14.1

Chapter 14Trip Distribution

In th is chapt er , th e mechanics of the second crime t ravel dem and m odeling st age -t r ip d ist r ibu t ion - is expla ined. T rip distribu tion is a m odel of th e num ber of tr ips th atoccur between each or igin zone and each des t ina t ion zone. It u ses the p red ict ed number oft r ips or igin a t in g in each or igin zon e (t r ip product ion model) a nd the predict ed number oft r ips endin g in each dest ina t ion zone (tr ip a t t r action model). Thus, t r ip d ist r ibu t ion is amodel of t r avel between zones - tr ips or links. The m odeled t r ip dist r ibut ion can then becompared to the actua l d is t r ibu t ion to see whether the model p roduces a reasonableapproximat ion .

Theoret ica l Backgroun d

Th e t heoret ical backgr oun d beh ind t he t r ip d ist r ibu t ion m odule is pr esen ted firs t . Next , the specific pr ocedures and t est s a re discussed wit h the m odel bein g illus t ra ted wit hda ta from Ba lt imore County.

Lo g ic o f t h e Mo d e l

Tr ip dis t ribu t ion usua lly occu r s t h rough an a lloca t ion model t ha t sp lit s t r ip s fromeach origin zone into distinct destina tions. Tha t is, th ere is a m at rix which r elat es thenumber of t r ips origin a t ing in ea ch zone t o th e n umber of t r ips en din g in ea ch zone. F igure14.1 illust ra tes a typica l ar rangemen t . In t h is mat r ix, th ere are a number of or igin zones,M, an d a number of dest ina t ion zones, N. The origin zones include a l l the dest ina t ionzones bu t may also include some addit iona l ones. The r easons t ha t there would bed ifferen t numbers of zones for the or igin and des t ina t ion models a re tha t cr ime da ta foroth er jur isd ictions a re not a vailable bu t tha t a sizea ble n umber of crim es tha t occur red inthe study jur isd iction are commit ted by individua ls who lived t hose oth er jur isd ictions. If itwere possible to obta in crime da ta for the City of Balt imore, t hen it would be pr eferable tohave th e same number of zones for both the or igin file an d t he dest ina t ion file.

For exam ple, with the Balt imore Coun ty dat a tha t a re being us ed t o illus t ra te themodel, th ere are 325 dest ina t ions zones for Balt imore Coun ty while th e or igin zonesin clu de both the 325 in Ba lt im ore Cou nty a nd 207 more from the adjacen t City ofBa lt imore. As chapt er 12 poin ted out , the study a rea sh ould exten d beyond t he m odelingarea un t il the origins of at lea st 95% of a ll t r ips en din g in the s tudy a rea a re counted.

Each cell in the mat r ix in dica tes the number of trips t ha t go from each or igin zoneto each dest in a t ion zon e. To use the exa mple in figure 14.1, t here were 15 t r ips from zon e1 t o zone 2, 21 t r ips from zone 1 t o zone 3, and so fort h . Note t ha t the t r ips a reasym met r ical; tha t is, t r ips in one d irection are differ en t than t r ips in the opposit edir ect ion . To use the table, t here were 15 t r ips from zon e 1 to zon e 2, bu t only 7 t r ips fromzone 2 to zone 1.

14.2

The t r ips on the diagona l ar e intra-zonal t r ips , t r ips tha t origina te and end in thesam e zone. Again, to use th e exam ple below, th ere were 37 tr ips th at both originat ed andended in zon e 1, 53 t r ips tha t both or igin a ted and ended in zon e 2, a nd so for th .

In su ch a model, cons tancy is m ainta ined in tha t the number of t r ips origina t ingfrom a ll or igins zones must equa l the number of t r ips end ing in a ll des t ina t ion zones . Th isis t he fun da men ta l ba lancing equ a t ion for a t r ip d ist r ibu t ion . In equ a t ion form, it isexpressed as:

M N

G O i = G D j (14.1)I=1 j=1

wh er e t he origins, O i, a re summed over M origin zones while t he dest ina t ions , D j, a r esu mmed over N dest ina t ion zones. To use the exa mple below, the t ota l number of originsis equa l to th e tota l nu mber of dest ina t ions, an d is equa l to 43,240

Figure 14.1

Exam ple Crime Origin-De stin ation Matrix

The ba lancing equ a t ion is im plemen ted in a ser ies of st eps t ha t include m odelingthe number of cr imes or igina t ing in ea ch zone, add ing in t r ips origina t ing from out side t hestudy a rea (ext erna l t r ips), a nd sta t is t ica lly ba la ncin g t he or igin s and dest in a t ion s so tha t

14.3

equ a t ion 14.1 h olds . This was done in the t r ip gen er a t ion st age. But , it is es sen t ia l tha tthe st ep sh ould have been completed for the t r ip dist r ibut ion to be implem ented.

Ob se rv e d a n d P r e di ct e d D is tri bu ti on s

Ther e a re t wo tr ip d ist r ibu t ion mat r ices tha t need to be dist inguish ed. The firs t isthe observed (or empir ica l) d is t ribu t ion . Th is is the actua l number of t r ip s t ha t a r eobser ved t raveling bet ween each origin zone and each dest ina t ion zone. In gener a l, withcr ime data , such an empir ica l d is t ribu t ion wou ld be obta ined from an a r r es t r ecord wherethe residen ce (or a r rest ) loca t ion of each offender is list ed for each cr ime t ha t the offenderwas cha rged with . In t h is case, th e residen ce/a r rest loca t ion would be consider ed t heorigin wh ile t he cr ime loca t ion would be cons idered the dest ina t ion.

In chapt er 12, it was m ent ioned t ha t there is alwa ys un cer ta int y as t o the t rueor igin loca t ion of a cr im e in ciden t , whether the offender actua lly t r aveled from theresidence locat ion t o th e cr ime loca t ion or even wh et her the offender wa s a ctu a lly living a tthe residence loca t ion . But absen t any a lt erna t ive evidence, a meanin gfu l d is t r ibu t ion canst ill be obta ined by sim ply t rea t ing the r esidence locat ion as a n approximate origin .

The obser ved d ist r ibu t ion is calcula ted by sim ply en umer a t ing the number of t r ipsby each origin-dest ina t ion combina t ion . This is somet imes ca lled a t rip link (or t r ip pa ir ). There are not any special st a t ist ics oth er than a simple two-way cross-classifica t ion table.

