Word Count: 6850(text) + 7 (tables/figures) x 250 = 8600 equivalent words

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1 ESTIMATING SURVEY WEIGHTS WITH MULTIPLE CONSTRAINTS USING ENTROPY OPTIMIZATION METHODS Hllel Bar-Gera Department of Industral Engneerng and Management Ben-Guron Unversty of the Negev P O Box 653, Beer-Sheva, 845, Israel Tel: ; Fax: Emal: bargera@bguacl Karthk Kondur (correspondng author) Department of Cvl and Envronmental Engneerng Arzona State Unversty, Room ECG252 Tempe, AZ Tel: (48) ; Fax: (48) Emal: karthkkondur@asuedu Bhargava Sana Department of Cvl and Envronmental Engneerng Arzona State Unversty, Room ECG252 Tempe, AZ Tel: (48) ; Fax: (48) Emal: bsana@asuedu Xn Ye Department of Cvl and Envronmental Engneerng Arzona State Unversty, Room ECG252 Tempe, AZ Tel: (48) ; Fax: (48) Emal: xnye@asuedu Ram M Pendyala Department of Cvl and Envronmental Engneerng Arzona State Unversty, Room ECG252 Tempe, AZ Tel: (48) ; Fax: (48) Emal: rampendyala@asuedu Word Count: 685(text) + 7 (tables/fgures) x 25 = 86 equvalent words November 28

2 ABSTRACT Household travel surveys are the man source of nformaton for understandng ndvdual travel behavor and also for conductng a varety of travel analyses, from reportng descrptve statstcs to calbraton and valdaton of advanced travel forecastng models A challenge often faced by transportaton professonals s to accurately expand the survey households to represent the populaton The problem becomes even more complcated when dealng wth household travel surveys, because the goal s to fnd household weghts such that the dstrbutons of the characterstcs n the weghted sample not only match gven dstrbutons of households but also those of persons Ths paper presents an Entropy Maxmzaton methodology to estmate household survey weghts to match the exogenously gven dstrbutons of the populaton, ncludng both households and persons The paper also presents a Relaxed Formulaton to deal wth cases when constrants are not feasble and convergence s not acheved The methodology s appled to a large geography - Marcopa County regon, Arzona, and a small geography blockgroup, and estmaton results are presented Estmaton results show that the Strct Formulaton can be used to estmate the weghts when constrants mposed by dstrbutons of populaton characterstcs are feasble Relaxed formulaton can be used to estmate weghts when the constrants are nfeasble such that dstrbutons of the populaton characterstcs are satsfed to wthn reasonable lmts Keywords: Entropy Maxmzaton, Convex Optmzaton, Survey Expanson, Survey Data Weghts, Fttng Dstrbutons 2

3 INTRODUCTION Household travel surveys are a fundamental cornerstone of travel analyses, provdng data for a range of applcatons from basc descrptve statstcs to calbraton and valdaton of advanced forecastng models A maor concern n these surveys, lke many other types of surveys, s the proper representaton of the entre populaton by the sample of respondents Two man tools are typcally appled to address ths ssue: a) proper random samplng as part of the survey desgn; and b) assocaton of a weght to each response In many surveys, where each ndvdual s consdered as a separate response unt, the determnaton of weghts can be accomplshed n a relatvely smple manner The entre populaton s dvded nto subgroups, for example by gender, or by a combnaton of gender and age Then the weght of each subgroup s smply the rato between the proporton of the subgroup n the populaton and ther proporton among survey respondents Ths approach assumes that nformaton about the proporton of each subgroup n the entre populaton s known exogenously to the survey, for example from a census Choosng whch characterstcs to control, and how to dvde the populaton nto subgroups, requres proper consderaton Yet, once these choces are made, the weghts are determned by a straghtforward closed form computaton When exogenous nformaton s avalable on the margnal dstrbuton of each control varable separately, rather than the ont dstrbuton of the combnaton of all of the varables of nterest, the choce of weghts may be slghtly more complcated Determnng weghts for travel surveys s even more challengng because the response unt s a household, and not an ndvdual Therefore, weghts are appled to each household Typcally, separate exogenous nformaton exsts about the dstrbuton of household characterstcs (eg number of members, resdence type) and about the dstrbuton of person characterstcs In order to apply the smplstc weghtng scheme descrbed above, the dstrbuton should be gven by complete household structure For example, one structure could be: three member household n a suburban apartment, ncludng a Caucasan female age 3-4; a 3

4 Hspanc male age 2-3; and a chld age 5-2 In most practcal cases, dstrbutons by complete household structure are not avalable or not relevant The goal s therefore to fnd a weght for each household so that the dstrbutons of characterstcs n the weghted sample match the exogenously gven dstrbutons n the populaton, for both household characterstcs as well as person characterstcs Ths s not a trval goal Unlke the smple weghts descrbed above, there does not seem to be any closed form computaton for ths problem, and an teratve process s probably necessary In addton, n some cases, a perfect match cannot be obtaned, and t s only possble to get as close as possble to the target dstrbutons Ths may happen f the exogenous data s nconsstent, for example: 2 households wth three members; household wth four members; 5 females; and 4 males; suggestng a total populaton of, from the household nformaton and 9, from the person nformaton (gender) at the same tme It may also be a result of more subtle reasons, as dscussed later n the paper Fnally, n most cases, the same target dstrbutons can be obtaned by many dfferent sets of weghts For example, f there are two dentcal households, characterstc dstrbutons are determned by the sum of the weghts of the two households, and the breakdown of the weghts between the two households can be chosen arbtrarly In ths smple case, t s natural to assume that both households should receve the same weght The purpose of ths paper s to dscuss the problem of fndng weghts for household travel surveys and present a soluton usng a formal mathematcal methodology The proposed methodology can be appled to estmate household weghts rrespectve of the technque used for sample desgn Followng the background secton, we start wth a mathematcal formulaton of the condtons for matchng the exogenous dstrbuton, n the form of a set of lnear equatons We use ths formulaton to dscuss the cases where a perfect match cannot be found, and the cases when many dfferent sets of weghts lead to the same dstrbuton Next we propose Entropy maxmzaton as a method to dentfy the most reasonable set of weghts gven the exogenous 4

