smulated by means of a traffc assgnment model on the generc lnk l and f the vector of the predcted lnk traffc flows f l ; then we have: f = H d (1) where H s the assgnment matrx whch s estmated by means of an assgnment models. Let us assume to know the true O-D travel demand d. Because of dfferent errors (due to modellng approxmatons, measurement errors, temporal varaton of traffc flows etc) the observed lnk flow fˆ l dffers from the smulated one f l by the so-called assgnment error ε l, that s: fˆ l = f l + ε l (2) By consderng both (1) and (2) we can wrte the relatonshp between the observed lnks flow vector fˆ and travel demand d: fˆ = Hd+ ε (3) where ε s a vector of random resduals. The varance var(ε l ) can be obtaned by means of expermental relatonshps. Equaton (3) wll be used n the followng to descrbe the drect and ndrect O-D traffc counts estmaton based methods. In ths work, two startng O-D matrces estmated by means of tradtonal methods have been consdered: ODD = obtaned by drect statstcal estmaton procedures ODM= estmated by a system of travel demand models These matrces have been calbrated wth reference to the network of Salerno (a mddle szed town of Southern Italy) by usng a set of TC observed on the lnks of the consdered networks (both car and transt networks) and dfferent methods.

3 Four dfferent method to obtan the startng O-D matrces have been consdered: ODDF = drect estmaton of the ODD matrces wth traffc counts ODMF= drect estmaton of the ODM matrces wth traffc counts ODMC= ndrect estmaton of the O-D matrces by calbratng the demand models coeffcents wth traffc counts ODMCF= drect estmaton of the matrces obtaned by the ODMC method 2.1 Drect estmaton of O-D matrx wth traffc counts: the multstep method The drect matrces calbraton methods gve an updated O-D matrce by combnng a startng O-D matrces (target) wth a set of TC. In ths work we propose to employ the GLS (Generalsed Least Square Estmator) proposed by Cascetta (1984) n an heurstc method we have called Multstep procedure. The multstep method apply repeatedly the GLS estmator to the startng O-D matrx and t searches for the matrx that smulates (by means of the assgnment model) the lnk traffc flows wth the lowest devaton wth respect to the observed ones The GLS Estmator Let dˆ be a startng estmate of the travel demand for the -th o-d par. Ths estmate, apart from the estmaton methods, dffers from the true O-D vector by the unknown term θ : dˆ =d +θ (4) Traffc counts can be effcently used to reduce these errors and to obtan statstcally better performances by means of the GLS estmator that s defned as: * T 1 ˆ T 1 d = arg mn D( d) = ( d dˆ) V ( d d) + (ˆ f Hd) W (ˆ f Hd) (5) d 0 where V and W are the varance and varance-covarance matrces of the random resduals vectors θ and ε respectvely. To reduce the complexty of the problem a smpler form of the estmator (5) can be wrtten as: 2 ( ) ( ) = + 2 d dˆ fˆ l h ld d arg d 0mn (6) var( θ l ε ) var( l ) An estmate d* obtaned by solvng problem (6) can be consdered as the travel demand vector that mnmze the sum of the square devatons (dstances) wth respect to both the startng estmate and the observed flows. Both the devatons are weghted by the respectve varance terms of the resdual terms. In ths way, the worse s the startng estmate d (resp. f l ) the less s ts weght, so that the var(ε ) (resp. var(f l )) s hgher. The varance terms can be evaluated by consderng the varaton coeffcents: var( ε) C v flow = ; E [] fˆ var( θ ) Cv d = (7) E dˆ []