The second d ist r ibu t ion , however , is a model of the t r ip d ist r ibu t ion mat r ix. This isusu a lly ca lled th e predicted distr ibut ion. In th is case, a simple model is used toappr oximate the actua l empirical dist r ibut ion . The t r ips origina t ing in ea ch or igin zone area llocat ed to dest ina t ion zones usu a lly on the ba sis of bein g dir ectly pr opor t iona l toa t t ract ion s and in versely propor t ion a l t o cost s (or im pedance).

Thus, a model of the t r ip d ist r ibu t ion is produced tha t approximates the actu a l,em pir ical dist r ibu t ion . Ther e a re a number of rea sons why th is would be u seful - to be ableto apply the model t o a differen t da ta set from which it was ca libra ted, t o use the model foreva lua t ing a policy inter ven t ion , or to use t he m odel for forecas t ing fu ture crim e t r ipdis t r ibu t ion . But , wha tever the reason , it has to be rea lized tha t the model is not theobserved dis t r ibu t ion . There will a lways be a difference between the obs erved dis t r ibu t ionfrom which a model is const ru cted a nd t he r esulting predicted distr ibut ion of th e model. Itis useful t o compare the observed and predict ed model because th is a llows a test of thevalidity of the impeda nce funct ion . But , ra rely, if ever, will the pr edicted dist r ibut ion beiden t ical t o th e em pir ical d ist r ibu t ion.

Anoth er way to th ink of th is is th at th e actu al distr ibut ion of crime tr ips is complex,represen t in g a la rge number of differen t decis ion s on the par t of offen ders who do notnecessar ily use the same decision logic. The m odel, on the oth er hand, is a sim plea llocat ion on t he ba sis of th ree or , somet imes, four var iables . Almost by definit ion , it willbe much simpler than the rea l dist r ibut ion . Still, the simple model can often capt ure themost impor tan t character ist ics of the actua l dist r ibut ion . Hen ce, modeling can be an

14.4

extr emely useful an a lyt ica l exercise tha t a llows oth er types of ques t ions t o be asked tha ta re not possible wit h ju st the obs erved dis t r ibu t ion .

The Gravi ty Model

A model tha t is usua lly used for t r ip dis t r ibu t ion is tha t of the gravity fun ction , anapp lica t ion of Newt on’s funda men ta l law of a t t r action (Oppen heim , 1980; F ield a ndMa cGregor, 1987; Or tuzar and Willu msen , 2001). Much of the discuss ion below is a lsorepea ted in cha pt er 9 on journey t o cr ime since ther e is a common t heoret ical basis . In theoriginal Newtonian form ulat ion, the a tt ra ction, F, between t wo bodies of respective massesM 1 an d M 2, sepa ra ted by a d ist ance D, will be equa l t o

M1 M2

F = g ----------------- (14.2) D2

where g is a const an t or scalin g factor wh ich en su res tha t the equ a t ion is ba lanced inter ms of th e m ea su rem en t un it s (Oppenheim , 1980). As we a ll know, of cour se , g is thegravita t iona l cons tan t in the Newton ian formula t ion . The numera tor of the funct ion is theattraction t erm (or , a lt erna t ively, the a t t r act ion of M2 for M1) wh ile the denomin a tor of theequ a t ion, d 2, indica tes tha t the a t t r act ion between the two bod ies fa lls off as a funct ion oftheir squared d is t ance. It is an im pedance (or resist ance) term.

Soc ia l Applications o f the Gravi ty Concept

The gra vity model ha s been the bas is of many app lica t ions t o human societies a ndhas been applied t o social int eract ions sin ce the 19 t h cen tury. Ra venstein (1895) andAndersson (1897) applied the concep t to the ana lys is of migra t ion by a rgu ing tha t thetendency t o migra te between regions is in versely propor t ion a l t o the squared dis t ancebet ween the r egions . Reilly’s ‘law of ret a il gravita t ion’ (1929) applied the Newt onia ngra vity model directly an d su ggest ed t ha t reta il t r avel between two centers would bepropor t ion a l t o the product of their popula t ion s and in versely propor t ion a l t o the square ofthe dist ance sepa ra t ing them:

P i P j

I ij = "----------------- (14.3) Dij

2

where I ij is th e int eraction between center s I an d j, P i an d P j ar e the respective populat ions,D ij is th e dista nce between t hem r aised to th e second power an d " is a ba lancing cons tan t . In the m odel, t he in it ia l popu la t ion, P i, is called a production while the second popula t ion ,P j, is called an attraction .

St ewar t (1950) and Zipf (1949) applied t he concept to a var iety of ph enomena(migra t ion , fr eigh t t r a ffic, in format ion ) usin g a sim plified form of the gr avity equa t ion

14.5

P i P j

I ij = K ----------------- (14.4) Dij

where the terms a re as in equa t ion 14.3 but the exponent of dist ance is on ly 1. Given apa r t icula r pa t t er n of in ter action for any type of goods , service or human activit y, anopt imal loca t ion of facilities sh ould be solvable.

In the St ewa r t /Zipf framework, the t wo P’s wer e both popu la t ion sizes . However , inmodern use, it is n ot n ecessa ry for t he productions a nd a t t r actions t o be iden t ical u n it s(e.g., P i cou ld be popu lat ion while P j cou ld be em ploymen t ).

Tri ps as In te ra ct io n s

It should be obviou s tha t th is in teract ion equa t ion can be applied to t r ips from onearea (zon e) to another . Changing t he sym bols sligh t ly, t he tota l volu me of t r ips from apar t icu la r or igin zon e, i, t o a sin gle loca t ion , j, is dir ect ly propor t ion a l t o the product of thep roduct ions a t i and the a t t r act ions a t j, a nd in versely propor t ion to the im pedance (orcost ) of t r avel between the two zones

" P i $ Aj

T ij = --------------- (14.5) Dij

where P i a re t he productions for zone I, Aj a re t he a t t r actions zone j, " is a pr oductionconstan t , $ is an at tr action const an t, and D ij is the im pedance (cost ) of t r avel between zon eii a nd zone j.

Over time, the concept ha s been genera lized an d applied to ma ny different types oft ravel beha vior . For exam ple, Hu ff (1963) applied t he concept to reta il t r ade betweenzones in an u rban a rea using the genera l form of

S j8

Aij = " ----------------- (14.6) Dij

D

where Aij is the number of purchases in loca t ion j by r esiden t s of loca t ion I, S j is thea t t ractiveness of zone j (e.g., squ are foota ge of ret a il space), D ij is the dist ance betweenzones I and j, " is a const an t , 8 is the exponent of S j, and D is the exponent of dis t ance(Bossard, 1993). Dij

-D is somet imes ca lled an inverse d istance funct ion . This differ s fromthe t radit ion a l gravity fu nct ion by a llowin g t he exponents of the product ion from loca t ion I,t he a t t r act ion from loca t ion j, and t he dist ance bet ween zones t o var y.

Equa t ion 14.6 is a single const ra in t model in tha t only the a t t r act iveness of acommercia l zone is const r a ined , t ha t is the sum of a ll a t t r act ions for j mus t equa l the tota l

14.6

a t t raction in the r egion. Again , it can be gener a lized to all zones by, first , est imat ing thetota l t r ips gen er a ted from one zone, i, to an oth er zone, j,

P i8 Aj

J

T ij = " ----------------- (14.7) Dij

D

where T ij is t he in ter action bet ween two locat ions (or zones), P i is product ions of t r ip s fromzone I, Aj is t he a t t r activeness of zone j, D ij is t he dist ance bet ween zones I and j, 8 is theexponent of P i, J is the exponent of Aj, D is th e exponent of dista nce, an d " is a const an t .

Second, t he t ota l number of t r ips genera ted by a locat ion , I, to all dest ina t ions isobt a in ed by summin g over a ll des t in a t ion locat ions, j:

T i = " P i8 G (Aj

J/D ijD) (14.8)

and gener a lizing th is to a ll zones, we get:

" P i8 $ Aj

J

T ij = ------------ (14.9) D ij

D

where " is a const an t for the productions, P i8, but $ is a const an t for the a t t r actions, Aj

J.This type of funct ion is ca lled a double const ra in t model because t he equa t ion has t o becons t ra ined by the number of un it s in both the or igin and des t ina t ion loca t ions ; tha t is , thesum of P i over a ll loca t ions must be equa l to the tota l number of p roduct ions while the sumof Aj over a ll loca t ions must be equa l t o t he tot a l number of a t t r act ions. Ad justmen t s a r eusu a lly requ ired to ha ve the sum of individua l pr oductions a nd a t t r actions equ a l the t ota ls(usua lly est im ated in dependent ly).

Negat ive Exponent ia l Dis tance Fu nct ion

One of the pr oblems with the t r adit iona l gravity formulat ion is in t he measu rementof t r avel im peda nce (or cost ). For loca t ions sepa ra ted by sizea ble d ist ances in space, thegr avity for mula t ion can work proper ly. H owever , a s the dis t ance between loca t ion sdecreases, t he denomin a tor approaches in fin it y. Con sequent ly, a n a lt erna t ive expressionfor the in teract ion uses the nega t ive exponent ia l fu nct ion (Hägerst rand, 1957; Wilson ,1970).

T ji = Aj$ e (-"Dij) (14.10)

where T ji is t he a t t r action of loca t ion j for res iden t s of locat ion I , Aj is th e att ra ctiveness of

loca t ion j, D ij is th e dista nce between locat ions i and j, $ is the exponent of S j, and e is the

base of the na tura l loga r it hm (i.e ., 2.7183...). Der ived from pr in ciples of entropy

14.7

m axim ization, th e lat ter pa rt of th e equat ion is a negative exponent ial fun ction t ha t h as a

maximum value of 1 (i.e., e -0 = 1) (Wilson , 1970). This h as t he a dva ntage of making the

equa t ion more st able for in teract ion s between loca t ion s tha t a re close together . F orexample, Cliff an d H agget t (1988) used a nega t ive exponen t ia l gravity-type m odel todescr ibe t he diffusion of mea sles in to th e United St a tes from Ca nada and Mexico. It hasa lso been argued tha t the nega t ive exponen t ia l function gener a lly gives a bet t er fit tourba n t ravel pa t t er ns, pa r t icula r ly by au tomobile (Foot, 1981; Bossard, 1993;). Figure 14.2shows a typica l n ega t ive exponent ia l fu nct ion and one recommended for home-based workt r ips by the Tr anspor ta t ion Research Boa rd as a defau lt va lu e (NCH RP , 1995).

Note tha t by moving the dist ance term to th e numer a tor, s t r ictly speaking it nolonger is a n impeda nce term sin ce impedance increa ses with dis t ance. Rat her it is adiscount factor (or disincentive); th e int eraction is discoun ted with dista nce. Neverth eless,the term ‘im pedance’ is st ill u sed, pr im ar ily for h is tor ica l r easons.

There a re other dis t ance funct ion s, a s well. Chapter 9 explor ed some of these. F orexam ple, we ar e find ing tha t , for cr ime t r ips, th ese other funct ions m ay produce bet t erresu lts than the negat ive exponent ial (e.g., the lognorm al), prim ar ily because m any cr imesar e comm itted at short -to-moderat e dista nces.

Trave l Impedance

On e of the biggest adva nces in th is t ype of model h as been to increase the flexibilityof th e denomina tor. In th e tra ditiona l gra vity model, th e denomina tor is dista nce. This isa proxy for a discoun t factor (or cost ); the far ther two zones a re from each other , th e lesslikely t her e is to be in ter action bet ween them , a ll oth er th ings bein g equ a l. Conversely, thecloser two zones a re, t he m ore likely t her e is to be in ter action , a ll oth er th ings bein g equ a l.

Trave l Time

It has been rea lized, however, th a t dist ance is on ly an appr oximat ion to impeda nce.In rea l tr avel, tr avel t ime is a much bet t er indica tor of the cost of t r avel in t ha t time va r iesby the t ime of da y, da y of week , and other factors . For exa mple, t r avel across town in anymet ropolita n a rea is gener a lly a lot easier a t 3 in t he morn ing, say, than a t the pea kafternoon rush per iod. The d ifference in t r avel t ime can var y as m uch as t wo-to-threet im es between peak and off-peak hours. Usin g on ly dis t ance, h owever , t hese va r ia t ion s a renever picked u p becau se the dist ance between locat ions is invar ian t .

Th is rea liza t ion has led to the concep t of t ravel im pedance which, in t ur n, has led tothe concep t of travel cost. ‘Im peda nce’ is the resist ance (or discount ing) in t r avel betweentwo zones. Usin g tr avel t ime a s a n impeda nce var iable, th e longer it t akes to t r avelbet ween two zones , the les s lik ely t her e will be in ter action bet ween them , a ll oth er th ingsbeing equa l. Conversely, a sh or ter t r avel t ime leads t o grea ter int eract ion between zones,again , a ll other th ings being equ a l. Sim ilar ly, a t r avel rout e t ha t sh ort en s t r avel t ime will

Default Home-Based Work Trip Impedance(Source: National Cooperative Highway Research Program 365, 1995)

0.0000

0.0005

0.0010

0.0015

0.0020

0 4 8 12 16 20 24 28

Distance (miles)

Rel

ativ

e im

ped

anceFigure 14.2:

14.9

gen er a lly be select ed over one tha t t akes longer even if th e first one is longer in dis t ance. For exam ple, it ’s been documented t ha t people will change work loca t ions t ha t a re fa r therfrom their home if t r aveling to th e new work locat ion takes less t ime (e.g., t r aveling in the‘opposit e’ dir ect ion t o th e bu lk of t r a ffic; Wachs , Taylor, Levin e & On g, 1993).

If t r avel t ime is a crit ical componen t of t r avel, why then don’t offender s commitmore cr imes a t , say, 3 in t he morn ing than a t the pea k a ft ernoon t ravel t imes? Since th eimpeda nce is less a t 3 in t he morn ing than a t , say, 5 in t he aft ernoon , wouldn’t the modelpredict more t r ips occur r in g in the ea r ly morn in g h ours than actua lly occu r in those hours?Th e a nswer has to do wit h the n umer a tor of the gravit y equ a t ion and not just theden omina tor . At 3 in t he morn ing, yes, it is ea sier to t r avel between two loca t ions, at leas tby per sona l au tomobile (not by bus or t r a in when those services a re less frequen t ). But thea t t ract ion side of the equa t ion is also less st rong at 3 in t he morn ing. For a st reet robber ,ther e a re fewer poten t ia l ‘vict ims’ on t he st reet a t 3 in the m orn ing than in the la tea fternoon . For a residen t ial burgla r , th ere is more likely to be someone a t home while theyburgle a t n igh t than in the a ft ernoon. The t ravel t im e component is only one dim ension ofthe likelihood of t r avel between two loca t ions. The d ist r ibut ion of opport un ities a nd oth ercost s can a lt er the likelih ood cons iderably.

Never theless, sh ift ing to an impeda nce funct ion a llows a t r avel model to bet t erreplica te actua l t r avel con dit ion s. Most t r avel demand models used by t ranspor ta t ionpla nners use an im pedance funct ion , r a ther than a dis t ance funct ion .1 Dist ance would onlybe meaningfu l if the st anda rds wer e invar ian t with respect t o t ime (e.g., a m odel ca lcu lat edover an en t ir e year , 24 hours a day). As will be demonst ra ted in chapter 16 on networkassignment , a t r avel t im e ca lcu la t ion leads to a very d ifferen t network a lloca t ion than adist ance ca lcu lat ion . For exam ple, if dist ance is used a s a n impeda nce var iable, th en thesh ort est t r ips will r a rely t ake t he freewa ys becau se t r avel to an d from a freewa y usu a llyma kes a t rip longer th an a direct r out e between a n origin and a destinat ion. But as m ostpeople under st and, ta kin g a freewa y to tr avel a sizeable dist ance is usua lly a lot quickerthan t raver sin g an urba n ar t er ia l system wit h many t ra ffic light s, s top sign s, crossin gpedest r ians, cross t r a ffic from parking lots and shopping malls, a nd other urba n ‘obst acles’. Today, the use of dis t ance in t r avel demand modeling h as vir tua lly been dropped by m osttr an sport at ion plann ers.

Trave l Cost

An even bet t er concep t of impedance is tha t of travel cost (somet imes ca lledgeneralized cost) which incorpora tes r ea l an d per ceived cost s of t r avel between twoloca t ions . Travel t ime is one component of t r avel cos t in tha t there is an implicit cos t to thet r ip (e.g., an hour ly wage or pr ice as sign ed). In th is case, two differ en t individua ls willva lu e the t im e for a t r ip differen t ly dependin g on their hour ly ‘wage’. F or exa mple, for anin dividua l who pr ices h is /her t r avel a t $100 an hour , t he per min ute cost is $1.67. F oranoth er individua l wh o pr ices h is/her t r avel a t $12 an hour , the per minute cost is 20¢. These r elat ive pr ices a ssigned t o t r avel will su bst an t ially affect individua l choices in t r avelmodes a nd r ou tes. For in st ance, th ese t wo hypothet ica l individu a ls will pr obably use a

14.10

differen t t r avel mode in gett ing from an air por t to a h otel on a t r ip; the former willpr obably t ake a taxi wh er ea s t he la t t er will probably t ake a bu s or t r a in (if ava ilable).

Bu t cos t involves other d imens ions tha t need to be cons idered . There a re rea loper a t ing cost s in the use of a vehicle - fuel, oil, maint enance, insu rance. Man y tr avelst udies h ave su ggest ed tha t dr iver s in corpora te t hese cost s a s par t of their implicit hour lyt ravel pr ice (Or tuza r and Willumsen , 2001; 323-327). But , t here a re a lso rea l, ‘out -of-pocket ’ cost s such as parkin g or toll cost s. P arkin g is pa r t icu la r ly a major expense forint ra -urban dr iving behavior . In m any built-up bus iness dist r ict s, pa rkin g cost s can beconsiderable, for example as m uch a s $40 a day in m ajor m etropolita n center s. In mostbu sy commer cial a rea s, t her e a re some pa rking cost s, if only a t on-st reet pa rking met er s. Thus, a t r avel cos t model n eeds to in corpora te these rea l cos t s as the out -of-pocket cost smay overwh elm t he implicit va lue of the t r avel t ime. For exam ple, an offender who lives10 minu tes from the downtown a rea by ca r wou ld probably not dr ive in to the downtown tocommit a robbery since th a t individua l will ha ve to bear the pr ice of pa rkin g. Ther e arelot s of well known s tor ies tha t circu la te abou t bank robbers who a re caught because theyincur pa rking t icket s while commit t ing their cr ime. How often th is has occur red is notknown from any s tudy t ha t I’m aware, bu t the story lin e is cogn izan t of the actua l cos t s oft ravel tha t must be in cur red as par t of t r avel.

In addit ion to rea l cost s a re per ceived cost s. For t r ansit user s, pa r t icu lar ly, th eseper ceived cost s a ffect the ease a nd t ime of t r avel. One of the st anda rd quest ions in t r avelsu rveys for t r ansit user s is t he t ime it t akes to walk from t heir h ome to the nearest busst op or in t ra -ur ba n ra il system (if ava ilable) and from t he la st t r ansit st op to th eir fina ldestinat ion; th e longer it t ak es to access the tr an sit system, th e less likely an individua lwill use it . Similar ly, tr ansfers bet ween bus es or t r a ins decrea se t he likelihood of t r avel bytha t mode, a lmost in pr opor t ion to th e number of t r ansfer s. Th e r ea son is t he difficult y inget t ing ou t of one bus or t r a in and in to another . Bu t , the t ime between t ra ins adds animplicit t r avel cost ; the longer t he wait bet ween bus es, th e less likely tha t mode will beused by t raveler s. In sh ort , ease of access a nd conven ience ar e posit ive incent ives in usin ga mode or a rout e wh ile difficult y in accessin g it , lack of conven ience, an d even fear of bein gvulnerable to cr ime will decrease t he likelihood of using th a t mode or rou te.2

If the concept is expanded to tha t of an offender , t here a re other perceived cost s tha tmight a ffect t r avel. One obviou s one is the likelihood of bein g ca ugh t . I t may be easy forone offender to t r avel to a upscale, high visibility sh oppin g ar ea , but if there are manypolice and secur ity guards a roun d, t he in dividua l is m ore likely t o be cau ght . Hen ce, tha tlikelihood (or , more accura tely, an assumpt ion tha t the offender makes abou t tha tlikelihood sin ce he/she doesn’t rea lly k now what is the rea l lik elihood) is liable to affect thechoice of a dest ina t ion and, possibly, even a rou te.

Another percep tua l com ponent a ffect in g a likely choice is the reliabilit y of thet ransport a t ion mode. Man y offender s a re poor and don’t have expen sive, well maint a inedveh icles. If th e veh icle is n ot capa ble of h igher speeds or is even likely to break down wh ilean offence is bein g com mit ted, t ha t veh icle is not liable to be used in makin g a t r ip or thechoice of dest ina t ion may be altered. It is well kn own tha t many offender s st ea l vehicles

14.11

for use in a cr im e. F ears about not bein g iden t ified a re clea r ly a major factor in thosedecis ion s, bu t the reliabilit y of t heir own vehicles may a lso be a factor .

Thus, in sh ort , a more r ea list ic model of the incent ive or d isin cent ive to ma ke a t r ipbetween two loca t ion s requir es a complex fu nct ion tha t weight s a number of factorsa ffect in g t he cost of t r avel - t he t r avel t im e, implicit opera t in g cost s, ou t -of-pocket cost s,and per ceived cost s. Ma ny t ravel demand m odels u sed by Met ropolitan PlanningOrga niza t ion s use such a funct ion , u sua lly u nder the la bel of ‘genera lized cost ’. The morecomplex the p ricing s tructu re for pa rk ing and t r avel with in a met ropolit an a rea , t he morelikely a genera lized cost funct ion will p rovide a rea list ic m odel of t r ip dis t r ibu t ion .

Trave l Uti li ty

Th e fin a l concep t tha t is in t roduced in defin in g impedance is tha t of travel utility. ‘Ut ilit y’ is an in dividua l con cept , r a ther than a zon a l on e. Also, it is the flip side of cost(i.e., h igher cost is as sociat ed with less u t ility). A gener a lized cost funct ion ca lcu lat es t heobject ive and a vera ge perceived cost s of t r avel between two zones. But the u t ility of t r avelfor an ind ividua l is a funct ion of both those rea l cos t s and a number of ind ividua lcharacter is t ics tha t affect the va lue p laced on tha t t r avel. Thus , two ind ividua ls living inthe sa me zone (per haps even living next door to each other ) who t ravel to the sa medest ina t ion loca t ion may ‘pr ice’ their t r ip very differen t ly. Aside from income differenceswhich affect s t he avera ge hour ly ‘wage’, th ere may be differences du e to convenience,a t t r act iveness, or a host of other factors. Other factors a re more id iosyn cra t ic. Forexam ple, a t r ip by a gan g mem ber int o another gan g’s ‘tu r f’ might be expected t o increa sethe perceived cost s to the in dividua l of t r aveling t o tha t loca t ion , a bove and beyon d anyobjective cost factors . Alter na t ively, a t r ip t o a loca t ion wh er e a close friend or rela t ive isloca ted m ight decrea se t he per ceived cost of t r avel to tha t zone. In oth er words, t here areboth object ive cost s a s well a s subject ive cost s in t r avel between two zones .

The concept of u t ility ma y be less u seful for cr ime a na lysis t han for gener a l tr avelbehavior . For one th ing, since th e concept is individua l, it can on ly be ident ified byindividua l su rveys (Domen cich and McFa dden , 1975). For cr ime ana lysis , th is m akes itvir tua lly imposs ible t o use since it is very difficult to in ter view offender s, a t lea st in theUnit ed Sta tes. In addit ion , t he mathemat ics requir ed for a r t icu la t in g a u t ilit y fu nct ion aredifficu lt sin ce u t ilit y fu nct ion s have a very com plex for m, u sua lly involving t he bin omia l ormult inomia l logit fun ction wit h a Weibu ll er ror t er m. In the next cha pt er , br ief ment ion ismade of th is t ype of model.

But, for completeness sake, we need to un dersta nd t ha t t he likelihood ordis in cen t ive to t r avel between two loca t ion s is a funct ion of in dividua l ch aracter is t ics aswell as object ive t r avel cost componen ts.

Impedan ce Fun ct ion

Thus, for a zona l type m odel, we can leave the gra vity funct ion as a gener a lizedimpeda nce funct ion . For t r avel between any one zone and a ll other zones, we ha ve:

14.12

T i = " P i8 G (Aj

J/I ij) (14.11)

where the number of t r ips from zon e i t o a ll other zon es, j, is a funct ion of the product ion sa t zone i and t he r ela t ive a t t r action of any one zone, j, to th e impeda nce of th a t zone for i,I ij. The im peda nce function , I ij , is some declining funct ion of cost for t r avel between twozones. It does not h ave to be any par t icu lar form and can be (and u su a lly is) a non-linea rfun ction. The cost s can be in ter ms of dis t ance, tr avel t ime, speed (which is conver ted in tot ravel t ime) or gen er a l cost s. The gr ea ter the separa t ion bet ween two zones (i.e., th eh igher the impeda nce), the less likely ther e will be a t r ip between them . Gen er a lizing th isto a ll zones, we get:

T ij = " P i8 $ Aj

J I ij (14.12)

where P i is t he production capacity of zone I, Aj is t he a t t r action of zone j, Iij, is agenera lized funct ion tha t discounts the in teract ion with increas ing separa t ion in d is tance,t ime, or cost , " an d $ a re constan t s tha t a re applied to the product ion s and a t t r act ion srespectively, an d 8 an d J a re ‘fin e tun in g’ exponents of the product ion s and a t t r act ion srespectively. This is t he gra vity funct ion tha t we will est ima te in t he Crim eS tat t r ipdis t r ibu t ion model.

Alterna tive Mode ls: Inte rven ing Oppo rtun ities

There a re a lt erna t ive a lloca t ion s procedures to the gr avity m odel. One well knownone is tha t of interven ing opportun ities. St ouffer (1940) modified t he simple gravityfunct ion by a rgu ing tha t the a t t r act ion between two loca t ions was a funct ion not on ly of thecharacter ist ics of the relat ive a t t r act ions of two loca t ions, but of int ervening opport un itiesbet ween the loca t ions. H is h ypothesis “..assumes tha t ther e is n o necessa ry rela t ionsh ipbet ween mobility a nd d ist ance... tha t the number of persons going a given dis t ance isdirectly proport iona l to th e num ber of opport un ities at t ha t dista nce and inverselypr opor t iona l to th e number of in ter ven ing opport un it ies” (Stouffer , 1940, p . 846). Thismodel was u sed in the 1940s to expla in in ter st a te a nd in ter -coun ty migra t ion (Br ight andThomas, 1941; Isbell, 1944; Isa rd, 1979). Using th e gra vity type formulat ion , th is can bewritten as:

Sj$

Aji = " ----------------- (14.13) G(Sk

>) D ij 8

where Aji is t he a t t r action of loca t ion j by r es iden t s of locat ion I , S j is th e att ra ctiveness ofzone j, Sk is the a t t r act iveness of a ll other loca t ions t ha t a re interm ediate in d is tancebet ween loca t ions I a nd j, D ij is th e dista nce between zones I an d j, $ is the exponent of S j, >is the exponent of Sk , and 8 is t he exponen t of dis tance. While the in ter ven ingopport un it ies a re im plicit in equ a t ion 14.7 in the exponents, $ an d 8, and coefficient , ",equa t ion 14.13 ma kes the int ervening opport un ities explicit. The impor tance of the concept

14.13

is tha t t r avel between two loca t ions becomes a complex funct ion of the spa t ial environmentof nearby a reas and not ju st of the two loca t ion s.

In pr actice, in sp ite of it s m ore in tu it ive theoret ical m odel, the in ter ven ingopport un ities m odel does n ot improve pred ict ion much beyond t ha t of the gra vity modelsin ce it in clu des the a t t r act ion s associa ted wit h the dest in a t ion zon es. Also, it is a moredifficu lt model t o est im ate sin ce the a t t r act ion s of a ll other zon es must be considered foreach zon e pa ir (or igin -dest in a t ion combin a t ion ). Consequent ly, it is ra rely used in actua lpr actice (Or tuzar and Willu msen , 2001; Zhao et a l, 2001).

Another a lt erna t ive method was conducted by Porojan (2000) in applying thegravity model t o in t erna t iona l t r ade flow. He added a spa t ia l au tocor rela t ion componen t inaddit ion to im pedance and obt a in ed a sligh t ly bet t er fit than the pure gr avity fu nct ion .However, whether th is appr oach would improve th e fitt ing of int ra -regiona l cr ime t ravelpa t t erns in st ill unkn own. Nevert heless, th is and oth er appr oaches m ight improve th epr edicta bility of a gr avity function for in t ra -ur ba n crim e t ravel.

Method of Estimat ion

The Crim eS tat t r ip d ist r ibu t ion model implemen ts equ a t ion 14.12. Th e specificdet a ils a re discussed below, but the model is iter a t ive. The st eps a re as follows:

1.Depend ing on whether a singly const ra ined or doubly const ra ined model is to beest im ated, it st a r t s wit h a in it ia l guess of the va lu es for " or $ (or both for a doublyconst ra ined model). Table 14.1 illust ra tes th e thr ee models.

Ta ble 14 .1

Three Methods of Constraining the Gravi ty Model

S in g le c on s tra in t

Cons t ra in origins

T i j = " P i8 Aj

J I i j

Const ra in dest in a t ion s

T i j = P i8 $ Aj

J I i j

D o ub le c on s tra in t

Const ra in both or igin s and dest in a t ion s

T i j = " P i8 $ Aj

J I i j

14.14

2. The r out ine proceeds t o est imate t he va lue for each cell in the origin -dest in a t ion mat r ix (see figure 14.1 above) usin g t he exist in g est im ates for "an d $.

3. The rou t ine t hen su ms t he rows a nd colum ns in the mat r ix. Then ,depen ding on whether a single- or double-const ra int model is to be est ima tedand, if a s ingle-cons t ra in t , whet her origins or dest ina t ions a re t o be heldcons tan t , it then ca lcu la tes the ra t io of the summed va lue (row or column orboth) to the in it ia l row or column sum. The inverse of tha t r a t io is thesubsequent est im ate for " or $ (or bot h for a double-const ra in ed model).

4. The rout in e repea t s st eps 2 and 3 un t il the changes from one it era t ion to thenext a re ver y sm all.

5. Th e la st es t im ate of " or $ (or both for a double-const ra ined model) is takenas t he final values of th ese par am eters.

