ORIGIN DESTINATION DISAGGREGATION USING FRATAR BIPROPORTIONAL LEAST SQUARES ESTIMATION FOR TRUCK FORECASTING

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1 ORIGIN DESTINATION DISAGGREGATION USING FRATAR BIPROPORTIONAL LEAST SQUARES ESTIMATION FOR TRUCK FORECASTING Unversty of Wsconsn Mlwukee Pper No Ntonl Center for Freght & Infrstructure Reserch & Educton College of Engneerng Deprtment of Cvl nd Envronmentl Engneerng Unversty of Wsconsn, Mdson Author: Aln J. Horowtz Center for Urbn Trnsportton Studes Unversty of Wsconsn Mlwukee Prncpl Investgtor: Aln J. Horowtz Professor, Cvl Engneerng nd Mechncs Deprtment, Unversty of Wsconsn Mlwukee Februry 13, 009 1

2 Orgn Destnton Dsggregton Usng Frtr Bproportonl Lest Squres Estmton for Truck Forecstng Abstrct Ths workng pper descrbes group of technques for dsggregtng orgn-destnton tbles for truck forecstng tht mkes explct use of observed trffc on network. Sx models wthn the group re presented, ech of whch uses nonlner lest-squres estmton to obtn row nd column fctors for splttng trp totls from nd to lrger geogrphcl res nto smller ones. The technques re phlosophclly smlr to Frtr fctorng, lthough the soluton method s qute dfferent. The technques re tested on full-szed network for Northfeld, MN nd re found to found to work effectvely. Introducton nd Mthemtcl Underpnnngs It s often desrble to obtn hghly detled orgn-destnton tble for vehcles or commodtes, when only much more ggregted tble s vlble. These stutons typclly rse when survey dt re orgnzed nto frly lrge dstrcts (zp codes, ctes, countes or sttes) n order to preserve confdentlly or smply to provde menngful flow comprsons when the number of dt smples s lmted. Commercl vehcle nd freght dt, n prtculr, re prone to ths type of sptl ggregton. For the purposes of ths dscusson, the ggregted OD tble wll be sd to contn trp dt between dstrcts, whle the dsggregted OD tble wll be sd to contn trp dt between zones. Trdtonl prctce hs been to dsggregte dstrct-level orgn-destnton tble by fctorng t long ts rows nd columns, smultneously. Tht s: T = A B τ (1) where: = n orgn (row) n the dsggregrted (zonl) tble nd where s n element n the set of zones I; = destnton (column) n the dsggregted (zonl) tble nd where s lso n element n the set of zones I k = n orgn (row) n the ggregted (dstrct-level) tble nd k s n element n the set of dstrcts K; l = destnton (column) n the ggregted (dstrct-level) tble nd l s lso n element n the set of dstrcts K; T = the dsggregted orgn-destnton tble, zone-to-zone; A = row splt fctor for ech zone tble orgn; B = column splt fctor for ech zone tble destnton; τ l = the ggregted orgn-destnton tble, dstrct-to-dstrct. The sets of splts, A nd B, hve the effect of spredng lrge number of trps between n orgn nd destnton nto smller numbers of trps between, perhps, mny orgns nd mny destntons. Ech nd ech s ssocted wth one nd only one k or l, respectvely. So for

3 notton purposes, t s necessry to defne two further groups of sets: trck of the structurl reltonshp between the two tbles. Tht s, A k B l A L k nd B L l, whch keep L = the set of rows tht re ssocted wth row k n the ggregted dstrct tble; L = the set of columns tht re ssocted wth column l n the ggregted dstrct tble. Zones nest nto dstrcts nd no zone my occur n multple dstrcts. Ech dstrct tble ndex cn be computed s functon of gven zonl tble ndex. Ths s, when s known, then k cn be found by referencng the set, L. A k It should be recognzed tht ths trdtonl prctce gnores the possblty tht there re specl zone-to-zone nterctons tht re hdden n the ggregton t the dstrct-to-dstrct level. For exmple, lrge fctory mght shp to lrge wrehouse, cretng prtculrly lrge OD flow between two zones tht mght not be pprent by ust lookng t the flow between the two respectve dstrcts. Orgn-destnton tbles re often thought to be symmetrc over 4-hour perod for pssenger trvel; however, commodty flow tbles cnnot be ssumed to be symmetrc nd vehcle flow tbles, both pssenger nd freght, re rrely symmetrc for perods of tme shorter thn dy. Although the term orgn-to-destnton s used n ths dscusson, the procedures developed heren re eqully pplcble to producton-to-ttrcton flows for pssenger trvel nd producton-to-consumpton flows for commodtes. The mount of dt vlble to determne A nd B vres consderbly dependng upon the plnnng problem. Very often plnners wll clculte the splts from socoeconomc dt or by pplyng trp generton equtons, s they mght hve been prepred for trvel forecstng model. Another common method s to determne the splts by observng the mount of trvel n ech dsggregted zone, such s the zone s VMT (vehcle mles of trvel). Another possble dt source s trffc counts on ndvdul lnks. Indvdul trffc counts re dffcult to use drectly for determnng the splts becuse ny one count s not usully ssocted wth ny specfc zone. Indeed, the reltonshp between trffc count nd the number of trps tht re generted n nerby zone, s wll be seen, s qute complex. Recent work on estmton of orgn-destnton tbles from trffc counts hs drect mplctons for the OD tble dsggregton problem. One technque n prtculr, Frtr bproportonl lest-squres estmton, cn be sutbly modfed to crete needed row nd column splts. In prtculr, Frtr bproportonl estmton seeks the soluton of ths nonlner, lest-squres mnmzton problem (Horowtz, 005) to obtn sets of row nd column fctors to refne rough (or seed ) tble t the sme level of ggregton: * * mn P = w C s p + x y T z T () A I I I I where x = row (orgn) fctor for zone ; y = column (destnton) fctor for zone ; ( ) 1 x y 3