5 dstrbutons, leadng to a non-lnear convex optmzaton problem We also present a relaxed convex optmzaton problem to deal wth cases when a perfect match cannot be found Effcent convergng algorthms for both the non-relaxed and the relaxed problems are presented Smallscale numercal examples are used to llustrate the dscussed ssues Large-scale examples usng real survey data are then presented to demonstrate the practcalty of the proposed approaches 2 BACKGROUND 2 Survey Weghtng and Survey Expanson A challenge often faced by transportaton professonals wth travel survey data s the representaton of the survey data to embody the populaton Ths s frst acheved by a proper survey desgn and second, by assgnng approprate weghts to each response When an ndvdual s consdered as an ndependent response unt, the weghts for any subgroup may be estmated by the rato between proporton of the subgroup n the populaton and proporton of the subgroup n the sample A two step procedure used to expand the household weghts and estmate the ourney to work trps from the99 Boston Regonal Household-Based Travel Survey was presented by Harrngton and Wang (995) In another study usng the same dataset, Harrngton and Wang (995) expanded the household weghts usng a three step procedure to estmate the total trps made by all households The frst step nvolved basc expanson, followed by adustment of the expanded households to match the census household dstrbutons by land use zone level, and the fnal adustment accounted for the trps made by households that dd not turn n ther travel dares Ths smple method for generatng weghts cannot be used when only margnal dstrbutons of populaton characterstcs are avalable In the presence of margnal dstrbutons of each property, the weghts may be estmated usng teratve adustment methods Demng and Stephan (94) presented an teratve method to estmate the ont dstrbuton of the combnaton 5

6 of all characterstcs of nterest gven margnal dstrbutons of ndvdual characterstcs Ireland and Kullback (968) showed that the estmates of the ont dstrbuton from ths teratve procedure are Best Asymptotcally Normal (BAN) estmates, The study also showed that ont dstrbuton estmates mnmze the dscrmnaton nformaton subect to the margnal totals of each property and presented the convergence of the teratve procedure Fenberg (97) also presented a proof of convergence for ths teratve procedure usng a geometrc representaton Although these methods can be used n travel surveys to generate household weghts to match the exogenously gven margnal dstrbutons of household varables, the resultng person weghts (for the persons wthn a household) suffer from not matchng wth the gven exogenous person margnal dstrbutons In the presence of complete household structure margnal dstrbutons, the above methods can be used to generate household weghts to match the exogenously gven dstrbutons for both household and person propertes Ths data, however s not avalable n most cases and hence the need for an alternatve methodology that can produce household weghts whch are able to match both household and person level margnal dstrbutons The problem of estmatng weghts to match exogenously gven dstrbutons of multple dmensons of nterest s not ust lmted to Travel Surveys; ths problem s also experenced n the Consumer Expendture Survey and the Amercan Communty Survey Alexander and Roebuck (986) compared sx dfferent constraned mnmum dstance methods for estmatng household sample weghts such that the estmated weghts matched both household and person dstrbutons of nterest Asala (27) used a three-dmensonal rakng methodology for estmatng household and person weghts such that the nconsstency n the populaton forecasts of household attrbutes and person attrbutes of nterest s reduced In partcular, they were nterested n reducng the nconsstences n the estmates of the number of households (a household attrbute) and householders (a person attrbute), marred-couple households and spouses, unmarred partner households and unmarred partners, marred-couple subfamles and spouses n subfamles Ths paper presents an entropy maxmzng methodology for estmatng 6

7 weghts whch satsfy exogenously gven dstrbutons of varables under the dfferent dmensons of nterest The proposed methodology can be appled to estmate weghts for any survey sample wth two or more dmensons of nterest 22 Entropy Mechansm The Entropy maxmzaton methodology proposed n ths paper presents a way to estmate weghts that match the exogenously gven dstrbutons of the populaton ncludng both household and person level margnal dstrbutons Entropy maxmzaton prncples trace ther roots to statstcal thermodynamcs The development and the applcaton of Entropy maxmzaton technques have been conducted n the feld of transportaton n numerous studes One of the earlest efforts to use the prncples of Entropy maxmzaton n the feld of transportaton plannng was carred out by Wlson (967, 969, 97), n the estmaton of orgn-destnaton dstrbutons by gravty models A comprehensve dscusson of the Entropy formulaton and ts equvalence to gravty trp dstrbuton, Logt mode choce and the Logt stochastc traffc assgnment s presented n Oppenhem (995) Many transportaton researchers have nvestgated Entropy related models and, over the years, Entropy maxmzng technques have been used to develop models of trp dstrbuton, mode splt, and route choce Jornsten and Lundgren (989) presented the smlarty between the Entropy maxmzaton methodology and the tradtonal logt-type framework to model mode splts Further they presented that the logt model can be obtaned as a specal case of the general Entropy model Fang and Tsao (995) consdered the lnearly constraned Entropy maxmzaton problem wth quadratc cost and present a globally convergent algorthm whch was both robust and effcent The algorthm was appled to a problem of trp dstrbuton wth quadratc costs Akamatsu (997) showed the equvalence between an optmzaton problem wth lnk varables and the stochastc user equlbrum assgnment proposed by Daganzo and Sheff (977) The usual path flow based Entropy functon was decomposed nto a lnk flow based functon and the 7

8 lkeness between the decomposed form and the LOGIT assgnment were presented usng Markov propertes that form the bass of Dal s algorthm (Dal 97) Ross (989) proposed Entropy maxmzaton as a condton for the most lkely route flow soluton among all userequlbrum solutons A tme dependent combned model for trp dstrbuton and traffc assgnment was proposed by L et al (22) The orgn-destnaton matrx was estmated usng the observed Entropy value and mnmzng the total system travel tme Wang et al (26) studed the nhabtant trp dstrbuton patterns and presented a trp dstrbuton model based on Entropy optmzaton approach subect to typcal characterstc constrants based on orgn moments In general, these studes show that the Entropy maxmzaton methodology s capable of provdng solutons to constraned optmzaton problems across a wde range of applcatons n the feld of transportaton The problem of estmatng survey weghts can ndeed be formulated as a constraned optmzaton problem, where one s attemptng to mnmze the dfference between the weghted sample dstrbutons and known populaton dstrbutons across a set of control varables at both the household and person-levels Ths s the problem of nterest n ths paper 3 MODELING METHODOLOGY 3 Basc Condtons Formulaton A mathematcal formulaton, n the form of a set of lnear equatons, s developed to estmate household weghts whch match sample dstrbutons aganst the exogenously gven populaton dstrbutons A hypothetcal survey s presented and used to llustrate the methodology proposed n ths paper The data for the hypothetcal survey example s presented n Tables and 2 Table contans the household data and Table 2 presents the person characterstcs of the ndvduals belongng to the sample households Suppose that there are only two household characterstcs of nterest ownershp and locaton Each characterstc can take a set of characterstc values For example, ownershp of the household can take two values rented and 8

9 owned Smlarly, locaton of the household can be ether urban or suburban Two or more characterstcs of a household are combned to form a composte household type (CHT) The CHTs for the sample households are also shown n Table for the example data For example, RU represents a rentng household lvng n an urban area Table 2 contans person data from the example survey The table presents nformaton on the eleven persons resdng n the fve sample households Smlar to the household data, suppose there are two person characterstcs of nterest gender and ethncty The gender can ether be male or female and ethncty of the person can be Caucasan, Hspanc, or Asan The person characterstcs of nterest are combned to form a composte person type (CPT) The CPTs are shown n Table 2 for the example data For example, FC represents a female Caucasan The man component of the formulaton s the frequency matrx A Each column n A corresponds to a sample household Each row wthn a column gves the contrbuton from a sample household to a certan populaton characterstc Specfcally, hhc s represent household characterstc values and cpt s represent person characterstc values They are normally referred to as the control varables Each of the hhc s s ether one or zero dependng on whether a partcular household has a certan characterstc value or not The value cpt represents the number of persons wth the person characterstc belongng to household hhc hhc A = cpt cpt p q hhc hhc cpt cpt m mp m mq 9

10 The constructon of matrx A s llustrated wth the help of the hypothetcal survey data presented n Tables and 2 If only margnal totals by each populaton (household and person) characterstc are separately avalable, then matrx A s constructed Consder nformaton about household 3 (column 3 of matrx A ) The house s owned, therefore the element correspondng to the rented row s and that correspondng to the owned row s The house s located n an urban area, so the element correspondng to the suburban row s and that correspondng to the urban row s There are two males n the household (from Table 2) One s Caucasan and the other s Hspanc Therefore, the elements correspondng to the rows Male, Caucasan and Hspanc are two, one and one respectvely All the other elements n the thrd column are zeroes On the other hand, f the frequency dstrbuton for the composte household and person types obtaned by combnng more than one populaton characterstc s avalable then matrx A 2 s constructed Consderng household 3 agan, we wll llustrate the constructon of A 2 matrx Snce t s an owned house and located n an urban area, the element correspondng to the composte row OU row s and the elements correspondng to other household types RU, RS and OS wll be Out of the two persons belongng to the household, one s a Caucasan male and the other s a Hspanc male Therefore, the elements correspondng to the rows MC and MH are and respectvely Note that n the composte type representaton, there may be more than one person of the same type n the household For example there are two Caucasan females n household and the element correspondng to FC under column of household wll be equal to 2 The possblty of values above n matrx A hghlghts one of the man dfferences between travel surveys and other surveys