6. Once the parameter s have been est im ated, t he model ca n be applied to theca libra t ion da t a set or t o another da t a set . Note tha t t he parameter s a r e rowor column specific (or both ). Tha t is, in the ‘const ra in origins’ model, ther e isa sepa ra te coefficien t for each r ow. In the ‘const ra in dest ina t ions’ model,ther e is a sepa ra te coefficien t for each column. In the ‘const ra in both originsand dest ina t ions’, ther e is a sepa ra te coefficien t for each cell (row-columncombin a t ion ).

7. A compar ison can be made between the observed dis t r ibu t ion and thepr edicted (modeled) dist r ibut ion . Because m ost or igin-dest ina t ion mat r icesa re ver y large, the vast major ity of cells will have zer o in them . Thus, a chi-squ are test would be ina ppr opr iat e. Ins tead, a compa r ison of the trip lengthdis t r ibu t ion is made usin g t wo differen t st a t is t ics - a coin cidence ra t io a ndthe Komologorov-Sm irn ov Two-sa mple sta t istic. Deta ils a re provided below.

Cr i m eS t a t III Tri p D is tri bu t io n Ro u ti n e s

Next , we examine t he actua l tools tha t a re available in t he Crim eS tat t r ipdis t r ibu t ion m odule. Th e t ools a re illu st ra ted wit h exa mples from Balt imore County.

The Crim eS tat t r ip dist r ibut ion modu le includes one set up screen and five rout inestha t implemen t the m odel:

1. Calculate observed or ig in-des t inat ion d i s tribut ion . If th er e is a fileava ilable with the coord ina tes for individua l origin s a nd dest ina t ions (e.g.,an a r rest record), t h is rout in e will ca lcu la te the empir ica l t r ip dis t r ibu t ionmat r ix;

14.15

2. Calibrate impedance funct ion . If th er e is a file available wit h thecoordina tes for individua l or igins and des t ina t ions, t h is rou t ine will ca libra t ean empir ica l impedance funct ion .

3. S e tu p ori gi n-d e st in a ti on m o de l. This screen a llows t he user to define thepa rameters of a t r ip dist r ibut ion (or igin-dest ina t ion) model with eith er amathemat ica l or empir ica l impedance funct ion .

4. Ca li bra te o ri gi n-d e st in a ti on m o de l. This rout ine calibrat es thepa ramet er s of th e t r ip d ist r ibu t ion m odel (equ a t ion 14.12) us ing thepa rameters defined on t he set up pa ge.

5. Ap ply pre d ic te d ori gi n-d e st in a ti on m o de l . This rout ine applies th eest ima ted pa rameters t o a da ta set . The da ta set can be eith er theca libra t ion file or another file.

6. Compare observed and predic ted orig in-des t inat ion tr ip lengths . Th is r out ine compa res the t r ip len gths from t he obser ved (em pir ical) t r ipdistr ibut ion with t ha t pr edicted by th e model. Compa rison a re ma degraph ically, by a coincidence ra t io, by the Komologorov-Sm irnov Two-Sa mpletest, an d by a Chi squa re test on t he most frequent tr ip links.

Each of these r ou t ines a re described in det a il below. Figur e 14.3 shows a screensh ot of the t r ip dist r ibut ion modu le.

Descr ibe Orig in-Dest ination Tr ips

An empir ica l descr ip tion of t he actua l t r ip dis t ribu t ion ma t r ix can be made if t hereis a da ta set tha t includes in dividu a l or igin a nd des t ina t ion loca t ions. The u ser defines t heor igin loca t ion and the des t ina t ion loca t ion for each r ecord and a set of zones from which tocompa re t he individua l or igins a nd dest ina t ions. Th e r out ine m atches up each originloca t ion with the neares t zone, each des t ina t ion loca t ion with the neares t zone, andcalcula tes the number of t r ips from each origin zone t o each dest ina t ion zone. Th is is anobserved dist r ibut ion of t r ips by zone.

The steps in r un ning the model ar e as follows:

1. Calculate observed or ig in-des t inat ion tr ips . Check if a n empir ica lor igin-dest ina t ion t r ip dist r ibut ion is to be ca lcu lat ed.

2. Origi n file . The origin file is a list of origin zones wit h a sin gle pointrepr esen t ing the zone (e.g., the cen t roid). It m ust be inpu t as either thepr im ary or secondary file. Specify whether the da ta file is the pr im ary orseconda ry file.

Figure 14.3:

Trip Distribution Module

14.17

Or ig in ID . Specify the or igin ID va r ia ble in the da ta file (e.g., CensusTract ,Block, TAZ).

3. De stin ation fi le . The dest ina t ion file is a list of dest ina t ion zones wit h asin gle point represen t ing the zone (e.g., the cent roid). It must be in pu t aseit her the pr imary or seconda ry file. Specify whet her the da ta file is t hepr imary or seconda ry file.Specify the dest ina t ion ID va r iable in the da ta file(e.g., CensusTr act , Block , TAZ).

Note: all destinat ion ID’s should be in t he origin zone file and m ust ha ve the sam ena mes an d both should be cha ra cter (string) var iables.

4. Se le ct da ta file . The da ta set must have in dividua l or igin and dest in a t ionloca t ions. Each r ecord must have the X/Y coordina tes of an or igin loca t ionand t he X/Y coordina tes of a dest ina t ion locat ion . For exa mple, an a r rest filemight list in dividua l inciden t s wit h each in ciden t having a cr im e loca t ion(th e dest ina t ion) an d a residence or a r rest locat ion (th e origin ). Select t he fileth at ha s th e X an d Y coordina tes for t he origin and destina tion locat ions.Crim eS tat can rea d ASCII, dba se '.dbf', ArcView '.shp' and Ma pIn fo 'da t 'files. Select the tab and specify the type of file to be selected. Use the browsebut ton to sea rch for the file. If the file type is ASCII, select the type of da tasepa ra tor (comm a, sem icolon , space, ta b) an d t he number of columns.

Va r ia bl es . Defin e the file which conta in s the X and Y coordin a tes for boththe or igin (residence) and dest in a t ion (cr im e) loca t ion s.

C ol u m n . Select the va r ia bles for the X and Y coordin a tes respect ively forboth the or igin and dest ina t ion loca t ion s (e.g., Lon , La t , H om eX, H om eY,IncidentX, Incident Y.) Both loca t ions m ust be defined for the rou t ine t owork.