4 2.2 Indrect estmaton of O-D matrx wth traffc counts: NGLS estmator The ndrect matrces calbraton methods gve an updated vector of the travel demand models coeffcents by combnng a startng estmate of the coeffcents wth a set of TC. Once the model coeffcents have been calbrated then an updated O-D matrx can be estmated. The updated model leads to mproved O-D matrces estmates (Ottomanell, 2001). In ths framework we assume a travel demand model as a relatonshp between the travel demand matrx (vector) and a vector of land-use attrbutes SE, a vector of transportaton system attrbutes T and a vector of unknown model coeffcents vector β, that s: d = d(se, T, β) (8) or, smply, d = d(β) for a gven study area. Usually, estmates βˆ of the vector β can be acheved by usng the Maxmum Lkelhood (ML) approach by employng dsaggregate data collected by samplng surveys. Snce βˆ s an estmate of the true vector β for each element we can assume that: ˆ β = β + δ (9) where δ s a random resdual error. Thus, gven the stochastc equatons (3) and (9) we have: βˆ = β + δ ( β) ε f ˆ = Hd + As n the GLS O-D matrx estmator, we can mprove the model coeffcents calbraton by defnng and solvng the followng optmsaton problem: * T 1 T 1 β = arg mn ( β βˆ) D ( β βˆ) + (ˆ f Hd( β)) W (ˆ f Hd( β)) (10) β S where S s the feasblty set for the soluton; D and W are the varance-covarance matrces of the terms δ and ε respectvely. For practcal applcatons equaton (10) can be smplfed by neglectng the covarance terms of D and W and a smpler formulaton can be obtaned: 2 ( ) [ ] = = + 2 β k βˆ fˆ l h ld ( β ) k β arg mn Z( β) (11) k β S var( δ k ) l var( ε l) Statstcally, f the startng estmate βˆ s obtaned by samplng surveys then the vector β could be consdered as a Generalzed Least Square Estmator (NLGLS), conversely t could be assumed as a Bayesan estmator (Cascetta e Russo, 1997). The varance of the terms δ (weght of the coeffcent vectors dfference) n the eq. (11) can be evaluated as: var( δ) C v beta = (12) E [] βˆ

5 As shown by Ottomanell (2001), under correct assumptons on the weghts of the resdual terms, the NLGLS leads to good mprovements n lnk flows estmaton even f the startng estmate of the coeffcents s qute naccurate. A computer program has been developed to solve the problem (11). The nputs are the demand model, a startng estmate of β and a set of TC. The outputs are the soluton of the problem (11) and the O-D matrces for each transportaton mode consdered n the model. 3 EXPERIMENTAL ANALYSIS ON A REAL SIZED NETWORK The consdered O-D matrx has dmenson 63x63 (orgns x destnatons). The road network graph s consttuted by 526 nodes and 1147 lnks, whle the transt network graph has 3072 nodes and 5355 lnks (hyperpaths approach). The demand model was a classcal four-steps model based on random utlty theory wth 45 coeffcents to be calbrated. Comparsons have been performed by usng mean average error (MAE) and mean square error (MSE) statstcs. Frst, we compared the performances of the tradtonal demand estmaton methods. The statstcal performances of the SUE traffc assgnment by usng the O-D matrces wthout TC calbraton were: Table 1 Statstcal performances of the lnk flow estmaton by usng the startng matrces Car Network Transt Network ODM ODD ODM ODD MAE 216,3 214,2 139,1 221,8 MSE 75314, , , ,5 The performances of the tradtonal and TC based calbraton procedures have been evaluated n term of capablty of the O-D matrces to represent the lnk traffc flows. Thus, each matrces obtaned from the dfferent TC based methods have been assgned to the networks by means of a SUE traffc assgnment model and the smulated lnk flows have been compared wth the sets of the observed lnk flows. The comparsons has been carred