4 C = ground count for lnk drecton, wth ech drecton on two-wy lnks tbulted seprtely, nd s n element n the set of ll counted drectons A; T = number of trps between orgn nd destnton to be estmted; * T = seed trp tble; p = estmted proporton of trps between zones nd tht use lnk drecton (s determned by n equlbrum trffc ssgnment); I s the set of zones, = 1 to N or = 1 to N; A s the set of lnk drectons; w = lnk weght for lnk drecton ; z = the trp tble weght; nd s = scle fctor tht s ether set to 1 or selected utomtclly to scle the trp tble to produce the correct verge trffc count before optmzton. For exmple, ths equton mght be useful for pproxmtng pek-hour orgn-destnton tble, zone-to-zone, from pek-hour trffc counts nd from 4-hour orgn-destnton tble, lso zone-to-zone. Seed tbles re often bult from survey dt, behvorl trvel theory or expert udgment. The estmton fnds the best compromse set of orgn nd destnton fctors tht gves good greement wth trffc counts nd does not devte hugely from the seed tble. It s lso mthemtclly necessry to constrn the fctors to be greter thn zero, nd t s qute desrble n most crcumstnces to keep them wthn resonble bounds. If the ggregted, dstrct-level OD tble s perfect, then the followng reltonshp must hold: τ = T (3) A B Lk Ll However, t s entrely possble tht the dstrct-level OD tble s less thn perfect, becuse t too s subect to vrous dt collecton errors or ndequces n theory. In such cses, t my be pproprte to vod usng Equton 3 s strct constrnt. It s hghly lkely tht plnner cn fnd some dt to suggest how the dstrct-level OD tble mght be dsggregted, ustfyng the use of splts, A nd B, t lest tenttvely. However, trffc counts mght suggest tht dfferent splts re better for the purpose. Therefore, Equton 1 should be modfed to nclude the nformton comng from ll sources: T = sx A y B τ (4) where x nd y re emprcl modfers, somehow derved from trffc counts, of Thus, s, x nd y hve smlr purposes to the sme vrbles n Equton. A nd There re lmts s to how mny x s nd y s cn be estmted, gven the mount of dt vlble to the problem from the dstrct-level OD tble nd from the trffc counts. It s lso entrely possble tht gven zone s trffc (orgns or destntons) mght not trvel on ny of the counted lnks; n such cses the x s nd y s must defult to 1.0, wth ll the fctorng crred by the predetermned s, A s nd B s. Beyond Equton 3, zonl OD tble tht perfectly conserves trps must requre these reltonshps to be stsfed: B. 4

