Chapter 10
Journey to Crime Estimation
Th e J ou rn ey to Cr im e (J t c) r ou t i n e is a dis t a n ce -ba s ed m et h od wh ich m a k e s
est im a t es a bou t t h e lik ely r esid en t ia l loca t ion of a ser ia l offen der . It is a n a pplica t ion of
location th eory, a fr a m ewor k for id en t ifyin g op t im a l loca t ion s fr om a dis t r ibu t ion of
m a r k et s, s u pply ch a r a ct er is t ics , p r ices, a n d even t s. Th e followin g d is cu ssion gives som e
ba ckgr ou n d t o t h e t ech n iqu e. Th ose wis h in g t o sk ip t h is pa r t ca n go t o pa ge 10-19 for t h e
specifics of t h e J t c r ou t ine.
Loca tio n The ory
Loca t ion t h eory is con cern ed wit h one of t h e cent r a l iss u es in geogr a ph y. This
t h eor y at t em pt s t o find a n opt ima l loca t ion for a n y par t icu lar dist r ibut ion of a ct ivities,
popu la t ion, or even t s over a r egion (Ha gget t , Cliff a n d F r ey, 1977; Kr u eckeber g a n d
Silver s, 1974; St oph er a n d Meybu r g, 1975; Oppen h eim , 1980, Ch . 4; Boss a r d, 1993). In
clas sic loca t ion t h eor y, econ om ic r esour ces wer e a lloca t ed in r elat ion t o idea lized
r epr esen t a t ion s (An selin a n d Ma dden , 1990). Thu s, von Thü n en (1826) a n a lyzed t h e
dis t r ibu t ion of a gr icu lt u r a l la n d a s a fu n ct ion of t h e a ccessibilit y t o a sin gle popu la t ion
cen t er (wh ich wou ld be m or e exp en sive t owa r ds t h e cen t er ), t h e va lu e of t h e pr odu ct
pr odu ced (wh ich wou ld va r y by cr op), a n d t r a n spor t a t ion cost s (wh ich wou ld be m or e
expen sive far t h er from t h e cen t er ). In ord er t o ma xim ize pr ofit a n d m in im ize cost s, a
dis t r ibu t ion of a gr icu lt u r a l la n d u ses (or cr op a r ea s) em er ges flowin g ou t fr om t h e
popu la t ion cen t er a s a ser ies of concen t r ic r in gs . Weber (1909) a n a lyzed t h e dis t r ibu t ion of
in du st r ia l loca t ion s a s a fu n ct ion of t h e volu m e of m a t er ia ls t o be sh ip ped, t h e dis t a n ce
t h a t t h e goods h a d t o be sh ipp ed, a n d t h e u n it dis t a n ce cost of sh ipp in g; cons equ en t ly,
ind u st r ies becom e loca t ed in p a r t icu lar con cen t r ic zon es a r ou n d a cen t r a l city. Bur gess
(1925) an a lyzed t h e dist r ibu t ion of u r ba n la n d u ses in Ch icago an d d escribed con cent r ic
zon es of bot h in du st r ia l a n d r es id en t ia l u ses . Th eir t h eor y for m ed th e ba ck dr op for ea r ly
st u dies on t h e ecology of cr im in a l beh a vior a n d ga n gs (Th r a sh er , 1927; Sh a w, 1929).
In m or e m oder n u se, t h e loca t ion of per son s wit h a cer t a in n eed or be h a vior (t h e
‘dem a n d’ sid e) is iden t ified on a sp a t ia l pla n e a n d p la ces a r e select ed a s t o ma xim ize va lu e
a n d m in im ize t r a vel cost s. F or exa m ple, for a con su m er faced wit h t wo ret a il sh ops s ellin g
t h e sa m e pr odu ct , on e being closer but m or e expens ive wh ile t h e ot h er being fa r t h er but
less expen sive, t h e con su m er h a s t o t r a de off t h e valu e t o be gain ed a gain st t h e increa sed
t r a vel t im e r equ ir ed. In design in g facilit ies or p la ces of a t t r a ction (th e ‘su pp ly’ sid e), th e
dis t a n ce bet ween ea ch possible fa cilit y loca t ion a n d t h e loca t ion of t h e r eleva n t popu la t ion
is com pa r ed t o t h e cost of loca t in g n ea r t h e fa cilit y. F or exa m ple, given a dis t r ibu t ion of
con s u m er s a n d t h eir p r op en s it y t o s pen d , s u ch a t h eor y a t t em p t s t o loca t e t h e op t im a l
pla cem en t of r et a il st or es, or , given t h e dist r ibut ion of pa t ient s, t h e t h eor y at t em pt s t o
loca t e t h e opt ima l placemen t of m edica l fa cilit ies.
10.1
P r e d i c t i n g Lo c a t io n s f ro m a D i s t ri b u t io n
On e can a lso r ever se t h e logic. Given t h e dist r ibu t ion of dem a n d, t h e t h eory cou ld
be a pp lied t o est im a t e a cent r a l loca t ion fr om w h ich t r a vel dist a n ce or t im e is m in im ized.
On e of t h e ea r liest u ses of t h is logic wa s t h a t of J ohn Sn ow, who was in t er est ed in t h e
ca u ses of ch olera in t h e m id-19t h cen t u r y (Cliff a n d H a gget t , 1988). He post u lat ed t h e
t h eor y th a t wa t er wa s t h e m a jor vect or t r a n sm itt ing t h e ch olera ba ct er ia. Aft er
in ves t iga t in g wa t er s ou r ces in t h e Lon d on m et r op olit a n a r ea a n d con clu d in g t h a t t h er e wa s
a r ela t ion sh ip bet ween con t a m in a t ed wa t er a n d ch oler a cas es, he wa s a ble t o confirm h is
t h eor y by a n ou t br ea k of ch oler a ca ses in t h e Soh o dis t r ict . By p lot t in g t h e dis t r ibu t ion of
t h e case s a n d lookin g for wa t er sou r ces in t h e cen t er of t h e dist r ibu t ion (es se n t ia lly, t h e
center of minimum dista nce; see cha pter 4), he foun d a well on Broad St reet t ha t was, in
fact , cont a m in a t ed by seepa ge from n ea r by sewer s. Th e well wa s closed a n d t h e ep idem ic
in Soh o r eceded. In cid en t ly, in plot t in g t h e in cid en t s on a m a p a n d look in g for t h e cen t er of
t h e dis t r ibu t ion , S n ow a pplied t h e sa m e logic t h a t h a d been followed by t h e Lon don
Met r opolita n P olice Depa r t m en t wh o h a d developed t h e fa m ou s ‘pin ’ m a p in t h e 1820s.
Th eor et ica lly, th er e is an opt ima l solut ion t h a t m inim izes t h e dist a n ce bet ween
dem a n d a n d s u pp ly (Ru sh t on, 1979). However , comp u t a t ion a lly, it is a n a lm ost im poss ible
t a sk t o define, r equ ir in g t h e en u m er a t ion of ever y poss ible com bin a t ion . Con sequ en t ly in
pr a ct ice, a ppr oxim a t e, t h ou gh su b-opt im a l, solu t ion s a r e obt a in ed t h r ou gh a va r iet y of
m et h ods (E ver et t , 1974, Ch . 4).
Trave l Dem an d Modeling
A su b-set of locat ion t h eory m odels t h e t r a vel beh a vior of in dividu a ls. It a ctu a lly is
t h e con ver se. If loca t ion t h eor y a t t em pt s t o a lloca t e pla ces or sit es in r ela t ion t o bot h a
su pp ly-sid e a n d d em a n d-sid e, t r a vel dem a n d t h eory a t t em pt s t o model h ow ind ividu a ls
t r a vel bet ween pla ces, given a pa r t icu lar con st ellat ion of t h em . One con cept t h a t h a s been
fr equ en t ly u sed for t h is pu r pose is th a t of t h e gravity fun ction , a n a pplica t ion of Newt on ’s
fu n d a m en t a l la w of a t t r a ct ion (Op pen h eim , 1980). In t h e or igin a l N ewt on ia n for m u la t ion ,
th e att ra ction, F, between t wo bodies of respective masses M 1 an d M 2 , sepa r a t ed by a
dist a n ce D, will be equ a l t o
M1 M2
F = g ----------------D2
(10.1)
wh er e g is a con st a n t or s calin g fa ctor wh ich en su r es t h a t t h e equ a t ion is ba la n ced in
t er m s of th e m ea su r em en t u n it s (Op pen h eim , 1980). As w e a ll kn ow, of cour se , g is t h e
gr a vit a t ion a l con s t a n t in t h e N ewt on ia n for m u la t ion . Th e n u m er a t or of t h e fu n ct ion is t h e
attraction t er m (or , a lt er n a t ively, t h e a t t r a ct ion of M 2 for M 1 ) wh ile t h e den om in a t or of th e
equ a t ion, d 2 , in dica t es t h a t t h e a t t r a ct ion bet ween t h e t wo bod ies fa lls off a s a fu n ct ion of
t h eir squ ared d is t a n ce. It is a n im p ed a n ce t er m .
10.2
S o c i a l Ap p li c a ti o n s o f t h e G ra v i ty Co n c e p t
Th e gra vit y model ha s been t h e bas is of m a n y app lica t ion s t o h u m a n societies a n d
h a s been a pplied t o social int er a ct ion s sin ce t h e 19 t h cen t u r y. Ra ven st ein (1895) a n d
An d er s son (1897) a p plied t h e con cep t t o t h e a n a lys is of m igr a t ion by a r gu in g t h a t t h e
t en den cy t o m igr a t e bet ween r egion s is in ver sely pr opor t ion a l t o t h e squ a r ed dis t a n ce
bet ween t h e r egions . Reilly’s ‘la w of r et a il gra vit a t ion ’ (1929) ap plied t h e N ewt onia n
gra vit y model directly an d su ggest ed t h a t r et a il t r a vel bet ween t wo cen t er s would be
pr opor t ion a l t o t h e pr odu ct of t h eir popu la t ion s a n d in ver sely pr opor t ion a l t o t h e squ a r e of
t h e dist a n ce sepa r a t ing t h em :
Pi Pj
T ij = " ----------------D ij2
(10.2)
wh er e T ij is th e int eraction between center s i an d j, P i an d P j a r e t h e r es pective
popu la t ions , D ij is th e dista nce between t hem r aised to th e second power an d " is a
ba la n cing cons t a n t . In t h e m odel, t h e in it ia l popu la t ion, P i , is called a prod u ction wh ile t h e
se con d p opu la t ion, P j, is called an attraction .
St ewa r t (1950) a n d Zipf (1949) a pplied t h e con cept t o a wide var iety of ph en om en a
(migr a t ion , fr eigh t t r a ffic, excha n ge of in for m a t ion ) us in g a sim plified for m of t h e gr a vit y
equ a t ion
Pi Pj
T ij = " ----------------D ij
(10.3)
wh er e t h e t er m s a r e a s in equ a t ion 10.2 bu t t h e exp on en t of dis t a n ce is on ly 1. In doin g s o,
t h ey ba sica lly link ed locat ion t h eory wit h t r a vel beh a vior t h eory. Given a pa r t icula r
pa t t er n of in t er a ct ion for a n y t yp e of goods, s er vice or h u m a n a ct ivit y, a n opt im a l loca t ion
of facilit ies sh ould be s olvable.
In t he Stewar t/Zipf fra mework , th e two P’s were both populat ion sizes and,
t h er efor e, t h eir su m s h a d t o be equ a l. H owever , in m oder n u se, it ’s n ot n ecessa r y for t h e
pr odu ction s a n d a t t r a ction s t o be iden t ical u n it s (e.g., P i cou ld be popu lat ion wh ile P j cou ld
be em ploym en t ).
Th e t ot a l volum e of pr odu ct ion s (tr ips) fr om a sin gle loca t ion , i, is est ima t ed by
su m m in g over a ll d es t in a t ion locat ion s, j:
T i = K P i G (P j/D ij)
(10.4)
j
Over time, the concept ha s been genera lized an d applied to ma ny different types of
t r a vel beha vior . For exam ple, Hu ff (1963) a pplied t h e con cept t o r et a il t r a de bet ween
zon es in a n u r ba n a r ea u sin g t h e gen er a l for m of
10.3
Aj$
T ij = " ----------------D ij8
(10.5)
wh er e T ij is t h e n u m ber of pu r ch a ses in loca t ion j by r esid en t s of loca t ion i, Aj is t h e
a t t r a ctiven es s of zon e j (e.g., squ a r e foota ge of r et a il sp a ce), D ij is t h e dist a n ce bet ween
zones i an d j, $ is t h e expon en t of S j, a n d 8 is th e exponent of dista nce, an d " is a con s t a n t
(Boss a r d, 1993). Dij-8 is som et im es ca lled a n in v erse d ist an ce fun ction. Th is is a single
con strain t m odel in t h a t only t h e a t t r a ctiven ess of a com m er cial zone is con st r a in ed, t h a t is
t h e su m of a ll a t t r a ct ion s for j m u st equ a l t h e t ot a l a t t r a ct ion in t h e r egion .
Aga in , it can be gen er a lized t o all zones by, fir st , est im a t in g t h e t ota l t r ips
gen er a t ed from one zone, i, t o an oth er zon e, j,
P i D Aj$
T ij = " ----------------D ij 8
(10.6)
wh er e T ij is t h e in t er a ction bet ween t wo locat ions (or zones), P i is pr od uct ion s of t r ip s fr om
loca t ion/zone i, Aj is t h e a t t r a ctiven es s of locat ion/zone j, D ij is th e dista nce between zones i
an d j, $ is t h e expon en t of S j, D is t h e expon en t of H i , 8 is th e exponent of dista nce, an d " is
a con st a n t .
Secon d, t h e t ota l n u m ber of t r ips gen er a t ed by a locat ion , i, to a ll des t in a t ion s is
obt a in ed by s u m m in g over a ll d es t in a t ion locat ion s, j:
T i = " P i D G (Aj$/D ij 8)
(10.7)
j
Th is differ s fr om t h e t r a dit ion a l gr a vit y fu n ct ion by a llowin g t h e expon en t s of th e
pr odu ct ion fr om loca t ion i, t h e a t t r a ct ion fr om loca t ion j, an d t h e dist a n ce bet ween zon es t o
var y. Typica lly, th ese expon en t s a r e ca libra t ed on a kn own sa m ple befor e being a pplied t o
a for ecas t sa m ple a n d t h e loca t ion s a r e u su a lly mea su r ed by zon es. Th u s, r et a iler s in
decid in g on t h e loca t ion of a n ew st or e ca n u se t h is t yp e of m odel t o ch oose a sit e loca t ion t o
opt im ize t r a vel beh a vior of pa t r ons ; th ey will, t ypically, obt a in da t a on a ctu a l sh oppin g
t r ip s by cu st om er s a n d t h en ca libr a t e t h e m odel on t h e da t a , est im a t in g t h e exp on en t s of
a t t r a ct ion a n d dis t a n ce. Th e m odel ca n t h en be u sed t o pr edict fu t u r e sh oppin g t r ip s if a
facilit y is bu ilt a t a pa r t icula r loca t ion.
Th is t ype of fu n ct ion is ca lled a d ou ble con st ra in t m odel beca u se t h e ba la n cing
con s t a n t , K, h a s t o be con s t r a in ed by t h e n u m ber of u n it s in bot h t h e or igin a n d
dest in a t ion loca t ion s; t h a t is , t h e su m of P i over a ll loca t ion s m u st be equ a l t o t h e t ot a l
n u m ber of pr odu ct ion s wh ile t h e su m of P 2 over a ll loca t ion s m u st be equ a l t o t h e t ot a l
nu mber of at tr actions. Adjust ment s ar e usua lly required to ha ve the sum of individua l
pr odu ct ion s a n d a t t r a ct ion s eq u a l t h e t ot a ls (u su a lly est im a t ed in depen den t ly).
10.4
Th e equa t ion ca n be gener a lized t o ot h er t ypes of t r ips a n d differ en t m et r ics can be
su bst it u t ed for dis t a n ce, s u ch a s t r a vel t im e, effor t , or cost (Isa r d, 1960). F or exa m ple, for
com m u t in g t r ip s, u su a lly em p loym en t is us ed for a t t r a ct ion s, fr equ en t ly s ub-d ivid ed in t o
r et a il a n d n on -r et a il em ploym en t . In a ddit ion , for pr odu ct ion s, m edia n h ou seh old in com e
or ca r own er s h ip per cen t a ge is u sed a s a n a dd it ion a l p r od u ct ion va r ia ble. E qu a t ion 10.7
ca n be gen er a lized t o in clu de a n y t yp e of pr odu ct ion or a t t r a ct ion va r ia ble (10.8 a n d 10.9):
T ij = "1 P i D "2 Aj$/D ij
8
(10.8)
T i = "1 P i D G ("2 Aj$/D ij 8)
(10.9)
wh er e T ij is t h e n u m ber of t r ips pr odu ced by locat ion i t h a t t r a vel t o locat ion j, P i is eith er a
sin gle va r ia ble a ssocia t ed wit h t r ip s pr odu ced fr om a zon e or t h e cr oss-pr odu ct of t wo or
m ore va r ia bles a ss ocia t ed wit h t r ips pr odu ced fr om a zone, Aj is eit h er a sin gle va r ia ble
a ssociat ed wit h t r ips a t t r a ct ed t o a zon e or t h e cr oss-pr odu ct of t wo or m or e var iables
a ss ocia t ed wit h t r ips a t t r a cte d t o a zon e, D ij is eit h er t h e dis t a n ce bet ween t wo loca t ion s or
a n oth er va r ia ble m ea su r in g t r a vel effor t (e.g., tr a vel t im e, t r a vel cost ), D, $, a n d 8 a r e
exponen t s of th e r es pective t er m s, "1 is a const an t a ssociated with th e productions t o
en su r e t h a t t h e su m of t r ip s pr odu ced by a ll zon es equ a ls t h e t ot a l n u m ber of t r ip s for t h e
region (usu ally estimat ed independent ly), an d "2 is a con s t a n t a ss ocia t ed wit h t h e
a t t r a ct ion s t o en su r e t h a t t h e su m of t r ip s a t t r a ct ed t o a ll zon es equ a ls t h e t ot a l n u m ber of
t r ips for t h e r egion. Wit h out h a ving t wo cons t a n t s in t h e equ a t ion , usu a lly conflict in g
est im a t es of K will be obt a in ed by ba la n cin g t h e equ a t ion a ga in st pr odu ct ion s or
a t t r a ct ion s. The s u m m a t ion over a ll dest ina t ion loca t ion s, j (equ a t ion 10.9), produ ces t h e
t ota l n u m ber of t r ips from zon e i.
In te rv e n in g Op po rt un it ie s
St ou ffer (1940) m odified t h e sim ple gr a vit y fu n ct ion by a r gu in g t h a t t h e a t t r a ct ion
bet ween t wo locat ions wa s a fun ction n ot on ly of t h e cha r a cte r ist ics of t h e r ela t ive
a t t r a ctions of t wo locat ion s, bu t of in t er ven in g opp ort u n it ies bet ween t h e loca t ion s. H is
h ypot h es is “..a ss u m es t h a t t h er e is n o necess a r y r ela t ions h ip bet ween m obilit y a n d
dis t a n ce... t h a t t h e n u m ber of per son s goin g a given dis t a n ce is dir ect ly pr opor t ion a l t o t h e
n u m ber of oppor t u n it ies a t t h a t dis t a n ce a n d in ver sely pr opor t ion a l t o t h e n u m ber of
int er venin g opport u n ities”(St ou ffer , 1940, p. 846). This m odel wa s u sed in t h e 1940s t o
exp la in in t er st a t e a n d in t er coun t y m igr a t ion (Br igh t a n d Th om a s, 1941; Isbell, 1944; Isa r d,
1979). Usin g t h e gr a vit y t ype form u la t ion , we can wr it e t h is a s:
Aj$
T ji = " ----------------G(Ak >) D ij 8
(10.10)
wh er e T ji is t h e a t t r a ction of loca t ion j by r es ide n t s of locat ion i, Aj is th e att ra ctiveness of
zone j, Ak is t h e a t t r a ct iven ess of a ll ot h er loca t ion s t h a t a r e interm ediate in d is t a n ce
bet ween loca t ions i a n d j, D ij is th e dista nce between zones i an d j, $ is t h e expon en t of S j, >
is t h e expon en t of S k , 8 is th e exponent of dista nce, an d " is a con s t a n t . Wh ile t h e
10.5
in t er ven in g oppor t u n it ies a r e im plicit in equ a t ion 10.5 in t h e exp onen t s, $ an d 8, a n d
coefficien t , K, equ a t ion 10.10 m a k es t h e in t er ven in g op por t u n it ies exp licit . Th e im por t a n ce
of t h e con cept is t h a t t h e in t er a ct ion bet ween t wo loca t ion s becom es a com plex fu n ct ion of
t h e spa t ia l en vir on m en t of n ea r by a r ea s a n d n ot ju st of t h e t wo loca t ion s.
Urban Transp ortation Modeling
Th is typ e of m od el is in cor por a t ed as a for m a l s tep in t h e u r ba n t r a n sp or t a t ion
pla n n ing pr ocess, im plem en t ed by most r egion a l plan n ing or gan izat ion s in t h e Un ited
St a t es a n d elsewh er e (St oph er a n d Meybu r g, 1975; Kr u eckeber g a n d S ilver s, 1974; Field
an d MacGregor, 1987). The step, called trip d istribu tion , is lin k ed t o a five st ep m odel.
F ir st , da t a a r e obta in ed on t r a vel beh a vior for a va r iet y of tr ip p u r poses. Th is is u su a lly
don e by sa m plin g hous eh olds a n d a sk in g ea ch m em ber t o keep a t r a vel dia r y docu m en t in g
a ll t h eir t r ips over a t wo or t h r ee da y period. Tr ips a r e a ggr egat ed by individua ls a n d by
h ou seh old s. F r equ en t ly, t r ip s by d iffer en t pu r poses a r e sepa r a t ed. Secon d, t h e volu m e of
t r ips pr odu ced by a n d a t t r a cted t o zones (ca lled t r a ffic a n a lysis zon es) is est im a t ed, usu a lly
on t h e ba sis of t h e n u m ber of h ou seh old s in t h e zon e a n d som e in dica t or of in com e or
pr iva t e vehicle own er sh ip. Third , tr ips pr odu ced by each zon e a r e dist r ibut ed t o every
ot h er zon e u su a lly u sin g a gr a vit y-t yp e fu n ct ion (equ a t ion 10.9). Th a t is , t h e n u m ber of
t r ips pr odu ced by ea ch or igin zone a n d en din g in ea ch d es t in a t ion zon e is est im a t ed by a
gr a vit y m od el. Th e d is t r ibu t ion is ba s ed on t r ip pr od u ct ion s , t r ip a t t r a ct ion s , a n d t r a vel
‘r esist a n ce’ (mea su r ed by t r a vel dis t a n ce or t r a vel t im e). Four t h , zon e-to-zon e t r ips a r e
a lloca t ed by mode of t r a vel (ca r , bus, wa lkin g, et c); a n d, fift h , tr ips a r e a ssign ed t o
pa r t icula r r out es by t r a vel m ode (i.e., bu s t r ips follow differ en t r out es t h a n pr iva t e veh icle
t r ips ). Th e a dva n t a ge of t h is p r ocess is t h a t t r ips a r e a llocat ed a ccord in g t o or igins,
dest ina t ion s, dist a n ces (or t r a vel t imes ), modes of t r a vel a n d r ou t es. Since all zon es a r e
m odeled sim u lt a n eou sly, a ll in t er m edia t e dest in a t ion s (i.e ., in t er ven in g oppor t u n it ies) a r e
incorpora ted into th e model. Chapt ers 11-17 present a crime tr avel deman d model.
Al te r n a t i v e D i s ta n c e D e c a y F u n c t i o n s
On e of t h e pr oblems with t h e t r a dit ion a l gr a vit y for m u lat ion is in t h e m ea su r em en t
of t r a vel r esist a n ce, eit h er dist a n ce or t ime. F or loca t ion s sep a r a t ed by sizeable dist a n ces
in s pa ce, th e gra vit y for m u lat ion ca n wor k p r oper ly. However , as t h e dist a n ce bet ween
loca t ion s d ecr ea ses , t h e d en om in a t or a pp roa ch es in fin it y. Con sequ en t ly, a n a lt er n a t ive
expr ession for t h e in t er a ction h a s been pr oposed wh ich u ses t h e n ega t ive expon en t ia l
fu n ct ion (H ä ger st r a n d, 1957; Wils on , 1970).
Aji = S j$ e (-"D ij)
(10.11)
wh er e Aji is t h e a t t r a ction of loca t ion j for r es ide n t s of locat ion i, S j is th e att ra ctiveness of
loca t ion j, D ij is th e dista nce between locat ions i an d j, $ is t h e expon en t of S j, e is th e base
of t h e n a t u r a l logar it h m (i.e., 2.7183...), a n d " is an em pir ica lly-der ived expon en t .
Somet imes known as en tropy m axim ization , th e lat ter pa rt of th e equat ion includes a
n ega t ive exp onen t ia l fun ction wh ich h a s a m a xim u m va lu e of 1 (i.e., e -0 = 1). This ha s th e
10.6
a dva n t a ge of m a k in g t h e equ a t ion m or e st a ble for in t er a ct ion s bet ween loca t ion s t h a t a r e
close t ogeth er . For exa m ple , Cliff a n d H a gget t (1988) u se d a n ega t ive exp onen t ia l gr a vit yt ype m odel t o des cr ibe t h e diffu sion of m ea sles in t o t h e Un ite d St a t es from Ca n a da a n d
Mexico. It h a s a lso been a r gued t h a t t h e n egat ive expon en t ial fu n ct ion gener a lly gives a
bet t er fit t o u r ba n t r a vel pa t t er n s, pa r t icu lar ly by au t om obile (F oot , 1981; Bossa r d, 1993;
NCH RP , 1995).
Ot h er fu n ct ion s h a ve also be u sed t o describe t h e dist a n ce decay - n egat ive lin ea r ,
n or m a l d is t r ibu t ion , logn or m a l d is t r ibu t ion , qu a dr a t ic, P a r et o fu n ct ion , s qu a r e r oot
expon en t ial, an d so for t h (H a gget t a n d Ar n old, 1965; Taylor , 1970; E ldr idge an d J on es,
1991). La t er in t h e ch a pt er , we will exp lor e sever a l d iffer en t m a t h em a t ica l for m u la t ion s
for descr ibin g t h e dis t a n ce deca y. On e, in fa ct , d oes n ot n eed t o u se a m a t h em a t ica l
fu n ct ion a t a ll, bu t cou ld em p ir ica lly d es cr ibe t h e d is t a n ce d eca y fr om a la r ge d a t a s et a n d
u t ilize th e described va lues for pr edict ion s. The u se of m a t h em a t ica l fu n ct ion s h a s evolved
ou t of bot h t h e N ewt on ia n t r a d it ion of gr a vit y a s well a s va r iou s loca t ion t h eor ies wh ich
u sed t h e gra vit y fu n ct ion . A m a t h em a t ica l fu n ct ion m a kes sen se u n der t wo con dit ion s: 1)
if t r a vel is un ifor m in a ll d ir ect ion s; a n d 2) a s a n a pp roxim a t ion if t h er e is in a dequ a t e d at a
fr om wh ich t o ca libr a t e a n em p ir ica l fu n ct ion . Th e fir s t a ss u m pt ion is u su a lly wr on g s in ce
ph ysica l geogra ph y (i.e., ocea n s, r iver s, m ou n t a ins ) a s well as a sym m et r ica l st r eet
n et wor k s m a k e t r a vel ea sie r in som e dir ect ions t h a n oth er s. As we sh a ll see below, th e
dist a n ce decay is quit e irr egula r for jou r n ey to cr ime t r ips a n d would be bet t er described by
a n em pir ica l, r a t h er t h a n m a t h em a t ica l fu n ct ion .
In sh ort , t h er e is a lon g hist ory of r esea r ch on both t h e loca t ion of pla ces a s well a s
t h e lik elih ood of int er a ction bet ween t h es e pla ces, wh et h er t h e in t er a ction is fr eigh t
m ovem en t , la n d pr ices or in divid u a l t r a vel beh a vior . Th e gr a vit y m odel a n d va r ia t ion s on
it h a ve been u sed t o descr ibe t h e in t er a ct ion s bet ween t h ese loca t ion s.
Trav e l Be h av io r of Crim in als
J o u r n e y t o Cr im e T ri p s
Th e a pp licat ion of tr a vel beh a vior t h eor y t o cr im e h a s a sizea ble h ist ory a s w ell.
Th e a n a lysis of dis t a n ce for jour n ey t o cr im e t r ips wa s a pp lied in t h e 1930s by Wh it e
(1932), who noted t h a t pr oper t y cr ime offen der s gener a lly t r a veled far t h er dist a n ces t h a n
offend er s com m it t in g cr im es a ga in st people, a n d by Lot t ier (1938), who an a lyzed t h e r a t io
of ch a in s t or e bur gla r ies t o t h e n u m ber of ch a in s t or es by zon e in Det r oit. Tu r n er (1969)
a n a lyzed d elin qu en cy beh a vior by a dis t a n ce decay t r a vel fun ction sh owing h ow more cr im e
t r ips t en d t o be close t o th e offen der ’s h ome wit h t h e frequ en cy dr oppin g off wit h dis t a n ce.
P h illips (1980) is, app a r en t ly, th e firs t t o u se t h e t er m jou rney to crim e is d es crib in g t h e
t r a vel dist a n ces t h a t offen der s m a k e t h ough H a r r ies (1980) n ote d t h a t t h e a ver a ge
dis t a n ce t r a veled h a s evolved by t h a t t im e in t o a n a n a logy wit h t h e jou r n ey t o wor k
st a t ist ic.
Rh odes a n d Con ly (1981) expa n ded on t h e concep t of a crim inal com m ute a n d
sh owed h ow r obber y, bur gla r y an d r a pe pa t t er n s in t h e Dist r ict of Colum bia followed a
10.7
dis t a n ce decay pa t t er n . LeBea u (1987a ) an a lyzed t r a vel dis t a n ces of r a pe offend er s in Sa n
Diego by vict im-offen der r elat ion sh ips a n d by met h od of a ppr oa ch . Boggs (1965) a pplied
t h e in t er ven in g opp ort u n it ies m odel in a n a lyzing t h e dist r ibu t ion of crim es by a r ea in
r ela t ion t o th e dist r ibu t ion of offend er s. Ot h er em pir ical descrip t ion s of jou r n ey t o cr im e
dis t a n ces a n d ot h er t r a vel beh a vior pa r a m et er s h a ve be en st u died by Blu m in (1973),
Cu r t is (1974), Repet t o (1974), P yle (1974), Ca pon e a n d Nich ols 1975), Ren ger t (1975),
Sm ith (1976), LeBeau (1987b), an d Ca n t er a n d La r kin (1993). It h a s gener a lly been
a ccep t ed t h a t pr oper t y cr im e t r ip s a r e lon ger t h a n per son a l cr im e t r ip s (LeBe a u , 1987a ),
t h ou gh except ion s h a ve been n ot ed (Tur n er , 1969). Also, it wou ld be expect ed t h a t
a ver a ge t r ip dis t a n ces will va r y by a n u m ber of fact or s: cr im e t yp e; m et h od of oper a t ion ;
t im e of da y; a n d, even , t h e va lu e of th e pr oper t y r ea lized (Ca pon e a n d Nich ols , 1975).
Mode lin g th e Offe n de r Se arc h Area
Concept u a l wor k on t h e t ype of m odel h a ve been m a de by Br a n t in gh a m a n d
Br a n t ingh a m (1981) wh o a n a lyzed t h e geom etry of crim e a n d con cept u a lized a crim in a l
sea r ch a r ea , a geogra ph ica l ar ea m odified by th e spa t ial dist r ibut ion of poten t ial offen der s
a n d pot en t ia l t a r get s, t h e a wa r en ess spa ces of pot en t ia l offen der s, a n d t h e exch a n ge of
in for m a t ion bet ween p ot en t ia l offen d er s . In t h is sen s e, t h eir for m u la t ion is sim ila r t o t h a t
of St ou ffer (1940), wh o descr ibed in t er ven in g op por t u n it ies, t h ou gh t h eir ’s is a beh a vior a l
fra mework . An import an t concept developed by th e Bra nt ingha m’s is th at of decreased
cr im in a l a ct ivit y n ea r t o a n offen d er ’s h om e ba s e, a s or t of a s a fet y a r ea a r ou n d t heir n ea r
n eigh bor h ood. P r esu m a bly, offen der s, p a r t icu la r ly t h ose com m it t in g p r oper t y cr im es, go a
lit t le wa y from t h eir h ome ba se so a s t o decr ea se t h e lik elih ood t h a t t h ey will get cau gh t .
Th is w a s n ote d by Tu r n er (1969) in h is s t u dy of delin qu en cy in P h ila delph ia . Th u s, t h e
Br a n t in gh a m ’s post u la t ed t h a t t h er e wou ld be a sm a ll sa fet y a r ea (or ‘bu ffer ’ zon e) of
r elat ively lit t le offen der a ct ivity n ea r t o t h e offen der ’s ba se loca t ion ; beyon d t h a t zon e,
h owever , t h ey p ost u la t ed t h a t t h e n u m ber of cr im e t r ip s wou ld decr ea se a ccor din g t o a
dis t a n ce d eca y m odel (t h e exa ct m a t h em a t ica l for m wa s n ever specifie d, h owever ).
Cr im e t r ip s m a y n ot even begin a t a n offen der ’s r esid en ce. Rou t in e a ct ivit y t h eor y
(Coh en a n d F elson , 1979; 1981) su ggest s t h a t cr ime opport u n ities a ppea r in t h e a ct ivities
of ever yd a y life. Th e r ou t in e pa t t er n s of wor k , s h oppin g, a n d leis u r e a ffect t h e con ver gen ce
in t im e a n d p la ce of would be offen der s, s u it a ble t a r get s, a n d a bs en ce of gua r dia n s. Ma n y
cr imes m a y occu r wh ile a n offen der is t r a veling fr om on e a ct ivity t o a n ot h er . Thu s,
m odelin g cr im e t r ips a s if t h ey a r e r efer en ced r ela t ive t o a r esiden ce is n ot n ecess a r ily
goin g t o lea d t o be t t er pr edict ion .
Th e m a t h em a t ics of jou r n ey t o cr im e h a s been m odeled by Ren ger t (1981) u sin g a
m odified gen er a l opp ort u n it ies m odel:
P ij = K U i Vj f(D ij)
(10.12)
wh er e P ij is t h e pr oba bilit y of a n offen der in loca t ion (or zon e) i com m itt ing a n offen se a t
loca t ion j, U i is a m ea su r e of t h e n u m ber of cr ime t r ips pr odu ced a t loca t ion i (wh a t Renger t
called em issiveness), Vj is a m ea su r e of th e n u m ber of crim e t a r get s (a t t r a ct iven ess) a t
10.8
loca t ion j, a n d f(D ij) is a n u n sp ecified fun ction of t h e cost or effor t expen ded in t r a velin g
fr om loca t ion i to loca t ion j (dist a n ce, tim e, cost ). He did n ot t r y to oper a t ion a lize eith er
t h e pr odu ction sid e or t h e a t t r a ction sid e. Never t h eles s, con cept u a lly, a cr im e t r ip would
be exp ected t o in volve both elem en t s a s well a s t h e cost of t h e t r ip.
In sh ort , t h er e h a s been a gr ea t dea l of res ea r ch on t h e t r a vel beh a vior of crim in a ls
in comm itting acts a s well as a nu mber of sta tistical form ulat ions.
P re dic tin g th e Loca tio n of Se rial Offen de rs
Th e jou r n ey to cr ime for m u lat ion , as in equa t ion 10.9, h a s been u sed t o est ima t e t h e
origin locat ion of a ser ia l offen der ba sed on t h e dist r ibu t ion of crim e in ciden t s. Th e logic is
t o plot t h e dis t r ibu t ion of t h e in cid en t s a n d t h en u se a pr oper t y of t h a t dis t r ibu t ion t o
est im a t e a lik ely or igin loca t ion for t h e offen der . In spect in g a pa t t er n of cr im es for a
cen t r a l loca t ion is a n in t u it ive id ea t h a t police d ep a r t m en t s h a ve u s ed for a lon g t im e. Th e
dis t r ibu t ion of in ciden t s d escribes a n a ctivit y a r ea by a n offend er , wh o lives s omewh er e in
th e center of th e distr ibut ion. It is a sam ple fr om t h e offen d er ’s a ct ivit y s pa ce. U sin g t h e
Br a n t ingh a m ’s t er m inology, th er e is a s ea r ch a r ea by an offen der with in wh ich t h e cr imes
a r e com m itt ed; most likely, t h e offen der a lso lives with in t h e sea r ch a r ea .