out wth respect to a hold-out sample (TC not used n the calbraton procedures) and to a calbraton lnk flows set (TC used n the calbraton procedures). In the reported results fve dfferent sets of calbraton TC that have been bult by a random samplng appled to the whole TC set have been consdered. Assgnments have been conducted to both the car and transt network; for the sake of edtng, n ths paper only the car network wll be taken nto account and, consequently only the car mode matrx n the mornng peak perod. The O-D matrces have been updated by usng the GLS estmator. The multstep method has been appled to the ODM, ODD ODMCF matrces yeldng to an acceptable soluton after sx teratons. Three dfferent values for the varance terms (weght) of the lnk flows have been here consdered: 1, 10 and 50; the weght of the demand vector dfferences n the eqn. (6) has

6 been assumed equal to 1. Results obtaned for the ODMF (correcton wth TC of ODM matrces) are shown n Tab. 1. Table 1. Statstcal performances for the ODMF car network assgnment (GLS-Multstep method) (weght_f=1; weght _d=1) HOLD-OUT SAMPLE CALIBRATION FLOWS SET observed #1 #2 #3 #4 #5 #1 #2 #3 #4 #5 Startng MAE 200,2 277,7 205,3 233,9 237,0 221,0 198,6 219,5 211,3 210,4 MAE step1 245,1 235,0 223,3 215,1 243,7 94,3 103,0 106,2 108,5 106,1 MAE step2 250,9 240,4 228,8 203,1 240,2 66,4 73,3 88,0 90,5 102,7 MAE step3 253,9 266,3 226,7 187,9 252,2 58,6 70,6 89,8 84,2 87,2 MAE step4 251,5 246,7 232,3 200,1 249,8 60,8 60,8 84,6 89,8 103,2 MAE step5 259,3 247,1 232,5 196,1 254,1 59,2 70,1 77,5 78,0 86,4 MAE step6 273,2 267,1 236,8 198,3 248,7 58,7 64,1 85,8 90,7 78,0 Startng MSE 67681, , , , , , , , , ,0 MSE step , , , , , , , , , ,6 MSE step , , , , ,5 9250, , , , ,2 MSE step , , , , ,1 8898, , , , ,4 MSE step , , , , ,1 9074,4 7815, , , ,7 MSE step , , , , ,3 8313,4 9779, , , ,4 MSE step , , , , ,2 9318,7 9164, , , ,2 As t can be seen from Tab. 1, wth respect to the hold-out-samples, the set #4 leads to mprovement of the startng performance. The comparson wth respect to all the TC sets used n the GLS-multstep procedure shows an mprovement of the statstcal performances (both MAE and MSE). In the followng table are reported the results obtaned wth the ODMC method. In ths case the NLGLS estmator descrbed n the eq. (11) had been employed by consderng Cv beta =0.7 and Cv flows = Table 2. Statstcal performances for the ODMC car network assgnment (NLGLS) (Cv beta =0.7 and Cv flows =0.001) HOLD-OUT SAMPLE CALIBRATION FLOWS SET observed #1 #2 #3 #4 #5 #1 #2 #3 #4 #5 Startng MAE 200,2 277,7 205,3 233,9 237,0 221,0 198,6 219,5 211,3 210,4 Endng MAE 232, ,5 203,6 246,6 193, ,9 205,0 188,6 Startng MSE 67681, , , , , , , , , ,0 Endng MSE 80111, , , , , , , ,0 Also n ths case better performances are obtaned wth respect to the calbraton s, whle mprovements wth respect to the hold-out-samples are obtaned wth the sets #2 and #4. The ODMC has been calbrated wth the GLS-multstep method and the obtaned matrx ODMCF has been assgned to the network. The statstcal performances are reported n the followng table:

7 Table 3. Statstcal performances for the ODMCF car network assgnment (weght_f =1; weght_d=1) HOLD-OUT SAMPLE CALIBRATION FLOWS SET observed #1 #2 #3 #4 #5 #1 #2 #3 #4 #5 Startng MAE 200,2 277,7 205,3 233,9 237,0 221,0 198,6 219,5 211,3 210,4 MAE step1 239,8 237,3 244,5 197,4 240,8 77,4 74,3 85,4 96,1 92,2 MAE step2 251,1 249,7 247,1 203,5 235,7 61,2 62,2 76,8 97,1 83,6 MAE step3 236,3 248,3 256,5 197,9 233,1 70,8 59,9 67,9 86,9 81,3 MAE step4 243,7 254,5 255,0 188,1 244,9 58,1 57,8 69,6 82,7 76,7 MAE step5 254,9 258,3 243,9 188,9 248,7 64,7 58,1 81,8 77,2 85,0 MAE step6 253,3 255,9 242,7 180,4 249,9 59,8 61,2 77,5 76,5 77,8 Startng MSE 67681, , , , , , , , , ,0 MSE step , , , , , , , , , ,8 MSE step , , , , ,1 7866,0 9063, , , ,6 MSE step , , , , , ,0 8332, , , ,2 MSE step , , , , ,3 8980,5 8038, , , ,5 MSE step , , , , , ,8 8223, , , ,6 MSE step , , , , ,5 9968,0 8621, , , ,3 Ths method s more effectve than all the prevous snce we see better reducton of the performance ndcator values at the same step and the lowest value of the MAE (TC set #4) To have a more comprehensve and summarsed vew of the analyss, for each method the average values of the performances ndcators computed wth respect to the fve TC sets have been consdered and reported n the Table 4 both for car and transt networks. Table 4. Statstcal performance ndcators average values Transt Network Car Network weghts Hold out sample MAE MSE MAE MSE Calbraton Hold out sample Calbraton weghts Hold out sample Calbraton Flows set Hold out sample Calbraton ODD 209,0 222, , ,1 226,1 210, , ,9 ODM 154,0 134, , ,2 230,8 212, , ,0 ODDF weght_f=1 weght _d=1 ODMF Cv_f=0,1 Cv_d=1 ODMC Cv_f=0,1 Cv_beta=0,7 weght _f=1 ODMCF weght _d=1 100,4 103, ,3 weght _f= ,8 weght _d=1 97,8 100, , ,6 weght_f=50 weght _d=1 123,2 131, , ,1 Cv_f=0,001 cv_beta=0,7 118,9 126, ,3 weght _f= ,3 weght _d=1 217,6 90, , ,8 213,5 97, , ,4 239,4 193, , ,3 232,0 85, , ,2 As t can be seen ODMF method n the average performs better concernng both transt and car demand consderng values obtaned for hold out samples.

8 4 CONCLUSIONS In ths work has been shown how TC based methods can be employed n dfferent way to mprove, adjust or update avalable O-D matrces. In general, all the methods lead to mprovement of the startng smulaton capablty of the traffc assgnment model. Obvously, the fnal performances depend on the qualty of the startng matrx (on the avalable data, n general) and the effects of the weghts, n ths case, can be negatve. The multstep methods based on GLS estmator show smlar performances. The NGLS gves lower performances because of the stronger effects of the weght (see Ottomanell, 2001). Better performances have been obtaned for the transt network assgnment. The obtaned results are very nterestng also consderng the sze of the network. REFERENCES D Gang M., Ottomanell M. (2000). Calbratng path choce models on congested networks wth traffc counts. Proceedng of the 8 th Meetng of the EURO WG on Transportaton. Rome, Italy Cascetta, E., Russo, F. (1997). Calbratng Aggregate Travel Demand Model wth Traffc Counts: Estmators and Statstcal Performance. Transportaton, 24: 271:293 Cascetta, E.(1984). Estmaton of trp matrces from traffc counts and survey data: a generalzed leas squares estmator. Transportaton Research, 18B: D Gang M. (1989) Una valutazone delle prestazon statstche degl estmator delle matrc O/D che combnano rsultat d ndagn e/o modell con contegg de fluss d traffco. Rcerca Operatva n. 51: Maher, M.J. (1983). Inference on trp matrces from observatons on lnk volumes: A bayesan statstcal approach. Transportaton Research, 17B: Ottomanell M. (2001). Effects of data accuracy n aggregate travel demand models calbraton wth traffc counts. In Mathematcal Methods and Optmzaton on Transportaton Systems (Pursula M. & Nttymak J. eds.), Chapter 14 th. Kluwer Academc Publsher Yang H., Ida Y. and Sasak T. (1992)., Estmaton of Orgn-Destnaton Matrces from Lnk Traffc Counts on Congested Networks, Transportaton Research, 26B, vol. 6, pp Van Zuylen, J. G., Wllumsen, L.G. (1982). The most lkely O-D matrx estmated from traffc counts. Transportaton Research, 14B: