1

Crim e Stat Ve rs ion 3.1 Update Note s

These notes provide in format ion on ly on changes to Cr imeSta t III since the releaseof version 3.0. The upda ted program and complete documenta t ion are found on theCr imeSta t download page a t :

h t tp://www.icpsr .umich .edu/cr imesta t

Fixes an d Im prove m e n ts to Ve rs ion 3.0

The following fixes and improvements in version 3.0 have been made.

1. For any ou tpu t file, the program now checks tha t a pa th which is definedactua lly exist s.

2. The Geomet r ic Mean ou tpu t in the “Mean center and standard dist ance”rout ine under Spa t ia l Descr ipt ion now a llows weighted va lues. It is definedas (Wikipedia , 2007a):

N

iGeomet r ic Mean of X = GM(X) = Ð ( X ) (Up.1)Wi 1/(EWi)

i=1

N

iGeomet r ic Mean of Y = GM(Y) = A (Y ) (Up.2)Wi 1/(EWi)

i=1

where A is the product t erm of each poin t va lue, i (i.e., the va lues of X or Y

iare mult iplied t imes each other ), W is the weight used (defau lt=1), and N isthe sample size (Ever it t , 1995). The weight s have to be defined on thePr imary File page, either in the Weight s field or in the In tensity field (butnot both together ).

The equa t ion can be eva lua ted by logar ithms.

i i 1 G[W *Ln(X )]

1 1 2* 2 2* NLn[GM(X)] = ---- [ W *Ln(X ) + W Ln(X ) + ..+ W Ln(X ) ] = ------------------ (Up.3)

i i GW GW

i i 1 G[W *Ln(Y )]

1 1 2* 2 2* NLn[GM(Y)] = ---- [ W *Ln(Y ) + W Ln(Y ) + ..+ W Ln(Y ) ] = ----------------- (Up.4)

i i GW GW

2

GM(X) = e (Up.5)Ln [GM(X)]

GM(Y) = e (Up.6)Ln [GM(Y)]

The geomet r ic mean is the ant i-log of the mean of the logar ithms. If weight sa re used, then the logar ithm of each X or Y va lue is weighted and the sum ofthe weighted logar ithms are divided by the sum of the weight s. If weight sa re not used, then the defau lt weight is 1 and the sum of the weight s willequa l the sample size. The geomet r ic mean is outpu t as par t of the Mcsdrout ine and has a ‘Gm’ prefix before the user defined name.

The geomet r ic mean is used when un it s a re mult ipled by each other (e.g., astock’s va lue increases by 10% one year , 15% the next , and 12% the next )(Wikipedia , 2007a). One can’t just t ake the simple mean because there is acumula t ive change in the unit s. In most cases, th is is not r elevan t to poin t(incident ) loca t ions since the coordina tes of each incident a re independentand are not mult iplied by each other . However , the geomet r ic mean can beusefu l because it fir st conver t s a ll X and Y coordina tes in to logar ithms and,thus, has the effect of discount ing ext reme values.

3. Also, the Harmonic Mean ou tpu t in the “Mean cen ter and standard distance”rout ine under Spa t ia l Descr ipt ion now a llows weighted va lues. It is definedas (Wikipedia , 2007b):

i GW

Harmonic mean of X = HM(X) = ------------------- (Up.7)

i i G [W /(X )]

i GW

Harmonic mean of Y = HM(Y) = ------------------- (Up.8)

i iG [W /(Y )]

iwhere W is the weight used (defau lt=1), and N is the sample size. Theweight s have to be defined on the Pr imary File page, either in the Weight sfield or in the In tensity field (bu t not both together ).

In other words, the harmonic mean of X and Y respect ively is the inverse ofthe mean of the inverse of X and Y respect ively (i.e., t ake the inverse; takethe mean of the inverse; and inver t the mean of the inverse). If weight s a reused, then each X or Y va lue is weighted by it s inver se and the numera tor isthe sum of the weight s. If weight s a re not used, then the defau lt weight is 1and the sum of the weight s will equa l the sample size. The harmonic meanis ou tpu t as par t of the Mcsd rou t ine and has a ‘Hm’ prefix before the userdefined name.

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Typica lly, ha rmonic means a re used in ca lcu la t ing the average of ra tes, orquant it ies whose va lues a re changing over t ime (Wikipedia , 2007b). Forexample, in ca lcu la t ing the average speed over mult iple segments of equa llength (see chapter 16 on Network Assignment ), the ha rmonic mean shou ldbe used, not the ar ithmet ic mean . If there are two adjacent road segments,each one mile in length and if a ca r t r avels over the fir st segment 20 milesper hour (mph) bu t over the second segment a t 40 mph, the average speed isnot 30 mph (the ar ithmet ic mean), but 26.7 mph (the harmonic mean). Thecar takes 3 minutes to t r avel the fir st segment (60 minutes per hour t imes 1mile divided by 20 mph ) and 1.5 minutes to t r avel the second segment (60minutes per hour t imes 1 mile divided by 40 mph). Thus, the tota l t ime tot ravel the two miles is 4.5 minutes and the average speed is 26.7 mph (60minutes per hour t imes 2 miles divided by 4.5 minutes).

Aga in , for poin t (inciden t ) loca t ions, the ha rmonic mean would normally notbe relevant since the coordina tes of each of the inciden t s a re independent . However , since the harmonic mean is weighted more heavily by the smallerva lues, it can be usefu l to discount cases which have ou t lying coordina tes.

4. The test st a t ist ic for the Linear Nearest Neighbor index on the DistanceAnalysis I page now gives the cor rect probability level.

5. Severa l fixes have been made to the Cr ime Demand model rou t ines:

A. In the “Make predict ion” rou t ine under the Tr ip Genera t ion module ofthe Cr ime Travel Demand model, the ou tpu t var iable has beenchanged from “Predict ion” to “ADJ ORIGINS” for the or igin model and“ADJ DEST” for the dest ina t ion model.

B. In the “Calcu la te observed or igin-dest ina t ion t r ips” rou t ine under the“Descr ibe or igin -dest ina t ion t r ips” of the Tr ip Dist r ibu t ion module ofthe Cr ime Travel Demand model, the ou tpu t var iable is now ca lled“FREQ”.

C. Under the “Setup or igin-dest ina t ion model” page of the Tr ipDist r ibut ion module of the Cr ime Travel Demand, there is a newparameter defin ing the min imum number of t r ips per cell. Typica lly,in the gravity model, many cells will have small predicted values (e.g.,0.004). In order to concent ra te the predicted va lues, the user can seta min imum level. If the predicted value is below th is min imum, therout ine au tomat ica lly set s a zero (0) va lue with the remainingpredicted va lues being re-sca led so tha t the tota l number of predictedt r ips remains constan t . The defau lt va lue is 0.05.This parameter should be used cau t iously, however , as ext remeconcent ra t ion can occur by merely ra ising th is va lue. Because thenumber of predicted t r ips remains constan t , set t ing a minimum tha t

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is too h igh will have the effect of increasing a ll eligible va lues (i.e.,those grea ter than the min imum) substan t ia lly beyond the min imum. For example, in one run where the min imum was set a t 5, the re-sca led min imum va lue was 13.3.

D. For the Network Assignment rout ine, the prefix for the network loadoutput is now VOL.

E . In defin ing a t r avel network either on the Measurement Parameterspage or on the Network Assignment page, if the network is defined assingle direct iona l, then the “From one way flag” and “To one way flag”opt ions a re blanked ou t .

6. Crim e Trave l De m an d P roje ct Dire ctory Utility . The Cr ime TravelDemand module is a complex model tha t involves many differen t files. Because of th is, we recommend tha t the separa te st eps in the model bestored in separa te director ies under a main project directory. While the usercan save any file to any directory with in the module, keeping the inpu ts andoutput files in separa te director ies can make it easier to iden t ify files as wellas examine files tha t have a lready been used a t some la ter t ime.

A new project directory u t ility t ab under the Cr ime Travel Demand modulea llows the crea t ion of a master directory for a project and four separa te sub-director ies under the master directory tha t cor respond to the four modelingstages. The user pu t s in the name of a project in the dia logue box and poin t sit to a pa r t icu la r dr ive and dir ectory loca t ion (depending on the number ofdr ives ava ilable to the user ). For example, a project directory might be ca lled“Robber ies 2003” or “Bank robber ies 2005”. The u t ility then crea tes th isdirectory if it does not a lready exist and crea tes four sub-director iesundernea th the project directory:

Tr ip genera t ionTr ip dist r ibu t ionMode splitNetwork assignment

The user can then save the differen t ou tpu t files in to the appropr ia tedirector ies. Fur ther , for each sequen t ia l st ep in the cr ime t ravel demandmodel, the user can easily find the outpu t file from the previous st ep whichwould then become the input file for the next st ep.

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Ne w Rou tin e s

Baye s ian J ou rn e y to Crim e Modu le

The Bayesian J ourney to Cr ime module (Bayesian J t c) a re a set of tools forest imat ing the likely residence loca t ion of a ser ia l offender . It is an extension of thedistance-based J ourney to cr ime rout ine (J t c) which uses a typica l t r avel dist ance funct ionto make guesses about the likely residence loca t ion . The extension involves the use anor igin-dest ina t ion mat r ix which provides in format ion about the par t icu la r or igins ofoffenders who commit ted cr imes in par t icu la r dest ina t ions.

F ir st , the empir ica l theory behind the Bayesian J t c rou t ine will be descr ibed. Then ,the da ta requirements will be discussed. Fina lly, the rou t ine will be illust ra ted some datafrom Balt imore County.

Baye s ian P robabili ty

Bayes Theorem is a formula t ion that r ela tes the condit iona l and margina lprobability dist r ibut ions of random var iables. The m arginal probability dist r ibu t ion is aprobability independent of any other condit ions. Hence, P(A) and P(B) is the margina lprobability (or just pla in probability) of A and B respect ively.

The conditional probability is the probability of an event given tha t some otherevent has occur red. It is wr it t en in the form of P(A| B) (i.e., even t A given tha t even t B hasoccur red). In probability theory, it is defined as:

P (A and B)P(A| B) = ----------------- (Up.9)

P(B)

Condit iona l probabilit ies can be best be seen in cont ingency tables. Table Up.1below show a possible sequence of counts for two var iables (e.g., t aking a sample of personsand count ing their gender - male = 1; female = 0, and their age - older than 30 = 1; 30 oryounger = 0). The probabilit ies can be obta ined just by count ing:

P(A) = 30/50 = 0.6P(B) = 35/50 = 0.7P(A and B) = 25/50 = 0.5P(A or B) = (10 + 5 + 25)/50 = 0.8P(A| B) = 25/35 = 0.71P(B| A) = 25/30 = 0.83

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Table Up.1:

Exam ple of De te rm in in g P robabilit ie s by Cou n tin g

A has A hasNOT Occur red Occur red TOTAL

B has NOTOccur red 10 5 15

B hasOccur red 10 25 35

TOTAL 20 30

However , if four of these six ca lcu la t ions a re known, Bayes Theorem can be used tosolve for the other two. Two logica l t erms in probability are the ‘and’ and ‘or ’ condit ions. Usua lly, the symbol c is used for ‘or ’ and 1 is used for ‘and’, but wr it ing it in words makesit easier to understand. The following two theorems define these.

1. The probability tha t either A or B will occur is

P(A or B) = P(A) + P(B) - P(A and B) (Up.10)

2. The probability tha t both A and B will occur is:

P(A and B) = P(A) * P(B| A) = P(B)*P(A| B) (Up.11)

Bayes Theorem rela tes the two equiva len t s of the ‘and’ condit ion together .

P(B) * P(A| B) = P(A) * P(B| A) (Up.12)

P(A) * P(B| A)P(A| B) = ------------------- (Up.13)

P(B)

The theorem is somet imes ca lled the ‘inverse probability’ in tha t it can inver t twocondit iona l probabilit ies:

P(B) * P(A| B)P(B| A) = -------------------- (Up.14)

P(A)

By plugging in the va lues from the example in table Up.1, the reader can ver ify tha tBayes Theorem produces the correct resu lt s (e.g., P(B| A) = 0.7 * 0.71/0.6 = 0.83).

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Baye s ian In fe re n ce

In the st a t ist ica l in terpreta t ion of Bayes Theorem, the probabilit ies a re est imates ofa r andom va r iable. Let 2 be a parameter of in terest and let X be some da ta . Thus, we canexpress Bayes Theorem as:

P(X| 2) * P(2)P(2| X) = ------------------- (Up.15)

P (X)

In terpret ing th is equa t ion , P(2| X) is the probability of 2 given the da ta , X. P(2) isthe probability tha t 2 has a cer t a in dist r ibut ion and is oft en ca lled the prior probability. P (X| 2) is the probability tha t the da ta would be obta ined given tha t 2 is t rue and is oft enca lled the likelihood function (i.e., it is the likelihood tha t the da ta will be obta ined giventhe dist r ibu t ion of 2). Fina lly, P(X) is the margina l probability of the da ta , the probabilityof obta in ing the da ta under a ll possible scenar ios; essen t ia lly, it is the da ta .

We can rephrase th is equa t ion in logica l t erms:

Likelihood of Pr iorThe probability tha t obta in ing the da ta probability2 is t rue given the given 2 is t rue * of 2da ta , X = -------------------------------------------------- (Up.16)

Pr ior probability of X

In other words, th is formula t ion a llows an est imate of the probability of a par t icu larparameter , 2, to be upda ted given new informat ion . Since 2 is the pr ior probability of anevent , given some new da ta , X, Bayes Theorem can be used to upda te the est imate of 2. The pr ior probability of 2 can come from pr ior studies, an assumpt ion of no differencebetween any of the condit ions a ffect ing 2, or an assumed mathemat ica l dist r ibu t ion . Thelikelihood funct ion can a lso come from empir ica l studies or an assumed mathemat ica lfunct ion . Ir respect ive of how these a re in terpreted, the resu lt is an est imate of theparameter , 2, given the evidence, X. This is ca lled the posterior probability (or poster iordist r ibu t ion).

A poin t tha t is often made is tha t the pr ior probability of obta in ing the da ta (thedenomina tor of the above equa t ion) is not known or can’t easily be eva lua ted. The da ta a rewhat we obta in from some da ta ga ther ing exercise (either exper imenta l or obta ined fromobservat ions). Thus, it ’s not easy to est imate it . Consequent ly, oft en the numera tor on ly isused for est imate the poster ior probability since

P(2| X) % P(X| 2) * P(2) (Up.17)

where % means ‘propor t iona l to’. In some sta t ist ica l methods (e.g., the Markov Cha inMonte Car lo simula t ion , or MCMC), the parameter of in terest is est imated by thousands ofrandom simula t ions using approximat ions to P(X| 2) and P(2) respect ively.

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Application of Baye s ian In fe re n ce to J ou rn e y to Crim e An alys is

We can apply Bayes Theorem to the journey to cr ime methodology. In the J ourneyto Cr ime (J t c) method, we want to make a guess (est imate) about where a ser ia l offender isliving. The J t c method produces an est ima te based on an assumed t ravel dist ance funct ion(or , in more refined uses of the method, t r avel t ime). Tha t is, we assume tha t an offenderfollows a typica l t r avel dist ance funct ion . This funct ion can be est imated from pr iorstudies (Canter and Gregory, 1994; Canter , 2003) or from crea t ing a sample of knownoffenders - a ca libra t ion sample (Levine, 2004) or from assuming tha t every offenderfollows a par t icu la r mathemat ica l funct ion (Rossmo, 1995; 2000). Essen t ia lly, it ’s a pr iorprobability for a par t icu la r loca t ion , 2. Tha t is, it is a guess about where the offender lives(the dist r ibu t ion of 2). We assume the offender we’re in terested in a r rest ing is following anexist ing t ravel dist ance model. It is the equiva len t of P(2) in equa t ion Up.15 above.

However , we can add addit iona l in format ion from a sample of known offenderswhere both the cr ime loca t ion and the residence loca t ion are known. We would obta in th isin format ion from the ar rest records, each of which will have a cr ime loca t ion defined(which we’ll ca ll a ‘dest ina t ion’) and a residence loca t ion (ca lled an ‘or igin’). If we thenassign these loca t ions to a set of zones, we can crea te a mat r ix tha t rela tes the or igin zonesto the dest ina t ion zones (figure Up.1). This is ca lled an origin-destination mat r ix (or a t r ipdist r ibu t ion mat r ix).

In th is figure, the number indica te the number of cr imes tha t were commit ted ineach dest ina t ion zone tha t or igina ted (i.e., the offender lived) in each or igin zone. Forexample, t aking the fir st row in figure Up.1, there were 37 cr imes tha t were commit ted inzone 1 and in which the offender a lso lived in zone 1; there were 15 cr imes commit ted inzone 2 in which the offender lived in zone 1; however , there were on ly 7 cr imes commit tedin zone 2 in which the offender lived in zone 1; and so for th .

Note two th ings about the mat r ix. Fir st , the number of or igin zones can be (andusua lly is) grea ter than the number of dest ina t ion zones because cr imes can or igina teou tside the study area . Second, t he marginal t ot a ls have to be equal. That is, t he numberof cr imes commit ted in a ll dest ina t ion zones has to equa l the number of cr imes or igina t ingin a ll or igin zones.

We can t rea t t h is in format ion a s t he likelihood est ima te for t he J ou rney to Cr imeframework. Tha t is, if we have a cer ta in dist r ibu t ion of inciden t s commit ted by apar t icu lar ser ia l offender , we can use th is mat r ix to make an est imate of the likely or iginzones tha t th is offender came from, independen t of any assumpt ion about t r avel distance.In other words, th is mat r ix is equiva len t to the likelihood funct ion in equa t ion Up.15,which is repea ted below:

P(X| 2) * P(2)P(2| X) = ----------------- repea t (Up.15)

P (X)

Figure Up.1: Crime Origin-Destination Matrix

1

1

2

3

2 3

4

4

5

M

5 N

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

Crime destination zoneC

rime

orig

in z

one

37

53

81

12

24

92

7

12

4

0

6 1 0

G

G

84

178

1466

12 9

14

2115

5

8

10

7 28

43

4

7

2

3

3

4

6

10

12

33

15

14

153 99 110276 1245 812

711

346

1050

43,240

N ot e t h a t t h e ou t p u t of t h e r ou t in e lis t s P (O -D ) a n d P (O -D | J t c) r a t h er t h a n P (O ) a n d1

P (O | J t c). H ow ever , t h is w ill be ch a n ged in t h e n ext ver s ion .

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Thus, we can improve our est imate of the likely loca t ion of a ser ia l offender byupda t ing the est imate from the J t c method, P(2), with informat ion from an empir ica lly-der ived likelihood est imate, P(X| 2). Bu t , wha t abou t the denomina tor , P (X)? Essen t ia lly,it ’s the spa t ia l dist r ibu t ion of a ll cr imes ir respect ive of which par t icu lar model or scenar iowe’re explor ing. One can th ink of it as a general dist r ibu t ion of offenders ir respect ive ofwhere any par t icu la r offender has commit ted cr imes (I ca ll it the ‘round up the usua lsuspect s’ dist r ibu t ion).

I’m going to change the symbols a t th is poin t so the J tc r epresent s the dist ance-based J ourney to Cr ime est imate, O r epresen t s an est imate based on an or igin-dest ina t ionmat r ix, and O| J tc r epresen t s the par t icu lar or igins associa ted with cr imes commit ted inthe same zones as tha t iden t ified in the J tc est imate. Therefore, there are th ree differen test imates of where an offender lives:

1. An est imate of the residence loca t ion of a single offender based on theloca t ion of the inciden t s tha t th is person commit ted and an assumed t raveldist ance funct ion , P(J t c);

2. An est imate of the residence loca t ion of a single offender based on a genera ldist r ibu t ion of a ll offenders, ir r espect ive of any par t icu la r dest ina t ions forinciden t s, P(O). Essen t ia lly, th is is the dist r ibu t ion of or igins ir respect ive ofthe dest ina t ions; and

3. An est imate of the residence loca t ion of a single offender based on thedist r ibu t ion of offenders given the dist r ibu t ion of inciden t s commit ted by thesingle offender , P(O| J tc).