F or exa m ple, Ca n t er (1994) sh ows h ow t h e a r ea defin ed by t h e dis t r ibu t ion of th e
‘J a ck t h e Ripper ’ m u r der s in t h e ea st en d of Lon don in t h e 1880s in clu ded t h e key s u spect s
in t h e ca se (t h ou gh t h e ca se wa s n ever solved). Kin d (1987) a n a lyzed t h e in cid en t loca t ion s
of th e ‘York shire Ripper’ who comm itted th irteen mu rders a nd seven at tem pted mu rders in
n or t h ea st E n gla n d in t h e lat e 1970s a n d ea r ly 1980s. Kind a pplied t wo differ en t
geogra ph ical cr it er ia t o est im a t e t h e r es ide n t ia l loca t ion of th e offen der . Fir st , h e
est im a t ed t h e cen t er of m in im u m dis t a n ce. S econ d, on t h e a ssu m pt ion t h a t t h e loca t ion s of
t h e m u r der s a n d a t t em pt ed m u r der s t h a t wer e com m it t ed la t e a t n igh t wer e clos er t o t h e
offen der ’s r esid en ce, h e gr a ph ed t h e t im e of t h e offen se on t h e Y axis a ga in st t h e m on t h of
t h e yea r (ta k en a s a pr oxy for len gt h of da y) on t h e X axis a n d p lott ed a t r en d lin e t h r ough
t h e da t a t o a ccou n t for sea sona lity. Bot h t h e cen t er of m inim u m dist a n ce a n d t h e m u r der s
com m it t ed a t a la t er t im e t h a n t h e t r en d lin e p oin t ed t owa r d s t h e Leed s/Br a d for d a r ea ,
ver y clos e t o wh er e t h e offen der a ct u a lly lived (in Br a dford).
R os sm o Mo de l
Rossm o (19 93; 1995) h a s a da pt ed loca t ion t h eor y, pa r t icu la r ly t r a vel beh a vior
m odelin g, t o ser ia l offen der s. In a ser ies of pa per s (Ros sm o, 1993a ; 1993b; 1995; 1997) h e
out lin ed a m a t h em a t ical a pp r oach t o iden t ifyin g t h e h ome ba se locat ion of a ser ia l
offen der , given t h e d is t ribu t ion of t h e in cid en t s. Th e m a t h em a t ics rep res en t a for m u la t ion
of t h e Br a n t in gh a m a n d Br a n t in gh a m (1981) s ea r ch a r ea m od el, d is cu s sed a bove in wh ich
t h e sea r ch beh a vior of a n offend er is s een a s followin g a dis t a n ce decay fu n ction wit h
decr ea sed a ct ivit y n ea r t h e offen der ’s h om e ba se. H e h a s pr odu ced exa m ples sh owin g h ow
t h e m odel ca n be a pplied t o ser ia l offen der s (Ros sm o, 1993a ; 1993b; 1997).
10.9
The model ha s four steps (what he called crim ina l geographic targetin g):
1.
F ir st , a r ect a n gu la r st u dy a r ea is defin ed t h a t ext en ds be yon d t h e a r ea of th e
inciden t s com m itt ed by th e ser ial offen der . The a vera ge dista n ce bet ween
poin t s is t a k en in bot h t h e Y an d X dir ect ion . H a lf t h e Y in t er -poin t dis t a n ce
is a dd ed t o th e m a xim u m Y va lu e a n d s u bt r a cted fr om t h e m in im u m Y
va lu e. H a lf t h e X in t er -p oin t d is t a n ce is a dd ed t o t h e m a xim u m X va lu e a n d
su bt r a ct ed from t h e m inim u m X valu e. These a r e bas ed on pr oject ed
coordina tes; presum ably, th e directions would have to be adjusted if
sph er ica l coor din a t es wer e u sed. Th e r ect a n gu la r st u dy d efin es a gr id fr om
which column s an d rows can be defined.
2.
F or ea ch grid cell, t h e Ma n h a t t a n dist a n ce t o ea ch inciden t loca t ion is t a ken
(see ch a pt er 3 for defin it ion ).
3.
For ea ch Man h a t t a n dist a n ce fr om a gr id cell t o an incident loca t ion , MDij,
on e of t wo fu n ct ion s is evalu a t ed:
A.
If t h e Man h a t t a n dist a n ce, MD ij, is less t h a n a specified bu ffer zon e
r a diu s, B, th en
T
P ij = A {k [ (1-N)(B g-f ) / (2B - | x i - xc| + | yi - yc| )g ] }
(10.13)
c=1
wh er e P ij is t h e r esu lt a n t of offend er in t er a ction for gr id cell, i; c is t h e
in cid en t n u m ber , s u m m in g t o T; N = 0; k is a n em pir ica lly det er m ined
con s t a n t ; g is a n em p ir ica lly d et er m in ed exp on en t ; a n d f is a n
em pir ica lly det er m ined expon en t .
Th e Gr eek let t er , A, is t h e pr odu ct sign , in dica t in g t h a t t h e r esu lt s for
each grid cell-inciden t dist a n ce, MD ij, a r e m u ltiplied together across
all incidents, c. This equa tion r educes to
T
P ij = A {k (1-0)(B g-f ) / (2B - | x i - xc| + | yi - yc| )g }
(10.14)
c=1
KBg-f
= A ---------------------------------c=1
(2B - | x i - xc| + | yi - yc| )g
T
P ij
(10.15)
Wit h i n t h e bu ffe r r egion , t h e fu n ct ion is t h e r a t i o of a con s t a n t , k ,
t im es t h e r a diu s of th e bu ffer , B, r a is ed t o a n ot h er const a n t (g-f),
10.10
divid ed by t h e differ en ce bet ween t h e dia m et er of th e cir cle (2B) a n d
t h e Man h a t t a n dist a n ce, MD ij, ra ised to a const an t, g. This is a n on linear fu n ct ion .
B.
If t h e Man h a t t a n dist a n ce, MD ij, is gr ea t er t h a n a specified buffer
zon e r a diu s, B, th en
T
P ij = A {k [ N / (| x i - xc| + | yi - yc| )f }
(10.16)
c=1
wh er e P ij is t h e r es u lt a n t of offen der in t er a ction for gr id cell, i, a n d
in cid en t loca t ion , j; c is t h e in cid en t n u m ber , s u m m in g t o T; N = 1; k is
a n em pir ica lly det er m ined con st a n t (t h e sa m e a s in equa t ion 10.15
above); an d f is a n em pir ically det er m in ed expon en t (th e sa m e a s in
equ a t ion 10.15 a bove).
Again , t h e Gr eek let t er , A, indica t es t h a t t h e r esu lt s for ea ch gr id cellinciden t dist a n ce, MD ij, a r e m u lt iplie d t ogeth er a cross a ll in ciden t s, c.
This equat ion r educes to
T
P ij = A { k [ 1/(| x i - xc| + | yi - yc| )f }
(10.17)
c=1
k
= A { -------------------------------- }
c=1
(| x i - xc| + | yi - yc| )f
T
P ij
(10.18)
Ou t side of t h e buffer r egion , th e fu n ct ion is a con st a n t , k, divided by
t h e dista n ce, MD ij, r a is ed t o a n exp on en t , f. It is a n in ver se dis t a n ce
fu n ct ion a n d d r op s off r a p id ly wit h d is t a n ce
4.
F ina lly, for ea ch grid cell, i, t h e fu n ct ion s evalu a t ed in s t ep 3 a bove ar e
sum med over all incidents.
F or bot h t h e ‘wit h in bu ffer zon e’ (n ea r t o hom e ba se) a n d ‘ou t sid e bu ffer zon e’ (far
from h ome ba se) fu n ctions, t h e coefficien t , k, a n d exponen t s, f an d g, a r e em pir ically
det er m in ed. Th ough h e doesn ’t dis cus s h ow th ese a r e calcu la t ed, t h ey a r e pr esu m a bly
est im a t ed from a sa m ple of kn own offend er locat ion s wh er e t h e dist a n ce to ea ch in ciden t is
k n own (e.g., ar r es t r ecor ds ).
Th e r es u lt is a su r face m odel in dica t in g a likelih ood of th e offen der r es idin g a t t h a t
loca t ion. H e des crib es it a s a pr obabilit y su r face, bu t it is a ctu a lly a density s u r fa ce. S in ce
t h e pr oba bilit y of int er a ct ion bet ween a n y on e grid cell, i, a n d a n y on e inciden t , j, ca n n ot be
gr ea t er t h a n 1, t h e s u r fa ce a ct u a lly in d ica t es t h e p r od u ct of in d ivid u a l lik elih ood s t h a t t h e
10.11
offend er u ses t h a t locat ion a s t h e h ome ba se. To be a n a ctu a l pr obabilit y fu n ction, it would
h a ve to be re-sca led so t h a t t h e su m of t h e grid cells wa s equ a l to 1.
Th e secon d fu n ction - ‘out sid e t h e bu ffer zon e’ (equa t ion 10.16) is a cla ss ic gr a vit y
fu n ct ion , s im ila r t o equ a t ion 10.5 excep t t h er e is n o a t t r a ct ion d efin it ion . It is t h e d is t a n ce
decay pa r t of t h e gr a vit y fu n ction. Th e firs t fun ction, equ a t ion 10.13, is a n in crea sin g
cu r vilinea r fu n ct ion designed t o m odel t h e a r ea of decrea sed a ct ivity n ea r t h e offen der ’s
h om e bas e.
S tr en gt h s a n d w ea k n esses of th e Ros sm o m od el
Th e Ros sm o m odel h a s bot h st r en gt h s a n d wea k n esses. F ir st , t h e m odel h a s som e
t h eor et ica l ba sis u t ilizin g t h e Br a n t in gh a m a n d Br a n t in gh a m (1981) fr a m ewor k for a n
offend er s ear ch ar ea a s well as t h e ma t h em a t ics of t h e gra vit y model an d dist inguish es t wo
t ypes of t r a vel beha vior - n ea r t o h om e a n d far t h er fr om h om e. Secon d, th e m odel does
r epr esen t a sys t em a t ic a ppr oa ch t owa r ds id en t ifyin g a lik ely h om e ba se loca t ion for a n
offen d er . By eva lu a t in g ea ch gr id cell in t h e s t u dy a r ea , a n in d ep en d en t es t im a t e of t h e
likelih ood is obta in ed, wh ich can t h en be in t egr a t ed in t o a con t in u ous su r face wit h a n
int er pola t ion gra ph ics r ou t ine.
Th er e a r e pr oblem s w it h t h e pa r t icula r for m u la t ion, h owever . Fir st , t h e exclu sive
u se of Ma n h a t t a n dis t a n ces is qu est ion a ble. U n less t h e st u dy a r ea h a s a st r eet n et wor k
t h a t follows a u n ifor m grid, m ea su r ing dist a n ces h or izon t a lly a n d vert ica lly ca n lead t o
over es t im a t ion of t r a vel d is t an ces ; fu r t h er , t h e m or e t h e la you t differ s fr om a n or t h -s ou t h
a n d ea st -west orien t a t ion, t h e gr ea t er t h e dist ort ion. S in ce m a n y u r ba n a r ea s d o not h a ve
a u n ifor m grid st r eet layou t , th e m et h od will n ecessa r ily lead t o overes t ima t ion of t r a vel
dista nces in places where there ar e diagona l or irr egular str eets. 1
Secon d, t h e u se of a pr odu ct t er m , A, com p lica t es t h e m a t h em a t ics . Th a t is , t h e
t ech n ique eva lua t es t h e dist a n ce fr om a pa r t icu lar grid cell, i, t o a pa r t icu lar inciden t
loca t ion , j. It t h en m u ltiplies th is result by all oth er results. Since the P values are
a ct u a lly den sit ies, which ca n be grea t er t h a n 1.0, t h e pr ocess, if st r ict ly a pplied, wou ld be a
com poun din g of pr oba bilit ies with overes t ima t ion of t h e likelih ood for grid cells close t o
in cid en t loca t ion s a n d u n d er es t im a t ion of t h e lik elih ood for gr id cells fa r t h er a wa y. In t h e
descript ion of t h e m et h od, however, Rossm o a ct u a lly m en t ion s su m m ing t h e t er m s. Thu s,
t h e su bs t it u t ion of a s u m m a t ion s ign , G, for t he product sign would help th e mat hema tics.
A th ir d pr oblem is in t h e dis t a n ce d eca y fu n ct ion (equ a t ion 10.16). Th e u se of an
in ver se dis t a n ce ter m h a s p r oblem s a s t h e dist a n ce bet ween t h e gr id cell loca t ion, i, a n d
t h e inciden t loca t ion , j, decr ea ses. F or some t ypes of cr imes , th er e will be lit t le or n o buffer
zone a r oun d t h e offen der ’s h ome ba se (e.g., ra pes by a cqua in t a n ces). Cons equ en t ly, t h e
bu ffer zon e r a diu s, B, wou ld a ppr oa ch 0. H owever , t h is wou ld ca u se t h e m odel t o be com e
u n st a ble s in ce th e in ver se dis t a n ce ter m will a pp r oach in finit y.
F our t h , t h e u se of a m a t h em a t ical fun ction t o descr ibe t h e dist a n ce decay, wh ile
ea sy t o defin e, p r oba bly over sim plifies a ct u a l t r a vel beh a vior . A m a t h em a t ica l fu n ct ion t o
10.12
describe dist a n ce decay is an a ppr oxim a t ion t o a ct u a l tr a vel beha vior . It a ssu m es t h a t
t r a vel is equ a lly likely in ea ch d ir ection , t h a t t r a vel dis t a n ce is u n ifor m ly ea sy (or difficult )
in ea ch dir ect ion , a n d t h a t , s im ila r ly, op por t u n it ies a r e u n ifor m ly dis t r ibu t ed. F or m ost
u r ba n a r ea s, t h ese con dit ion s would n ot be tr u e. Few cit ies for m a per fect grid (Salt Lak e
Cit y is, of cou r se, a n except ion ), t h ou gh m ost cit ies h a ve sect ion s t h a t a r e gr id ed. Bot h
ph ysica l geogra ph y limit t r a vel in cert a in d irection s a s does t h e h ist or ica l st r eet st r u ct u r e,
wh ich is oft en der ived fr om ea r lier com m u n it ies. A m a t h em a t ica l fu n ct ion does n ot
consider th is str uctu re, but ra th er assu mes th at th e ‘impedance’ in a ll directions is
u n ifor m .
Th is la t t er cr it icism , of cou r se, wou ld be t r u e for a ll m a t h em a t ica l for m u la t ion s of
t r a vel dis t a n ce. Ther e a r e cor r ection s t h a t can be m a de t o ad jus t for t h is. F or exa m ple, in
t h e u r ba n t r a vel dem a n d t ype m odel, t r ip d ist r ibu t ion bet ween loca t ions is est im a t ed by a
gr a vit y m odel, bu t t h en t h e dis t r ibu t ed t r ip s a r e con st r a in ed by, fir st , t h e t ot a l n u m ber of
t r ips in t h e r egion (est im a t ed sepa r a t ely), second , by m ode of t r a vel (bu s v. s in gle dr iver v.
dr iver s plu s pa ssen ger s v. wa lk , et c.), a n d, t h ir d, by t h e r ou t e st r u ct u r e u pon wh ich t h e
t r ips a r e event u a lly a ssign ed (Kru eckeber g an d Silvers, 1974; St oph er a n d Meybur g, 1975;
F ield an d Ma cGr egor , 1987). Calibra t ion a t a ll st a ges aga ins t kn own da t a set s en su r es
t h a t t h e coefficien t s a n d exponen t s fit ‘r ea l world ’ da t a a s closely as p ossible. It would t a k e
t h ese t ypes of m odifica t ion s t o m a ke t h e t r a vel dist r ibut ion t ype of m odel post u lat ed by
Rossm o a n d ot h er s be m or e r ea list ic.
Fifth , th e model im poses m a t h em a t ica l rigidity on t h e da t a . Wh ile th ere a r e two
d iffer en t fu n ct ion s t h a t cou ld va r y fr om p la ce t o p la ce, t h e p a r t icu la r t yp e of d is t a n ce d eca y
fu n ct ion m igh t a ls o va r y. S pecifyin g a s t r ict for m for t h e t wo equ a t ion s lim it s t h e
flexibilit y of a pplyin g t h e m odel t o differ en t t yp es of cr im e or t o pla ces wh er e t h e dis t a n ce
deca y d oes n ot follow t h e for m specified by Rossm o.
A sixth problem is tha t opport un ities for comm itting crimes - th e att ra ctiveness of
loca t ion s , a r e n ever m ea s u r ed . Th a t is , t h er e is n o en u m er a t ion of t h e op por t u n it ies t h a t
wou ld exis t for a n offen d er n or is t h er e a n a t t em p t t o m ea s u r e t h e s t r en gt h of t h is
a t t r a ct ion . Ins t ea d, th e sea r ch a r ea is inferr ed st r ict ly fr om t h e dist r ibut ion of inciden t s.
Beca u se t h e dist r ibut ion of offen der opport u n ities would be expect ed t o var y fr om pla ce t o
pla ce, t h e m odel wou ld n eed t o be r e-ca libr a t ed a t ea ch loca t ion . In t h is sen se, bot h t h e
Ca n t er m odel a n d m y jou r n ey t o cr im e m odel (bot h descr ibed below) als o sh a r e t h is
wea kn ess. It is un der st a n da ble in t h a t vict im/ta r get oppor t u n ities a r e difficu lt t o defin e a
priori sin ce t h ey ca n be in t er pr et ed differ en t ly by in divid u a ls . N ever t h eless, a m or e
com plet e t h eor y of jou r n ey t o cr im e beh a vior wou ld h a ve t o in cor por a t e som e m ea su r e of
op por t u n it ies , a poin t t h a t bot h Br a n t in gh a m a n d Br a n t in gh a m (1981) a n d Ren ger t (1981)
h a ve ma de.
F in a lly, t h e ‘bu ffer zon e’ concep t is bu t one in t er pr et a t ion of th e t en den cy of m a n y
cr im es n ot t o be com m it t ed clos e t o t h e h om e loca t ion . Th er e a r e ot h er in t er p r et a t ion s t h a t
a r e a pplica ble. F or exa m ple, t h e dis t r ibu t ion of cr im e oppor t u n it ies is oft en n ot clos e t o t h e
h ome locat ion , eit h er . Ma n y cr im es occur in com m er cial a r ea s. In m ost Am er ican cit ies,
r esiden t ial a r ea s a r e n ot loca t ed in com m er cial a r ea s. Thu s, t h er e will u su a lly be a
10.13
dis t a n ce bet ween a r esid en t ia l loca t ion a n d a n ea r by cr im e oppor t u n it y. Th is does n ot
im p ly a n yt h in g a bou t a ‘s a fet y zon e’ for t h e offen d er bu t , in s t ea d , m a y illu s t r a t e t h e
dis t r ibu t ion of t h e oppor t u n it ies. If we cou ld m a p t h e t r a vel dis t a n ce of, sa y, sh oppin g
t r ips , we wou ld p r obably fin d a sim ila r dis t r ibu t ion t o th a t seen in m ost of jou r n ey t o cr im e
st u die s (a n d illu st r a t ed below).
Th e con cept of a ‘bu ffer zon e’ is a h ypot h esis, n ot a cert a in t y. The la n gu a ge of it is
so a ppea lin g t h a t m a n y people be lieve it t o be t r u e. Bu t , t o dem on st r a t e t h e exis t en ce of a
‘bu ffer zon e’ wou ld r equ ire in t er viewing offen der s (or offen der s wh o h a ve been a r r est ed)
a n d dem on st r a t ing t h a t t h ey did not com m it crim es n ea r t h eir r esiden ce even t h ou gh t h er e
wer e opport u n ities (i.e., th ey va lued sa fet y over oppor t u n ity). To m y kn owledge, t h er e h a s
n ot been a st u dy t h a t dem on st r a t ed t h is yet. Ot h er wise, on e ca n n ot dist ingu ish bet ween
t h e ‘bu ffer zon e’ h ypot h esis a n d t h e dist r ibut ion of a vailable opport u n ities. Th ey ma y ver y
well be t h e sa m e t h in g.
Ca n te r Mo de l
Ca n t er ’s gr ou p in Liver pool (Ca n t er a n d Ta gg, 1975; Can t er a n d La r kin , 1993;
Ca n t er a n d Sn ook, 1999; Ca n t er , Coffey a n d H u n t ley, 2000) h a ve m odified t h e dis t a n ce
deca y fu n ct ion for jou r n ey t o cr im e t r ip s by u sin g a n ega t ive exp on en t ia l t er m , in st ea d of
t h e in ver se dis t a n ce. Their Dragn et pr ogr a m u ses t h e n ega t ive exp on en t ia l fu n ct ion
Y = " e (-$ Dij/ P )
(10.19)
wh er e Y is t h e lik elih ood of an offen der t r a velin g a cert a in dis t a n ce to comm it a crim e,, D ij
is t h e dist a n ce (from a h ome ba se loca t ion t o an in ciden t sit e), " is a n a r bit r a r y con st a n t , $
is t h e coefficien t of th e dis t a n ce (a n d, h en ce, a n expon en t of e ), P is a n or m a liza t ion
const an t, and e is t h e ba se of t h e n a t u r a l logar it h m . Th e m odel is sim ilar t o equa t ion 10.11
except , lik e Ros sm o, it does n ot in clu de t h e a t t r a ct iven ess of t h e loca t ion .
Us ing t h e logic t h a t m ost cr imes a r e com m itt ed n ea r t h e offen der ’s h om e bas e,
Ca n t er , Coffey an d H u n t ley (2000) us e a five st ep pr oces s t o est im a t e a sea r ch s t r a t egy:
1.
Th e st u dy ar ea is defin ed by a r ect a n gle t h a t is 20% lar ger in a r ea t h a n t h a t
defin ed by t h e m in im u m a n d m a xim u m X/Y poin t s. A gr id cell st r u ct u r e of
13, 300 cells is im posed over t h e r ect a n gle. E a ch gr id cell is a r efer en ce
locat ion , i.
2.
A decay coefficient is select ed. In equa t ion 10.19, t h is wou ld be th e
coefficien t , $ , for t h e dist a n ce ter m , D ij, bot h of wh ich a r e expon en t s of e .
Un like Rossm o, Can t er u ses a ser ies of decay coefficient s from 0.1 to 10 to
est im a t e t h e sen sit ivit y of t h e m odel. Th e equ a t ion in dica t es t h e lik elih ood
wit h wh ich a n y loca t ion is lik ely t o be t h e h om e ba se of t h e offen der ba sed on
one in ciden t .
10.14
3.
Beca u se d iffer en t offen der s h a ve d iffer en t sea r ch a r ea s, t h e m ea su r ed
dis t a n ces for ea ch cell a r e divided by a n orm a liza t ion coefficien t , P , t h a t
a djus t s a ll offen ses t o a com pa r a ble ra n ge. Can t er u ses t wo differ en t t ypes
of n or m a liza t ion fu n ct ion : 1) m ea n int er -poin t dist a n ce bet ween a ll offen ses
(a cr os s a gr ou p of offen d er s ); a n d 2) t h e QRa n ge, wh ich is a n in d ex t h a t
t a kes int o a ccou n t a sym m et r y in t h e or ient a t ion of t h e inciden t s.
4.
F or ea ch r efer en ce cell, i, t h e dis t a n ce bet ween ea ch gr id cell a n d ea ch
incident locat ion is evalua ted with th e fun ction a nd t he sta nda rdized
likelih oods a r e su m m ed t o yield a n est im a t e of locat ion pot en t ia l.
5.
A search cost ind ex is defined by t h e pr oport ion of t h e st u dy ar ea t h a t h a s t o
be sear ch ed t o find t h e offend er. By ca libra t ing th e model agains t kn own
ca ses, a n est ima t e of sea r ch efficiency is obt a ined .
Ad dit ion a l m od ifica t ion s ca n be a dd ed to t h e fu n ct ion s t o m a k e t h em m or e flexible
(Ca n t er , Coffey a n d H u n t ley, 2000). F or exa m ple, ‘s t ep s’ a r e d is t an ces nea r t o h om e wh er e
offen der s a r e n ot lik ely t o a ct wh ile ‘p la t ea u s’ a r e con st a n t dis t an ces nea r t o h om e wh er e
t h er e is t h e h igh est lik elih ood of a ct in g. F or exa m ple, Ca n t er a n d La r k in (1993) fou n d a n
a r ea a r ou n d ser ial offen der s’ h om es of a bou t 0.61 mile in r a diu s wit h in wh ich t h ey were
less likely to comm it crimes.
Ca n t er a n d S n ook (1999) pr ovide es t im a t es of t h e sea r ch cost (or efficiency)
a ssociat ed wit h var iou s dist a n ce coefficient s. For exa m ple, wit h t h e kn own h om e bas e
loca t ions of 32 bu r gla r s, a $ of 1.0 yielded a m ea n sea r ch cost of 18.06%; t h a t is, on a vera ge,
only 18.06% of th e st u dy a r ea h a d t o be sea r ched t o fin d t h e loca t ion of 32 bu r gla r s in t h e
ca libr a t ion sa m ple. Clea r ly, for som e of t h em , a la r ger a r ea h a d t o be sea r ch ed wh ile for
ot h er s a sm a ller a r ea ; t h e a ver a ge wa s 18.06%. Con ver sely, t h e m ea n sea r ch cost in dex for
24 r a pis t s wa s 21.10% a n d for 37 m u r der er s 28.28%. Th ey fur t h er explor ed t h e m a r gina l
in cr ea se in loca t in g offen der s by in cr ea sin g t h e p er cen t a ge of t h e s tu dy a r ea t h a t ha d t o be
sea r ch ed. They fou n d for t h eir t h r ee sa m ples (bur gla r y, r a pe, homicide) th a t m or e t h a n
h a lf t h e offen der s cou ld be loca t ed wit h in 15% of t h e a r ea sea r ch ed.
Th e Ca n t er m odel is differ en t fr om t h e Ros sm o m odel is t h a t it su ggest s a sea r ch
st r a t egy by t h e police for a ser ia l offen der r a t h er t h a n a pa r t icula r loca t ion. Th e st r en gt h
of it is to in dica t e h ow n a r r ow a n a r ea t h e p olice s hou ld con cen t r a t e on in or der t o op tim ize
find in g a n offen der . Clea r ly, in m ost cas es , only a sm a ll a r ea n eed s be sea r ched.
S tr en gt h s a n d w ea k n esses of th e Ca n ter m od el
Th e m odel h a s bot h st r en gt h s a n d wea k n esses. F ir st , t h e m odel p r ovides a sea r ch
st r a t egy for la w en for cem en t . By exa m in in g wh a t t ype of fun ction bes t fits a cert a in t ype
of cr ime, police ca n t a r get t h eir s ea r ch effor t s m or e efficient ly. The m odel is rela t ively ea sy
t o imp lemen t a n d is pr a ct ica l. Secon d, th e m a t h em a t ica l for m u lat ion is st a ble. Un like t h e
inver se dist a n ce fu n ct ion in t h e Rossm o m odel, equa t ion 10.19 will n ot h a ve problems
a ssocia t ed wit h dis t a n ces t h a t a r e clos e t o 0. F u r t h er , t h e m odel d oes pr ovide a sea r ch
10.15
st r a t egy for ident ifyin g an offen der . It is a u seful t ool for law en for cem en t officer s,
pa r t icu lar ly a s t h ey fr a m e a sea r ch for a ser ial offen der .
Th er e a r e a lso wea kn esses t o t h e m odel. Fir st , it lacks a t h eor et ica l bas is. Can t er ’s
r esea r ch h a s p r ovided a gr ea t dea l in t er m s of u n der st a n din g t h e a ctivit y sp a ces of ser ia l
offen der s (Can t er a n d La r kin , 1993; Can t er a n d Gr egor y, 1994; Can t er , 1994; H odge an d
Ca n t er , 2000). H owever , t h e em pir ica l m odel u sed is st r ict ly pr a gm a t ic. Secon d,
m a t h em a t ica lly, it imp oses t h e n egat ive expon en t ial fu n ct ion with ou t con sider ing ot h er
dist a n ce decay models. In t h e Dragn et pr ogra m , th e decay fu n ct ion is a s t r ing of 20
n u m ber s s o th a t , in t h eory, a n y fu n ction can be exp lor ed. H owever , t h e defau lt is a
n ega t ive exp on en t ia l. Th e n ega t ive exp on en t ia l h a s been u sed in m a n y t r a vel beh a vior
st u dies (F oot , 1981; Bos sa r d, 1993), bu t it does n ot a lwa ys pr odu ce t h e best fit . La t er on ,
I’ll sh ow exam ples of t r a vel beha vior wh ich sh ow a dist inctly non-monotonic fu n ct ion , even
beyon d a h om e bas e ‘bu ffer zon e’. Wh ile t h e m odel ca n be ad a pt ed t o be m or e flexible by
differ en t exp on en t s a n d in clu din g s t eps a n d pla t ea u s, for exa m ple, it is st ill t ied t o t h e
n ega t ive expon en t ia l form . Th u s, t h e m odel m ight work in som e loca t ion s, bu t m a y fa il in
ot h ers ; a u ser ca n ’t eas ily adjust t h e model to m a ke it fit n ew da t a .
Th ir d, t h e coefficien t of t h e n ega t ive exp onen t ia l, ", is defined ar bitr ar ily. In t he
Dragn et pr ogra m , it is us u a lly set a s 0.5. While th is en su r es t h a t t h e r esu lt n ever exceed
1.0 for a n y on e in cid en t , t h er e is a lim it on t h e loca t ion p ot en t ia l s u m m a t ion s in ce t h e t ot a l
pot en t ia l is a fun ction of t h e n u m ber of in ciden t s (i.e., it will be h igh er for m ore in ciden t s).
Th u s, t h e u se of " en ds u p being a r bitr a r y. It wou ld h a ve been bet t er if t h e coefficient
wer e ca libra t ed a gain st a kn own sa m ple.
F ou r t h , a n d fin a lly, a ls o sim ila r t o t h e Ros sm o m odel (a n d t o m y J t c m odel below),
cr im in a l op por t u n it ies (or a t t r a ct ion s ) a r e n ever m ea s u r ed , bu t a r e in fer r ed fr om t h e
pa t t er n of crim e in ciden t s. As a pr a gm a t ic t ool for in for m in g a police sea r ch, one cou ld
a r gue t h a t t h is is not im port a n t . However , in a differ en t loca t ion , th e dist a n ce coefficient
is lia ble to differ a s is t h e sea r ch cost ind ex. It wou ld n eed t o be re-ca libra t ed ea ch t ime.
Never t h eles s, t h e Ca n t er m odel is a u seful t ool for p olice depa r t m en t a n d ca n h elp
sh a pe a sea r ch st r a t egy. It is differ en t fr om t h e ot h er loca t ion m odels in t h a t it is n ot
focu sed so m u ch on t h e bes t p red ict ion for a loca t ion of a n offen der (t h ou gh t h e s um m a t ion
discuss ed a bove in st ep 4 ca n yield t h a t ) a s it does in defin ing wh er e t h e sea r ch sh ou ld be
optimized.
Ge o gra ph ic P ro fi li ng
J ou r n ey t o cr im e est im a t ion sh ou ld be dis t in gu is h ed fr om geographical profiling.
Geogr a ph ical pr ofiling involves u n der st a n din g t h e geogra ph ical sea r ch p a t t er n of crim in a ls
in r ela t ion t o t h e s pa t ia l d is t r ibu t ion of p ot en t ia l offen d er s a n d p ot en t ia l t a r get s , t h e
a wa r en ess s pa ces of pot en t ia l offen der s in clud in g t h e la belin g of ‘good’ t a r get s a n d cr im e
a r ea s, a n d t h e in t er ch a n ge of in for m a t ion be t ween pot en t ia l offen der s wh o m a y m odify
t h eir a wa r en ess spa ce (Br a n t in gh a m a n d Br a n t in gh a m , 1981). Accor din g t o Ros sm o:
10.16
“...Geogra ph ic pr ofilin g focu ses on t h e pr oba ble spa t ial beh a viou r of t h e offen der
wit h in t h e con t ext of t h e loca t ion s of, a n d t h e s pa t ia l r ela t ion s h ip s bet ween , t h e
var iou s crim e sit es. A psychologica l pr ofile pr ovides in sight s in t o a n offen der ’s
likely m ot iva t ion , beha viou r a n d lifest yle, an d is t h er efor e dir ect ly con n ect ed t o
h is/h er spa t ial a ct ivity. Ps ych ologica l an d geogra ph ic pr ofiles th u s a ct in t a n dem t o
h elp in vest iga t ors develop a pict u r e of t h e per son r espon sible for t h e crim es in
qu es t ion” (Ross m o, 1997).
In ot h er words , geogra ph ic pr ofiling is a fr a m ewor k for un der st a n ding h ow an
offen der t r a vers es a n a r ea in sea r ch ing for vict ims or t a r gets ; t h is, of n ecessit y, involves
u n der st a n din g th e social en vir on m en t of a n a r ea , th e wa y th a t t h e offen der u n der st a n ds
t h is en vir on m en t (t h e ‘cognit ive m a p’) a s well as t h e offen der ’s m ot ives.
On t h e oth er h a n d, jou r n ey t o cr im e es t im a t ion follows a m u ch s im pler logic
in volvin g t h e dis t a n ce dim en sion of t h e spa t ia l p a t t er n in g of a cr im in a l. It is a m et h od
a imed a t est ima t ing t h e dist a n ce t h a t ser ial offen der s will t r a vel t o com m it a cr ime a n d, by
implicat ion, the likely locat ion from which t hey star ted th eir crime ‘tr ip’. In short, it is a
st r ict ly st a t ist ica l ap pr oa ch t o est ima t ing t h e r esiden t ial wh er ea bou t s of a n offen der
compa red to un dersta nding the dyna mics of serial offenders.
It r em a in s a n em pir ica l qu est ion wh et h er a con cept u a l fr a m ewor k , s u ch a s
ge ogr a p h ic p r ofilin g, ca n p r e dict b et t e r t h a n a s t r i ct ly s t a t is t ica l fr a m e wor k .
Un der st a n din g of a p h en omen a , su ch a s s er ia l m u r der s, s er ia l r a pis t s, a n d s o fort h , is a n
im p or t a n t r es ea r ch a r ea . We s eek m or e t h a n ju s t st a t is t ica l p r ed ict ion in bu ild in g a
kn owledge base. H owever, it d oesn ’t n ecessa r ily follow t h a t u n der st a n din g produ ces bet t er
pr edictions. In m a n y a r ea s of h u m a n a ctivit y, st r ictly st a t ist ical m odels a r e bet t er in
pr edict in g t h a n exp la n a t or y m odels . I will r et u r n t o t h is poin t la t er in t h e sect ion .
Th e Cr i m eS t a t J o u rn e y to Crim e Ro u tin e
Th e jou r n ey to cr ime (J t c) r ou t ine is a dia gnost ic designed t o a id police depa r t m en t s
in t h eir in vest iga t ion s of ser ia l offen der s. Th e a im is t o est im a t e t h e lik elih ood t h a t a
ser ial offen der lives at a n y par t icu lar loca t ion . Usin g th e loca t ion of inciden t s com m itt ed
by t h e ser ia l offen der , t h e pr ogr a m m a k es st a t ist ical gu esses a t wh er e t h e offend er is lia ble
t o live, ba sed on t h e sim ila r it y in t r a vel p a t t er n s t o a kn own sa m ple of ser ia l offen der s for
t h e sa m e t ype of cr ime. Th e J tc r ou t ine bu ilds on t h e Rossm o (1993a; 1993b; 1995)
fra m ework , bu t ext en ds it s m odelin g cap a bilit y.
1.
A grid is over laid on t op of t h e st u dy ar ea . This grid can be eith er imp or t ed
or ca n be gen er a t ed by Crim eS tat (see cha pter 2). The grid represents t he
en t ir e st u dy a r ea . Un like Rossm o or Ca n t er a n d S n ook , t h er e is n o opt im a l
st u dy ar ea . The t ech n ique will model t h a t wh ich is defin ed. Thu s, t h e u ser
h a s t o select a n a r ea in t elligen t ly.
2.
Th e r ou t in e ca lcu la t es t h e dis t a n ce bet ween ea ch in cid en t loca t ion
com m it t ed by a ser ia l offen der (or gr ou p of offen der s wor k in g t oget h er ) a n d
10.17
ea ch cell, defin ed by th e cen t r oid of t h e cell. Rossm o (1993a; 1995) u sed
in dir ect (Man h a t t a n ) dist a n ces. H owever , t h is would be a pp r opr ia t e only
wh en a city fa lls on a u n ifor m grid. The J tc r ou t in e a llows bot h d ir ect a n d
in dir ect dis t a n ces. In m ost ca ses, d ir ect dis t a n ces wou ld be t h e m ost
a pp r opr ia t e choice a s a police depa r t m en t would n orm a lly locat e origin a n d
dest ina t ion loca t ion s r a t h er t h a n pa r t icu lar r ou t es t h a t a r e t a ken (see
below).
3.
A d is t an ce d eca y fu n ct ion is ap plied to ea ch gr id cell-in cid en t pa ir a n d s um s
t h e va lu es over a ll inciden t s. Th e u ser h a s a choice wh et h er t o model t h e
t r a vel dist a n ce by a m a t h em a t ica l fu n ct ion or a n em pir ica lly-der ived
fu n ct ion .
4.
Th e r esu lt a n t of t h e dist a n ce decay fu n ction for ea ch gr id cell-in ciden t pa ir
a r e su m m ed over a ll inciden t s t o pr odu ce a lik elih ood (or den sit y) est im a t e
for ea ch gr id cell.
In bot h ca ses, t h e pr ogr a m ou t pu t s t h e t wo r esu lt s: 1) t h e gr id cell wh ich h a s
t h e pea k likelih ood est im a t e; an d 2) th e lik elih ood est im a t e for ever y cell.
Th e la t t er ou t p u t ca n be s a ved a s a S u rfer ® for W in d ow s ‘d at ’, ArcView
S pat ial A n alyst © ‘a sc’, ASCII ‘gr d’, ArcView ® ‘.sh p’, M apIn fo® ‘.m if’ ,
Atlas*GIS ™ ‘.bn a ’ file or a s a n Ascii grid ‘gr d’ file wh ich ca n be r ea d by m a n y
GIS p a cka ges (e.g., AR C/ IN FO ® , Vertical Mapper © ). These files can a lso be
r ea d by oth er GIS p a cka ges (e.g., M a pt it ud e).
