Origin-Destination Table Disaggregation Using Fratar Biproportional Least Squares Estimation

Transcription

1 Orgn-Destnton Tble Dsggregton Usng Frtr Bproportonl Lest Squres Estmton Aln J. Horowtz (Correspondng Author) Professor, Center for Urbn Trnsportton Studes, Unversty of Wsconsn Mlwukee, PO Box 784, Mlwukee, WI 53201, voce: , fx: , e-ml: November 2, 2009 Word Count: 5744 words + 5 fgures + 2 tbles Pper Number

2 Horowtz 2 Orgn-Destnton Tble Dsggregton Usng Frtr Bproportonl Lest Squres Estmton Abstrct: Ths pper descrbes group of technques for dsggregtng orgn-destnton tbles for trvel forecstng tht mkes explct use of observed trffc on network. Fve 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 work effectvely.

3 Orgn-Destnton Tble Dsggregton Usng Frtr Bproportonl Lest Squres Estmton INTRODUCTION AND PREVIOUS WORK It s often desrble to obtn hghly detled orgn-destnton (OD) 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. Ths subect hs gotten recent ttenton n the professonl lterture wth the relese of the Freght Anlyss Frmework 2 (FAF 2 ) nd subsequent nterest expressed by mny plnners n dsggregtng commodty flows from the 114 domestc regons to smller zones, such s countes. In pper tht drectly ddress the FAF 2 dsggregton problem, Run nd Ln (1) dentfed two prevlng methods of OD tble dsggregton nd present ther own sttstcl method of usng economc dt. Run nd Ln observed tht the proportonl weghtng method s most often chosen by plnners. Exmples of the proportonl weghtng method n the freght plnnng lterture bound, for exmple studes by Fsher, Ang-Olson nd L (2), Sorrtn nd Smth (3), Vswnthn, et l. (4), nd Rownsk, Ope nd Spsovc (5). These studes prncplly use economc dt, wth n occsonl use of ggregted trffc sttstcs, to crete the dsggregton fctors. 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 pplcton of the proportonl weghtng method 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) kl where: = n orgn (row) n the dsggregted (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 zonl tble orgn; B = column splt fctor for ech zonl tble destnton; τ kl = the ggregted orgn-destnton tble, dstrct-to-dstrct.

4 Horowtz 4 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 A B for notton purposes, t s necessry to defne two further sets, L k nd L l, whch keep trck of the structurl reltonshp between the two tbles. Tht s, A L k B L l = the set of rows tht re ssocted wth row k n the ggregted dstrct tble; = 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 A k cn be found by referencng the set,. L 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 nerly symmetrc over 24-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 possble 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), s llustrted by Rownsk, Ope nd Spsovc (5) nd Bttelle (6). Bttelle (6) used zonl VMT to crete the FAF 2 trffc ssgnments. Rownsk, Ope nd Spsovc (5) concluded tht zonl VMT ws helpful n cretng dsggregted truck OD mtrx n New Jersey. The use of zonl VMT dt to supplement socoeconomc nformton s good de, but t s lso problemtcl. To cte n extreme exmple, zone wth lrge nterstte hghwy could hve huge VMT even though the zone s otherwse desolte. A potentlly rcher dt source s trffc counts on ndvdul lnks, whch would be needed nywy to clculte zonl VMT. However, ndvdul 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 qute complex. However, to properly nclude trffc counts n cretng row nd column fctors, there s need to pull certn key concepts from the feld of estmtng synthetc OD tbles usng ground counts. The erlest methods dte from the lte 1970s nd mny mprovements hve been proposed snce. A very complete revew of the lterture s t reltes to sttc methods through 1998 hs been wrtten by Abrhmsson (7). Severl such methods re derved from the concept of lest-squres estmton, whch s the steppng-off pont for the work presented here.

5 Horowtz 5 Some recent work on estmton of orgn-destnton tbles from trffc counts by Horowtz (8) 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 to obtn sets of row nd column fctors to refne rough (or seed ) tble t the sme level of ggregton: 2 * * mn P = w C s p + 2 x y T z T ( 1 x y ) (2) A I I I I where x = row (orgn) fctor for zone ; y = column (destnton) fctor for zone ; 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 = set of zones, = 1 to N or = 1 to N; A = 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. Ths technque ws lter extended to dynmc OD tbles by Horowtz nd Dn (9). For exmple, ths equton mght be useful for pproxmtng pek-hour orgndestnton tble, zone-to-zone, from pek-hour trffc counts nd from 24-hour orgndestnton 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, sy no smller thn 0.2 nd no lrger thn 5. If n ggregted, dstrct-level OD tble s perfect, then the followng reltonshp must hold: τkl = T (3) A B Lk Ll 2

6 Horowtz 6 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. Prevous reserch nd prctce suggests tht plnner should be ble to fnd some socoeconomc 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) kl where x nd y re emprcl modfers, somehow derved from trffc counts, of A nd B. Thus, s, x nd y hve smlr purposes to the sme vrbles n Equton 2. In mny plnnng stutons socoeconomc dt nd trffc dt would be to some extent redundnt wth ech other, so well constructed dsggregton should fnd x s nd y s tht re close to 1. Plcng tght bounds on the x s nd y s would tend to strengthen the contrbuton of the socoeconomc dt to the computed zonl OD tble. 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. The strength of the soluton depends upon how much trffc dt re vlble nd the szes of the two OD tbles. There re 2N vrbles, where N s the number of zones n the zonl OD tble. For exmple, f the dstrct-level OD tble hs 12 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 272 zones,.e., hlf the number of dt tems, but probbly somewht 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 fve of these wys re dscussed n ths pper. Model I: Dstrct-level OD tble s pproxmte Model II: Dstrct-level OD tble s perfect Model III: Dstrct-level OD tble s pproxmte, OD s re ffected by trp utlty Model IV: Dstrct-level OD tble s perfect, OD s re ffected by trp utlty Model V: Dstrct-level OD tble s pproxmte, some zone-to-zone flows re specl MODEL FORUMULATIONS Model I: Dstrct-Level OD Tble Is Approxmte Model I s 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

7 Horowtz 7 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 τ + τ x A y B kl z kl s A x 0, I y 0, I I I 2 k K l K or s possble user opton to lend strength to ny socoeconomc dt, x y mn x x mx mn y y mx, I, I A k L B l L x A y where both x mn nd y mn re greter thn zero. 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 beforehnd, so the effects re best evluted fter trl optmzton hs been completed. The scle fctor, s, hs been retned from Equton 2. A scle fctor cn help elmnte systemtc errors n dt collecton or dust for dfferent sets of unts. For exmple, t s concevble, though not recommended, 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 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 Perfect The lest squres estmton for ths model tres to mtch ground counts whle fttng the dstrctlevel OD tble exctly. B τ kl 2 (5) mn = w C s 2 P p x A y B τkl (6) A I I where k nd l re tken to be functons of nd, respectvely. These constrnts must hold. τ kl = s x A y B τ, k K, l K (6b) A B Lk Ll kl x 0, I

8 Horowtz 8 y 0, I Ths method cn be mplemented wthn the mthemtcs of Model I, f sutbly lrge vlue of z (see Equton 2 nd prevous secton) s selected to ssure tht the frst constrnt s stsfed. Externl Sttons n Models I nd II 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 (7) G = 1, otherwse (7b) Ths sme vrble should be ntroduced n the mplementton of ny of the models descrbed here. Models III nd IV: OD s Are Affected by Trp Utlty Model IV (perfect dstrct-level tble) cn be treted s specl cse of Model III (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: U V kl = utlty of trvel from zone to zone ; nd = 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 pproxmted by tkng weghted verge of ll zoneto-zone utltes. Thus, x A B Lk Ll V kl = (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,, mght be needed for zone prs from prtculrly lrge dstrcts: F

9 Horowtz 9 U F = e V, (9) kl e nd k nd l re functons of nd. Model III s obectve functon cn be obtned by slghtly enhncng Model I: mn P = w C s p τ + τ x A y B F kl z kl s A I I 2 k K l K A k L B l L x A y B Fτkl (10) Prmeters of U could be dopted from trp dstrbuton model for the regon, or t could be obtned drectly through the optmzton process. Model IV could be mplemented by settng z to lrge number. Both Models III nd IV re mld deprtures form the proportonl weghtng prdgm, s they nclude nformton bove nd beyond wht cn be gotten from smply lookng t orgn zones by themselves or destnton zones by themselves. One would surmse tht Models III nd IV would work best when zones re lrge (for exmple, countes) reltve to verge trps lengths. Model V: 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. Model V represents mor deprture from the proportonl weghtng prdgm. 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 for specl zone prs: H H H = 1, f the ntercton between nd s specl, nd (11) = 0, otherwse. (11b) 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 I the obectve functon becomes: 2 mn P = z k K l K A τ w C kl s L s A k I I B l L A B p A B ( x y + H M ) ( x y + H M ) τ kl 2 τ kl 2 + (12) nd, x y H M 0, + whch llows M to be ether postve or negtve. It s dffcult to mgne how specl zonl nterctons could otherwse be dscerned from the trdtonl pproch (tht s wth zonl

10 Horowtz 10 socoeconomc nd/or VMT dt) wthout consderble locl knowledge, specl surveys or trffc counts. 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 n ntl zone-to-zone OD tble. The sole reson for the ntl 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 n ntl zone-to-zone OD tble s Equton 1. The ntl zone-to-zone OD tble does not drectly contrbute to the creton of orgn nd destnton fctors, but the ntl tble s retned n the method of successve verges (MSA) process nd cn slghtly nfluence ssgned volumes, dependng upon the number of MSA tertons. 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 (10,11). An ccurte trffc ssgnment needs ccurte lnk delys tht requre the correct lodngs, whch cn only be found for congested networks fter szble number of MSA tertons. The lgorthm selected for the soluton of the lower-level mnmzton problem s the grdent proecton method wth PARTAN. Serches n the grdent proected drecton re stopped when the step sze, η, decreses beyond: η < θ 2N (13) where θ s sutbly smll number nd 2N s the number of zones. 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. Trls on bg networks reveled tht speed of convergence could be n ssue becuse rther lrge optmzton problem s solved t ech MSA terton. Therefore, cre ws tken to fne-tune the optmzton step by mplementng nlytcl dfferentton nd prllel processng.

11 Horowtz 11 Dstrct-toDstrct OD Tble Intl 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 COMPUTATIONAL TESTS, ALL VEHICLE OD TABLE ESTIMATION IN NORTHFIELD The Northfeld, MN network ws selected for testng the computtonl propertes of OD tble dssggregton. Northfeld ws the smllest of three networks tested n the erler study by Horowtz (8). The network s ust 4.7 mles cross (est-west) for ts longest dmenson. Northfeld hs populton of bout 17,000 people. These current tests nvolved pssenger, commercl nd freght vehcles n sngle-clss trffc ssgnment. The Northfeld network s shown n Fgure 1. It hs 29 zones nd 12 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, dstrct boundres re ncluded s thck blue lnes whle zonl centrods nd externl sttons re shown s lrge green dots. Dstrctng for externl sttons, whch do not hve logcl polygons, s shown by thck blck lnes.

12 Horowtz 12 FIGURE 2 Northfeld Test Network nd Internl Dstrctng The ggregted OD tble ws creted by grvty model t the zone-to-zone level wth home-bsed-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 (12). 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:

13 Horowtz 13 All lnk weghts, w, were set to 1; 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; Equton 9 for Models III nd IV set U = -0.1t, where t s the congested nterzonl trvel tme. All x s nd y s were constrned to be between 0.2 nd 5. All smultons nd optmztons used re spred equlbrum trffc ssgnment (13), 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, propertes whch re benefcl to the process of estmtng row nd column fctors. 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, s wll be seen lter. Lnk delys were clculted usng opertonl nlyss methods from the 2000 Hghwy Cpcty Mnul for both sgnlzed nd unsgnlzed ntersectons. The tme perod of the smultons ws full 24-hours, ssgned sttclly. Computtonl results were obtned for the frst four models (I through IV) nd the ffth model (some specl flows) ws checked for resonbleness. A sttstcl summry of the frst four models 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 orgnl zone-tozone OD tble ws not chnged durng the smulton nd the dstrct OD tble ws bult by ggregtng the orgnl zone-to-zone OD tble. The verge ggregted OD flow ws 554 vehcles. The ordnry 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 (41 by 41) OD tble nsted of dstrct-to-dstrct (11 by 11) OD tble. A mor contrbutor to the error of the smulton ws n verge of pproxmtely 1200-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 some routes wth slght dvntges n free trvel tme nd were not mong those counted.

14 Horowtz 14 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 III dffers from Model I by mkng grvty-type dustments. Tble 1 shows tht Model II preserves the dstrctto-dstrct 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 I. Even the 9.2 trp error n the ggregted OD tble for Model III s not lrge. It s lkely tht Northfeld s too smll cty to show tht zonl-level, grvty-type dustment re sgnfcnt over nd bove the grvty model ssumptons lredy embedded n the dstrct-to-dstrct OD tble. The zonl-level grvty model effects re smply too subtle gven the need to lmt the number of MSA tertons, the need to set optmzton convergence crter, errors nherent wthn the trffc counts nd devtons n the trffc counts from theory. As n exmple, Fgure 3 shows mp of the computed destnton fctors from Model I. These destnton fctors hve been multpled by the scle fctor, s. Drker red htchng ndctes zones tht hve destnton fctors between 0.2 nd 0.4 nd the drker green htchng sgnfes destnton fctors between 3.2 nd 4. Brown zones re neutrl. 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 green zones s red zones n ech dstrct. The results for orgn fctors nd for Model I were smlr. It s dffcult to further nterpret Fgure 3 wthout consderble locl knowledge. A vsul comprson of lnk counts to ssgned volumes dd not provde ny ddtonl nsghts s to how the models could be further mproved. The scle fctor, s, ws selected by the lgorthm to be between 1.12 nd 1.17, dependng upon the model. The 1996 Quck Response Freght Mnul (14) sttes tht commercl vehcles mke up 10.5% of trffc on urbn prncpl rterls, so these scle fctors re bt greter thn wht would be expected f they were only ccountng for the bsence of trucks n the dstrct-todstrct OD tble. The reson for the lrger scle fctors thn expected s not entrely obvous, but mght be ttrbutble to prmeters from NCHRP Report #365 not beng exctly pproprte for Northfeld. All of the fctors chnged from ther orgnl vlues of x nd y of 1.0, nd ll of the fctors fell esly wthn the constrnts of beng between 0.2 nd 5.0. These ntl tests of Models I to IV demonstrte tht they cn gve plusble results, but these tests do not necessrly demonstrte tht the results re ccurte. Model V ws not tested s fully s the other four models becuse t ws not possble to dentfy sets of zone prs pror tht needed specl ttenton n Northfeld. When rndom zone prs were selected for Model V, both the ft to ground counts nd to the dstrct-level OD tble mproved n ech cse, but not to n nterestng degree. Lettng the lgorthm select the zone prs s not computtonlly fesble.

15 Horowtz FIGURE 3 Destnton Fctors for Model I t Zones (Shded Ares) nd t Externl Sttons (Dots) (Drker Green Indctes Lrger Fctors, Drker Red Indctes Smller Fctors nd Brown 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 I of the prevous tests. Retn the orgn nd destnton fctors. 2. Buld dstrct OD tble from the zone-to-zone OD tble by smply ddng up zone prs. 3. Assgn the zone-to-zone OD tble to the network nd obtn ssgned lnk volumes. 4. Set the ground count on ll rterl lnks equl to the ssgned lnk volumes, rtfclly. 5. Usng the dstrct-to-dstrct OD tble nd the rtfcl ground counts, compute new zone-to-zone OD tble.

16 Horowtz Compre the orgnl nd new orgn nd destnton fctors. Ths procedure resulted n 527 ground counts on the network tht were perfectly consstent wth the zone-to-zone OD tble. For these tests, the OD tble weght, z, ws left t 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. Tble 2 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-todstrct tble ws 623 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 (0.2 to 5.0) to determne of these bounds were nhbtng the estmton process. The results n Tble 2 for the less-constrned optmzton, lthough brely mproved, ndcte tht the constrnts were not serous ssue n fndng orgn nd destnton fctors. TABLE 2 Summry of Computtonl Tests, Artfcl Input Dt Model Averge Ground Count Averge Assgned Lnk Volume n Volumes RMS Dfference I Constrned I Unbounded 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.

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

18 Horowtz Recovered Destnton Fctors Orgnl Destnton Fctors FIGURE 5 Sctter Chrt Showng the Computed Destnton Fctors Agnst the Known Destnton Fctors The tests of Tble 2 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. It s lkely tht more MSA tertons or tghter optmzton convergence crter would strghten the sctter chrts even more. Of course, there s lwys trdeoff between computtonl precson nd the need for mngeble executon tmes whle recognzng the mount of error nherent n the source dt. CONCLUSIONS 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.

19 Horowtz 19 These methods developed n ths pper re prmrly 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 detled plnnng purposes. The methods mke effectve use of trffc count dt for dsggregtng OD tbles wthout resortng to cretng zonl VMT sttstcs, whch cn dstort the reltonshp between zone s level of ctvty nd trffc wthn tht zone. The choce mong the set of models lrgely rests on the needs of the study, but the smplest model (I) worked well for Northfeld. It gve ssgned volumes tht were respectbly close to trffc counts nd dd not serously dstort the dstrct-level OD tble. Models III nd IV, whch ncluded grvty-type ssumptons fled to mprove upon Models I nd II. The tests presented here mnly delt wth pssenger trvel, but the most nterest n OD tble dsggregton n recent yers reltes to freght. Therefore, ddtonl tests re needed on full-scle freght networks. ACKNOWLEDGMENTS Ths study ws funded by the Center for Infrstructure Reserch nd Educton, ntonl unversty reserch center of the US Deprtment of Trnsportton. REFERENCES 1. Mnyn Run nd Je Ln, Genertng County-Level Freght Dt Usng Freght Anlyss Frmework (FAF2.2) for Regonl Truck Emssons Estmton, pper presented t Trnsport Chcgo, June 5, Mchel Fscher, Jeffrey Ang-Olson nd Anthony L, Externl Urbn Truck Trps Bsed on Commodty Flows: A Model, Trnsportton Reserch Record, #1707, 2000, pp Jose A. Sorrtn nd Robert L. Smth, Development of Sttewde Truck Trp Forecstng Model Bsed on Commodty Flows nd Input-Output Coeffcents, Trnsportton Reserch Record, #1707, 2000, pp Krshnn Vswnthn, Dnel Begn, Vdy Mysore, nd Nnd Srnvsn, Dsggregtng Freght Anlyss Frmework Verson 2 Dt for Flord: Methodology nd Results, Trnsportton Reserch Record, #2049, 2008, pp Jkub Rownsk, Ker Ope nd Lzr N. Spsovc, Development of Method to Dsggregte the 2002 FAF2 Dt Down to the County Level for New Jersey, pper presented t the 87th Annul Meetng of the Trnsportton Reserch Bord, , Jnury Mks Alm, Edwrd Fekpe, Mohmmed Med, FAF2 Freght Trffc Anlyss, June 27, 2007, reports7/ndex.htm#toc 7. Torgn Abrhmsson, Estmton of Orgn-Destnton Mtrces Usng Trffc Counts A Lterture Survey, Interntonl Insttute for Appled Systems Anlyss, IR , My Aln J. Horowtz, Tests of Fmly of Trp Tble Refnements for Quck Response Trvel Forecstng, Trnsportton Reserch Record, Journl of the Trnsportton Reserch Bord, Number 1921, 2005, pp

20 Horowtz Aln J. Horowtz nd Lyl Dn, Tests of Dynmc Extensons to Fmly of Trp Tble Refnements Methods, Trnsportton Reserch Record, Journl of the Trnsportton Reserch Bord, #2003, 2007, pp Y. Chen, Blevel Progrmmng Problems: Anlyss, Algorthms nd Applctons, PhD Thess, Unversty of Montrel, C. S. Fsk, Trp Mtrx Estmton from Lnk Trffc Counts: The Congested Network Cse, Trnsportton Reserch B, Vol 23B, No. 5, 1989, pp Wllm Mrtn nd Nncy A. McGuckn, Trvel Estmton Technques for Urbn Plnnng, NCHRP Report #365, Aln J. Horowtz, Computtonl Issues n Incresng the Sptl Precson of Trffc Assgnments, Trnsportton Reserch Record Journl, #1777, 2001, pp Cmbrdge Systemtcs, et. l., The Quck Response Freght Mnul, Federl Hghwy Admnstrton, DOT-T-97-10, September 1996.