11 A = Re nted Owned Hhold Urban Suburban Male 2 Female Caucasan Person 2 Hspanc Asan A 2 2 = RU RS Hhold OU OS FC MC FH Person MH FA MA It must be noted that the household and person composte types are created by combnng household characterstcs and person characterstcs respectvely Combnng both household and person characterstcs together to form a household structure type would be prohbtve For the example scenaro, there are sx composte person types and assumng there are at most persons n a household, one could form a mllon household structures ( 6 ) Let m = p + q be the total number of rows n A, and let B be the column vector whch contans the exogenous nformaton about the dstrbuton of household characterstcs (eg number of members, resdence type, combned household types) and about the dstrbuton of person characterstcs These dstrbuton of household and person characterstcs of nterest may be obtaned from data sources such as a populaton census The problem then s to fnd a column vector W (weghts vector) satsfyng the constrants specfed by column vector B: A W = B and w =, 2 n () Here, w represents the weght attrbuted to each observaton (household) n the sample The weghts are estmated such that the weghted sum of all the households n the sample would match the exogenously gven dstrbutons of household and person characterstcs Ths formulaton of the constrants as a set of lnear equatons wth non-negatvty requrements provdes mportant nsghts nto the nature of the problem In partcular, whle n some cases, ths set of constrants may have a unque soluton, n many cases there wll ether be

12 nfnte number of possble solutons or there wll be no soluton at all Usually, the number of households s much larger than the number of constrants, leadng to an underdetermned system of equatons wth an nfnte set of feasble solutons These constrants can be embedded nto a lnear programmng problem, wth an artfcally chosen lnear obectve functon, and solved (at least n prncple) by any general lnear programmng method, such as the Smplex Dfferent artfcally chosen obectve functons are lkely to lead to many dfferent solutons The solutons found by the Smplex method, whle satsfyng the condtons on margnal dstrbutons, are corner solutons and potentally unsutable as survey weghts The weghts estmated wll be a combnaton of zero weghts and non-zero weghts The number of non zero-weghts n any corner soluton wll be equal (at most) to the number of constrants, meanng that the weght of most households wll be zero Ths knd of weghtng scheme may be undesrable for survey weghtng even though they satsfy the margnal dstrbutons Lnear programmng theory can also be used to analyze the condtons under whch the problem s nfeasble In the followng dscusson, two formulatons are presented to address these ssues Frst, a Strct Formulaton s presented for the case when the constrants are underdetermned Then a Relaxed Formulaton s presented that can deal wth cases where there s no feasble soluton for the set of orgnal constrants 32 Strct Formulaton In order to choose the most reasonable set of weghts subect to the constrants n equaton () an Entropy maxmzaton approach s proposed, where the Entropy s defned as E = n = w [ w / w ) ] ln( (2) Optmzaton of ths functon leads to a soluton where all the w s are close to the respectve w s In order to gve equal mportance to all canddates from the survey and not ntroduce any ndvdual bas, all the w s are assumed to be Ths reduces our problem to the 2

13 followng mnmzaton problem Please note that for convenence, the formulatons from ths pont forward are presented as mnmzng the negatve of the Entropy functon subect to: then be, Mnmze E = w [ w ) ] n = ln( (3) A W = B (4) w =, 2 n (5) Ths s a strctly convex problem, and therefore has a unque soluton The Lagrangan wll L = E + m n λ b aw (6) = = The frst order condton for mnmzaton gves the followng equatons: m L / w = ln( w ) λ a = (7) = w = exp m = a ( λ ) m λ a = exp (8) = Defne, c exp( λ ) (9) From equaton (8), w = m = c a () The Lagrangan dual functon ( λ, w( λ) ) L s then maxmzed and the frst order partal dervatve of the Lagrangan functon wth respect to λ returns the constrant n n m n L / λ = ( ( E)/ w )( w / λ ) + b aw + λl a w / λ = () = = l= = 3

14 w m / λ = a exp λ a (2) = n L / λ = b a w = (3) = The soluton can be obtaned usng a coordnate-by-coordnate search algorthm In ths algorthm a sngle varable λ s chosen at a tme and the problem reduces to a sngle dmenson problem of optmzng the Lagrangan wth respect to the chosen varable All varables are consdered n a sequence, one after the other and ths process s repeated teratvely untl convergence s acheved The sngle varable optmzaton can be vewed as movng from λ to λˆ Defne, ρ = exp( ˆ λ ) / exp( λ ) (4) Then from equatons (9) and (), w a wˆ = ρ (5) From equaton (3) the optmal pont wth respect to λ s where, n a w = = ˆ b (6) n = a a w ρ = b (7) Equaton (7) s a polynomal n ρ and has a unque soluton because the left hand sde s a monotoncally ncreasng functon from zero to nfnty Its soluton can be obtaned usng the Newton-Raphson method The procedure for estmatng weghts usng the Strct Formulaton s summarzed n a step by step manner below: Step : Set teraton number k = 4

15 Step 2: If k =, start wth an ntal set of λ s and correspondng w s (from equatons (9) and ()) else, use the values from the (k-) th teraton Step 3: Set = Step 4: Obtan ρ from Newton-Raphson usng equaton (7) for current, wth the latest (updated) set of w s Step 5: Update all the w s usng equaton (5) Step 6: Update λ for current usng equaton (4) Step 7: Increment by and f s less than or equal to m, go to Step 4; else, go to Step 8 Step 8: If the Average Absolute Relatve Error between the gven and calculated exogenous dstrbuton s less than a small postve value (such as ), then convergence s acheved and the algorthm ends If not, ncrement k by and go to Step 2 33 Relaxed Formulaton As dscussed n the ntroducton, there may be cases where a perfect match between the weghted sums and the exogenous dstrbutons of populaton characterstcs cannot be found because of nfeasblty n the constrants The ssues of nfeasblty can be addressed by usng the proposed relaxed convex optmzaton Each of the constrants s relaxed usng a relaxaton factorγ and a new vector representng the "chosen" margnals Bˆ s created The chosen margnal bˆ s obtaned as follows: ˆ = b γ (8) b where b s the orgnal margnal The obectve functon from the Strct Formulaton s modfed by addng a new term nvolvng ths relaxaton factor The problem now becomes, 5