5 x A A L k B L l = 1, for ll dstrct rows, k y B = 1, for ll dstrct columns, 1 However, perfectly-conservng zonl OD tble my not be desred becuse of dt collecton errors, so these reltonshps could be consdered ust pproxmte for mny stutons. The strength of the soluton depends upon how much trffc dt re vlble nd the szes of the two OD tbles. There re N vrbles, where N s the number of zones n the zonl OD tble. For exmple, f the dstrct-level OD tble hs 1 dstrcts nd f there re 400 trffc counts, then there re 544 dt tems n the estmton ( ). Ths mens tht the estmton cn relbly expnd ths tble to t most 7 zones,.e., hlf the number of dt tems, but probbly lot less. The loctons of the 400 trffc counts mtter. More splts cn be estmted when the trffc counts re spred evenly throughout the regon nd where the countng sttons re locted on mor rods. There re number of dfferent wys to formulte the estmton methodology, nd sx of these wys re dscussed n ths pper, long wth some extensons: Model I: Dstrct-level OD tble s perfect Model II: Dstrct-level OD tble s pproxmte Model III: Dstrct-level OD tble s perfect, OD s re ffected by trp utlty Model IV: Dstrct-level OD tble s pproxmte, OD s re ffected by trp utlty Model V: Dstrct-level OD tble s pproxmte, lnk-to-lnk flows re vlble Model VI: Dstrct-level OD tble s pproxmte, some zone-to-zone flows re specl Model I: Dstrct-Level OD Tble Is Perfect The lest squres estmton for ths model tres to mtch ground counts whle fttng the dstrctlevel OD tble exctly. mn P = w C s p x A y B τ (5) A I I where k nd l re tken to be functons of nd, respectvely. These constrnts must hold. τ = s x A y B τ, k K, l K (5b) A B L k Ll x 0, I y 0, I Ths method cn be mplemented wthn the mthemtcs of Model II, f sutbly lrge vlue of z (see next secton) s selected to ssure tht the frst constrnt s stsfed. 5

6 As prctcl mtter, ll of the models must ccount for the presence of externl sttons on the network. There cn be no ntrzonl trps wthn externl sttons. Ths s perhps mnor detl, but t cn be hndled by ntroducng nother fctor G (ner where A nd B pper n ll expressons) whch pples to ll zones wthn the set E of externl sttons, where E I. Thus, G = 0, f = nd E (6) G = 1, otherwse (6b) Ths sme vrble should be ntroduced n the mplementton of ny of the models descrbed here. The scle fctor, s, hs been retned from Equton. A scle fctor cn help elmnte systemtc errors n dt collecton or dust for dfferent sets of unts. For exmple, t s concevble tht n ggregted freght OD tble cn be gven n unts of tons of whle the lnk volumes cn be gven n unts of trucks. At the surfce, s ppers to be entrely redundnt. However, upper nd lower bound constrnts plced on x s nd y s cn sometmes mke t desrble to keep them close to 1, nd vlue of s 1 llows ths to hppen more redly. Model II: Dstrct-Level OD Tble Is Approxmte Model II s more consstent wth pst prctce n OD tble estmton from ground counts, where the seed tble s consdered to be, t best, rough pproxmton of relty. Assumng tht the dstrct-level OD tble s pproxmte provdes some flexblty to the estmton nd recognzes tht there my be serous nconsstences between the ground count dt nd the survey dt tht were used to buld the dstrct-level OD tble. mn P = w C s p + x A y B z τ s A I I τ x A y B τ (7) k K l K I I x 0, I y 0, I Agn, k nd l re functons of nd. Judgment s to whch re most ccurte, ether ground counts or ggregted OD flows, s expressed by the set of lnk-drecton weghts, w, nd by the sole tble weght, z. The ctul effects of these weghts re not obvous nd the effects re best evluted fter the optmzton hs been completed. Model III nd IV: OD s Are Affected by Trp Utlty Model III (perfect dstrct-level tble) cn be treted s specl cse of Model IV (pproxmte dstrct-level tble). Both models ntroduce the de from grvty model of trp dstrbuton or logt model of destnton choce tht there s less lkelhood of trp between pr of zones f there s consderble sptl seprton between them. Sptl seprton s mesured by trveler s utlty, whch s lmost lwys ncresngly more negtve or less postve s trp dstnce ncreses. Most trvel forecstng models clculte vlue of utlty prmrly from the trvel tme between the two zones. Defne: 6

7 U = utlty of trvel from zone to zone ; nd V = utlty of trvel from dstrct k to dstrct l; Utlty n these cses re determnstc nd cn be obtned drectly from the trffc network. The dstrct-to-dstrct utlty of trvel my be found by tkng weghted verge of ll zone-to-zone utltes. Thus, x A B Lk Ll V = (8) x A y B A k L B l L A y B U whch would need to be recomputed repetedly s the set of x s nd y s become better known. If the A s nd B s re known frly well, then ntlly, x = 1 nd y = 1. Assumng logt or mxmum-entropy reltonshp for destnton choce, then the followng correcton, F, mght be needed for zone prs from prtculrly lrge dstrcts: U F = e V, (9) e nd k nd l re functons of nd. Wthn the frmework of typcl trvel forecstng model, utlty would lkely be clculted from the trvel tme between two zones. Tht s, U = φt (10) where t = the trvel tme between zones nd, nd φ = negtve constnt. Model IV s obectve functon cn be obtned by slghtly enhncng Model II: mn P = w C s p + x A y B F z τ s A I I τ x A y B Fτ (11) k K l K I I φ could be dopted from trp dstrbuton model for the regon, or t could be obtned drectly through the optmzton process. Model III could be mplemented by settng z to lrge number. Model V: Dstrct-level OD Tble Is Approxmte, Lnk-to-Lnk Flows re Avlble Model III could be further expnded to nclude nformton comng from toll rod trnsponders. Reders could be plced throughout the communty, nd t would then be possble to gn better understndng of the movement of trffc. Reder loctons re nether orgns nor destntons, but re ponts n between. Dt collected from mny reders could gretly enhnce 7

8 OD tble dsggregton beyond wht could be ccomplshed through ground counts lone, becuse these dt embody sptlly precse flows. Snce not every vehcle hs trnsponder, ny reder locton lso needs to be count locton, so reder loctons re subset of lnks n the set of counted lnks, R A. Reder loctons would be ordered by tme of dy, so: g = the frst reder on trp, closest to the orgn; nd h = the second reder on trp, closest to the destnton. It s possble for vehcle to trverse mny reders long sngle trp, but ll reder dt need to be orgnzed nto prs of loctons to be comptble wth method known s select lnk nlyss wthn trvel forecstng model. So further defne: gh p = estmted proporton of trps pssng between lnk drectons g nd h tht hve ther orgn n nd ther destnton n (s determned by n equlbrum trffc ssgnment); nd Q gh = expnded, smpled number of trps between lnk drectons g nd h, s scertned by reder nd count dt. mn P = r g R h R A Q w C gh s s I I I I p gh p x A y B τ x A y B τ + z k K l K τ s I I x A y B τ + (1) where r s weght on the trnsponder component. Model VI: Dstrct-Level OD Tble Is Approxmte, Some Zone-to- Zone Flows Are Specl In some crcumstnces t mght be necessry to ccount for known hgh nterctons between specfc prs of zones. The number of such zone prs must be kept to ust few so s not to overwhelm the estmton process. Therefore, t s ssumed tht these zone prs cn be dentfed n dvnce, even f the ctul level of ntercton s unknown. Such nterctons cn be ncorported nto the model by defnng msk H for specl zone prs: H = 1, f the ntercton between nd s specl, nd (13) H = 0, otherwse. (13b) The ddtve dustment for specl trps between n OD pr, M, cn then be nserted nto ny of the prevous models. For exmple, for Model II the obectve functon becomes: mn P = z k K l K A τ w C s s I I I I A B p ( x y + H M ) A B ( x y + H M ) τ τ + (14) 8

9 nd M 0, f t s ssumed tht ths vrble s used only to ncrese the number of trps for zone pr, whch would be typcl. But more generlly, x y + HM 0, whch llows M to be ether postve or negtve. Further Vrtons Constrnts on Zonl Fctors. Addtonl constrnts on x nd y could be mposed to keep the results wthn resonble bounds throughout the optmzton nd n the results. These constrnts overrde the nonnegtvty constrnts. Thus, x y mn x x mx mn y y mx, I (15), I where both x mn nd y mn re greter thn zero. Sngle Stton OD Survey. Dependng upon the vlblty of dt, further ddtons could be mde to the methodology. For exmple, t mght be possble to nclude sngle-stton orgndestnton (SSOD) survey dt nto the obectve functon. An SSOD survey s typclly conducted on rurl rod, perhps t wegh stton or t rest re. Ech drver s sked to gve hs/her orgn nd destnton mong other nformton. SSOD surveys re comprtvely nexpensve. Unfortuntely, SSOD dt re hevly bsed towrd long trps nd towrd trps tht ust hppen to be pssng by tht prtculr wegh stton or rest re. However, the select lnk nlyss feture n trvel forecstng models cn emulte the results from n SSOD survey wth ll ts ssues, so SSOD survey dt cn be ncluded n the obectve functon wthout dstorton. Multclss Trffc Assgnments for Trucks. Truck routng n urbn networks would lkely be nfluenced by congeston on streets tht crry both freght nd pssenger trffc, wth the lrgest number of vehcle servng pssengers. To properly hndle ths cse, t s necessry to mplement the estmton methodology wthn multclss equlbrum trffc ssgnment. Ground counts would be provded only for trucks, but the p rry would be nfluenced by the presence of pssenger vehcles tht excerbte delys for trucks long streets nd t ntersectons. Blevel Soluton Algorthm The OD dsggregton problem, ny model, s solved by embeddng t wthn trvel forecstng frmework, s llustrted n Fgure 1. Two seprte nput OD tbles must be provded: the dstrct-to-dstrct OD tble nd seed zone-to-zone OD tble. The reson for the seed zone-to-zone tble s to obtn trffc ssgnment tht cn be used to compute the p rry nd to obtn n ntl set of delys on lnks nd t ntersectons. A good source of seed zone-to-zone OD tble s Equton 1. The seed zone-to-zone OD tble does not drectly 9

10 contrbute to the creton of orgn nd destnton fctors, but the seed tble s retned n the MSA vergng process nd cn slghtly nfluence ssgned volumes. Blevel lgorthms smlr to Fgure 1 must be used when the OD tble hs not been fully determned t the pont of ntl trffc ssgnment nd congeston s present on the network. An ccurte trffc ssgnment requres ccurte lnk delys tht requre the correct lodngs, whch cn only be found for congested networks fter szble number of MSA tertons. Dstrct-toDstrct OD Tble Seed Zone-to- Zone OD Tble All-or-Nothng Trffc Assgnment MSA Volume Avergng, OD Tble Avergng nd Dely Clcultons Fnd Orgn nd Destnton Fctors New Zone-to- Zone OD Tble FIGURE 1. Blevel Algorthm for Solvng the OD Tble Dsggregton Problem Computtonl Consdertons Models I through V re constrned nonlner mnmzton problems n N dmensons. Beyond dmensonlty, computton tme nd memory usge re lso hevly nfluenced by the number of lnk drectons on the network, whch hs drect mpct on the number of nonzero elements n the p rry. The sze of the ggregted (dstrct-level) OD tble hs lttle effect on computtonl effcency. Memory usge s lrgely dctted by the p rry, even when stored n mnner tht squeezes out ll zero elements. Computton tests presented n ths pper were done wthn 3-bt Wndows envronment, whch hs prctcl memory lmt of 3 GB. For exmple, moderteszed metropoltn network wth 000 zones nd n verge of 50 counted lnks used between ech zone pr n multpth trffc ssgnment would hve nonzero p rry sze of 10 9 cells, whch would more thn fll ny 3-bt Wndows computer. 10

11 Computton tme cn lso be n ssue, s lrge problems could tke dys or even weeks of computton tme on stndrd desktop computer. In order to reduce computton tme, the expermentl softwre hs been wrtten to smultneously use ll processors n multprocessor computer. Substntl prllel processng power s now vlble n ffordble desktop computers tht come equpped wth four, eght nd even 16 processors. The lgorthm for soluton s the grdent proecton method wth PARTAN. Serches n the grdent proected drecton re stopped when the step sze, η, decreses beyond: η < θ N (16) where θ s sutbly smll number nd N s the number of vrbles. The optmzton s termnted when the reltve chnge n the obectve functon between PARTAN steps s smller thn nother smll rbtrry number, determned through trl nd error process. Computtonl Tests, All Vehcle OD Tble Estmton n Northfeld The Northfeld, MN network ws selected for testng the computtonl propertes of OD tble dssggregton. These tests nvolved pssenger, commercl nd freght vehcles n sngleclss trffc ssgnment. The Northfeld network s shown n Fgure 1. It hs 9 zones nd 1 externl sttons, whch were orgnzed nto 11 dstrcts. Externl sttons were treted smlrly to zones n the tests. Becuse ll streets were ncluded n the network, there were 819 lnks, but ust 60 lnk drectons hd trffc counts. The number of ground counts s undesrbly less thn the number of vrbles. In Fgure 1, zone boundres re ncluded s thn blck lnes. The spder-lke structures show the reltonshp between zones nd dstrcts. 11

12 463 FIGURE. Northfeld Test Network The ggregted OD tble ws creted by grvty model t the zone-to-zone level wth homebsed-work, home-bsed-nonwork, nd nonhome-bsed trp purposes for pssengers, then ggregted to the dstrct level. The zone-to-zone OD tble ws retned for comprson purposes. All prmeters were tken from NCHRP Report #365 for pssenger trvel. Becuse the ggregted OD tble omtted ny consderton of freght or commercl vehcles, substntl dsgreement wth the ground counts ws ntcpted. So not only would the model be expected to dsggregte the OD tble, but t would lso be expected to correct for errors nherent n the ggregted OD tble cused by omttng mny trucks. Zonl chrcterstcs were vlble tht could hve permtted the creton of frly good sets of zone splts, A nd B, but there ws prtculr nterest n seeng wht cruder set of zone splts would ccomplsh. So for these tests ll A s nd B s were set to the recprocl of the number of zones n ther respectve dstrcts. Models I through IV pply to the Northfeld cse. Optmzton prmeters were set s follows: All lnk weghts, w, were set to 1; 1

13 The OD tble weght, z, ws set to 100 for Models II nd IV nd 10,000 for Models I nd III; There ws pror sclng of the OD tble,.e., s 1, to t lest ccount for the omsson of trucks from the dstrct-to-dstrct OD tble; All x s nd y s were constrned to be between 0. nd 5. All smultons nd optmztons used re spred equlbrum trffc ssgnment, whch lods trffc t lmost ll ntersectons nd dspenses wth centrod connectors, whch re common devces n trvel forecstng networks. Ths ssgnment method s ble to ssgn the vst morty of ntrzonl trps to the network nd s hghly multpth. Equlbrum ws cheved by runnng 40 tertons of the method of successve verges (MSA), whch s more thn n most trvel forecstng pplctons, but not suffcent to reduce convergence error to neglgble mount. The tme perod of the smultons ws full 4-hours. A sttstcl summry of the four models (I through IV) nd n ordnry smulton re shown on Tble 1. Dt re gven for the 60 lnk drectons wth ground counts. TABLE 1. Summry of Computtonl Tests, Inexct Input Dt Model Averge Ground Count Averge Assgned Lnk Volume RMS Dfference n Volumes RMS Dfference n Aggregte OD Tble I II III IV Smulton The RMS dfference n the OD tble for the smulton ws zero becuse the zone-to-zone OD ws not chnged durng the smulton nd the dstrct OD tble ws bult by ggregtng the zone-to-zone OD tble. The verge ggregted OD flow ws 554 vehcles. The smulton performed very poorly n mtchng ground counts n relton to Models I to IV, even though t hd the dvntge of beng provded zone-to-zone OD tble (41 by 41) nsted of dstrct-to-dstrct (11 by 11) OD tble. A mor contrbutor to the error of the smulton ws n verge of pproxmtely 100-vehcle systemtc underestmte of ll ground counts; presumbly mny of these were trucks. Another possblty for the underestmte s tht the smulton does not hve enough congeston to crete dverson due to equlbrum effects nd trffc s unrelstclly beng kept on routes wth slght dvntges n free trvel tme nd were not counted. Models III nd IV dd slghtly worse thn Models I nd II, whch ws unexpected gven tht the dstrct-to-dstrct OD tble ws creted wth grvty model nd Model IV dffers from Model II by mkng grvty-type dustments. Tble 1 shows tht Model I preserves the dstrct-todstrct OD tbles (to wthn one-hlf of trp), wth only slght ncrese n the RMS dfference between the forecst nd the ground counts over Model II. Even the 9. trp error n the ggregted OD tble for Model 4 s not lrge. 13

14 As n exmple, Fgure 3 shows mp of the computed destnton fctors from Model II. These destnton fctors hve been multpled by the scle fctor, s. Drker red htchng ndctes zones tht hve destnton fctors between 0. nd 0.4 whch the drker blue htchng hs destnton fctors between 3. nd 4. The spder-lke structures hve been emphszed to show the reltonshp between zones nd dstrcts. A zone could hve lrge destnton fctor becuse t hs more ctvty, overll, thn ts compnon zones or becuse t hs ctvtes tht generte dsproportonte mount of trvel not ccounted for n the NCHRP Report #365 prmeters, such s truck trvel. The mp confrms ntuton by hvng bout s mny blue zones s red zones n ech dstrct. The results for orgn fctors nd for Model II were smlr. It s dffcult to further nterpret Fgure 3 wthout consderble locl knowledge. The scle fctor, s, ws selected by the lgorthm to be between 1.1 nd 1.17, dependng upon the model. The 1996 Quck Response Freght Mnul (Cmbrdge Systemtcs, 1996) sttes tht commercl vehcles mke up 10.5% of trffc on urbn prncpl rterls, so these scle fctors re somewht greter thn wht would be expected f they were only ccountng for the bsence of trucks n the dstrct-to-dstrct OD tble. All of zones chnged from ther orgnl vlues of x nd y of 1.0, nd ll of the fctors fell esly wthn the constrnts of Equton 15. These ntl tests of Models I to IV demonstrte tht they cn gve plusble results, but these tests do not demonstrte tht the results re ccurte. 14

15 463 FIGURE 3. Destnton Fctors for Model IV t Zones (Shded Ares) nd t Externl Sttons (Dots) (Drker Blue Indctes Lrger Fctors, Drker Red Indctes Smller Fctors nd Mgent s Neutrl) To better guge the ccurcy of the models n reconstructng n underlyng zone-to-zone OD tble, these steps were performed: 1. Crete resonble zone-to-zone OD tble for Northfeld, n ths cse by doptng the output tble from the lst terton of Model II of the prevous tests. Retn the orgn nd destnton fctors.. Buld dstrct OD tble from the zone-to-zone OD tble. 3. Assgn the zone-to-zone OD tble to the network nd obtn lnk volumes. 4. Set the ground count on ll rterl lnks equl to the computer lnk volumes. 5. Usng the dstrct-to-dstrct OD tble nd the ground counts, compute new zone-tozone OD tble. 6. Compre the orgnl nd new orgn nd destnton fctors. 15

16 Ths procedure resulted n 57 ground counts on the network. For these tests, the OD tble weght, z, ws left t from 100, even though there were greter number of ground counts thn the erler tests. It should be noted tht the nput zone-to-zone tble ws computed from Equton 4, so the reltonshp between the zonl nd dstrct-level OD tbles perfectly dheres to the theory, nd the computed ground counts re consstent wth both OD tbles. Tble shows tht the optmzton, s expected, s fndng soluton tht s very close to n exct ft to both the dstrct-to-dstrct OD tble nd the ground counts. The verge OD flow n the dstrct-to-dstrct tble ws 63 vehcles, so the errors n mtchng the ggregted OD tble s ust 1% nd the error n mtchng ground counts s less thn 3%. The remnng smll dfferences between ssgned volumes nd ground counts n the OD tble re ttrbuted mnly to convergence error of the equlbrum trffc ssgnment lgorthm. The test ws repeted by elmntng the bounds on x s nd y s (Equton 15) to determne of these bounds were nhbtng the estmton process. The results n Tble for the unconstrned optmzton, lthough slghtly mproved, ndcte tht the constrnts were not serous ssue n fndng orgn nd destnton fctors. TABLE. Summry of Computtonl Tests, Artfcl Input Dt Model Averge Ground Count Averge Assgned Lnk Volume n Volumes RMS Dfference II Constrned II Unconstrned RMS Dfference n Aggregte OD Tble The most nterestng outputs of these tests re shown n the sctter chrts of Fgures 4 nd 5, whch compre the results of the optmzton wth the known orgn nd destnton fctors. The orgnl nd computed sets of fctors compre very well to ech other. 16

17 Recovered Orgn Fctors Orgnl Orgn Fctors FIGURE 4. Sctter Chrt Showng the Computed Orgn Fctors Agnst the Known Orgn Fctors 17

18 Recovered Destnton Fctors Orgnl Destnton Fctors FIGURE 5. Sctter Chrt Showng the Computed Destnton Fctors Agnst the Known Destnton Fctors The tests of Tble nd Fgures 4 nd 5 were delzed so tht they could be redly nterpreted. They do not represent the sternest test of the models nd soluton lgorthm. However, these tests ndcte tht f the dt ft the theory of the model, f there s consstency between the ggregted OD tble nd ground counts nd f there re suffcent number of ground counts, lest squres optmzton cn do very well n estmtng orgn nd destnton fctors. Conclusons nd Recommendtons Ths pper outlnes severl optmzton models tht cn dsggregte orgn destnton tbles by usng nformton from ground counts. Four of the optmzton models were tested on rel dt from Northfeld, MN nd were found to work effectvely. One of these optmztons models ws tested on relstc, but rtfcl dt, on the sme Northfeld network nd ws found to be ble to ccurtely reproduce known underlyng orgn nd destnton fctors n ground counts. These methods developed n ths pper re ntended for commercl vehcle or freght forecsts becuse orgn-destnton dt for these flows re often ggregted to level where they re no longer useful for plnnng purposes. Addtonl tests re needed on full-scle freght networks. 18

19 Acknowledgements Ths study ws funded by the Center for Infrstructure Reserch nd Educton, ntonl unversty reserch center of the US Deprtment of Trnsportton. References Aln J. Horowtz, Tests of Fmly of Trp Tble Refnements for Quck Response Trvel Forecstng, Trnsportton Reserch Record Journl, #191, 005, pp Cmbrdge Systemtcs, et. l., The Quck Response Freght Mnul, Federl Hghwy Admnstrton, DOT-T-97-10, September