5.
F igu r e 10.1 sh ows t h e logic of t h e r out in e a n d figu r e 10.2 sh ows t h e J our n ey t o
Crime (J tc) screen. There ar e two par ts t o th e rout ine. First, th ere is a calibrat ion m odel
wh ich is u se d in t h e em pir ically-der ived dis t a n ce fu n ction . Secon d, t h er e is t h e J our n ey t o
Cr im e (J t c) m od el it s elf in wh ich t h e u s er ca n s elect eit h er t h e a lr ea d y-ca libr a t ed dis t a n ce
fun ction or t h e m a t h em a t ical fu n ction . Th e em pir ically-der ived fun ction is, by far , t h e
ea siest t o us e a n d is , cons equ en t ly, th e defau lt choice in Crim eS tat. Th e discu ss ion of it is
on p. 35. H owever , t h e m a t h em a t ica l fu n ct ion ca n be u sed if t h er e is in a dequ a t e d at a t o
const ru ct a n empirical dista nce decay fun ction or if a pa rt icular form is desired.
D is ta n ce Mo de lin g Us in g Ma th e m at ic al F u n ct io n s
We’ll s t a r t by illu s t r a t in g t h e u s e of t h e m a t h em a t ica l fu n ct ion s beca u s e t h is h a s
been t h e t r a dit ion a l way t h a t dist a n ce decay ha s been exam ined . The Crim eS tat J t c
r out in e a llows t h e u se r t o defin e dist a n ce deca y by a m a t h em a t ical fu n ction .
P r ob ab ili ty Di st an c e Fu n c ti on s
Th e u ser select s on e of five pr oba bilit y d en sit y d is t r ibu t ion s t o defin e a lik elih ood
t h a t t h e offen der h a s t r a veled a pa r t icu la r dis t a n ce t o com m it a cr im e. Th e a dva n t a ge of
h a ving five fun ctions, a s opp osed t o on ly on e, is t h a t it pr ovides m ore flexibilit y in
descr ibin g t r a vel beh a vior . Th e t r a vel dis t a n ce dist r ibu t ion followed will va r y by cr im e
t yp e, t im e of da y, m et h od of
10.18
Figure 10.1:
Journey to Crime Interpolation Routine
Tra
vel
Primary file:
Crime locations
dem
and
fun
ctio
n
Reference grid
Figure 10.2:
Journey to Crime Screen
op er a t ion , a n d n u m er ou s ot h er va r ia bles . Th e five fu n ct ion s a llow a n a pp r oa ch t h a t ca n
sim u lat e m or e a ccu r a t ely t r a vel beha vior u n der differ en t con dit ion s. Ea ch of t h ese h a s
pa r a m et er s t h a t ca n be m odified, a llowin g a ver y la r ge n u m ber of possibilit ies for
descr ibin g t r a vel beh a vior of a cr im in a l.
Figur e 10.3 illust ra tes th e five types. 2 Defa u lt va lu es ba sed on Ba lt im or e Cou n t y
h a ve been pr ovided for ea ch . The u ser , however , ca n ch a n ge th ese a s n eeded.
Br iefly, th e five fu n ct ion s a r e:
Li n ea r
Th e sim plest t ype of dist a n ce m odel is a lin ea r fu n ct ion . This m odel post u lat es t h a t
t h e lik elih ood of com m it t in g a cr im e a t a n y p a r t icu la r loca t ion declin es by a con st a n t
a m ou n t with dist a n ce fr om t h e offen der ’s h om e. It is h igh est n ea r t h e offen der ’s h om e but
dr ops off by a con st a n t a m ou n t for ea ch u n it of dis t a n ce u n t il it fa lls t o zer o. Th e for m of
th e linear equat ion is:
f(d ij) =
A +B*d ij
(10.20)
wh er e f(d ij) is t h e lik elih ood t h a t t h e offen der will com m it a cr im e a t a pa r t icu la r loca t ion ,
i, define d h er e a s t h e cen t er of a gr id cell, d ij is t h e dist a n ce bet ween t h e offen der ’s
r es id en ce a n d loca t ion i, A is a slope coefficien t wh ich d efin es t h e fall off in d ist a n ce, an d B
is a con st a n t . It would be expect ed t h a t t h e coefficient B wou ld h a ve a n ega t ive sign sin ce
t h e lik elih ood sh ou ld declin e wit h dis t a n ce. Th e u ser m u st pr ovide va lu es for A an d B .
Th e defa u lt for A is 1.9 a n d for B is -0.06. Th is fu n ct ion a ssu m es n o bu ffer zon e a r ou n d t h e
offend er ’s r esiden ce. Wh en t h e fun ction r ea ches 0 (th e X a xis), th e r out in e a u t oma t ically
su bst it u t es a 0 for t h e fu n ct ion .
N e g a t i v e Ex p on e n t i a l
A sligh t ly m ore comp lex fun ction is t h e n ega t ive expon en t ia l. In t h is t ype of model,
t h e lik elih ood is a lso h igh es t n ea r t h e offen der s h ome a n d d r ops off wit h dis t a n ce.
However, the decline is at a const an t rate of decline, t h u s dr oppin g quickly nea r t h e
offen der ’s h ome u n t il is a pp r oach es zer o likelih ood. Th e m a t h em a t ical form of t h e n ega t ive
expon en t ia l is
f(d ij) =
A*e
-B*d ij
(10.21)
wh er e f(d ij) is t h e lik elih ood t h a t t h e offend er will com m it a crim e a t a pa r t icula r locat ion , i,
define d h er e a s t h e cen t er of a gr id cell, d ij is t h e dis t a n ce bet ween ea ch r efer en ce loca t ion
10.21
Figure 10.3:
Journey to Crime Travel Demand Functions
Five Mathematical Functions
Trucated Negative
Exponential
Normal
Relative Density Estimate
10
8
Lognormal
6
Linear
4
2
Negative
Exponential
0
0
4
8
12
Distance from Crime
16
20
a n d ea ch cr im e loca t ion, e is t h e ba se of t h e n a t u r a l loga r it h m , A is t h e coefficien t a n d B is
a n exp on en t of e . The u ser inp u t s valu es for A - t h e coefficient , an d B - t h e expon en t . The
defau lt for A is 1.89 a n d for B is -0.06. This fun ct ion is sim ilar t o t h e Can t er m odel
(equ a t ion 10.19) except t h a t t h e coefficien t is calibr a t ed. Also, like t h e lin ea r fun ction, it
a ssu m es n o buffer zon e a r ou n d t h e offen der ’s r esiden ce.
N or m a l
A n or m a l d is t ribu t ion a ss um es th e p ea k lik elih ood is at som e op tim a l d is t an ce fr om
t h e offen der ’s h om e bas e. Thu s, t h e fu n ct ion r ises t o t h a t dist a n ce a n d t h en declines . The
r a t e of in crea se pr ior t o th e opt im a l dis t a n ce an d t h e r a t e of decrea se from t h a t dis t a n ce is
sym m et r ical in both dir ection s. Th e m a t h em a t ical for m is:
Zij
=
(d ij - Mean D)
------------------Sd
(10.22)
-0.5*Zij2
1
f(d ij) = A * -------------------- * e
S d * SQRT(2 B)
(10.23)
wh er e f(d ij) is t h e lik elih ood t h a t t h e offen der will com m it a crim e a t a pa r t icula r loca t ion, i
(defin ed h er e a s t h e cen t er of a gr id cell), d ij is t h e dis t a n ce bet ween ea ch r efer en ce loca t ion
a n d ea ch cr im e loca t ion, Mea n D is t h e m ea n dis t a n ce inpu t by t h e u se r , S d is th e s ta n da r d
devia t ion of dist a n ces, e is t h e bas e of t h e n a t u r a l logar ith m , an d A is a coefficient . The
u se r in pu t s va lu es for Mea n D, S d , an d A. The defau lt va lues a r e 4.2 for t h e m ea n dist a n ce,
Mea n D, 4.6 for t h e st a n da r d d evia t ion, S d , a n d 29.5 for t h e coefficie n t , A.
By ca r efu lly scaling t h e pa r a m et er s of t h e m odel, th e n or m a l dist r ibut ion ca n be
a da pt ed t o a dist a n ce decay fu n ct ion with a n increa sin g likelihood for n ea r dist a n ces a n d a
decr ea sin g lik elih ood for fa r dis t a n ces. F or exa m ple, by ch oosin g a st a n da r d devia t ion
grea t er t h a n t h e m ea n (e.g., Mea n D = 1,S d = 2), t h e dis t r ibu t ion will be sk ewed t o t h e left
beca u se t h e left t a il of t h e n or m a l dist r ibut ion is not eva lua t ed. The fun ct ion becom es
similar t o th e model postu lated by Bran tingham an d Bran tingham (1981) in t ha t it is a
sin gle fu n ct ion wh ich descr ibes t r a vel beh a vior .
L og n or m a l
Th e lognorm a l fu n ct ion is sim ilar t o t h e n or m a l except it is m or e sk ewed, eith er t o
t h e left or t o t h e r igh t . It h a s t h e poten t ial of sh owing a very r a pid increa se n ea r t h e
offen der ’s h om e bas e with a m or e gra du a l decline fr om a loca t ion of pea k likelihood (see
F igu r e 10.3). It is a lso sim ilar t o t h e Bra n t ingh a m a n d Br a n t ingh a m (1981) m odel. The
ma th emat ical form of th e fun ction is:
1
f(d ij) = A * ----------------------------- *
d 2 ij * S d * S Q R T (2 B)
-[ ln (d 2 ij) - M e a n D ] 2 / 2 *s d 2
e
10.23
(10.24)
wh er e f(d ij) is t h e lik elih ood t h a t t h e offend er will com m it a crim e a t a pa r t icula r locat ion , i,
define d h er e a s t h e cen t er of a gr id cell, d ij is t h e dis t a n ce bet ween ea ch r efer en ce loca t ion
a n d ea ch cr im e loca t ion, Mea n D is t h e m ea n dis t a n ce inpu t by t h e u se r , S d is th e s ta n da r d
devia t ion of dist a n ces, e is t h e bas e of t h e n a t u r a l logar ith m , an d A is a coefficient . The
u se r in pu t s M ea n D, S d , an d A. Th e defa u lt valu es a r e 4.2 for t h e mea n dist a n ce, Mean D,
4.6 for t h e st a n da r d d evia t ion, S d , a n d 8.6 for t h e coefficien t , A. Th ey wer e ca lcu la t ed fr om
t h e Ba lt im or e Cou n t y da t a (see t a ble 10.3).
T r u n c a t e d N e g a t i v e Ex p on e n t i a l
Th e t r u n ca t ed n ega t ive exp on en t ia l is a join ed fu n ct ion m a de u p of t wo dis t in ct
m a t h em a t ical fu n ction s - t h e lin ea r a n d t h e n ega t ive exp onen t ia l. For t h e n ea r dis t a n ce, a
posit ive lin ea r fun ction is d efined, s t a r t in g a t zer o likelih ood for dis t a n ce 0 an d in crea sin g
t o d p , a locat ion of pea k like lih ood. Th er eu pon , t h e fun ction follows a n ega t ive expon en t ia l,
declin in g qu ickly wit h dis t a n ce. The t wo ma t h em a t ical fun ctions m a k in g up t h is s plin e
fu n ct ion a r e
Lin ea r :
f(d ij) = 0 + B*d ij = B*d ij
Nega t ive
-C*d ij
E xponen t ia l: f(d ij) = A*e
for d ij $ 0, d i j# d p
(10.25)
for Xi > d p
(10.26)
wh er e d ij is t h e dis t a n ce fr om t h e h om e ba se, B is t h e slop e of t h e lin ea r fu n ct ion a n d for
t h e n egat ive expon en t ial fu n ct ion A is a coefficient a n d C is a n expon en t . Since t h e
n ega t ive exp onen t ia l only s t a r t s a t a pa r t icula r dis t a n ce, d p , A, is a ssu m ed t o be th e
intercept if t h e Y-a xis wer e t r a n sposed t o t h a t dist a n ce. Similar ly, th e slope of t h e lin ea r
fu n ct ion is es t im a t ed fr om t h e p ea k dis t a n ce, d p , by a pea k like lih ood fu n ction. Th e defau lt
va lu es a r e 0.4 for t h e pea k dis t a n ce, d p , 13.8 for t h e pea k lik elih ood, a n d -0.2 for t h e
expon en t , C. Again , th ese wer e ca lcu lat ed wit h Balt imore Coun t y dat a (see t a ble 10.3)
Th is fu n ct ion is t h e closest a ppr oxim a t ion t o t h e Rossm o m odel (equ a t ion s 10.13
a n d 10.16). H owever , it differ s in sever a l m a t h em a t ica l p r oper t ies. F ir st , t h e ‘n ea r h om e
ba se ’ fun ction is lin ea r (equ a t ion 10.25), ra t h er t h a n a n on-lin ea r fun ction (equ a t ion 10.13).
It a ss u m es a sim ple in cre a se in t r a vel lik elih oods by dist a n ce fr om t h e h ome ba se , u p t o
t h e edge of t h e sa fet y zon e. 3 Secon d, t h e dis t a n ce deca y p a r t of t h e fu n ct ion (equ a t ion
10.26) is a n ega t ive expon en t ia l, r a t h er t h a n a n in ver se dis t a n ce fu n ct ion (equ a t ion 10.13);
con sequ en t ly, it is m or e st a ble when dist a n ces a r e very close t o zero (e.g., for a cr ime wh er e
t h er e is n o ‘n ea r h ome ba se ’ offset ).
Ca li b ra t i n g a n Ap p r o pr ia t e P r o b a b il it y D i st a n c e F u n c t i o n
Th e m a t h em a t ics a r e r ela t ively st r a igh t for wa r d. H owever , h ow does on e kn ow
wh ich dist a n ce fu n ction t o us e? The a n sw er is t o get som e da t a a n d ca libr a t e it . It is
im p or t a n t t o obt a in d a t a fr om a s a m ple of k n own offen d er s wh er e bot h t h eir r es id en ce a t
t h e t ime t h ey com m itt ed crim es a s well as t h e cr ime loca t ion s a r e kn own . This is ca lled
10.24
t h e calibration d ata set. E a ch of t h e m od els ar e t h en t es t ed aga in st t h e ca libr a t ion da t a
set u sin g an a ppr oa ch sim ilar t o t h a t explain ed below. An er r or a n a lysis is con du ct ed t o
det er m ine wh ich of t h e m odels best fits t h e da t a . Fin a lly, th e ‘best fit’ m odel is us ed t o
es t im a t e t h e lik elih ood t h a t a pa r t icula r ser ia l offen der lives a t a n y one loca t ion. Th ough
t h e pr ocess is t ediou s, on ce t h e pa r a m et er s a r e ca lcu la t ed t h ey ca n be u sed r epea t edly for
predictions.
Beca u se ever y ju r is dict ion is un iqu e in t er m s of t r a vel p at t er n s, it is im p or t a n t t o
ca libr a t e t h e p ar a m et er s for t h e p ar t icu la r ju r is dict ion . Wh ile t h er e m a y be s om e
sim ila r it ies be t ween cit ies (e.g., E a st er n “cen t r a lized” cit ies v. West er n “a u t om obile” cit ies),
t h er e a r e a lwa ys u n iqu e t r a vel p a t t er n s defin ed by t h e popu la t ion size, h is t or ica l r oa d
pa t t er n , an d ph ysica l geogra ph y. Con sequ en t ly, it is necessa r y to ca libra t e t h e pa r a m et er s
a n ew for ea ch n ew cit y. Idea lly, th e sa m ple sh ould be a la r ge en ough so t h a t a r elia ble
est im a t e of t h e pa r a m et er s ca n be obt a in ed. F u r t h er , t h e a n a lyst sh ou ld ch eck t h e er r or s
in ea ch of t h e m od els to en su r e t h a t t he bes t ch oice is us ed for t h e J t c r ou t in e. H owever ,
once it h a s been com plet ed, t h e pa r a m et er s ca n be r e-us ed for m a n y yea r s a n d on ly
periodically re-checked.
Data S et from Ba ltimore Cou nty
I’ll illu st r a t e with da t a fr om Balt imore Coun t y. The s t eps in ca libra t ing t h e J tc
par am eters were as follows:
1.
49,083 ma t ch ed a r r est a n d in ciden t r ecor ds from 1992 t h r ou gh 1997 wer e
obt a in ed in or der t o pr ovide da t a on wh er e t h e offen der lived in r ela t ion t o
th e crime locat ion for which t hey were ar rested. 4
2.
Th e da t a set wa s ch ecked t o en su r e t h a t t h er e wer e X an d Y coordin a t es for
bot h t h e a r r est ed ind ividu a l’s r esiden ce loca t ion a n d t h e cr ime in ciden t
loca t ion for wh ich t h e individu a l was being ch a r ged. The da t a wer e clean ed
t o elim ina t e du plica t e r ecor ds or en t r ies for wh ich eith er t h e offen der ’s
r esiden ce or t h e inciden t loca t ion wer e m issin g. The fina l da t a set h a d
41,424 recor ds. Th er e wer e m a n y mu ltiple r ecor ds for t h e sa m e offen der
sin ce an in dividu a l ca n com m it m ore t h a n one cr im e. In fact , m ore t h a n h a lf
t h e r ecor ds in volved ind ividu a ls wh o wer e list ed t wo or m or e t imes . The
dist r ibut ion of offen der s by th e n u m ber of offen ses for wh ich t h ey were
ch a r ged is seen in Ta ble 10.1. As wou ld be exp ect ed, a sm a ll pr opor t ion of
in divid u a ls a ccou n t for a sizea ble pr opor t ion of cr im es; a ppr oxim a t ely 30% of
t h e offen der s in t h e da t a ba se a ccou n t ed for 56% of t h e inciden t s.
3.
Th e da t a wer e im por t ed in t o a s pr ea ds h eet , bu t a da t a ba se pr ogr a m cou ld
equ a lly h a ve been u sed. F or ea ch r ecor d, th e dir ect dist a n ce bet ween t h e
a r r est ed individu a l’s r esidence an d t h e cr ime inciden t loca t ion was
ca lcu la t ed. Ch a pt er 2 pr esen t ed t h e for m u la s for ca lcu la t in g d ir ect
dist a n ces bet ween t wo loca t ion s a n d a r e r epea t ed in en dn ot e 5.5
10.25
Ta ble 10 .1
Nu m be r o f Offe n d e rs an d Offe n se s i n B al ti m or e Cou n ty : 1993-1997
J o u r n e y t o Cr im e D a t a b a se
Num ber of
Number of
Pe rcent of Num ber of Pe rcent of
Offe n s e s
Ind iv id u als
Offe n de rs
Incide nts
Incide nts
1
18,174
70.0%
18,174
43.9%
2
4,443
17.1%
8,886
21.5%
3
1,651
6.4%
4,953
12.0%
4
764
2.9%
3,056
7.4%
5
388
1.5%
1,940
4.7%
6-10
482
1.9%
3,383
8.2%
11-15
61
0.2%
757
1.8%
16-20
10
<0.0%
175
0.4%
21-25
3
<0.0%
67
0.2%
26-30
0
<0.0%
0
0.0%
30+
1
<0.0%
33
<0.0%
__________________________________________________________________
25,977
41,424
4.