Therefore, we can use Bayes Theorem to crea te an est imate tha t combinesin format ion both from a t ravel dist ance funct ion and an or igin -dest inat ion ma tr ix(equa t ion Up.18):1

P(O| J tc) * P(J tc)P(J tc| O) = ------------------------ (Up.18)

P (O)

Th e Baye s ian J ou rn e y to Crim e Estim ation Modu le

The Bayesian J ourney to Cr ime est imat ion module is made up of two rout ines, onefor diagnosing wh ich J ou rney to Cr ime method is best and one for applying tha t method toa par t icu la r ser ia l offender . Figure Up.2 shows the layout of the module.

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Data pre paration for Baye s ian J ou rn e y to crim e e s tim ation

There are th ree da ta set s tha t a re required and one opt iona l da ta set . The th reerequired ones a re:

1. The incident s commit ted by a single offender for which an est imate will bemade of where tha t individua l lives;

2. A J ourney to Cr ime t ravel distance funct ion tha t est imates the likelihood ofan offender commit t ing cr imes a t a cer t a in dist ance (or t r avel t ime if anetwork is used); and

3. An or igin-dest ina t ion mat r ix.

The four th , opt iona l da ta set is a diagnost ics file of mult iple known ser ia l offendersfor which both their r esidence and cr ime loca t ions a re known.

S er ia l offen d er d a t a

For each ser ia l offender for whom an est imate will be made of where tha t per sonlives, the da ta set should include the loca t ion of the incident s commit ted by the offender .The da ta a re set up as a ser ies of records in which each record represen t s a single event . On each da ta set , there are X and Y coordina tes ident ifying the loca t ion of the incident sth is person has commit ted (Table Up.2).

Journey to Crime travel function

The Journey to Crime travel function (Jtc) is an estimate of the likelihood of an offendertraveling a certain distance. Typically, it represents a frequency distribution of distances traveled, thoughit could be a frequency distribution of travel times if a network was used to calibrate the function withthe Journey to crime estimation routine. It can come from an a priori assumption about travel distances,prior research, or a calibration data set of offenders who have already been caught. The “CalibrateJourney to Crime function” routine (on the Journey to Crime page under Spatial modeling) can be used toestimate this function. Details are found in chapter 10 of the CrimeStat manual.

The BJtc routine can use two different travel distance functions: 1) An already-calibrateddistance function; and 2) A mathematical formula. Either direct or indirect (Manhattan) distances can beused though the default is direct (see Measurement parameters).

Or igin -d est in a t ion m a t r ix

The or igin-dest ina t ion mat r ix rela tes the number of offenders who commit cr imes inone of N zones who live (or igina te) in one of M zones, simila r to figure Up.1 above. It canbe crea ted from the “Calcu la te observed or igin-dest ina t ion t r ips” rou t ine (on the ‘Descr ibeor igin-dest ina t ion t r ips’ page under the Tr ip dist r ibut ion module of the Cr ime TravelDemand model).

Bayesian Journey-to-Crime Modeling Screen Figure Up.2:

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Table Up.2:

Minimum Information Required for Serial Offender Data:Example for Offender Who Committed Seven Incidents

ID UCR INCIDX INCIDYTS7C 430.00 -76.494300 39.284600TS7C 440.00 -76.450900 39.318500TS7C 630.00 -76.460600 39.315700TS7C 430.00 -76.450700 39.318100TS7C 311.00 -76.449700 39.316200TS7C 440.00 -76.450300 39.317800TS7C 341.00 -76.448200 39.312300

Dia gn ost i cs fi l e for Ba yesia n J t c r ou t in e

The four th da ta set is used for est imat ing which of severa l pa rameters is best a tpredict ing the residence loca t ion of ser ia l offenders in a par t icu la r ju r isdict ion . Essen t ia lly,it is a set of ser ia l offenders, each record of which has in format ion on the X and Ycoordina tes of the residence loca t ion as well as the cr ime loca t ion . It is simila r to tableUp.2 above except tha t it involves in format ion on mult iple offenders, each of whomcommit ted mult iple even t s. For example, offender T7b commit ted seven inciden t s whileoffender S8a commit ted eigh t incident s.

The a im of the diagnost ics file is to provide in format ion to the ana lyst about whichof severa l pa rameter s (to be descr ibed below) are best a t guessing where an offender lives. The assumpt ion is tha t if a par t icu lar parameter was best with the K offenders in adiagnost ics file in which the residence loca t ion was known, then the same parameter willa lso be best for a ser ia l offender for whom the residence loca t ion is not known.

How many ser ia l offenders a re needed to make up a diagnost ics file? There is nosimple answer to th is. Clear ly, the more, the bet t er since the a im is to iden t ify whichparameter is most sensit ive with a cer t a in level of precision and accuracy. I used 88offenders in my diagnost ics file (see below). Cer ta in ly, a min imum of 10 would benecessary. But , more would cer ta in ly be more accura te. Fur ther , the offender records usedin the diagnost ics file should be simila r in other dimensions to the offender tha t is beingt racked. However , th is may be impract ica l. In the example da ta set , I combined offenderswho commit ted differen t types of cr imes. As I show below, differen t cr imes appear todist inguish the parameters tha t should be used. But , th is does r equire many sets of knownser ia l offenders.

Once the da ta set s have been collected, they need to be placed in an appended file,with one ser ia l offender on top of another . Each r ecord has to represen t a single inciden t . Fur ther , the records have to be a r ranged sequent ia lly with a ll the records for a single

14

offender being grouped together . The rou t ine au tomat ica lly sor t s the da ta by the offenderID. But , to be su re tha t the resu lt is consisten t , the da ta shou ld be prepared in th is way.

The st ructure of the records is seen in the example in table Up.3. At the minimum,there is a need for an ID field, and the X and Y coordina tes of both cr ime loca t ion and theresidence loca t ion. Thus, in the example, a ll the records for the fir st offender (Num 1) a retogether ; a ll the records for the second offender (Num 2) a re together ; and so for th . The IDfield is any st r ing var iable. In table Up.2, the ID field is labeled “Num #”, but any labelwould be acceptable as long as it is consist ent (i.e., a ll the records of a single offender a retogether ).

In addit ion to the ID field, the X and Y coordina tes of both the cr ime and residenceloca t ion must be included on each record. In table Up.3, the cr ime loca t ion coordina tes a reca lled IncidX and IncidY respect ively while the residence loca t ion coordina tes a re ca lledHomeX and HomeY respect ively. Again , any label is acceptable as long as the columnloca t ions in each record are consist en t . As with the J ourney to Cr ime ca libra t ion file, otherfields can be included.

Logic of th e Rou tin e

The rou t ine is divided in to two par t s (under the “Bayesian J ourney to Cr imeEst imat ion” page of “Spa t ia l Modeling”):

1. Diagnost ics for J ourney to Cr ime methods; and

2. Est imate likely or igin loca t ion of a ser ia l offender .

The “diagnost ics” rou t ine takes the diagnost ics ca libra t ion file and est imates anumber of methods for each ser ia l offender in the file and test s the accuracy of eachparameter aga inst the known residence loca t ion. The resu lt is a compar ison of thedifferen t methods in terms of accuracy in predict ing both where the offender lives as wellas min imizing the dist ance between where the method predict s the most likely loca t ion forthe offender and where the offender actua lly lives.

The “est imate” rout ine a llows the user to choose one method and to apply it to theda ta for a single ser ia l offender . The resu lt is a probability sur face showing the resu lt s ofthe method in predict ing where the offender is liable to be living.

15

Table Up.3:

Exam ple Re cords in Baye s ian J ou rn e y to Crim e Diagn ostics File

Offender ID HomeX HomeY IncidX IncidYNum 1 -77.1496 39.3762 -76.6101 39.3729Num 1 -77.1496 39.3762 -76.5385 39.3790Num 1 -77.1496 39.3762 -76.5240 39.3944Num 2 -76.3098 39.4696 -76.5427 39.3989Num 2 -76.3098 39.4696 -76.5140 39.2940Num 2 -76.3098 39.4696 -76.4710 39.3741Num 3 -76.7104 39.3619 -76.7195 39.3704Num 3 -76.7104 39.3619 -76.8091 39.4428Num 3 -76.7104 39.3619 -76.7114 39.3625Num 4 -76.5179 39.2501 -76.5144 39.3177Num 4 -76.5179 39.2501 -76.4804 39.2609Num 4 -76.5179 39.2501 -76.5099 39.2952Num 5 -76.3793 39.3524 -76.4684 39.3526Num 5 -76.3793 39.3524 -76.4579 39.3590Num 5 -76.3793 39.3524 -76.4576 39.3590Num 5 -76.3793 39.3524 -76.4512 39.3347Num 6 -76.5920 39.3719 -76.5867 39.3745Num 6 -76.5920 39.3719 -76.5879 39.3730Num 6 -76.5920 39.3719 -76.7166 39.2757Num 6 -76.5920 39.3719 -76.6015 39.4042Num 7 -76.7152 39.3468 -76.7542 39.2815Num 7 -76.7152 39.3468 -76.7516 39.2832Num 7 -76.7152 39.3468 -76.7331 39.2878Num 7 -76.7152 39.3468 -76.7281 39.2889....Num Last -76.4320 39.3182 -76.4297 39.3172Num Last -76.4880 39.3372 -76.4297 39.3172Num Last -76.4437 39.3300 -76.4297 39.3172Num Last -76.4085 39.3342 -76.4297 39.3172Num Last -76.4083 39.3332 -76.4297 39.3172Num Last -76.4082 39.3324 -76.4297 39.3172Num Last -76.4081 39.3335 -76.4297 39.3172

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Diagn ostics for J ou rn e y to Crim e Me th ods

The following applies to the “diagnost ics” rou t ine on ly.

Data In pu t

The user input s the four required da ta set s.