16 n m Mnmze F = w [ ln( w ) ] + γ [ ln( γ ) ] = = α (9) subect to: A W = Bˆ (2) w =, 2 n (2) bˆ = b γ =, 2 m (22) γ =, 2 m (23) Entropy s agan used as a penalty functon to obtan relaxaton factors ( γ s) as close to as possble, where the factors α provde flexblty to control constrant mportance relatve to each other and relatve to the orgnal Entropy term The Lagrangan L can then be wrtten as, L = F + m n λ b γ aw (24) = = The frst order condton for mnmzaton s appled and the partal dervatve of the Lagrangan L wth respect to w wll gve the same equatons as (7) and (8) derved durng the development of the Strct Formulaton Frst order partal dervatve of the Lagrangan functon wth respect to λ wll return the constrant expressed as, L / γ = α ln( γ ) + b λ = (25) For notatonal convenence we defne p as b /α, therefore λ from equaton (25) can be p γ = exp( p λ ) = c (26) The Lagrangan dual functon ( λ, w( λ), γ ( λ) ) L s maxmzed and the frst order partal dervatve of the Lagrangan dual functon wth respect to λ wll return the constrant 6

17 n L / λ = b γ a w = (27) = ρ s defned as was done n equaton (4) and then, p γˆ = γ ρ (28) A non-lnear equaton n ρ s obtaned Ths can be solved as before usng the Newton- Raphson method Ths equaton also has a unque soluton because the left hand sde s monotoncally ncreasng from zero to nfnty whle the rght hand sde s decreasng from nfnty to zero n = a w ρ a = b γ ρ p (29) The procedure for estmatng the weghts usng the Relaxed Formulaton s summarzed n a step by step manner below: Step : Set teraton number k = Step 2: If k =, start wth an ntal set of λ s and correspondng Step 3: Set = else, use the values from the (k-) th teraton w s (from equaton (8)); Step 4: Obtan ρ from Newton-Raphson usng equaton (29) for current, wth the latest (updated) set of w s Step 5: Update all the w s usng equaton (5) Step 6: Update γ for the current usng equaton (28) Step 7: Update λ for current usng equaton (4) Step 8: Increment by and f s less than or equal to m, go to Step 4; else, go to Step 9 Step 9: If the Average Absolute Relatve Error between the gven and calculated exogenous dstrbuton s less than a small postve value (such as ), then 7

18 convergence crteron s met and algorthm ends If not, ncrement k by and go to Step 2 4 DATA DESCRIPTION As a case study, the Entropy based methodology s llustrated usng Census 2 data The obectve s to expand the fve percent Publc Use Mcrodata Sample (PUMS) to represent the populaton resdng n the county The exogenous margnal dstrbutons of household and person level characterstcs are obtaned from the Census 2 summary fles The methodology was frst appled to represent the total populaton resdng n Marcopa County The algorthm was then appled to a blockgroup wthn the county Ths was done to llustrate the applcablty of the algorthm to geographes of dfferent szes and provde gudance on mplementaton of the algorthm The performance of the algorthm for small geography (blockgroup) and large geography (county) s documented and observatons from ths evaluaton exercse are presented As mentoned n secton 3, the constrants can ether be margnal dstrbutons of household and person characterstcs or ont dstrbutons of composte household and person characterstcs The composte ont dstrbutons of populaton characterstcs tend to mpose more control on the estmaton of household weghts and therefore produce household weghts for the sample that more closely represent the populaton under consderaton In ths study the Entropy algorthm s appled by mposng ont dstrbutons of both household and person characterstcs as constrants The ont dstrbutons of household and person characterstcs are not readly avalable from Census data However, Census 2 summary fles provde margnal dstrbutons of household and person level characterstcs whch can be used to generate ont dstrbutons of composte household and person characterstcs The Iteratve Proportonal Fttng (IPF) procedure descrbed n Beckman et al (996) was used to generate the ont dstrbutons of composte characterstcs 8

19 The algorthm was appled to a large scale and a small scale geography usng the same set of household and person characterstcs of nterest The household propertes used to generate composte household ont dstrbutons nclude household type, household sze and total household ncome Person gender, age, and race were used to generate composte person ont dstrbutons Table 3 shows the household and person characterstcs used n ths study, and lstng of categores for each populaton characterstc used n the example exercse 5 MODEL ESTIMATION RESULTS Ths secton presents estmaton results for the large and small geography consdered n ths paper Frst, results are presented for the entre regon of Marcopa County whch also consttutes the large geography analyzed n ths study Then, the estmaton results are presented for a blockgroup whch comprses the small geography analyzed The estmaton results for the blockgroup hghlghts potental ssues wth usng the Strct Formulaton for small geographes Results are then presented for the blockgroup usng the Relaxed Formulaton The algorthm was coded n MATLAB The ont dstrbutons of composte person and household characterstcs used as constrants n the algorthm were estmated usng the IPF procedure presented n Beckman et al (996) The followng measures were used to montor the progress of the algorthm towards convergence and evaluate the ft of the estmated weghts a) Absolute Relatve Error (ARE): The Absolute Relatve Error (ARE) measures the devaton of the estmated weghts from the constrant under consderaton The ARE e for a populaton characterstc s defned as: e = a, b w b where e = absolute relatve error for the composte populaton characterstc 9