Th e r ecor ds wer e sort ed in t o su b-group s ba se d on differ en t t ypes of cr im es .
F or t h e Balt imore Coun t y exa m ple, eleven cat egor ies of cr ime in ciden t wer e
u se d. Ta ble 10.2 p r es en t s t h e cat egories wit h t h eir r es pective sa m ple sizes.
Of cou r se, ot h er su b-gr ou ps cou ld h a ve been id en t ified. E a ch su b-gr ou p wa s
sa ved a s a sepa r a t e file. Th e sa m e r ecor ds can be p a r t of m u lt iple files (e.g.,
a r ecor d cou ld be in clud ed in t h e ‘a ll r obber ies’ file a s well a s in t h e
‘com m er cial r obber ies’ file). All r ecor ds wer e inclu ded in t h e ‘a ll cr imes ’ file.
5.
F or ea ch t ype of cr ime, t h e file wa s gr ou ped in t o dist a n ce int er vals of 0.25
m iles each. This in volved t wo st eps. F irs t , th e dist a n ce bet ween t h e
offen der ’s r es ide n ce an d t h e crim e loca t ion wa s s ort ed in a scen din g ord er .
Secon d, a frequ en cy distr ibut ion wa s con du ct ed on t h e dist a n ces a n d group ed
int o 0.25 mile in t er vals (oft en ca lled bin s). Th e degr ee of pr ecision in
dis t a n ce wou ld depen d on t h e size of t h e da t a set . F or 41,426 r ecor ds,
qu a r t er m ile bins wer e a ppr opr iat e.
6.
F or ea ch t ype of crim e, a n ew file wa s cr ea t ed wh ich in clud ed only t h e
fr equ en cy d is t ribu t ion of t h e d is t an ces br ok en down in t o qu a r t er m ile
dis t a n ce int er va ls, d i .
7.
In or der t o com pa r e differ en t t ypes of cr imes , which will h a ve differ en t
fre qu en cy dist r ibu t ions , t wo new va r ia bles wer e crea t ed. Fir st , t h e
frequ en cy in t h e in t er va l wa s con ver t ed in t o th e per cent a ge of a ll cr im es of in
ea ch in t er va l by dividin g t h e frequ en cy by th e t ota l n u m ber of in ciden t s, N ,
10.26
a n d m u lt ip lyin g by 100. Secon d, t h e dis t a n ce in t er va l wa s a dju st ed. Sin ce
t h e in t er va l is a r a n ge wit h a st a r t in g dist a n ce an d a n en din g
Ta ble 10 .2
B a l ti m o r e Co u n t y F i le s U s e d f o r Ca l ib ra t i o n
Cr im e T y p e
All crimes
H omicide
Ra pe
As sa u lt
Robber y (a ll)
Comm er cial r obber y
Ba n k r obber y
Bu r gla r y
Mot or veh icle t h eft
La r ceny
Ar s on
Sam ple S ize
41,426
137
444
8,045
3,787
1,193
176
4,694
2,548
19,806
338
dis t a n ce but h a s been iden t ified by sp r ea ds h eet pr ogr a m a s t h e begin n in g
dis t a n ce on ly, a s m a ll fr a ction, r epr esen t in g t h e m idp oin t of t h e in t er va l, is
a dded t o t h e dist a n ce int er val. In our ca se, since each in t er val is 0.25 miles
wid e, t h e a dju st m en t is h a lf of t h is , 0.125. E a ch n ew file, t h er efor e, h a d fou r
va r ia bles: t h e in t er va l d is t a n ce, t h e a dju st ed in t er va l d is t a n ce, t h e fr equ en cy
of in cid en t s wit h in t h e in t er va l (t h e n u m ber of ca ses fa llin g in t o t h e
in t er va l), a n d t h e per cent a ge of a ll cr im es of t h a t t ype w it h in t h e in t er va l.
8.
Us ing t h e r egres sion pr ogra m in t h e cr ime t r a vel dem a n d m odel (see cha pt er
12), a ser ies of r egr ession equ a t ion s wa s set u p t o m odel t h e fr equ en cy (or
th e percent age) as a fun ction of dista nce. In th is case, I used our rout ines,
but other st at istical packa ges could equa lly ha ve been u sed. Again, becau se
com pa r ison s bet ween differ en t t ypes of cr im es wer e of in t er es t , t h e
per cen t a ge of cr imes (by type) wit h in a n int er val wa s u sed a s t h e depen den t
var iable (a n d wa s defined a s a per cen t a ge, i.e., 11.51% wa s r ecor ded a s
11.51). Five equa t ion s t est ing ea ch of t h e five m odels wer e set u p.
Li n ea r
F or t h e lin ea r fu n ct ion , t h e t es t wa s
P ct i = A + Bd i
(10.27)
10.27
wh er e P ct i is t h e per cent a ge of a ll crim es of t h a t t ype fallin g in t o int er va l i, d i
is t h e dis t a n ce for in t er va l i, A is t h e in t er cept , a n d B is t h e slop e. A a n d B
a r e es t im a t ed dir ect ly from t h e r egr es sion equ a t ion.
N e g a t i v e Ex p on e n t i a l
F or t h e n egat ive expon en t ial fu n ct ion , th e var iables h a ve to be tr a n sfor m ed
t o est im a t e t h e pa r a m et er s. Th e fun ction is
P ct i = A * e
-B*d i
(10.28)
A n ew va r ia ble is defin ed wh ich is t h e n a t u r a l loga r it h m of t h e per cen t a ge of
a ll crim es of t h a t t ype fallin g in t o th e in t er va l, ln (Pct i ). This t er m wa s t h en
r egr es se d a ga in st t h e dist a n ce int er va l, d i .
ln (P ct i ) = K - B*d i
(10.29)
H owever , s in ce t h e or igin a l equ a t ion h a s been t r a n sfor m ed in t o a log
fu n ct ion , B is t h e coefficien t a n d A can be ca lcu la t ed dir ect ly fr om
ln (P ct i ) = ln (A) - B*d i
A=e
(10.30)
K
(10.31)
If t h e per cen t a ge in a n y bin wa s 0 (i.e., P ct i = 0), th en a value of -16 was
t a ken sin ce t h e n a t u r a l logar ith m of 0 ca n n ot be solved (it a ppr oxim a t es -16
a s t h e per cen t a ge a ppr oa ch es 0.0000001).
N or m a l
F or t h e n or m a l fu n ct ion , a m or e com plex tr a n sfor m a t ion m u st be us ed. The
n orm a l fu n ction in t h e m odel is
P ct i =
2
1
-0.5 *Z ij
A * ----------------------- * e
S d * SQRT(2 B)
(10.32)
First , a st an dar dized Z va r ia ble for t h e dist a n ce, d i , is crea t ed
Zi =
(d i - Mean D)
------------------Sd
(10.33)
where Mean D is th e mean dista nce and S d is t h e st a n da r d devia t ion of
dist a n ce. These a r e ca lcu lat ed from t h e or igina l da t a file (before cr ea t in g t h e
file of freq u en cy dis t r ibu t ion s). Secon d, a n orm a l t r a n sform a t ion of Z is
con st r u ct ed wit h
10.28
2
1
-0.5 *Z ij
N or m a l(Zi ) = -------------------- * e
S d * SQRT(2 B)
(10.34)
F in a lly, th e n orm a lized va r ia ble is r egr essed a ga in st t h e per cent a ge of a ll
crim es of t h a t t ype fallin g in t o th e in t er va l, Pct i wit h n o con s t a n t
P ct i = A* N or m a l(Zi )
(10.35)
A is est ima t ed by th e r egres sion coefficient .
L og n or m a l
F or t h e lognorm a l fun ction , a n oth er com ple x t r a n sform a t ion m u st be d one.
Th e logn or m a l fu n ct ion for t h e per cen t a ge of a ll cr im es of a t yp e for a
pa r t icula r dis t a n ce int er va l is
1
A * -------------------------*
d 2 ij * S d * SQRT (2B)
P ct i =
e
-[ln (d 2 i) - Mea n D ]2 / 2 *S d 2
(10.36)
Th e t r a n sfor m a t ion ca n be cr ea t ed in s t eps. F irs t , cr ea t e L
ln (d i 2 )
L =
(10.37)
Secon d, cr ea t e M
M = (l - Mean D)2
(10.38)
Th ird , cr ea t e O
m
O = --------(2*S d 2 )
(10.39)
F our t h , crea t e P by r a isin g e t o t h e O t h p ower .
P =e
O
(10.40)
F ift h , cr ea t e t h e logn or m a l con ver sion , Ln or m a l
1
Ln or m a l(d i ) = A * ----------------------------- * P
d 2 ij * S d * SQRT(2 B)
(10.41)
F in a lly, th e lognorm a l var ia ble is r egr essed a ga in st t h e per cent a ge of a ll
cr im es of t h a t t yp e fa llin g in t o t h e in t er va l, P ct i wit h n o con s t a n t
10.29
P ct i = A* Ln or m a l(d i )
(10.42)
A is est ima t ed wit h t h e r egres sion coefficient .
T r u n c a t e d N e g a t i v e Ex p on e n t i a l
F or t h e t r u n cat ed n ega t ive exp onen t ia l fun ction , t wo m odels wer e set u p.
Th e fir st a pplied t o t h e dis t a n ce r a n ge fr om 0 t o t h e dis t a n ce a t wh ich t h e
per cent a ge (or frequ en cy) is h igh es t , Ma xd i . Th e secon d a pp lied t o all
dis t a n ces gr ea t er t h a n t h is dis t a n ce
Lin ea r :
P ct i = A + Bd i for d ij $ 0, d i j# Ma xd ij
Nega t ive
-C*di
E xp on en t ia l: P ct i = A* e
for d i j> Ma xd ij
(10.43)
(10.44)
To u se t h is fu n ct ion , th e u ser specifies t h e dist a n ce a t wh ich t h e pea k
lik elih ood occu r s, d p (t h e p ea k d ist an ce) a n d t h e va lu e for t h a t pea k
likelihood, P (th e p ea k lik elih ood ). For t h e n ega t ive exp onen t ia l fun ction , t h e
u se r sp ecifies t h e exp onen t , C.
In or der t o splice t h e t wo equa t ion s t ogeth er (t h e spline), th e Crim eS tat
t r u n ca t ed n ega t ive exp on en t ia l r ou t in e st a r t s t h e lin ea r equ a t ion a t t h e
origin and ends it a t t he highest value. Thus,
A=0
(10.45)
B = P /d p
(10.46)
where P is the peak likelihood an d d p is t h e pea k d ist a n ce.
Th e expon en t , C, ca n be est ima t ed by tr a n sfor m ing t h e depen den t var iable,
P ct i , a s in t h e n ega t ive exp onen t ia l a bove (equ a t ion 10.28) an d r egr es sin g t h e
n a t u r a l log of t h e p er cen t a ge (ln (P ct i ) aga in st t h e dist a n ce int er va l, d i , on ly
for t h os e in t er va ls t h a t ar e gr ea t er t h a n t h e p ea k dis t a n ce. I h a ve fou n d t ha t
est im a t in g t h e t r a n sform ed equ a t ion wit h a coefficien t , A in
P ct i = A * e
-C*d i
(10.47)
ln (P ct i ) = Ln (A) - C*d i
(10.48)
gives a bet t er fit t o th e equ a t ion . H owever , t h e u ser n eed only in pu t t h e
exponen t , C, in t h e J t c rou t in e a s t h e coefficien t , A, of t h e n ega t ive
exp on en t ia l is ca lcu la t ed in t er n a lly t o pr odu ce a dis t a n ce va lu e a t wh ich t h e
pea k like lih ood occur s. Th e for m u la is:
10.30
ln (P ) + C*(d p - d i )
A=e
(10.49)
wh er e P is t h e pea k likelih ood, d p is t h e dist a n ce for t h e pea k like lih ood, C is
an exponent (assu med to be positive) an d d i is t h e dis t a n ce in t er va l for t h e
h is t ogr a m .
9.
On ce t h e pa r a m et er s for t h e five m odels h a ve been est ima t ed, th ey ca n be
com pa r ed t o see wh ich on e is best a t pr edict in g t h e t r a vel beh a vior for a
pa r t icu lar t ype of cr ime. It is t o be expect ed t h a t differ en t t ypes of cr imes
will h a ve d iffer en t op tim a l m od els an d t h a t t he p ar a m et er s will a ls o va r y.
Exam ples from B altimore County
Let ’s illu st r a t e wit h t h e Ba lt im or e Cou n t y d a t a . F igu r e 10.4 sh ows t h e fr equ en cy
dist r ibut ion for a ll t ypes of cr ime in Balt imore Coun t y. As can be seen , at t h e n ea r est
dis t a n ce in t er va l (0 t o 0.25 m iles wit h a n a ssign ed ‘a dju st ed’ m id poin t of 0.125 m iles),
a bou t 6.9% of a ll cr imes occu r with in a qua r t er m ile of t h e offen der ’s r esiden ce (it can be
seen on t h e Y-a xis). H owever, for t h e n ext in t er val (0.25 to 0.50 miles with a n a ssign ed
m idp oint of 0.375 m iles ), alm ost 10% of a ll crim es occur a t t h a t dis t a n ce (9.8%). In
su bsequ en t int er vals, h owever, t h e per cen t a ge decr ea ses, a lit t le less t h a n 6% for 0.50 to
0.75 m iles (wit h t h e m idp oin t bein g 0.625 m iles), a lit t le m ore t h a n 4% for 0.75 t o 1 mile
(th e m idp oint is 0.875 m iles ), an d s o fort h .
Th e bes t fit t in g st a t ist ical fun ction wa s t h e n ega t ive expon en t ia l. Th e pa r t icula r
equ a t ion is
P ct i = 5.575 *e
-0.229*d i
(10.50)
Th is is sh own with t h e solid lin e. As can be seen , th e fit is good for m ost of t h e dist a n ces,
t h ough it u n der est im a t es a t close t o zer o dist a n ce an d over est im a t es from a bout a h a lf m ile
t o a bou t fou r m iles. Th er e is on ly sligh t eviden ce of decr ea sed a ct ivit y n ea r t o t h e loca t ion
of t h e offen der .
H owever , t h e dist r ibu t ion va r ies by t ype of crim e. Wit h t h e Ba lt im ore Coun t y da t a ,
pr oper t y cr imes , in gener a l, occu r fa r t h er a wa y th a n per sona l cr imes . The t r u n ca t ed
n ega t ive exp on en t ia l gen er a lly fit pr oper t y cr im es bet t er , len din g s u ppor t for t h e
Br a n t in gh a m a n d Br a n t in gh a m (1981) fr a m ewor k for t h es e t yp es . F or exa m p le, la r cen y
offen der s h a ve a definit e sa fet y zon e a r ou n d t h eir r esiden ce (figu r e 10.5). Fewer t h a n 2%
of la r ceny t h eft s occu r wit h in a qu a r t er m ile of t h e offen der ’s r es ide n ce. However , t h e
per cent a ge ju m ps t o about 4.5% from a qu a r t er m ile t o a h a lf. The t r u n cat ed n ega t ive
exp on en t ia l fu n ct ion fit s t h e d at a r ea son a bly well t h ou gh it over es t im a t es fr om a bou t 1 t o
3 miles and u nderest imat es from a bout 4 to12 miles.
10.31
Figure 10.4:
Journey to Crime Distances: All Crimes
Negative Exponential Distribution
Number of trips for distance bin
10%
Fitted function
Percent of all crimes
8%
6%
4%
Negative exponential
2%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.5:
Journey to Crime Distances: Larceny
Truncated Negative Exponential Function
5.0%
Number of trips for distance bin
Fitted function
Percent of all larceny thefts
4.0%
Truncated negative exponential
3.0%
2.0%
1.0%
0.0%
0
10
5
20
15
Distance from offender's home (miles)
25
Sim ilar ly, motor vehicle t h eft s sh ow decrea sed a ct ivity n ea r t h e offen der ’s r esiden t ,
t h ou gh it is less pr on ou n ced t h a n la r cen y t h eft . F igu r e 10.6 sh ows t h e dis t r ibu t ion of
m ot or veh icle t h eft s a n d t h e t r u n ca t ed n ega t ive exp on en t ia l fu n ct ion wh ich wa s fit t o t h e
da t a . As ca n be s een , t h e fit is r ea son a bly good t h ough it t en ds t o un der est im a t e m idd le
r a n ge dis t a n ces (a ppr oxim a t ely 3-12 m iles).
Som e t yp es of cr im e, on t h e ot h er h a n d, a r e ver y d ifficu lt t o fit . F igu r e 10.7 sh ows
t h e dist r ibut ion of ba n k r obber ies. Pa r t ly beca u se t h er e wer e a lim ited n u m ber of ca ses
(N=176) a n d p a r t ly beca u se it ’s a com ple x pa t t er n , t h e t r u n cat ed n ega t ive exp onen t ia l ga ve
t h e best fit, bu t n ot a pa r t icu lar ly good one. As can be seen , th e lin ea r (‘n ea r h om e’)
fu n ct ion u n der es t im a t es som e of t h e n ea r dis t an ce lik elih ood s wh ile t h e n ega t ive
exponen t ia l dr ops off too quickly; in fact , t o ma k e t h is fu n ction even pla u sib le, t h e
r egr es sion wa s r u n only u p t o 21 m iles (ot h er wis e, it u n der es t im a t ed even m ore ).
F or som e cr im es, it wa s ver y d ifficu lt t o fit a n y s in gle fu n ct ion . F igu r e 10.8 sh ows
t h e fr equ en cy distr ibut ion of 137 homicides wit h t h r ee fu n ct ion s being fit t ed t o t h e da t a t h e t r u n ca t ed n ega t ive exp on en t ia l, t h e logn or m a l, a n d t h e n or m a l. As ca n be seen ea ch
fun ction fit s on ly some of t h e da t a , bu t n ot a ll of it .
Te s t i n g fo r Re s i d u a l Er ro r s in t h e Mo d e l
In sh or t , t h e five m a t h em a t ica l fu n ct ion s a llow a u ser t o fit a va r iet y of dis t a n ce
decay dist r ibut ion s. Ea ch of t h e m odels will pr edict some pa r t s of t h e dist r ibut ion bet t er
t h a n ot h er s. Con sequ en t ly, it is im por t a n t t o con du ct a n er r or a n a lysis t o det er m in e wh ich
model is ‘best’. In an error an alysis, th e residua l error is defined as
Residu a l er r or = Yi - E(Yi )
(10.51)
wh er e Yi is t h e obs er ved (a ct u a l) lik elih ood for dis t a n ce i a n d E (Yi ) is t h e lik elih ood
pr edict ed by th e m odel. If r a w n u m ber s of inciden t s a r e u sed, t h en t h e likelih oods a r e t h e
n u m ber of in ciden t s for a pa r t icula r dis t a n ce. If t h e n u m ber of in ciden t s a r e con ver t ed in t o
pr opor t ion s (i.e., pr oba bilit ies), t h en t h e lik elih oods a r e t h e pr opor t ion s of in cid en t s for a
pa r t icula r dis t a n ce.