1. Any pr imary file with an X and Y loca t ion. A suggest ion is to use one of thefiles for the ser ia l offender , bu t th is is not essen t ia l;

2. A gr id tha t will be over la id on the study area . Use the Reference File underDa ta Setup to define the X and Y coordina tes of the lower -left and upper -r igh t corners of the gr id as well as the number of columns;

3. A J ourney to Cr ime t ravel funct ion (J tc) tha t est imates the likelihood of anoffender commit t ing cr imes a t a cer ta in dist ance (or t ravel t ime if a networkis used);

4. An or igin-dest ina t ion mat r ix; and

5. The diagnost ics file of known ser ia l offenders in which both their residenceand cr ime loca t ions a re known.

Me th ods Te ste d

The “diagnost ics” rout ine compares seven methods for est imat ing the likely loca t ionof a ser ia l offender :

1. The J tc dist ance method, P(J tc);

2. The genera l cr ime dist r ibu t ion based on the or igin-dest ina t ion mat r ix, P(O). Essen t ia lly, th is is the dist r ibu t ion of or igins ir respect ive of the dest ina t ions;

3. The dist r ibu t ion of or igins based on ly on the inciden t s commit ted by theser ia l offender , P(O| J tc);

4. The product of the J t c est imate (1 above) and the dist r ibu t ion of or iginsbased on ly on the inciden t s commit ted by the ser ia l offender (3 above),P(J tc)*P(O| J tc). Th is is the numera tor of the Bayesian funct ion (equa t ionUp.18), the product of the pr ior probability t imes the likelihood est imate;

5. The simple average of the J t c est imate (1 above) and the dist r ibu t ion ofor igins based on ly on the dist r ibu t ion of inciden t s commit ted by the ser ia l

P r op os ed by D r . S h a w -p in M ia ou , C ollege S t a t ion , T X2

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offender (3 above), P(J tc) + P(O| J tc). Th is is an a lt erna t ive to the productt erm (4 above) ;2

6. The fu ll Bayesian est imate as indica ted in equa t ion Up.18 above (method 4above divided by method 2 above), P(Bayesian); and

7. The center of min imum distance, Cmd. Previous research has indica ted tha tthe cen ter of min imum of distance produces the least er ror in min imizing thedistance between where the method predict s the most likely loca t ion for theoffender and where the offender actua lly lives (Levine, 2004).

In te rpolate d Grid

For each ser ia l offender in tu rn and for each method, the rou t ine over lays a gr idover the study area. The gr id is defined by the Reference F ile parameters (under DataSetup; see chapter 3). The rou t ine then in terpola tes each input da ta set in to a probabilityest imate for each gr id cell with the sum of the cells equa ling 1.0 (with in th ree decimalplaces). The manner in which the in terpola t ion is done var ies by the method:

1. For the J t c method, P(J tc), the rout ine in terpola tes the selected dist ancefunct ion to each gr id cell t o produce a density est ima te. The densit ies a r ethen re-sca led so tha t the sum of the gr id cells equa ls 1.0 (see chapter 10);

2. For the genera l cr ime dist r ibut ion method, P(O), the rou t ine sums up theinciden t s by each or igin zone from the or igin-dest ina t ion mat r ix andin terpola tes tha t using the normal dist r ibu t ion method of the single kerneldensity rou t ine (see chapter 9). The density est imates a re conver ted toprobabilit ies so tha t the sum of the gr id cells equa ls 1.0;

3. For the dist r ibu t ion of or igins based on ly on the inciden t s commit ted by theser ia l offender , from the or igin-dest ina t ion mat r ix the rou t ine ident ifies thezone in which the incident s occur and reads on ly those or igins associa tedwith those dest ina t ion zones. Mult iple incident s commit ted in the sameor igin zone are counted mult iple t imes. The rou t ine adds up the number ofincident s counted for each zone and uses the single kernel density rou t ine toin terpola te the dist r ibu t ion to the gr id (see chapter 9). The densityest imates a re conver ted to probabilit ies so tha t the sum of the gr id cellsequa ls 1.0;

4. For the product of the J t c est imate and the dist r ibu t ion of or igins based on lyon the inciden t s commit ted by the ser ia l offender , the rou t ine mult iples theprobability est imate obta ined in 1 above by the probability est imate obta ined

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in 3 above. The probabilit ies a re then re-sca led so tha t the sum of the gr idcells equa ls 1.0;

5. For the simple average of the J t c est imate and the dist r ibu t ion of or iginsbased on ly on the inciden t s commit ted by the ser ia l offender , the rou t ineadds the probability est imate obta ined in 1 to the probability est imateobta ined in 3 above and divides by two. The product probabilit ies a re thenre-sca led so tha t the sum of the gr id cells equa ls 1.0;

6. For the fu ll Bayesian est imate as indica ted in equa t ion Up.18 above, therout ine takes the product est imate (4 above) and divides it by the genera lcr ime dist r ibut ion est imate (2 above). The resu lt ing probabilit ies a re thenre-sca led so tha t the sum of the gr id cells equa ls 1.0; and

7. Fina lly, for the cen ter of min imum distance est imate, the rou t ine ca lcu la testhe cen ter of min imum distance for each ser ia l offender in the “diagnost ics”file and ca lcu la tes the dist ance between th is st a t ist ic and the loca t ion wherethe offender is actua lly residing. This is used only for the dist ance er rorcompar isons.

Note in a ll of the probability est imate (excluding 7), the cells a re conver ted toprobabilit ies pr ior to any mult iplica t ion or division . The resu lt s a re then re-sca led so tha tthe resu lt ing gr id is a probability (i.e., a ll cells sum to 1.0).

Ou tpu t of Rou tin e

For each offender in the “diagnost ics” file, the rou t ine ca lcu la tes th ree differen tsta t ist ics for each of the methods:

1. The est imated probability in the cell where the offender actua lly lives. Itdoes th is by, fir st , iden t ifying the gr id cell in which the offender lives (i.e.,the gr id cell where the offender ’s residence X and Y coordina te is found) and,second, by not ing the probability associa ted with tha t gr id cell;

2. The percent ile of a ll gr id cells in the ent ire gr id tha t have to be searched tofind the cell where the offender lives based on the probability est imate from1, ranked from those with the h ighest probability to the lowest . Obviously,th is percen t ile will va ry by how large a reference gr id is used (e.g., with avery la rge r eference gr id, the percen t ile where the offender actua lly lives willbe small whereas with a small reference gr id, the percen t ile will be la rger ). Bu t , since the purpose is to compare methods, the actua l percen tage shou ldbe t rea ted as a rela t ive index. The resu lt is sor ted from low to h igh so tha tthe smaller the percent ile, the bet t er . For example, a percent ile of 1%indica tes tha t the probability est imate for the cell where the offender lives iswith in the top 1% of a ll gr id cells. Conversely, a percent ile of 30% indica tes

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tha t the probability est imate for the cell where the offender lives in with inthe top 30% of a ll gr id cells; and

3. The dist ance between the cell with the h ighest probability and the cell wherethe offender lives.

Table Up.4 illust ra tes a typica l probability ou tpu t for four of the methods (there aretoo many to display in a single table). Only five ser ia l offenders a re shown in the table.

Table Up.4:Sam ple Ou tpu t of P robabili ty Matrix

Percentilefor

Percentilefor

Percentilefor

Percentilefor

Offender P(Jtc) P(Jtc) P(O|Jtc) P(O|Jtc) P(O) P(O) P(Jtc)*P(O|Jtc) P(Jtc)*P(O|Jtc)

1 0.001169 0.01% 0.000663 0.01% 0.000270 11.38% 0.002587 0.01%

2 0.000292 5.68% 0.000483 0.12% 0.000377 0.33% 0.000673 0.40%

3 0.000838 0.14% 0.000409 0.18% 0.000153 30.28% 0.001720 0.10%

4 0.000611 1.56% 0.000525 1.47% 0.000350 2.37% 0.000993 1.37%5 0.001619 0.04% 0.000943 0.03% 0.000266 11.98% 0.004286 0.04%

Table Up.5 illust ra tes a typica l dist ance ou tpu t for four of the methods. Only five ser ia loffenders a re shown in the table.

Table Up.5:Sam ple Ou tpu t of Distan ce Matrix

Distance for

Offender Distance(Jtc) Distance(O|Jtc) Distance(O) P(Jtc)*P(O|Jtc)

1 0.060644 0.060644 7.510158 0.060644

2 6.406375 0.673807 2.23202 0.840291

3 0.906104 0.407762 11.53447 0.407762

4 3.694369 3.672257 2.20705 3.672257

5 0.423577 0.405526 6.772228 0.423577

Thus, these th ree indices provide in format ion about the accu racy of the method. Anidea l method will have a h igher probability in the cell where the offender lives than anyother cell, will be with in a very small percen tage of a ll gr id cells (ranked from those withthe h ighest probability to the lowest ), and will have a very small dist ance between the cellwith the h ighest probability and the cell where the offender actua lly lives.

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A poor indica tor , on the other hand, will have a much lower probability in the cellwhere the offender lives than any other cell, will have a h igh percen tage of a ll gr id cellswith probabilit ies grea ter than th is cell (r anked from those with the h ighest probability tothe lowest ), and will have a large dist ance between the cell with the h ighest probabilityand the cell where the offender actua lly lives.

Ou t p u t m a t r ices

The “diagnost ics” rou t ine ou tpu t s two separa te mat r ices. The probability est imates(numbers 1 and 2 above) a re presented in a separa te mat r ix from the dist ance est imates(number 3 above). The user can save the tot a l ou tpu t as a t ext file or can copy and past eeach of the two output mat r ices in to a spreadsheet separa tely. We recommend thecopying-and-past ing method (in to a spreadsheet ) as it will be difficu lt to line up differ ingcolumn widths for the two mat r ices and summary t ables a t the bot tom of each .

S u m m a r y S t a t i st i cs

The “diagnost ics” rout ine will a lso provide summary in format ion a t the bot tom ofeach mat r ix. There a re summary measures and counts of the number of t imes a methodhad the h ighest probability or the closest dist ance from the cell with the h ighest probabilityto the cell where the offender actua lly lived; t ies between methods a re counted as fract ions(e.g., two t ied methods a re given 0.5 each; three t ied methods a re give 0.33 each). For theprobability mat r ix, these st a t ist ics include:

1. The mean (probability or percen t ile);2. The median (probability or percen t ile);3. The st andard devia t ion (probability or percen t ile);4. The number of t imes the J t c est imate produces the h ighest probility;5. The number of t imes the O| J tc est imate produces the h ighest probability;6. The number of t imes the O est imate produces the h ighest probability;7. The number of t imes the product t erm est imate produces the h ighest

probability;8. The number of t imes the average term est imate produces the h ighest

probability; and9. The number of t imes the Bayesian est imate produces the h ighest probability.

For the dist ance mat r ix, these st a t ist ics include:

1. The mean dist ance;2. The median dist ance;3. The standard devia t ion dist ance;4. The number of t imes the J t c est imate produces the closest dist ance;5. The number of t imes the O| J tc est imate produces the closest dist ance;6. The number of t imes the O est imate produces the closest dist ance;

N ot e: t h e gr id is con t r olled by t h e R efer en ce F ile3

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7. The number of t imes the product t erm est imate produces the closestdist ance;

8. The number of t imes the average term est imate produces the closestdist ance;

9. The number of t imes the Bayesian est imate produces the closest dist ance;and

10. The number of t imes the cen ter of min imum distance produces the closestdist ance.

These st a t ist ics, especia lly the summary measures, should indica te which of themethods produces the best accuracy, defined in terms of h ighest probability (for theprobability mat r ix), closest dist ance (for the dist ance mat r ix), and efficiency, defined interms of the smallest search area to loca te the ser ia l offender .

Wh ich is th e Most Accu rate J ou rn e y to Crim e Estim ation Me th od?

To illust ra te the use of the Bayesian J t c diagnost ics rou t ine, the records of 88 ser ia loffenders who had commit ted cr imes in Ba lt imore County between 1993 and 1997 havebeen compiled. The number of incident s commit ted by these offenders var ied from 3 to 33and included a range of differen t cr ime types (la rceny, burgla ry, robbery, veh icle theft ,a r son , bank robbery).

P r oba bi l i t y Associa t ed w i t h Cel l w h er e Offen d er Lived

Table Up.6 presen t s the mean , median , and standard devia t ion of the var iousmethods for the probability mat r ix (the probability associa ted with the cell where theoffender lives). The probabilit ies a re small since there were 9000 cells in the gr id tha t wasused for the Bayesian J t c diagnost ics rou t ine. Of the six differen t parameter s used, the3

product t erm, P(J tc)*P(O| J tc), had the h ighest average probability while the fu ll Bayesianparameter (which divides the product t erm by the genera l probability) had the secondhighest . As might be expected, the genera l probability term, P(O), had the lowestprobability since th is parameter does not consider any of the informat ion associa ted withthe par t icu la r ser ia l offender .

Looking a t the median probability, which is the va lue a t which ha lf the cases havehigher va lues and ha lf a re lower , again the product t erm had the h ighest probabilityfollowed by the fu ll Bayesian term while the genera l probability had the lowest . Thus, interms of est imat ing the loca t ion where the offender actua lly lives, the product t ermproduced the h ighest est imate and the fu ll Bayesian term the second h ighest .

The st andard devia t ion of the probability indica tes the var iability in the est imates. Even though the product t erm produced the best average and median probability, it a lsohad the h ighest va r iability. Tha t is, on average, the product t erm is either much more

22

accura te or much less accura te than the other methods. On the other hand, the genera lt erm, P(O), has the least va r iability even though it is the least accura te of the parameters(i.e., precise, bu t not very accura te).

In other words, in terms of predict ing the loca t ion of where the offender actua llylived, the product t erm was more accura te, both in t erms of the average and the median ,than the J t c method. Using in format ion about the loca t ion of other offenders whocommit ted cr imes in the same loca t ions as t he ser ia l offender in addit ion to an est ima tebased on an assumed t ravel dist ance funct ion produced a more accura te est imate than justusing the in format ion about an assumed t ravel dist ance funct ion .

Table Up.6:

Me asu re m en t P aram e te rs for th e P robability for th e Ce ll Wh e re th e Offe n de r Live s

Product of Average of

Statistic P(Jtc) P(O) P(O|Jtc) P(Jtc)*P(O|Jtc) P(Jtc) & P(O|Jtc) P(Bayesian)

Mean 0.00084 0.00025 0.00052 0.00176 0.00068 0.00134

Median 0.00082 0.00027 0.00051 0.00166 0.00067 0.00119

Std. Dev. 0.00044 0.00009 0.00020 0.00114 0.00030 0.00097

Gr id P er cen t i l e Associa t ed w i th Cel l w h er e Offen d er Lived

Simila r to the probability would be the efficiency of the search . The gr id percen t ileis the percentage of cells in the ent ire gr id tha t have probabilit ies h igher than tha tassocia ted with the cell where the offender lives. This would indica te wha t percen tage ofthe ent ire gr id would need to be searched in order to loca te the cell where the offenderactua lly lives (i.e., a smaller percen t ile is bet ter ). Obviously, the exact percen tage will varyby the size of the gr id; with a large gr id, the percen t iles will genera lly be smaller than witha smaller gr id. St ill, as a rela t ive index, the percen t ile dist r ibu t ion is usefu l. Table Up.7presen t s the mean , median , and standard devia t ion of the var ious methods for the gr idpercen t ile dist r ibu t ion .

The smallest percent ile was obta ined with the average term, [P(J tc)+P(O| J tc)]/2,where only 4.1% of the gr id would need to be searched to find the offender , with the secondlowest percent ile being the product t erm, where 4.2% of the gr id would need to besearched. As would be expected, the worst percent ile was the genera l est imate where16.7% of the gr id would need to be searched.

Severa l measures have about equa l accuracy in terms of the median percent ile. TheJ tc est imate, P(J tc), the product t erm, the average t erm, and the fu ll Bayesian term show alow search percen t ile for ha lf the cases. Tha t is, they are about equa l in terms of accuracyfor ha lf the sample. Looking a t the standard devia t ion , the methods are rela t ively close interms of var iability though the genera l t erm is sligh t ly worse.

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In other words, in terms of sea rch efficiency, the product and average terms producea more efficien t sea rch than the J t c method and show less var iability across the individua lcases. Aga in , using in format ion about the loca t ion of other offenders who commit tedcr imes in the same places in addit ion to an est imate based on a t r avel dist ance funct ionproduces bet t er accuracy than just using an est imate based on a t r avel dist ance funct ion .

Table Up.7:

Me asu re m e n t P aram e ters of Grid P e rce n tile s for th e Ce ll Wh e re th e Offe n de r Live s

Product of Average of

Statistic P(Jtc) P(O) P(O|Jtc) P(Jtc)*P(O|Jtc) P(Jtc)& P(O|Jtc) P(Bayesian)

Mean 4.6% 16.7% 4.6% 4.2% 4.1% 4.6%

Median 0.1% 11.1% 0.2% 0.1% 0.1% 0.1%

StdDev 13.1% 15.5% 13.0% 12.8% 12.8% 13.2%

Dist a n ce bet w een Cel l w i t h High est Lik el ih ood a n d Cel l w h er eOffen d er Lived

Fina lly, the th ird type of measure to be used for compar ison is the dist ancebetween the cell with the h ighest probability and the cell where the offender actua llylived (in miles). Table Up.8 presen t s the mean , median , and standard devia t ion of thevar ious methods for the distance mat r ix. In addit ion to the six methods, the distancebetween the cen ter of minimum distance (Cmd) and the cell where the offender lived isincluded; ea r lier research poin ted to the cen ter of min imum distance between the mostaccura te est imate of where the offender lived (Levine, 2004).

Consisten t with the ear lier da ta , the cen ter of min imum did show the lowestaverage dist ance in predict ing the loca t ion of the ser ia l offender , 2.62 miles. However ,the product t erm had a mean a lmost as low, 2.64. Again , as would be expected, thedistance est imate for the genera l method, D(O), had the grea test average dist ance.

However , the product and average terms showed the lowest median dist anceer ror , even lower than the cen ter of min imum distance. Tha t is, in ha lf the cases, theproduct or the average had a lower er ror than any other method. Thus, in terms of anfinding a ser ia l offender , either the product or the average will be more accura te thanother methods for about ha lf the cases. Aga in , the genera l method had the leastaccuracy in terms of dist ance.

F ina lly, in terms of var iability, the cen ter of min imum distance has the least whereas thefu ll Bayesian t erm has the most . The product t erm est imate and the J t c est imate had about thesame amount of var iability.

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In other words, in terms of predict ing a loca t ion which is closest to where the offenderactua lly lived, the cen ter of min imum distance produced the least average er ror and the lowestvar iability of any other method. However , the product t erm produced an average er ror a lmostas low as the cen ter of min imum distance bu t a median er ror tha t was the lowest . The productt erm uses both in format ion about the or igins of offenders who commit ted cr imes in the sameloca t ions as the ser ia l offender as well as a t r avel distance funct ion . Fina lly, the J t c method,which on ly uses the t r avel dist ance funct ion , is less accura te than the cen ter of minimumdistance, the product t erm, and the average term.

Table Up.8:Me asu re m e n t P aram e te rs for th e Dis tan ce Be tw e e n th e

Ce ll w ith th e P e ak P robability an d th e Ce ll Wh e re th e Offe n de r Live s(Miles)

Distance for Distance for

Statistic D(Jtc) D(O) D(O|Jtc) P(Jtc)* P(O|Jtc) [P(Jtc) + P(O|Jtc)]/2 D(Bayesian D(Cmd)

Mean 2.78 8.21 3.12 2.64 2.70 3.23 2.62

Median 0.66 7.34 1.37 0.51 0.51 0.87 0.79

Std Dev 4.70 4.80 4.66 4.70 4.81 4.91 4.52

In conclusion , the product method appears to be the most usefu l using th ree differen tcr it er ia for accuracy. The Bayesian product method produces an est imate tha t has abou t thesame average er ror in terms of dist ance as the cen ter of minimum distance (the cur ren t ‘goldstandard’ for an est imate), bu t produced less er ror for about ha lf the cases. It a lso has theadvantage over the cen ter of min imum distance of producing an est imate for the cell where theoffender lives, where it shows the h ighest probability of any method both in terms of the meanand the median . Fina lly, the product t erm (as well as the average term) a re consist en t ly moreaccura te than the J t c method. Combin ing in format ion about the or igins of other offenders whocommit ted cr imes in the same loca t ions with an est ima te based on a t ravel dist ance funct ionproduces more accuracy than just using the t r avel dist ance funct ion .

The Bayesian approach , par t icu lar ly the product t erm (the numera tor of the fu llBayesian formula) appears to substan t ia lly reduce the er ror in finding a ser ia l offender over theJ tc method and, even , the cen ter of min imum distance (which is just a poin t est imate).

An ana lysis of the individua l cases a lso confirms th is. The product t erm produced ah igher probability in 74 of the 88 cases (count ing t ies). The J t c method produced a h igherprobability in only one case, which was about the same as the condit iona l case (one case) or thegenera l probability (two cases). The fu ll Bayesian term produced the h ighest probability in tencases.

In terms of the dist ance er ror measure, the cen ter of min imum distance produced thelowest average er ror for 30 of the 88 cases while the condit iona l t erm (O| J tc) had the lowester ror in 17.9 cases (including t ies). The product t erm produced a lower average dist ance er ror

25

for 9.5 cases (including t ies) and the J t c est imate produced lower average distance er rors in 8.2cases (aga in , including t ies). The rest were spread across the other methods with .

In other words, using in format ion about the or igin loca t ion of other offenders appears toimprove the accuracy of the J t c method. The resu lt is an index (the product t erm) tha t is a lmostas good as the cen ter of minimum, bu t one tha t is more usefu l since the cen ter of minimumdistance is on ly a single poin t .

Estim atin g Ru le Se ts for Diffe re n t Type s of Crim e

The above analysis looked a t a single index over a ll the test cases. An a lt erna t iveapproach is to ca tegor ize the cases in to discrete ca tegor ies and then determine which measureproduces the greatest accuracy. Two ca tegor iza t ion var iables were examined. The fir st was thenumber of cases tha t each offender commit ted. Unfor tuna tely, th is did not demonst ra te aconsisten t pa t t ern . The second was the cr ime type, which did show a consisten t pa t t ern .

Each of the ser ia l offender was ca tegor ized in to a crim e type, defined as the type ofoffense tha t the offender most ly commit ted. Few offenders commit ted on ly a single type of cr ime,but there was a predominant pa t t ern . One offender had an equa l number of incident s for twodifferen t types (robbery, la rceny); in th is case, the cr ime type was defined by the most ser ious ofthe cr imes.

The resu lt s were six cr ime type ca tegor ies: Larceny (45 cases); Assau lt (12 cases);Robbery (10 cases); Burgla ry (9 cases); Vehicle theft (10 cases); and other cr imes (2 cases). TableUp.9 shows the average probability for each cr ime type with the cell having the h ighestprobability being boldfaced for each cr ime type.

Table Up.9:

Crime Type Me asu re m e n t P aram e ters for th eP robability for th e Ce ll Wh e re th e Offe n de r Live s

Average of

Product P(Jtc) &

Crime Type Cases P(Jtc) P(O) P(O|Jtc) P(Jtc) *P(O|Jtc) P(O|Jtc) P(Bayesian)

Assault 12 0.00090 0.00029 0.00063 0.00184 0.00076 0.00142

Burglary 9 0.00139 0.00018 0.00070 0.00348 0.00104 0.00309

Larceny 45 0.00117 0.00024 0.00060 0.00259 0.00089 0.00201

Robbery 10 0.00088 0.00028 0.00062 0.00223 0.00075 0.00180

Vehicle theft 10 0.00076 0.00026 0.00054 0.00138 0.00065 0.00101

Other crimes 2 0.00123 0.00029 0.00069 0.00280 0.00096 0.00186

All incidents 88 0.00084 0.00025 0.00052 0.00176 0.00068 0.00134

The product t erm, P(J tc) * P(O| J tc), has the h ighest probability of any of the sixmeasures, as well as having the h ighest probability over a ll 88 cases. The fu ll Bayesian term,

26

which divides the product t erm by the genera l probability, P(O), has the second h ighestprobability. Thus, a t least with the probabilit ies, there isn’t any difference in the best methodfor each cr ime type. The product t erm is the most accura te. Note tha t J t c est imate has h igherprobability than the other est imates with the except ion of the product t erm. Again , addingin format ion abou t t he or igins of offender s who commit ted cr imes in the same loca t ion to the J t cest imate (which is based on a t r avel dist ance funct ion) has improved the overa ll accuracy.

Table Up.10 shows the average percen t ile of the gr id percen t ile dist r ibu t ion for the sixdifferen t methods over the cr ime types. Again , th is percen t ile is the average percen t of a ll gr idcells tha t have probabilit ies h igher than tha t in the cell where the offender lived. Since asmaller percen t ile is bet t er , for each cr ime type, the cell with the lowest average percen t ile isboldfaced.

With th is t able, the differen t cr ime types showed differen t pa t t erns. Overa ll, the averageterm had the lowest percen t ile (5.5%) with the product term being second lowest (5.6%). However , for the assau lt cases, the fu ll Bayesian had the lowest percen t ile. For the burgla rycases, the condit iona l probability, P(O| J tc), was the lowest . For the larceny, robbery, veh icletheft , and other cr imes, the average term had the lowest , though it was t ied with the productt erm and with the condit iona l probability for robbery and veh icle theft .

In other words, over a ll inciden t s the average t erm had the smallest sea rch percen t ilethough the product t erm was a lmost as efficien t . Note tha t both these methods a re moreefficien t than the J t c est imate with one except ion (for assau lt cases where the J t c method had asmaller percen t ile than the average and product t erms, though not the fu ll Bayesian).

Table Up.11 shows, for each cr ime type, the average dist ance between the cell with theh ighest probability and the cell where the offender lived. Since a smaller dist ance is bet t er , foreach cr ime type the cell with the lowest average dist ance is boldfaced.

Table Up.10:

Crim e Type Me asu rem e n t P aram e te rs for th e Grid P e rce n ti le for th e Ce ll Wh e re th e Offe n de r Live s

Average of

Product P(Jtc) &

Crime Type Cases P(Jtc) P(O) P(O|Jtc) P(Jtc) *P(O|Jtc) P(O|Jtc) P(Bayesian)

Assault 12 0.3% 9.0% 0.3% 0.3% 0.3% 0.6%

Burglary 9 0.03% 27.4% 0.03% 0.02% 0.02% 0.02%

Larceny 45 0.02% 15.0% 0.02% 0.02% 0.02% 0.02%

Robbery 10 15.1% 10.2% 5.2% 10.4% 9.7% 15.0%

Vehicle theft 10 0.8% 13.9% 1.0% 0.7% 0.7% 2.6%

Other crimes 2 0.02% 8.8% 0.1% 0.02% 0.02% 0.1%

All incidents 88 4.6% 16.7% 4.6% 4.2% 4.1% 4.6%

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Table Up.11:

Crime Type Me asu re m e n t P aram e ters for th e Dis tan ce Be tw e e n th e Ce ll w ith th e P e ak P robability an d th e Ce ll Wh e re th e Offe n de r Live s

(Miles)

Distance for

Distance for [P(Jtc) +

Crime Type Cases D(Jtc) D(O) D(O|Jtc) P(Jtc) *P(O|Jtc) P(O|Jtc)]/2 D(Bayesian) D(Cmd)

Assault 12 0.29 7.94 1.83 0.47 0.30 0.33 0.23

Burglary 9 2.00 9.32 3.34 1.65 1.65 3.01 2.45

Larceny 45 3.64 8.63 3.96 3.42 3.57 3.88 3.41

Robbery 10 3.54 7.92 2.41 3.61 3.64 3.98 2.92

Vehicle theft 10 2.35 6.36 1.75 2.14 2.15 3.05 2.21

Other crimes 2 0.51 5.88 1.38 0.25 0.25 4.06 0.41

All incidents 88 2.78 8.21 3.12 2.64 2.70 3.23 2.62

As ment ioned above, overa ll the cen ter of min imum distance had the smallest averagedistance between the cell with the h ighest probability and the cell where the offender lived withthe product t erm being a lmost as low (2.64 miles v. 2.62 miles). The cen ter of minimumdistance had the least average dist ance for the assau lt cases and for the la rceny cases, thoughwith the la t t er cr ime the product t erm was a lmost as low. For the burgla ry cases, the averageand product t erms had a lower average dist ance er ror , much lower than the cen ter of minimumdistance. For the robbery and veh icle theft cases, the condit iona l probability, D(O| J tc), had thesmallest average dist ance er ror .

If these cr ime type ru les a re used to select which method is to be applied for each case,then the average dist ance er ror would be reduced to 2.42 miles, compared to 2.62 miles for thecenter of min imum distance. In other words, some efficiency in the search area would beobta ined by select ing methods for differen t cr ime types, a t least in Ba lt imore County.

F igure Up.3 illust ra tes the differences in the search areas of the J t c (J t c), Bayesianproduct , cen ter of minimum distance (Cmd), and minimum cr ime type methods by const ruct ingcircles with radii equa l to the average er ror dist ance. Compared to the J t c method, the Bayesianproduct est imate had a sea rch area tha t is 9.8% smaller while the Cmd had a sea rch area tha t is11.6% smaller . However , the min imum cr ime type method had a sea rch area tha t is 24.3%smaller . Thus, in terms of efficiency, the min imum cr ime type reduced the search area bya lmost a quar ter . There is st ill considerable er ror in terms of ident ifying where the offenderlives, but the Bayesian method has reduced the search area while a cr ime type ru le set hasreduced it even more.

Of course, th is ana lysis is based on small sample sizes for most of the ca tegor ies. Itr emains to be seen whether these resu lt s hold up in subsequent studies. Never theless, it doessuggest tha t a more efficien t sea rch might be produced by select ing an appropr ia te method foreach cr ime type.

Figure Up.3:

D ick B lock of Loyola U n iver s it y of C h ica go h a s r ep lica t ed t h ese r esu lt s w it h d a t a on4

103 ser ia l offen d er s w h o com m it t ed r obber ies in C h ica go. H e fou n d t h a t t h e B a yes ia n

p r od u ct es t im a t e p r od u ced t h e h igh es t p r oba bilit ies , t h e sm a lles t sea r ch a r ea , a n d t h e

sm a lles t d is t a n ce bet w een t h e ce ll w it h t h e h igh es t p r oba bilit y a n d t h e ce ll w h er e t h e

offen d er a ct u a lly lived . H e a lso fou n d t h a t t h e B a yes ia n p r od u ct es t im a t e w a s

s ligh t ly m or e a ccu r a t e t h a n t h e cen t er of m in im u m d is t a n ce.