20 a, = value of element n matrx A correspondng to sample pont and populaton characterstc b = value of the populaton characterstc from the margnal dstrbuton w = weght attrbuted to sample pont from the prevous teraton In an deal stuaton, when the household weghts perfectly match the populaton dstrbutons, the ARE value s zero for all populaton characterstcs The Maxmum Absolute Relatve Error (MARE) s the maxmum of ARE for all populaton characterstcs In ths study, the progress of the algorthm was montored usng the MARE A value of MARE close to zero ndcates convergence of the algorthm for a partcular geography Although the ARE s a very good measure for montorng convergence, t can sometmes be msleadng For example a value of ARE equal to may be obtaned when the estmate s and the control s 2, and also when the estmated value s 2 and the control value s It can easly be seen that, the dfference n the frst example may be reasonable, but t may not be acceptable n the second example Therefore, although the ARE was used to montor the progress of the algorthm, t was not used to compare the estmated populaton characterstc dstrbutons aganst those obtaned from the IPF procedure b) Ch-squared: The Ch-squared test provdes a statstcally sound way to compare dscrete dstrbutons The Ch-squared statstc for estmated weghted sums wth populaton characterstcs s defned as: 2 χ = 2 a, w b b where χ 2 = Ch-squared statstc 2

21 a, = value of element n matrx A matrx correspondng to sample pont and populaton characterstc b = value of the populaton characterstc from the margnal dstrbuton w = weght attrbuted to sample pont from the last round of teraton Ths test can be used to compare the dstrbuton of populaton characterstcs from the estmated weghts aganst the dstrbuton obtaned usng the IPF procedure In an deal case, when the household weghts perfectly match the populaton dstrbutons, the Ch-squared value s zero In ths estmaton effort, a value of Ch-squared statstc close to zero was used as evdence to suggest smlarty between dstrbutons 5 Large Geography The algorthm was frst appled to the entre Marcopa County regon Convergence was acheved and t was observed that the weghted sums from the estmated weghts almost perfectly match the populaton characterstc constrants wthn computer precson lmts Fgure a shows the progress of the algorthm usng the Strct Formulaton Absolute Relatve Error and the Chsquared statstc were used to montor the progress of the algorthm Fgure a shows these measures usng a logarthmc scale on the Y axs and the teraton number on the X axs It can be seen that as the number of teratons ncreases, the measures approach a value close to zero The MARE observed after teratons was 47 x -3 whle the correspondng 2 value was x -3 The small value of MARE shows that the dstrbuton of the populaton characterstc obtaned usng the estmated weghts almost perfectly matches the ont household and person type constrants obtaned from the IPF procedure Ths observaton s further renforced by the very small value of the Ch-squared statstc suggestng that the dstrbutons are practcally dentcal The MARE and Ch-squared statstc measures ndcate that convergence was acheved usng the Strct Formulaton for the entre Marcopa County regon 2

22 52 Small Geography The lnes represented by S n Fgures b and c show the progress of the algorthm usng the Strct Formulaton for the blockgroup The MARE and Ch-squared statstc measures n the fgures are shown on the Y axs usng a logarthmc scale and the teraton number s shown on the X axs It can be seen from the fgures that the curve representng the MARE plateaus after 6 teratons and remans at a value of about 54 The Ch-squared statstc also plateaus after about 6 teratons and decreases at a very slow rate After teratons, the MARE observed for the Strct Formulaton was about 54 and the Ch-Squared value observed was about 38 The hgh value of MARE suggests that there s at least one populaton characterstc whose constrant was not satsfed, n other words a soluton was not found As dscussed earler, ths s caused when the gven ont household and person type constrants are nfeasble In ths partcular example we were actually able to verfy that the problem s nfeasble due to contradcton n the orgnal constrants The results from the IPF procedure provde a target total weghted sum of 752 for seven person non-famly households n the lowest ncome level category There are only two households n the survey belongng to ths composte type In one of these households, there are sx males 5-24 years old belongng to ethncty category 6 The target total for ths composte person type s 24, hence the maxmum possble weght for the frst household s 36 In the second household, there are two females 5-4 years old n the sxth ethncty category The target total for ths composte person type s 2 Hence the maxmum possble weght for the second household s In total the sum of the weghts of the two households n ths composte household type cannot exceed 46, about 6 tmes lower than the target total Clearly t s not possble to satsfy these three condtons smultaneously wth the gven sample of households A possble resoluton s to use the Relaxed Formulaton Fgures b and c plot the MARE and Ch-squared statstc for the Relaxed Formulaton usng dfferent levels of Relaxaton (R, R2, R3, R4, R5) The Relaxaton levels R, R2, R3, R4, and R5 were acheved by mantanng p from 22