20 Appendx I: Settng Up nd Runnng n OD Tble Dsggregton The OD tble dsggregton procedures (Models I, II, III, IV nd VI) re mplemented wthn n expermentl verson of the Quck Response System II (QRS II) softwre. Networks re prepred wth the Generl Network Edtor (GNE). Networks must be bult wth the QRSDynmcEx.dt pplcton schem. Networks bult usng the QRSDynmc.dt cn be upgrded to the correct schem usng Fle Append wthn GNE. Networks re prepred n the usul wy, wth ground counts entered for the ground count ttrbutes for one-wy nd two-wy street lnks. Lnks wthout ground counts should hve the ground count ttrbutes left t zero. Centrods nd externl sttons must hve unque nmes. Dstrcts re estblshed n the usul wy, wth ech zone belongng to sngle dstrct. Ech dstrct tg must hve unque nme. Intl orgn nd destnton splts (A s nd B s) re entered on centrods nd externl sttons n the Dstrct Shre: Orgn nd the Dstrct Shre: Destnton ttrbutes, respectvely. There should be no demogrphc dt on ny of the centrods nd no productons nd ttrctons t externl sttons. However, f the trffc ssgnment hs tme perod of greter thn 1 hour, then QRS II needs wy to determne the tme-of-dy of trvel. To ssure QRS II mkes resonbly good tme-of-dy ssumptons, t s necessry to plce very smll number of dwellng unts, retl employees, nd nonretl employees on one nd only one centrod (e.g., du = 0.1; re = 0.01; nre = 0.04). These vlues wll crete frctonl ntrzonl trps t ths centrod, but gves QRS II enough nformton to compute tme-of-dy dstrbuton of trffc. One dd fle s requred, AddDTrps.txt. Ths fle s gven n the sme formt s AddVTrps.txt, s descrbed n the QRS II Reference Mnul. AddDTrps.txt contns the dstrct-to-dstrct OD tble. Row nd column nmes re the nmes of the dstrct tgs wthn the network. Ths fle must be plced n the proect folder for the run,.e., the sme locton s Prm.txt. The proect folder s dsplyed on the QRS II mn wndow. An exmple AddDTrps.txt fle, orgnzed s sngle tble, s shown below. Externls Externls 3 Dstrct1 Dstrct Externls 1 Dstrct 3 Dstrct 4 Dstrct 6 Dstrct 5 Externls 4 Externls 5 END OF ROWS Externls Externls 3 Dstrct1 Dstrct Externls 1 Dstrct 3 Dstrct 4 Dstrct 6 Dstrct 5 Externls 4 Externls 5 END OF COLUMNS

21 END OF TABLES An optonl dd fle s AddODMsk.txt. Ths fle mplements Model VI by tellng QRS II whch specfc OD prs should be gven specl tretment. Row nd columns consst of centrod or externl stton nmes. Vlues re ether 1 (specl tretment) or 0 (no specl tretment). OD prs not shown re not gven specl tretment. AddODMsk.txt s n the sme formt s AddVTrps.txt. QRS II needs to be lerted to the presence of AddODMsk.txt on the Add Fles dlog box. AddODMsk.txt s plced n the proect folder. There re vrety of prmeter settngs for QRS II, dependng on wht must be ccomplshed. The prmeter settngs shown below on the Refnement dlog box re for sttc equlbrum trffc ssgnment, ll vehcle clsses t once. Prmeters not shown should be left t QRS II s defult or t the pproprte vlues for trvel forecst n tht prtculr network. Lest Squres Tble Dsggregton must be selected. Mnmum nd mxmum dustments bound the vlues of x s nd y s. The trp tble weght s the vlue of z. Ths vlue must be determned expermentlly. Mnmum Step Sze nd Obectve Precson re convergence crter. Smller numbers for these crter mke n optmzton run longer nd mke the results 1

22 more precse. Allow Unform Sclng of Pror Tble tells QRS II to fnd vlue for the scle fctor, s, before the optmzton commences. Otherwse, s wll be set equl to 1. Do Tme Adustment, Dsggregton nvokes Models III nd IV where the negtve of the Dsggregton Constnt s used to clculte the dustments wth Equton 15. QRS II produces severl reports of drect nterest. Among these re: ODFctors.txt AddATrps.txt AddRTrps.txt Output.dt Hstory.txt Contns the orgn fctors nd destnton fctors (x s nd y s) for ech zone. The frst vlue n the fle n the frst row s the scle fctor, s. Rows to N+1 contn the zonl fctors. These rows re ordered s n Nodelbl.txt. Msked OD dustments my pper t the end of ths fle. Contns the equlbrum verged zone-to-zone OD tble. As such, t combnes the results of mny optmztons. Contns the lst terton zone-to-zone OD tble. Ths tble s computed by pplyng the fctors n ODFctors.txt drectly to the dt n AddDTrps.txt. Contns the ssgned, equlbrum verged volumes on the network (lso found n LnkVols.txt). Contns optmzton sttstcs nd computton tme. It s possble to use the methods n ths pper to dsggregte n OD tble for sngle vehcle clss, but multclss ssgnment s necessry. These chnges must be mde to the bove procedure to get dsggregted tble for chosen clss: A seprte clss other thn the bse clss must be defned on the Multclss dlog box by gvng t clss letter (e.g., T ). The clss must be chosen on the Refnement dlog box where t sys Clsses. Dt for other clsses my be gven wthn n the usul wy, such s AddVTrps fles, AddPTrps.txt, or through demogrphcs vrbles on the network. No chnges re needed for AddDTrps.txt, except tht AddDTrps.txt s specfc to the chosen clss. For clss to hve dfferent pths, ts nputs to pth buldng must dffer n some mportnt wy from the bse clss (.e., clss specfc extr tmes n n AddETme fle).