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Con clu s ion of th e Evalu ation

The resu lt s of these compar ison suggest tha t the product t erm is probably the mostusefu l of a ll the measures, a t least with these da ta from Balt imore County. For a single guess ofwhere a ser ia l offender is living, the cen ter of min imum distance produced the lowest dist anceer ror . But , since it is on ly a poin t est imate, it cannot poin t to a search area where the offendermight be living. The product t erm, on the other hand, produced an average dist ance er rora lmost as small as the cen ter of minimum distance, bu t it produced est imates for other gr id cellstoo. Among a ll the J ourney to Cr ime measures, it had the h ighest probability in the cell wherethe offender lived and was among the most efficien t in terms of reducing the search area . Theaverage term was a lmost as good, but was less reliable than the product t erm.

Of course, each ju r isdict ion should re-run these diagnost ics to determine the mostappropr ia te measure. It is very possible tha t ju r isdict ions will have differen t resu lt s due to theuniqueness of their land uses, st r eet layout , and loca t ion in rela t ion to the cen ter of themet ropolit an a rea . Ba lt imore County is a suburb and the conclusions in a cen t ra l city or in arura l a r ea migh t be differen t .4

Estim ate Like ly Orig in Location of a Se ria l Offe n de r

The following applies to the Bayesian J t c “Est imate likely or igin of a ser ia l offender”rout ine. Once the “diagnost ic” rou t ine has been run and a prefer red method selected, the nextrou t ine a llows the applica t ion of tha t method to a single ser ia l offender .

Data In pu t

The user input s the th ree required da ta set s and a reference file gr id:

1. The inciden t s commit ted by a single offender tha t we’re in terested in ca tch ing. This m ust be the Pr imary File;

2. A J t c funct ion tha t est ima tes the likelihood of an offender commit t ing cr imes a t acer ta in dist ance (or t r avel t ime if a network is used);

3. An or igin-dest ina t ion mat r ix; and

4. The reference file a lso needs to be defined and should include a ll loca t ions wherecr imes have been commit ted (see Reference File).

30

Me th ods Te ste d

The Bayesian J t c “Est imate” rou t ine in terpola tes the inciden t s commit ted by the ser ia loffender to a gr id, a llowing the user to est imate where the offender is liable to live. There aresix differen t methods for est imat ing the likely loca t ion of a ser ia l offender tha t can be used. However , the user has to choose one of these:

1. The J tc dist ance method, P(J tc);

2. The genera l cr ime dist r ibu t ion based on the or igin-dest ina t ion mat r ix, P(O). Essen t ia lly, th is is the dist r ibu t ion of or igins ir respect ive of the dest ina t ions;

3. The dist r ibu t ion of or igins based on ly on the inciden t s commit ted by the ser ia loffender , P(O| J tc);

4. The product of the J t c est imate (1 above) and the dist r ibu t ion of or igins based on lyon the inciden t s commit ted by the ser ia l offender (3 above), P(J tc)*P(O| J tc). Th isis the numera tor of the Bayesian funct ion (equa t ion Up.18), the product of thepr ior probability t imes the likelihood est imate;

5. The weighted average of the J t c est imate (1 above) and the dist r ibu t ion of or iginsbased on ly on the dist r ibu t ion of inciden t s commit ted by the ser ia l offender (3above), P(J tc) + P(O| J tc). This is an a lt erna t ive to the product t erm (4 above). The user must select weight s for each of the est imates such tha t the sum of theweight s equa ls 1.0. The defau lt weight s a re 0.5 for each est imate; and

6. The fu ll Bayesian est imate as indica ted in equa t ion Up.18 above (method 4 abovedivided by method 2 above), P(Bayesian).

In te rpolate d Grid

For the method tha t is selected, the rou t ine over lays a gr id on the study a rea . The gr id isdefined by the reference file parameter s (see chapter 3). The rout ine then in terpola tes the inputda ta set (the pr imary file) in to a probability est imate for each gr id cell with the sum of the cellsequa ling 1.0 (with in th ree decimal places). The manner in which the in terpola t ion is donevar ies by the method chosen:

1. For the J t c method, P(J tc), the rout ine in terpola tes the selected dist ance funct ionto each gr id cell to produce a density est imate. The density est imates a reconver ted to probabilit ies so tha t the sum of the gr id cells equa ls 1.0 (see chapter10);

2. For the genera l cr ime dist r ibut ion method, P(O), the rou t ine sums up theinciden t s by each or igin zone and in terpola tes th is to the gr id using the normaldist r ibut ion method of the single kernel density rou t ine (see chapter 9). The

31

density est imates a re conver ted to probabilit ies so tha t the sum of the gr id cellsequa ls 1.0;

3. For the dist r ibu t ion of or igins based on ly on the inciden t commit ted by the ser ia loffender , the rou t ine iden t ifies the zone in which the inciden t occurs and readson ly those or igins associa ted with those dest inat ion zones in the or igin -dest inat ionmat r ix. Mult iple inciden t s commit ted in the same or igin zone are counted mult iplet imes. The rou t ine then uses the single kernel density rou t ine to in terpola te thedist r ibu t ion to the gr id (see chapter 9). The density est imates a re conver ted toprobabilit ies so tha t the sum of the gr id cells equa ls 1.0;

4. For the product of the J t c est imate and the dist r ibu t ion of or igins based only onthe incident s commit ted by the ser ia l offender , the rou t ine mult iples theprobability est imate obta ined in 1 above by the probability est imate obta ined in 3above. The product probabilit ies a re then re-sca led so tha t the sum of the gr idcells equa ls 1.0;

5. For the average of the J t c est imate and the dist r ibu t ion of or igins based only onthe inciden t s commit ted by the ser ia l offender , the rou t ine adds the probabilityest imate obta ined in 1 above by the probability est imate obta ined in 3 above. Thesum density est imates a re conver ted to probabilit ies so tha t the sum of the gr idcells equa ls 1.0; and

6. For the fu ll Bayesian est imate as indica ted in equa t ion Up.18 above, the rou t inetakes the product est imate (4 above) and divides it by the genera l cr imedist r ibu t ion est imate (2 above). The resu lt ing density est imates a re conver ted toprobabilit ies so tha t the sum of the gr id cells equa ls 1.0.

Note in a ll est imates, the cells a re conver ted to probabilit ies pr ior to any mult iplica t ionor division . The resu lt s a re then re-sca led so tha t the resu lt ing gr id is a probability (i.e., a ll cellssum to 1.0).

Ou tpu t of Rou tin e

Once the method has been selected, the rou t ine in terpola tes the da ta to the gr id cell andoutpu t s it a s a ‘shp’, ‘mif/mid’, or Ascii file for display in a GIS program. The t abu la r ou tpu tshows the probability values for each cell in the mat r ix and a lso indica tes which gr id cell has theh ighest probability est imate.

Accu m u lator Matrix

There is a lso an in termedia te ou tpu t , ca lled the accum ulator m atrix, which the user cansave. This list s the number of or igins iden t ified in each or igin zone for the specific pa t t ern ofinciden t s commit ted by the offender , pr ior to the in terpola t ion to gr id cells. Tha t is, in readingthe or igin-dest ina t ion file, the rou t ine fir st iden t ifies which zone each inciden t commit ted by theoffender fa lls with in . Second, it reads the or igin -dest inat ion ma tr ix and iden t ifies wh ich or igin

32

zones a re associa ted with inciden t s commit ted in the par t icu lar dest ina t ion zones. Fina lly, itsums up the number of or igins by zone ID associa ted with the inciden t dist r ibu t ion of theoffender . This can be usefu l for examining the dist r ibu t ion of or igins by zones pr ior toin terpola t ing these to the gr id.

Exam ple of th e Baye s ian J ou rn e y to Crim e Rou tin e

As an example, let ’s illust ra te the rou t ines with two differen t ser ia l offenders. F ir st ,let ’s look a t the genera l ou tpu t , which will be t rue for any ser ia l offender . Figure Up.4 present sthe probability ou tpu t for the genera l or igin model, tha t is for the or igins of a ll offendersir respect ive of where they commit their cr imes. It is a probability sur face in tha t a ll the gr idcells sum to 1.0. The map is sca led so tha t each bin cover s a probability of 0.0001. The cell withthe h ighest probability is h igh ligh ted in ligh t blue.

As seen , the dist r ibut ion is heavily weighted towards the cen ter of the met ropolit an a rea ,par t icu la r ly in the City of Ba lt imore. For the cr imes commit ted in Ba lt imore County between1993 and 1997 in which both the cr ime loca t ion and the residence loca t ion was known, about40% of the offenders resided with in the City of Ba lt imore and the bu lk of those living with inBalt imore County lived close to the border with City. In other words, as a genera l condit ion ,most offenders in Ba lt imore County live rela t ively close to the cen ter .

The genera l probability outpu t does not t ake in to considera t ion any in format ion aboutthe par t icu la r pa t t ern of an offender . Therefore, let ’s look specifica lly a t a par t icu la r offender . F igure Up.5 present s the dist r ibut ion of an offender who commit ted 14 offenses between 1993and 1997 before being caught and the residence loca t ion where the individua l lived whenar rested. Of the 14 offenses, seven were theft s (la rceny), four were assau lt s, two were robber ies,and one was a burgla ry. As seen , the inciden t s a ll occur red in the southeast corner of Ba lt imoreCounty in a fa ir ly concen t ra ted pa t t ern though two of the inciden t s were commit ted more thanfive miles away from the offender ’s residence.

The genera l probability model is not very precise since it assigns the same loca t ion to a lloffenders. In the case of offender S14A, the distance er ror between the cell with the h ighestprobability and the cell where the offender actua lly lived is 7.4 miles.