23 equaton (26) at,,,, respectvely for both person and household constrants Relaxaton R corresponds to the most relaxed case and R5 corresponds to the least relaxed case The results from R5 are very smlar to the strct formulaton results because of the very lttle amount of relaxaton allowed by the correspondng p value It can be seen from Fgure b that wth relaxaton, the value of MARE has decreased when compared to the Strct Formulaton The change n the value of MARE decreases wth ncrease n level of relaxaton The change n the value of MARE s greater for Relaxaton R compared to that of Relaxaton R5 whch almost overlaps the curve traced by the Strct Formulaton The fgure also shows that convergence was acheved for the Relaxed Formulaton for cases R, R2, and R3 whle the cases R4 and R5 exhbt behavor smlar to the Strct Formulaton It should be noted that convergence n the Relaxed Formulaton comes at the prce of alterng the constrants; as a result the dstrbuton of populaton characterstcs obtaned from the algorthm are further away from the target dstrbutons The MARE calculated n the Relaxed Formulaton s wth respect to the relaxed constrants and not usng the constrants from the IPF procedure Therefore the MARE for the Relaxed Formulaton only provdes a way to montor the progress of the algorthm and should not be used to measure the goodness of ft Goodness of ft under the Relaxed Formulaton can be evaluated usng the Ch-squared statstc Fgure c shows that the Ch-squared value does not mprove a lot wth relaxaton, especally for cases wth Relaxaton R and R2 Ths s expected because, when the constrants are relaxed, one s movng away from the obectve constrants; as a result, the Ch-squared statstc s larger compared to the Strct Formulaton Wth relaxaton R3 and R4, Ch-squared statstcs of 74 and 26 were estmated showng a margnal mprovement n ft wth relaxaton It s nterestng to note that for Relaxaton R5 the Ch-squared value s very smlar to that of the Strct Formulaton ndcatng that very small relaxatons may not contrbute much towards convergence and also there s no loss n the goodness of ft Alternate formulatons for relaxaton of constrants can further mprove the ft and s a good canddate for further research n the area 23

24 Fgure 2 shows the scatter plots for Relaxaton R, R3 and S wth weghted rato on the Y axs and the orgnal constrant on the X axs Weghted rato s defned as the rato of the weghted sum for the populaton characterstc to that of the orgnal constrant The values on both X and Y axes are plotted usng a logarthmc scale The scatter plots shown n the fgure correspond to three levels of relaxaton where R corresponds to hghest relaxaton, R3 corresponds to moderate relaxaton and Strct Formulaton corresponds to no relaxaton It was observed that the weghted ratos for the Relaxaton R3 are close to the Strct Formulaton whle that for Relaxaton R are very dfferent It can be seen that weghted ratos are larger for smaller values of orgnal constrants On the other hand, the weghted ratos are smaller for orgnal constrants wth hgher magntude It was also observed that the weghted ratos for relaxaton R vared from 4 x -5 to 423, for relaxaton R3 the varaton was between 7 and 29, and fnally for the strct formulaton the varaton was from to 2274 The ranges of weghted ratos corroborate our earler observaton showng slght mprovement n the Ch-squared statstc wth Relaxaton levels R3 and R4 It can therefore be seen that the Relaxed Formulaton can be used to estmate weghts when the constrants are nfeasble and stll be able satsfy the populaton characterstc constrants to wthn reasonable lmts The choce of level of relaxaton needs further exploraton and s another good canddate for research n the area 6 CONCLUSIONS Transportaton professonals are often faced wth the challenge of accurately expandng the survey households to represent the populaton The problem becomes even more complcated when dealng wth household travel surveys, because the goal s to fnd household weghts such that the dstrbutons of the populaton characterstcs from the estmated weghts should not only match gven dstrbutons of households but also that of persons An Entropy Maxmzaton 24

25 methodology s presented n ths paper that can not only match the exogenously gven dstrbutons of the households but also the person dstrbutons The model estmaton results provde valuable nsghts nto the actual formulaton and the accuracy of the household weghts for dfferent extents of geography The Strct Formulaton presented n ths paper can be used to estmate weghts when the constrants are feasble The resultng weghted sums almost perfectly match the dstrbutons of the populaton characterstcs Therefore, Strct Formulaton s approprate for geographes wth larger extents where the constrants are generally feasble On the other hand, smaller geographes are more lkely to suffer from nfeasble constrants In the case we examned, the results of the strct formulaton may be acceptable Addtonal flexblty can be obtaned by the relaxed formulaton as was observed wth moderate relaxaton levels Addtonal research s needed to evaluate the potental advantages of ths flexblty, and partcularly to study the relatonshp between the level of relaxaton, level of ft and the computaton tme 7 REFERENCES Akamatsu T (997) Decomposton of Path Choce Entropy n General Transport Networks Transportaton Scence 3(4): Alexander CH and Roebuck MJ (986) Comparson of Alternatve Methods for Household Estmaton Amercan Statstcal Assocaton Proceedngs of the Secton on the Survey Research Methods: 64-7 Asala ME (27) Weghtng and Estmaton Research Methodology and Results From the Amercan Communty Survey Famly Equalzaton Proect Federal Commttee on Statstcal Methodology Research Conference Beckman RJ, Baggerly KA & McKay MD (996) Creatng Synthetc Baselne Populatons Transportaton Research Part A 3(6):