Th e ch oice of ‘best m odel’ will depen d on wh a t pa r t of t h e dist r ibut ion is con sider ed
m ost im por t a n t . F igu r e 10.9, for exa m ple, s h ows t h e r esid u a l er r or s on veh icle t h eft for
t h e five fit t ed m odels. Th a t is, ea ch of t h e five models wa s fit t o th e pr opor t ion of veh icle
t h eft s by d is t a n ce in t er va ls (a s exp la in ed a bove). F or ea ch dis t a n ce, t h e dis cr epa n cy
bet ween t h e a ct u a l per cen t a ge of vehicle th eft s in t h a t int er val a n d t h e pr edict ed
per cen t a ge wa s ca lcu la t ed. If t h er e wa s a per fect fit , t h en t h e dis cr epa n cy (or r esid u a l) wa s
0%. If t h e a ct u a l per cen t a ge was grea t er t h a n t h e pr edict ed (i.e., th e m odel
u n der est ima t ed), t h en t h e r esidu a l was positive; if t h e a ct u a l was sm a ller t h a n t h e
pr edicte d (i.e., t h e m odel over es t im a t ed), th en t h e r es idu a l wa s n ega t ive.
10.34
Figure 10.6:
Journey to Crime Distances: Vehicle Theft
Truncated Negative Exponential Function
6%
Number of trips for distance bin
Fitted function
Percent of all vehicle thefts
5%
4%
Truncated negative exponential
3%
2%
1%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.7:
Journey to Crime Distances: Bank Robbery
Truncated Negative Exponential Function
6%
Number of trips for distance bin
Percent of all bank robberies
Fitted function
Truncated negative exponential
4%
2%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.8:
Journey to Crime Distances: Homicide
Normal, Lognormal, and Truncated Negative Exponential Functions
Number of trips for distance bin
14%
Fitted normal function
Fitted truncated negative exponential function
Fitted lognormal function
Percent of all homicides
12%
10%
Truncated negative exponential
8%
6%
Lognormal
4%
Normal
2%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.9:
Residual Error for Jtc Mathematical Models
Vehicle Theft
+20%
Actual Crimes - Predicted Crimes (%)
Negative
Exponential
Truncated
Negative
Exponential
0%
Normal
-20%
Linear
-40%
Lognormal
-60%
-80%
-100%
0
10
5
20
15
Distance from offender's home (miles)
25
Using CrimeStat for Geographic Profiling
Brent Snook, Memorial University of Newfoundland,
Paul J. Taylor, University of Liverpool, Liverpool
Craig Bennell, Carleton University, Ottawa
A challenge for researchers providing investigative support is to use
information about crime locations to prioritize geographic areas according to how
likely they are to contain the offender’s residence. One prescient solution to this
problem uses probability distance functions to assign a likelihood value to the
activity space around each crime location. A research goal is to identify the function
that assigns the highest likelihood to the offender’s actual residence, since this
should prove more efficient in future investigations.
CrimeStat was used to test of the effectiveness of two functions for a sample
of 68 German serial murder cases, using a measure known as error distance. The top
figures below illustrate the two functions used and the bottom figures portray the
corresponding effectiveness of the functions by plotting the percentage of the sample
‘located’ by error distance. A steeper effectiveness curve indicates that home
locations were closer to the point of highest probability and that, consequently, the
probability distance function was more efficient. In this particular test, no difference
was found between the two functions in their ability to classify geographic areas.
Negative Exponential
Probability of Location Offender's Home
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
Distance Decay Model
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Distance from Crime
Distance from Crime
100
100
90
90
80
80
Cumulative Percentage of Sample
Cumulative Percentage of Sample
Probability of Locating Offender's Home
Truncated Negative
Exponential
70
60
50
40
30
20
10
70
60
Cumulative Error Distance
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
45
50
55
Error Distance
60
65
70
75
80
85
90
800
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
800
Error Distance
Original Article: Taylor, P.J., Bennell, C., & Snook B. (2002) Problems of Classification in Investigative Psychology. Proceedings of
the 8th Conference of the International Federation of Classification Societies, Krakow, Poland
As ca n be s een in figu r e 10.9, t h e t r u n ca t ed n ega t ive exp on en t ia l fit t h e d a t a well
from 0 to about 5 miles, but t hen becam e poorer t ha n oth er models for longer dista nces.
Th e n ega t ive exp on en t ia l m odel wa s n ot a s good a s t h e t r u n ca t ed for dis t a n ces u p t o a bou t
5 m iles, bu t wa s bet t er for dis t a n ces beyon d t h a t poin t . Th e n or m a l d is t r ibu t ion wa s good
for dista nces from a bout 10 miles and far th er. The lognorm al was not pa rt icular ly good for
a n y dista n ces oth er t h a n a t 0 m iles, n or wa s t h e lin ea r .
Th e degr ee of pr edicta bilit y var ied by t ype of cr im e. F or s ome t ypes , pa r t icula r ly
pr oper t y cr imes , th e fit wa s r ea sona bly good. I obt a ined R 2 in t h e or der of 0.86 t o 0.96 for
bu r gla r y, robber y, ass a u lt , lar ceny, a n d a u t o th eft. For ot h er t ypes of crim e, pa r t icula r ly
violen t crim es, t h e fit w a s n ot ver y good wit h R 2 valu es in t h e or der of 0.53 (r a pe), 0.41
(a r son) an d 0.30 (h om icide). These R 2 va lu es wer e for t h e en t ir e dis t a n ce r a n ge; for a n y
pa r t icula r dis t a n ce, however , t h e pr edicta bilit y va r ied from ver y h igh t o very low.
In m odelin g dist a n ce decay wit h a m a t h em a t ical fun ction, a u ser h a s t o decide
wh ich pa r t of t h e dis t r ibu t ion is t h e m ost im por t a n t a s n o sim ple m a t h em a t ica l fu n ct ion
will n or m a lly fit a ll t h e d a t a (even a p pr oxim a t ely). In t h es e ca s es , I a s su m ed t h a t t h e n ea r
dist a n ces wer e m or e imp or t a n t (u p t o, say, 5 miles) an d, th er efor e, select ed t h e m odel
wh ich ‘best ’ fit t h ose dis t a n ces (see t a ble 10.2). H owever , it wa s n ot a lwa ys clea r wh ich
m odel wa s bes t , even wit h t h a t lim it ed crit er ia .
P r ob le m s w i th Ma th e m a ti ca l D is ta n ce D e ca y F u n ct io n s
Th er e a r e sever a l r ea son s t h a t m a t h em a t ica l m odels of dis t a n ce deca y d is t r ibu t ion s,
su ch a s illu st r a t ed in t h e J t c r ou t ine, do not fit d a t a very well. Fir st , as m en t ion ed ea r lier,
few cit ies h a ve a com plet ely sym m et r ical gr id s t r u ctu r e or even one t h a t is a pp r oxim a t ely
gr id-like (th er e a r e except ion s, of cou r se). Lim it a t ion s of ph ysical t opogr a ph y (m oun t a in s,
ocea n s, r iver s, lak es) as well as differ en t h ist or ica l developm en t pa t t er n s m a kes t r a vel
a sym m et r ical a r oun d m ost loca t ions .
Secon d, t h er e is popu la t ion den sit y. Sin ce m ost m et r opolit a n a r ea s h a ve m u ch
h igh er int en sit y of lan d u se in t h e cen t er (i.e., m or e a ct ivities a n d facilities), t r a vel t en ds t o
be dir ect ed t owa r ds h igh er la n d u se in t en sit y t h a n a wa y fr om t h em . F or or igin loca t ion s
t h a t ar e not dir ect ly in t h e cent er, t r a vel is more likely t o go t owa r ds t h e cent er t h a n awa y
from it .
Th is wou ld be tr u e of a n offen der a s well. If t h e per son wer e lookin g for eith er
per son s or pr oper t y as ‘t a r gets’, th en t h e offend er would be more likely t o t r a vel t owa r ds
t h e m et r opolita n cen t er t h a n a wa y fr om it. Since most m et r opolita n cen t er s h a ve st r eet
n et wor k s t h a t wer e la id ou t m u ch ea r lier , t h e st r eet n et wor k t en ds t o be ir r egu la r .
Con sequ en t ly, t r ip s will va r y by loca t ion wit h in a m et r opolit a n a r ea . On e wou ld exp ect
sh or t er t r ips by an offen der livin g close t o t h e m et r opolita n cen t er t h a n on e livin g fa r t h er
a wa y; sh or t er t r ips for offen der s living in m or e built -u p a r ea s t h a n in lower den sit y ar ea s;
sh ort er t r ips for offend er s in m ixed u se n eigh borh oods t h a n in st r ictly r esiden t ia l
n eigh borh oods ; an d s o fort h . Th u s, t h e dist r ibu t ion of t r ips of a n y sort (in our cas e, crim e
t r ips from a r esiden t ia l locat ion t o a crim e loca t ion ), will t en d t o follow a n ir r egu la r ,
10.40
dist a n ce decay type of dist r ibut ion . Simple m a t h em a t ica l models will not fit t h e da t a very
well an d will ma ke ma ny errors.
Th ir d, t h e select ion of a best m a t h em a t ica l fu n ct ion is pa r t ly depen den t on t h e
int erva l size us ed for t h e bin s. In t h e above exa m ples, an int erva l size of 0.25 miles was
u sed t o calcu la t e t h e fr equ en cy dis t r ibu t ion . Wit h a differ en t in t er va l s ize (e.g., 0.5 m iles),
h owever , a sligh t ly differ en t dis t r ibu t ion is obt a in ed. Th is effect s t h e m a t h em a t ica l
fu n ct ion t h a t is select ed a s well a s t h e p a r a m et er s t h a t ar e es t im a t ed . F or exa m p le, t h e
iss u e of wh et h er t h er e is a sa fety zon e n ea r t h e offend er ’s r esiden ce fr om wh ich t h er e is
decrea sed a ctivit y or n ot is pa r t ly dep en den t on t h e in t er va l size. Wit h a sm a ll int er va l,
t h e zone m a y be det ected wh er ea s wit h a sligh t ly lar ger in t er va l t h e su bt le dis t in ction in
measu red distan ces may be lost. On th e oth er ha nd, having a sma ller interval may lead to
u n r eliable est ima t es sin ce t h er e m a y be few ca ses in t h e int er val. Ha vin g a t ech n ique
depen d on t h e in t er va l s ize m a k es it vu ln er a ble t o m is -specifica t ion .
U s e s o f Ma th e m a ti ca l D is ta n ce D e ca y F u n ct io n s
Does t h is m ea n t h a t one s h ould n ot u se m a t h em a t ical dist a n ce fu n ctions? I wou ld
a r gu e t h a t u n der m ost cir cu m st a n ces, a m a t h em a t ica l fu n ct ion will give less pr ecis ion t h a n
a n em pir ica lly-der ived one (see below). However , th er e a r e t wo ca ses wh en a
m a t h em a t ica l model wou ld be ap pr opr iat e. Fir st , if t h er e is eith er n o da t a or ins u fficient
da t a t o model t h e em pir ical t r a vel dist r ibu t ion, t h e u se of a m a t h em a t ical m odel ca n ser ve
a s a n a ppr oxim a t ion . If t h e u ser h a s a good sen se of wh a t t h e dist r ibut ion looks like, t h en
a m a t h em a t ical m odel m a y be u sed t o ap pr oxim a t e t h e dist r ibu t ion . H owever , if a poor ly
defin ed fu n ct ion is select ed, t h en t h e select ed fu n ct ion m a y p r odu ce m a n y er r or s.
A s econ d ca s e wh en m a t h em a t ica l m od els of d is t a n ce d eca y wou ld be a p pr op r ia t e is
in t h eor y developm en t or a pplica t ion . Man y models of t r a vel beha vior , for exam ple,
a ss u m e a sim ple dis t a n ce decay t ype of fu n ction in ord er sim plify th e a llocat ion of t r ips
over a r egion . Th is is a com m on pr ocedu r e in t r a vel d em a n d m odelin g wh er e t r ip s fr om
ea ch of m a n y zon es a r e a ssign ed t o ever y ot h er zon e u sin g a gr a vit y t yp e of fu n ct ion
(St oph er a n d Meybur g, 1975; F ield an d Ma cGr egor , 1987). Even t h ou gh t h e m odel
pr odu ces er r or s beca u se it a ssu m es u n ifor m t r a vel beh a vior in a ll dir ect ion s, t h e er r or s a r e
cor r ected la t er in t h e m odelin g pr ocess by a dju st in g t h e coefficien t s for a llocat in g t r ips t o
pa r t icu la r r oa ds (t r a ffic a ssign m en t ). Th e m odel p r ovides a sim ple device a n d t h e er r or s
a r e cor r ect ed down t h e lin e. St ill, I wou ld a r gu e t h a t a n em pir ica lly-d er ived dis t r ibu t ion
will p rod uce fewer er r or s in a lloca t ion a n d, t h u s, r equ ir e les s a dju st m en t la t er on . E r r or s
can n ever h elp a m odel a n d it s bet t er t o get it m ore corr ect in it ia lly to ha ve t o ad jus t it
la t er on; th e a dju st m en t m a y be in a dequ a t e. Never t h eles s, t h is is com m on p r a ctice in
t r a n sp ort a t ion p la n n in g.
The J ou rn e y t o Crim e Ro u tin e Us in g a Math e m at ic al F orm u la
Th e J t c r ou t ine wh ich a llows m a t h em a t ica l modelin g is sim ple t o u se. Figur e 10.10
illu st r a t es h ow t h e u ser specifies a m a t h em a t ica l fu n ct ion . Th e r ou t in e r equ ir es t h e u se of
a gr id w h ich is define d on t h e r efer en ce file t a b of t h e pr ogra m (see cha pt er 3). Then , t h e
10.41
Figure 10.10:
Jtc Mathematical Distance Decay Function
u se r m u st sp ecify t h e m a t h em a t ical fu n ction a n d t h e pa r a m et er s. I n t h e figur e, t h e
t r u n ca t ed n egat ive expon en t ial is being defin ed. The u ser m u st inp u t valu es for t h e pea k
lik elih ood, t h e pea k dis t a n ce, an d t h e exp onen t (see equ a t ions 10.43 a n d 10.44 a bove). In
t h e figu r e, s in ce t h e ser ia l offen ses wer e a ser ies of 18 r obber ies, t h e pa r a m et er s for
r obber y ha ve been en t er ed int o t h e pr ogra m screen . The pea k likelihood wa s 9.96%
(en t er ed a s a wh ole nu m ber - i.e., 9.96); t h e dist a n ce a t wh ich t h is pea k likelihood occu r r ed
wa s t h e second dis t a n ce in t er va l 0.25-0.50 m iles (wit h a m id -poin t of 0.38 m iles); an d t h e
es t im a t ed exponen t wa s 0.177651. As m en t ioned a bove, t h e coefficien t for t h e n ega t ive
exponen t ia l pa r t of t h e equ a t ion is est im a t ed in t er n a lly.
Ta ble 10.3 gives t h e pa r a m et er s for t h e ‘best ’ m odels wh ich fit t h e da t a for t h e 11
t yp es of cr im e in Ba lt im or e Cou n t y. F or sever a l of t h ese (e.g., ba n k r obber ies), t wo or m or e
fu n ct ion s ga ve a pp roxim a t ely equ a lly good fit s . N ot e t h a t t hes e p ar a m et er s wer e
est im a t ed wit h t h e Ba lt im or e Cou n t y da t a . Th ey will n ot fit a n y ot h er ju r is dict ion . If a
u ser wis h es t o a pply t h is logic, t h en t h e pa r a m et er s sh ou ld be est im a t ed a n ew fr om
exist in g d a t a . N ever t h eless, on ce t h ey h a ve been ca libr a t ed, t h ey ca n be u sed for
predictions.
Th e r ou t in e ca n be ou t pu t t o ArcView , M apIn fo, Atlas*GIS , S u rfer for Win d ow s,
S pat ial A n alyst, a n d a s a n Ascii grid file wh ich ca n be r ea d by m a n y oth er GIS p a cka ges.
All bu t S u rfer for Win d ow s r equ ir e t h a t t h e r efer en ce grid be crea t ed by Crim eS tat.
D i s t a n c e Mo d e l i n g U s i n g a n E m p i r ic a l ly D e t e r m i n e d F u n c t i o n
An a lter n a t ive t o m a t h em a t ica l modelin g of dist a n ce decay is to emp irically describe
t h e jou r n ey to cr ime d ist r ibut ion a n d t h en u se t h is em pir ica l fu n ct ion t o est ima t e t h e
r es ide n ce locat ion. Crim eS tat h a s a t wo-dim en sion a l k er n el d en sit y r ou t in e t h a t ca n
ca libr a t e t h e d is t a n ce fu n ct ion if p r ovid ed da t a on t r ip or igin s a n d d es t in a t ion s . Th e logic
of ker n el dens ity est ima t ion wa s des cr ibed in cha pt er 8, an d won’t be rep ea t ed h er e.
E ss en t ia lly, a sym m et r ical fu n ction (th e ‘k er n el’) is pla ced over ea ch p oint in a dis t r ibu t ion.
Th e dis t r ibu t ion is t h en r efer en ced r ela t ive t o a sca le (a n equ a lly-s pa ced lin e for t wodim en sion a l k er n els a n d a gr id for t h r ee-dim en sion a l k er n els ) a n d t h e va lu es for ea ch
k er n el a r e su m m ed a t ea ch r efer en ce locat ion . See cha pt er 8 for det a ils.
Ca li b ra t e Ke r n e l D e n s i t y F u n c t i on
Th e Crim eS tat ca libr a t ion r ou t in e a llows a u ser t o d es cr ibe t h e d is t an ce d is t ribu t ion
for a sa m ple of jou r n ey t o cr im e t r ip s. Th e r equ ir em en t s a r e t h a t :
1.
Th e da t a set m u st h a ve t h e coordin a t es of both an origin locat ion a nd a
d es t in a t ion loca t ion ; a n d
2.
Th e r ecor ds of a ll or igin a n d d est in a t ion locat ion s h a ve been popu la t ed wit h
legit im a t e coordin a t e va lu es (i.e ., n o un m a t ch ed r ecor ds a r e a llowed).