On the other hand, the J t c method uses the dist r ibu t ion of the inciden t s commit ted by apar t icu lar offender and a model of a typica l t r avel dist ance dist r ibu t ion to est imate the likelyor igin of t he offender ’s residence. Figu re Up.6 shows the resu lt s of t he J t c probabilit y ou tpu t . The cell with the h ighest likelihood is h igh ligh ted in ligh t blue. As seen , th is cell is very close tothe cell where the actua l offender lived. The dist ance between the two cells was 0.34 miles.

With the J t c probability est imate, however , the area with a h igher probability (dark red)covers a fa ir ly la rge area indica t ing the rela t ive lack of precision of th is method. Of the 9,000cells in the gr id, the h ighest probability bin (0.0004 or h igher ) covers 802 gr id cells, or 8.9% ofthe study area . Never theless, the J t c est imate has produced a very good est imate of the loca t ionof the offender , as might be expected given the concent ra t ion of the inciden t s commit ted by th isperson .

Figure Up.4:

Figure Up.5:

Figure Up.6:

36

For th is same offender , F igure Up.7 shows the resu lt s of the condit iona l probability ofthe offender ’s residence loca t ion , tha t is the dist r ibu t ion of the likely or igin based on theor igins of offenders who commit ted cr imes in the same loca t ions as the S14A. Again , the mapis displayed with bins tha t cover a probability range of 0.0001 and the cell with the h ighestprobability is h igh ligh ted (in ligh t green). As seen , th is method has a lso produced a fa ir ly closeest imate, with the dist ance between the cell with the h ighest probability and the cell wherethe offender actua lly lived being 0.18 miles, about ha lf the er ror dist ance of the J t c method. Fur ther , the area with the h ighest probability (0.0004 or h igher ) covers on ly 508 of the 9000gr id cells, or 5.6% of the study a rea . Thus, the condit iona l probability est imate is not on lymore accura te than the J t c method, bu t a lso more precise (i.e., more efficien t in terms ofsea rch a rea ).

For th is same offender , figure Up.8 shows the resu lt s of the Bayesian product est imate,the product of the J t c probability and the condit iona l probability re-sca led to be a singleprobability (i.e., with the sum of the gr id cells equa l to 1.0). It is a Bayesian est imate becauseit upda tes the J t c probability est imate with the in format ion on the likely or igins of offenderswho commit ted cr imes in the same loca t ions (the condit iona l est imate). Again , the map binsizes cover a probability range of 0.0001 and the cell with the h ighest probability is h igh ligh ted(in da rk t an).

The dist ance er ror for th is method is 0.26 miles, not as precise as the condit iona lprobability est imate bu t more precise than the J t c est imate. Fur ther , th is method is lessprecise than the previous th ree methods as 936 of the 9000 cells, or 10.4% of the study area ,have probabilit ies h igher than the cell where the offender lived. In genera l, the productest imate is sligh t ly more precise, on average, than the J t c and condit iona l probability methods. But , in th is case, it is not .

F igure Up.9 shows the resu lt of the weighted average probability est imate. This is theaverage of the J t c probability and condit iona l probability; in th is case, the weight s for bothest imates were equa l (0.5). Aga in , the map bins represent probability ranges of 0.0001 and thecell with the h ighest likelihood is h igh ligh ted (in dark purple). The er ror dist ance between thecell with the h ighest probability and the cell where the offender actua lly lived was 0.26 miles,the same as with the product est imate. In th is case, the weighted average est imate was moreprecise than the product est imate with the number of cells having probabilit ies h igher thanthe cell where the offender lived being 638 out of 9000 cells, or 7.1% of the study area . It is notas precise as the condit iona l probability however .

F igure Up.10 shows the resu lt s of the fu ll Bayesian probability est imate. This methodtakes the Bayesian product est imate and divides it by the genera l or igin probability est imate. It is analogous to a risk measure tha t rela tes the number of event s to a baseline popula t ion . In th is case, it is the est imate of the probability of the upda ted J t c est imate rela t ive to theprobability of where offenders live in genera l. Again , the map bins represen t probabilityranges of 0.0001 and the cell with the h ighest likelihood is h ighligh ted (in dark yellow).

Figure Up.7:

Figure Up.8:

Figure Up.9:

Figure Up.10:

Figure Up.11:

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The fu ll Bayesian est imate produces an er ror of 0.34 miles, the same as the J t cest imate. However , it is less precise than any of the other methods as there a re 1005 ou t of the9000 gr id cells, or 11.2% of the study a rea , with higher probabilities.

Finally, the center of minimum distance (Cmd) is indicated on each of the maps with a grey cross. Inthis case, the Cmd is not as accurate as any of the other methods since it has an error distance of 0.58 miles.

In summary, all of the Journey to Crime estimate methods produce relatively accurate estimates ofthe location of the offender (S14A). Given that the incidents committed by this person were within a fairlyconcentrated pattern, it is not surprising that each of the methods produces reasonable accuracy.

But what happens if we take an offender who did not commit crimes in the same part of town? FigureUp.11 shows the distribution of an offender who committed 15 offenses (TS15A). Of the 15 offensescommitted by this individual, there were six larceny thefts, two assaults, two vehicle thefts, one robbery, oneburglary, and three incidents of arson. While the distribution of 13 of the offenses are within about a threemile radius, two of the incidents are more than eight miles away.

Only three of the Journey to Crime estimates will be shown. The general method produces an error of4.6 miles. Figure Up.12 shows the results of the Jtc method. Again, the maps bins are in ranges of 0.0001 andthe cell with the highest probability is highlighted. As seen, the cell with the highest probability is locatednorth and west of the actual offender’s residence. The distance error is 1.89 miles.

Figure Up.13 shows the result of the conditional probability estimate for this offender. Again, themapping bins represent probability ranges of 0.0001 and the cell with the highest probability is highlighted. Inthis case, the conditional probability method is less accurate than the Jtc method with a distance between thecell with the highest probability and the cell where the offender lived being 2.39 miles. However, this methodis also more precise than the Jtc method with 305 or the 9000 grid cells, or 3.4% of the study area, havingprobabilities higher than that in the cell where the offender lived.

Finally, figure Up.14 shows the results of the product probability estimate. Again, the mapping binsrepresent probability ranges of 0.0001 and the cell with the highest probability is highlighted. For this method,the error distance is only 0.47 miles, much less than the Jtc method. Further, it is smaller than the center ofminimum distance which has a distance error of 1.32 miles. Again, updating the Jtc estimate with informationfrom the conditional estimate produces a more accurate guess where the offender lives.

However, as with the previous example, the product estimate does this with less precision since 852 ofthe 9000 grid cells, or 9.5% of the study area, have probability levels higher than that in the cell where theoffender lived. As mentioned above, over all the 88 cases tested, the product estimate product the smallestsearch area of any of the Journey to Crime methods. However, in this case as well, it is less precise than theJtc method.

Figure Up.12:

Figure Up.13:

Figure Up.14:

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The “Estimate likely origin of a serial offender” routine allows the estimation of a probability gridbased on a single selected method. The user must decide which probability method to select and the routinethen calculates that estimate and assigns it to a grid. As mentioned above, the “diagnostics” routine should befirst run to decide on which method is most appropriate. In these 88 cases, the Bayesian product estimate wasthe most accurate of all the probability methods. But, we don’t really know whether it will be the mostaccurate for other jurisdictions. Differences in the balance between central-city and suburbs, the roadnetwork, and land uses may change the travel patterns of offenders.

Utility of the Bayesian Journey to Crime Estimation Method

Nevertheless, the Bayesian approach to Journey to Crime estimation does appear to improve theaccuracy of the method by integrating information based on the origins of offenders who committed crimes insimilar locations to that of the serial offender of interest. Using two sources of information improves both theaccuracy of the best guess (the cell with the highest probability) as well as the efficiency of the search. Whenthis is combined with differentiating serial offenders by their predominant crime, then even more accuracywill be obtained with the method.

In short, the Bayesian Jtc methodology is an improvement over the current methods and appears to beas good, and more useful, than the center of minimum distance. A caveat should be noted, however, in thatthe Bayesian method still has a substantial amount of error. The average distance error of even the bestapproach, which uses different methods for different types of crimes, was still 2.4 miles with an impliedsearch area of over 18 square miles. Much of this error reflects, I believe, the inherent mobility of offenders,especially those living in a suburb such as in Baltimore County. While adolescent offenders tend to commitcrimes within a more circumscribed area, the ability of an adult to own an automobile and to travel outside theresidential neighborhood is turning crime into a much more mobile phenomena than it was, say, 50 years agowhen only about half of American households owned an automobile.

Thus, the Bayesian approach to Journey to Crime estimation must be seen as a tool which produces anincremental improvement in accuracy and precision. Geographic profiling is but one tool in the arsenal ofmethods that police must use to catch serial offenders.

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References

Canter, David (2003). Dragnet: A Geographical Prioritisation Package. Center for Investigative Psychology, Department of Psychology, The University of Liverpool: Liverpool, UK.http://www.i-psy.com/publications/publications_dragnet.php.

Canter, D. and A. Gregory (1994). “Identifying the residential location of rapists”, Journal of the ForensicScience Society, 34 (3), 169-175.

Levine, Ned (2004). “Journey to crime Estimation”. Chapter 10 of Ned Levine (ed), CrimeStat III: A SpatialStatistics Program for the Analysis of Crime Incident Locations (version 3.0). Ned Levine & Associates,Houston, TX.; National Institute of Justice, Washington, DC. November.http://www.icpsr.umich.edu/crimestat. Originally published August 2000.

Rossmo, D. Kim (2000). Geographic Profiling. CRC Press: Boca Raton Fl.

Rossmo, D. Kim (1995). “Overview: multivariate spatial profiles as a tool in crime investigation”. In CarolynRebecca Block, Margaret Dabdoub and Suzanne Fregly, Crime Analysis Through Computer Mapping. PoliceExecutive Research Forum: Washington, DC. 65-97.

Wikipedia (2007a). “Geometric mean” http://en.wikipedia.org/wiki/Geometric_mean and “Weightedgeometric mean” http://en.wikipedia.org/wiki/Weighted_geometric_mean.

Wikipedia (2007b). “Harmonic mean” http://en.wikipedia.org/wiki/Harmonic_mean and “Weightedharmonic mean” http://en.wikipedia.org/wiki/Weighted_harmonic_mean.