Missi n g v a lu es . Identify wheth er th ere ar e any missing values for t hesefour fields (X an d Y coordina tes for both origin and destina tion locat ions). Bydefau lt , Crim eS tat will ignore r ecords wit h blank values in any of th e eligiblefields or records wit h non-numer ic values (e.g.,alph anumer ic character s, #,*). Blanks will a lways be excluded u n less t he user select s <none>. Therear e 8 possible options:

1. <bla nk> fields a re au toma t ically excluded. This is the defau lt2. <none> indicates tha t no records will be excluded. If th er e is a

bla nk field, Crim eS tat will tr eat it a s a 03. 0 is excluded4. –1 is excluded5. 0 and –1 ind ica tes t ha t both 0 an d -1 will be excluded6. 0, -1 an d 9999 in dicates tha t a ll th ree va lues (0, -1, 9999) will

be excluded

14.18

Any other numer ica l va lue can be t rea ted as a missing va lue by typ ing it(e.g., 99)Multiple numer ica l va lues can be tr ea ted a s m issing va lues bytypin g t hem, separa t in g each by commas (e.g., 0, -1, 99, 9999, -99).

Typ e o f coor d in a t e s ys t em a n d d a t a u n i t s . The coord ina te sys tem anddat a u nits a re listed for inform at ion. If th e coordina tes ar e in longitu des andla t itudes, t hen a spher ical syst em is being used a nd da ta un it s willau toma t ically be decimal degrees . If the coord ina te syst em is p rojected (e.g.,St a te P lane, Univer sa l Transver se Mer cat or – U TM), then da ta un it s cou ldbe either in feet (e.g., Sta te Pla ne) or meters (e.g., UTM.).

5. Ta ble o u tp u t. The fu ll or igin -des t ina t ion mat r ix is ou tpu t as a t able to thescreen including summ ar y file inform at ion a nd:

1. The origin zone (ORIGIN)2. Th e dest in a t ion zon e (DE ST)’3. The n umber of observed tr ips (FRE Q)

6. Save obse rv e d ori g in -de s tin ation tri ps . If specified, the full or igin-dest ina t ion mat r ix ou tpu t is saved a s a ‘dbf’ file named by the u ser . The fileout put includes:

1. The origin zone (ORIGIN)2. The dest ina t ion zone (DEST)3. The X coor dina te for the or igin zone (ORIGINX)4. The Y coor dina te for the or igin zone (ORIGINY)5. The X coor dina te for the dest ina t ion zone (DESTX)6. The Y coor dina te for the dest ina t ion zone (DESTY)7. The n umber of t r ips (FRE Q)

Note : each record is a un ique origin-dest ina t ion combina t ion . Ther eare M x N records wh er e M is the number of origin zones (includingth e extern al zone) an d N is the nu mber of destinat ion zones.

7. Save l inks . The top observed origin-destinat ion t rip links can be saved assepa ra te li ne object s for use in a GIS. Specify the outpu t file format(ArcView '.sh p', MapIn fo '.m if' or Atlas*GIS '.bna ') and t he file name.

Save top links . Because t he ou tpu t file is very lar ge (number of or igin zonesx n umber of dest in a t ion zon es), t he user can select a sub-set of zon ecombina t ions with the most observed t r ips. Indica t ing th e top K links willnar row the number down to the most im por tan t ones. The defau lt is the top100 or igin -dest in a t ion combin a t ion s. E ach outpu t object is a line from theor igin zone to the dest ina t ion zone with an ODT pr efix. The pr efix is placedbefore the outpu t file name. The line gr aphica l ou tpu t for each objectincludes:

14.19

1. An ID number from 1 to K, where K is the number of linksoutpu t (ID)

2. The fea ture pr efix (ODT)3. The origin zone (ORIGIN)4. The dest ina t ion zone (DEST)5. The X coor dina te for the or igin zone (ORIGINX)6. The Y coor dina te for the or igin zone (ORIGINY)7. The X coor dina te for the dest ina t ion zone (DESTX)8. The Y coor dina te for the dest ina t ion zone (DESTY)9. The n umber of observed tr ips for tha t combina t ion (FRE Q)10. The dist ance between the or igin zone and t he dest ina t ion zone.

S a v e p oi n t s. In t r a -zona l t r ip s (t r ip s in wh ich the or igin and des t ina t ion a rethe sa me zone) can be ou tpu t as separa te p oi nt objects a s an ArcView '.sh p',MapIn fo '.m if' or Atlas*GIS '.bna ' file . Aga in , t he top K poin t s a re outpu t(defau lt=100). Each ou tpu t object is a poin t r ep resen t ing an in t r a -zona l t r ipwit h an ODTPOINTS pr efix. The prefix is pla ced before t he out pu t filename.

The point graph ical out put for each object includes:

1. An ID number from 1 to K, where K is the number of linksoutpu t (ID)

2. The fea ture pr efix (POINTSODT)3. The origin zone (ORIGIN)4. The dest ina t ion zone (DEST)5. The X coor dina te for the or igin zone (ORIGINX)6. The Y coor dina te for the or igin zone (ORIGINY)7. The X coor dina te for the dest ina t ion zone (DESTX)8. The Y coor dina te for the dest ina t ion zone (DESTY)9. The n umber of observed tr ips for tha t combina t ion (FRE Q)

Exam ple of Observed Distribution from Balt imo re Coun ty

Figure 14.4 shows the outpu t of the top 1000 links for the obs erved t r ip dis t r ibu t ionfrom a sa mple of 41,974 records for inciden t s committ ed between 1993 an d 1997. Thezona l model used was t ha t of t r a ffic ana lysis zones (TAZ). These wer e discussed in cha pt er12. Because there a re a la rge number of lin ks (532 or igin zon es by 325 dest in a t ion zon es),the t op 1000 were t aken . These a ccoun ted for 19,615 cr ime t r ips (or 46.7% of all t r ips ). Ala rger number of links could have been selected, bu t the map would have become moreclu t t ered. Of the 19,615 t r ips tha t a re displa yed in the map, 7,913 or 40.3% are in t ra -zon a lt r ips . These were out pu t by the r out ine a s poin t s a nd h ave been displa yed a s circles wit hthe size propor t ion a l t o the number of t r ips. The remain in g 11,702 t r ip links were outpu tby t he rout in e as lines and are displa yed wit h the th ickness and st rengt h of color of theline bein g pr opor t iona l to th e number of t r ips .