26 Daganzo, CF & Sheff Y (977) On Stochastc Models of Traffc Assgnment Transportaton Scence : Demng WE & Stephan FF (94) On a Least Squares Adustment of Sampled Frequency Table When the Expected Margnal Totals are Known The Annals of Mathematcal Statstcs (4): Dal RB (97) A probablstc Multpath Traffc Assgnment Algorthm whch Obvates Path Enumeraton Transportaton Research 5: 83- Fang SC, & Tsao H-SJ (995) Lnearly-Constraned Entropy Maxmzaton Problem wth Quadratc Cost and Its Applcatons to Transportaton Plannng Problems Transportaton Scence 29(4): Fenberg SE (97) An Iteratve Procedure for Estmaton n Contngency Tables The Annals of Mathematcal Statstcs 47(3): Harrngton I & Wang C-Y (995) Adustng Household Survey Expanson Factors Ffth Natonal Conference on Transportaton Plannng Methods Applcatons -Volume II Harrngton I & Wang C-Y (995) Modfyng Transt Mode Share n Household Survey Expanson Transportaton Research Record 496: Ireland CT & Kullback S (968) Contngency Tables wth Gven Margnals Bometrka 55(): Jornsten KO & Lundgren JT (989) An Entropy-Based Modal Splt Model Transportaton Research Part B 23(5): L Y, Zlaskopoulos T & Boyce, D (22) Combned Model for Tme-dependent Trp Dstrbuton and Traffc Assgnment Transportaton Research Record 783: 98- Oppenhem N (995) Urban Travel Demand Modelng: From Indvdual Choces to General Equlbrum Wley-Interscence Publcaton Ross TF, McNel S & Hendrckson C (989) Entropy Model for Consstent Impact Fee Assessment Journal of Urban Plannng and Development 5(2):

27 Sheff Y (985) Urban Transportaton Networks: Equlbrum Analyss wth Mathematcal Programmng Methods Prentce Hall, Englewood Clffs Stopher PR & Metcalf HMA (996) Methods for Household Travel Surveys NCHRP Synthess of Hghway Practce 236, Transportaton Research Board, Washngton DC Wang D, Yao R & Jng C (26) Entropy Models of Trp Dstrbuton Journal of Urban Plannng and Development 32(): Wlson AG (967) A Statstcal Theory of Spatal Dstrbuton Models Transportaton Resrearch (3) Wlson AG (969) The Use of Entropy Maxmzng Models n the Theory of Trp Dstrbuton, Mode Splt and Route Splt Journal of Transport Economcs and Polcy 3: 8-26 Wlson AG (97) The Use of the Concept of Entropy n System Modellng Operatonal Research Quarterly 2(2):

28 Household Index TABLE Household Data for the Sample Survey Ownershp R - Rent, O - Own Locaton U - Urban, S - Suburban Composte Household Type R U RU 2 R S RS 3 O U OU 4 O S OS 5 O S OS Household Index TABLE 2 Person Data for the Sample Survey Person Index Gender M - Male, F - Female Ethncty C - Caucasan, H - Hspanc, A - Asan Composte Person Type F C FC 2 F C FC 2 M C MC 2 2 F A FA 3 M H MH 3 2 M C MC 4 F H FH 4 2 M H MH 5 M H MH 5 2 F A FA 5 3 F H FH

29 TABLE 3 Household and Person Characterstcs Used to Create the Jont Dstrbutons Descrpton Value Household Characterstcs Household Type Famly: Marred Couple Famly: Male Householder, No Wfe 2 Famly: Female Householder, No Husband 3 Non-famly: Householder Alone 4 Non-famly: Householder Not Alone 5 Household Sze Person 2 Persons 2 3 Persons 3 4 Persons 4 5 Persons 5 6 Persons 6 7 or more Persons 7 Household Income $ - $4,999 $5, - $24,999 2 $25, - $34,999 3 $35, - $44,999 4 $45, - $59,999 5 $6, - $99,999 6 $, - $49,999 7 Over $5, 8 Person Characterstcs Gender Male Female 2 Age Under 5 years 5 to 4 years 2 5 to 24 years 3 25 to 34 years 4 35 to 44 years 5 45 to 54 years 6 55 to 64 years 7 65 to 74 years 8 75 to 84 years 9 85 and more Ethncty Whte alone Black or Afrcan Amercan alone 2 Amercan Indan and Alask Natve alone 3 Asan alone 4 Natve Hawaan and Other Pacfc Islander alone 5 Some other race alone 6 Two or more races 7 29

30 FIGURE a Goodness of Ft Measure for the Large Geography usng Strct Formulaton of the Algorthm E+6 E+4 MARE χ2 Goodness of Ft Measure (Log Scale) E+2 E+ E-2 E-4 E-6 E-8 E- E-2 E-4 E E-8 E-2 Iteraton Number 3

31 FIGURE b Maxmum Absolute Relatve Error (MARE) for the Small Geography E+5 MARE - R MARE - R3 MARE - R2 MARE - R4 Maxmum Absolute Relatve Error (MARE) E+3 E+ E- E-3 E-5 E-7 E-9 E- MARE - R5 MARE - S E-3 E-5 Iteraton Number 3

32 FIGURE c Ch-squared Statstc for the Small Geography E+ E+9 E+8 χ2 - R χ2 - R3 χ2 - R5 χ2 - R2 χ2 - R4 χ2 - S E+7 Ch-squared Statstc E+6 E+5 E+4 E+3 E+2 E+ E+ E Iteraton Number 32

33 FIGURE 2 Plot of Relaxaton Rato versus Orgnal Constrants E+3 E+2 Rato - R Rato - R3 Rato - S E+ Weghted Rato E+ E-6 E-5 E-4 E-3 E-2 E- E+ E+ E+2 E- E-2 E-3 E-4 E-5 Orgnal Constrant 33