10.43
Ta ble 10 .3
J ou rn ey to C rime Mathe m at ic al Mode ls for Balt imor e Coun t y
P a r a m e t e r E s t i m a t e s fo r P e r c e n t a g e D i s t r ib u t i o n
(Sam ple Sizes in P ar enth eses)
ALL CRIMES
Nega t ive E xponen t ia l:
Coefficien t :
E xp on en t :
5.575107
0.229466
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
14.02%
0.38 miles
0.064481
HOMICID E
Tr u n ca t ed
Nega t ive E xponen t ia l:
RAP E
Lognor m a l:
Mea n
3.144959
St a n da r d Devia t ion 4.546872
Coefficient
0.062791
AS S AU LT
Tr u n ca t ed
Nega t ive E xponen t ia l:
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
27.40%
0.38 miles
0.181738
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
9.96%
0.38 miles
0.177651
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
4.9455%
0.625 miles
0.151319
ROBBERY
Tr u n ca t ed
Nega t ive E xponen t ia l:
COMMERCIAL ROBBERY
Tr u n ca t ed
Nega t ive E xponen t ia l:
10.44
Ta ble 10 .3 (con t inu ed)
BANK ROBBERY
Tr u n ca t ed
Nega t ive E xponen t ia l:
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
9.96%
5.75 miles
0.139536
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
20.55%
0.38 miles
0.162907
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
4.81%
0.63 miles
0.212508
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
4.76%
0.38 miles
0.193015
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
38.99%
0.38 miles
0.093469
BURGLARY
Tr u n ca t ed
Nega t ive E xponen t ia l:
AU TO TH E F T
Tr u n ca t ed
Nega t ive E xponen t ia l:
LAR CE N Y
Tr u n ca t ed
Nega t ive E xponen t ia l:
ARS ON
Tr u n ca t ed
Nega t ive E xponen t ia l:
10.45
Da ta S et D efin it ion
Th e st eps a r e r ela t ively ea sy. F ir st , t h e u ser defines a calibr a t ion da t a set wit h
both origin a n d d est in a t ion locat ion s. F igur e 10.11 illust r a t es t h is p r ocess. As wit h t h e
prima ry and seconda ry files, th e rout ine reads ArcView ‘sh p’, d B ase ‘dbf’, Ascii ‘t xt ’, a n d
M apIn fo ‘da t ’ files. For both t h e or igin loca t ion (e.g., t h e h om e r esiden ce of t h e offen der )
a n d t h e dest ina t ion loca t ion (i.e., t h e cr ime loca t ion ), th e n a m es of t h e var iables for t h e X
a n d Y coor din a t es m u st be iden t ified a s well a s t h e t ype of coor din a t e syst em a n d d a t a u n it
(see ch a pt er 3). In t h e exa m ple, t h e or igin loca t ion s h a s va r ia ble n a m es of H om eX a n d
H om eY an d t h e dest in a t ion loca t ion s h a s va r ia ble n a m es of In cid en t X an d In cid en t Y for
t h e X a n d Y coor din a t es of t h e t wo locat ion s r espectively. H owever , an y na m e is a ccept a ble
as long as th e two locat ions a re distinguished.
Th e u ser sh ould sp ecify wh et h er t h er e a r e a n y m iss in g valu es for t h ese fou r fields
(X a n d Y coor din a t es for both origin a n d d es t in a t ion locat ions ). By defa u lt , Crim eS tat will
ignore r ecor ds wit h bla n k va lu es in a n y of th e eligible fields or r ecor ds wit h n on-n u m er ic
valu es (e.g.,alph a n u m er ic ch a r a ct er s, #, *). Bla n ks will a lways be exclud ed u n less t h e u ser
select s <n on e>. Th er e a r e 8 possible opt ion s:
1.
2.
3.
4.
5.
6.
<bla n k > fields a r e a u t oma t ically exclud ed. Th is is t h e defau lt
<n one> in dica t es t h a t n o record s will be exclud ed. If th er e is a bla n k field,
Crim eS tat will tr eat it a s a 0
0 is exclud ed
–1 is exclud ed
0 a n d –1 ind ica t es t h a t bot h 0 an d -1 will be exclud ed
0, -1 a n d 9999 ind ica t es t h a t a ll t h r ee valu es (0, -1, 9999) will be exclud ed
Any ot h er n u m er ical va lu e can be t r ea t ed a s a m iss in g va lu e by t ypin g it (e.g.,
99)Mu lt iple n u m er ical va lu es can be t r ea t ed a s m iss in g valu es by t ypin g t h em , sep a r a t in g
ea ch by comm a s (e.g., 0, -1, 99, 9999, -99).
Th e pr ogr a m will ca lcu la t e t h e dis t a n ce bet ween t h e or igin loca t ion a n d t h e
dest ina t ion loca t ion for ea ch r ecor d. If t h e u n its a r e sph er ica l (i.e., lat /lon ), th en t h e
ca lcu lat ion s u se sp h er ica l geom et r y; if t h e u n its a r e pr oject ed (eit h er m et er s or feet ), th en
t h e ca lcu la t ion s a r e E u clidea n (see ch a pt er 3 for det a ils).
K er n el P a r a m et er s
Next , t h e u se r m u st define t h e k er n el pa r a m et er s for calibr a t ion. Th er e a r e five
ch oices t h a t h a ve t o be m a de (F igu r e 10.12):
1.
Th e m et h od of int er pola t ion . As wit h t h e t wo-dim en sion a l ker n el techn ique
descr ibed in ch a pt er 8, t h er e a r e five possible ker n el fu n ct ion s:
10.46
Figure 10.11:
Jtc Calibration Data Input
Figure 10.12:
Jtc Calibration Kernel Parameters
A.
B.
C.
D.
E.
N or m a l (t h e d efa u lt );
Qu a r t ic;
Tr ia n gu la r (conica l);
A nega t ive expon en t ia l (pea k ed); an d
A u n ifor m (fla t ) dis t r ibu t ion .
2.
Ch oice of ba n dwid t h . Th e ba n dwid t h is t h e wid t h of t h e ker n el fu n ct ion . F or
a n or m a l k er n el, it is t h e st a n da r d devia t ion of t h e n or m a l d is t r ibu t ion
wh er ea s for t h e oth er fou r ker n els (qua r t ic, t r ia n gu la r , nega t ive expon en t ia l,
a n d u n ifor m ), it is t h e r a diu s of t h e circle defined by t h e k er n el. As wit h t h e
t wo-dim en sion ker n el t ech n iqu e, t h e ba n dwid t h ca n be fixed in len gt h or
a da pt ive (var ia ble in len gt h ). However , for t h e one-dim en sion a l ker n el, t h e
fixed ba n dwidt h is t h e defau lt s ince an even est ima t e over a n equa l nu m ber
of in t er va ls (bin s) is desir a ble. I f th e fixed ba n dw idt h is s elect ed, t h e in t er va l
size m u st be s pecified a n d t h e u n it s (in m iles , kilomet er s, feet , m et er s, a n d
n a u t ical m iles ). The defa u lt is 0.25 m ile in t er va ls. I f th e a da pt ive
ba n dwidt h is select ed, th e u ser m u st ident ify th e m inim u m sa m ple size th a t
t h e ban dwidt h sh ou ld incor pora t e; in t h is ca se, th e ban dwidt h is widened
un til th e specified sam ple size is coun ted.
3.
Th e n u m ber of in t er pola t ion bin s. Th e bin s a r e t h e in t er va ls a long t h e
dis t a n ce scale (fr om 0 u p t o th e m a xim u m dis t a n ce for a jou r n ey t o cr im e
t r ip) a n d a r e u se d t o est im a t e t h e den sit y fun ction . Th er e a r e t wo ch oices .
F ir st , t h e u ser can sp ecify t h e n u m ber of in t er va ls (t h e defau lt choice wit h
100 in t er va ls ). In t h is ca se, t h e r ou t in e ca lcu la t es t h e m a xim u m dis t a n ce (or
lon gest t r ip ) bet ween t h e or igin loca t ion a n d t h e dest in a t ion loca t ion a n d
divides it by th e specified nu m ber of int er vals (e.g., 100 equ a l-sized
in t er va ls ). Th e in t er va l s ize is depen den t on t h e lon gest t r ip dis t a n ce
m ea su r ed. S econ d, t h e u ser ca n specify t h e dis t a n ce bet ween bin s (or t h e
int er val size). The defau lt choice is 0.25 m iles, but a n ot h er valu e ca n be
en t er ed. In t h is ca se , t h e r out in e cou n t s ou t in t er va ls of th e specified size
u n t il it r ea ch es t h e m a xim u m t r ip dist a n ce.
4.
Th e out pu t u n it s. Th e u ser sp ecifies t h e u n it s for t h e den sit y est im a t e (in
u n it s per m ile, k ilom et er , feet , m et er s, a n d n a u t ica l m iles).
5.
Th e out pu t calcula t ion s. Th e u ser sp ecifies wh et h er t h e out pu t r esu lt s a r e in
pr obabilit ies (th e defau lt ) or in den sit ies. For p r obabilit ies, t h e su m of a ll
ker n el est ima t es will equ a l 1.0. For den sit ies, th e su m of a ll ker n el
est ima t es will equ a l th e sa m ple size.
S a v ed C a l i b r a t i o n Fi l e
Th ird , th e u ser m u st defin e a n ou t pu t file t o sa ve th e em pir ica lly det er m ined
fu n ct ion . Th e fu n ct ion is t h en u s ed in es t im a t in g t h e lik ely h om e r es id en ce of a p ar t icu la r
10.49
fu n ct ion . Th e ch oices a r e t o sa ve t h e file a s a ‘dbf’ or Ascii t ext file. Th e sa ved file t h en ca n
be u sed in t h e J t c r ou t in e. F igu r e 10.13 illu st r a t es t h e ou t pu t file for m a t .
C a l i br a t e
Four th , th e calibrat e but ton r un s th e rout ine. A calibrat ion window appear s an d
in dica t es t h e pr ogress of t h e calcula t ions . Wh en it is fin ish ed, t h e u se r can view a gr a ph
illust r a t in g t h e es t im a t ed dis t a n ce decay fu n ction (Figu r e 10.14). Th e pu r pose is t o pr ovide
qu ick dia gn ost ics t o th e u ser on t h e fun ction a n d s elect ion of t h e k er n el pa r a m et er s. Wh ile
th e gra ph can be print ed, it is not a high qua lity print. If a h igh qua lity gra ph is needed,
t h e out pu t calibr a t ion file s h ould be im por t ed in t o a gr a ph ics pr ogra m .
E x a m p l e s fr o m B a lt i m o re C o u n ty
Let ’s illu st r a t e t h is m et h od by s howin g t h e r es ult s for t h e s am e d at a set s t h a t wer e
ca lcu la t ed a bove in t h e m a t h em a t ica l s ect ion (figu r es 10.4-10.8). In a ll ca ses, t h e n or m a l
k er n el fun ction wa s u se d. Th e ba n dw idt h wa s 0.25 m iles except for t h e ba n k r obber y da t a
set , which h a d only 176 ca ses, a n d t h e h om icide da t a set , which on ly h a d 137 ca ses;
beca u se of t h e sm a ll sa m ple sizes, a ba n dwid t h of 0.50 m iles wa s u sed for t h ese t wo da t a
set s. Th e in t er va l wid t h select ed wa s a dis t a n ce of 0.25 m iles bet ween bin s (0.5 m iles for
bank robberies an d homicides) an d probabilities were out put .
F igu r e 10.15 sh ows t h e ker n el est im a t e for a ll cr im es (41,426 t r ip s). A fr equ en cy
dis t r ibu t ion wa s ca lcula t ed for t h e sa m e n u m ber of in t er va ls a n d is overla id on t h e gr a ph .
It wa s select ed t o be com pa r a ble t o t h e m a t h em a t ica l fu n ct ion (see figu r e 10.4). N ot e h ow
clos ely t h e k er n el es t im a t e fit s t h e d a t a com p a r ed t o t h e n ega t ive exp on en t ia l
ma th emat ical fun ction. The fit is good for every value but t he peak value; th at is becau se
t h e ker n el aver a ges severa l in t er vals t ogeth er t o pr odu ce a n est ima t e.
F igu r e 10.16 sh ows t h e ker n el est ima t e for lar cen y th eft s. Again , th e ker n el
m et h od pr odu ces a m u ch closer fit a s a com pa r ison wit h figur e 10.5 will s h ow. Figu r e
10.17 s h ows t h e k er n el es t im a t e for veh icle t h efts. F igur e 10.18 sh ows t h e k er n el es t im a t e
for ba n k r obber ies a n d figu r e 10.19 sh ows t h e ker n el est im a t e for h om icides. An in spect ion
of t h ese gr a ph s sh ows h ow well t h e ker n el fu n ct ion fit s t h e da t a , com pa r ed t o t h e
m a t h em a t ica l fu n ct ion , even wh en t h e da t a a r e irr egula r ly spa ced (in vehicle th eft s, ban k
r obber ies, a n d h om icides). F igu r e 10.20 com pa r es t h e dis t a n ce deca y fu n ct ion s for
h om icides com m itt ed a gains t st r a n gers com pa r ed t o homicides com m itt ed a gains t kn own
victims.
In sh or t , t h e J t c ca libr a t ion r ou t in e a llows a m u ch clos er fit t o t h e da t a t h a n a n y of
t h e sim pler m a t h em a t ica l fu n ct ion s. While it ’s possible to produ ce a com plex
m a t h em a t ica l fu n ct ion t h a t will fit t h e da t a m or e closely (e.g., h igh er or der polyn om ia ls ),
t h e ker n el met h od is m u ch sim pler t o u se a n d gives a good a ppr oxim a t ion t o t h e da t a .
10.50
Figure 10.3:
Journey to Crime Travel Demand Functions
Five Mathematical Functions
Trucated Negative
Exponential
Normal
Relative Density Estimate
10
8
Lognormal
6
Linear
4
2
Negative
Exponential
0
0
4
8
12
Distance from Crime
16
20
Figure 10.14:
Jtc Calibration Graphic Output
Figure 10.15:
Journey to Crime Distances: All Crimes
Frequencies and Kernel Density Estimate
Number of trips for distance bin
Percent of all crimes
10%
Kernel density estimate
8%
6%
4%
2%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.16:
Journey to Crime Distances: Larceny
Frequencies and Kernel Density Estimate
5%
Number of trips for distance bin
Kernel density estimate
Percent of all larceny thefts
4%
3%
2%
1%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.17:
Journey to Crime Distances: Vehicle Theft
Frequencies and Kernel Density Estimate
5%
Number of trips for distance bin
Kernel density estimate
Percent of all vehicle thefts
4%
3%
2%
1%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.18:
Journey to Crime Distances: Bank Robbery
Frequencies and Kernel Density Estimate
10%
Number of trips for distance bin
Kernel density estimate
Percent of all bank robberies
8%
6%
4%
2%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.19:
Journey to Crime Distances: Homicide
Frequencies and Kernel Density Estimate
Number of trips for distance bin
Percent of all homicides
24%
Kernel density estimate
20%
16%
12%
8%
4%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Figure 10.20:
Journey to Crime Distances: Homicide by Victim Relationship
Frequencies and Kernel Density Estimate
Number of trips for distance bin
24%
Kernel density estimate
Victim known
(by victim relationship)
Percent of all homicides
20%
16%
12%
8%
Stranger
4%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Using Journey-To-Crime Routine for Journey-After-Crime Analysis
Yongmei Lu
Department of Geography
Southwest Texas State University
San Marcos, TX
The study of vehicle theft recovery locations can fill a gap in the knowledge
about criminal travel patterns. Although the journey-to-crime routine of CrimeStat
was designed to analyze the distance between offense location and offender’s
residential location, it can be used to describe the distance between vehicle theft
location and the corresponding recovery location.
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
Probability of recovering a stolen
vehicle by distance from vehicle
theft location
0.0
0
1.5
9
3.1
7
4.7
6
6.3
4
7.9
3
9.5
1
11
.10
12
.68
14
.27
15
.85
17
.44
19
.02
20
.61
22
.19
probability of recovery
There were more than 3000 vehicle thefts in the City of Buffalo in 1998.
Matching the offenses with vehicle recoveries in the same year, 1600 location pairs
were identified for a journey-after-vehicle-theft analysis. To evaluate the
randomness of the distances, 1000 groups of simulations were conducted. Every
group contains 1600 simulated trips of journey-after-vehicle-theft. The results
indicate that 1) short distances dominate journey-after-vehicle-theft, and 2) the
observed trips are significantly shorter than the random trips given the distribution
of possible vehicle theft and recovery locations.
distance from theft location (miles)
3.43
3.41
3.39
3.37
3.34
3.32
3.30
cumulative probability
0.00
3.28
0.20
0.00
3.25
0.40
0.01
3.23
0.01
3.21
0.60
3.19
0.02
3.17
0.80
3.14
1.00
0.02
3.12
1.20
0.03
3.10
0.03
3.08
probability of means
distance from vehicle theft location (miles)
Distribution of mean distances of
simulated vehicle theft-recovery
location pairs.
Using Journey to Crime for Different Age Groups of Offenders
Renato Assunção, Cláudio Beato, Bráulio Silva
CRISP, Universidade Federal de Minas Gerais , Brazil
CrimeStat offers a method for analysing the distance between the crime scene
and the residence of the offender using the journey to crime routine within the
spatial modeling module. We analysed homicide incidents in Belo Horizonte, a
Brazilian city of 2 million inhabitants, for the period January 1996 – December
2000. We used 496 homicide cases for which the police identified an offender who
was living in Belo Horizonte, and for which both the crime location and offender
residence could be identified. The cases were divided into three groups according to
the offender‘s age: 1) 14 to 24 (N=201); 2) 25 to 34 (N=176); and 3) 35 or older
(N=119). The journey to crime calibration routine was used to produce a probability
curve P(d) that gives the approximate chance of an offender travelling
approximately distance d to commit the crime.
We used the normal kernel, a fixed bandwidth of 1000 meters, 100 output
bins, and the probability (or proportion of all points) option, rather than densities.
This is to allow comparisons between the three age groups since they have different
number of homicides. We tested for each age group separately and directed the
output to a text file to analyse the three groups simultaneously.
The green, blue, and purple curves are associated with the 14-24, 25-34, 35+
year olds respectively. There are more similarities than differences between the
groups. Most homicides are committed near to the residence of the offenders with
between 60% t o 70% closer than one mile from their home. However, the curve does
not vanish totally even for large distances because there are around 15% of
offenders, of any age group, travelling longer than 3 miles to commit the crime. The
oldest offenders travel longer distances, on average, followed by the youngest group,
with the 25-34 year olds travelling the shortest distances.
0.06
0.04
0.02
0.0
probability
0.08
0.10
Journey to homicide probabilities in Belo Horizonte, Brazil
0
1
2
3
distance travelled (in miles)
4
The J ou rn e y t o Crim e Ro u tin e Us in g t h e Calib rat e d F ile
Aft er t h e dis t a n ce deca y fu n ct ion h a s been ca libr a t ed a n d sa ved a s a file, t h e file ca n
be us ed t o ca lcu lat e t h e likelih ood su r fa ce for a ser ial offen der . The u ser specifies t h e n a m e
of t h e a lr ea d y-ca libr a t ed dis t a n ce fu n ct ion (a s a ‘d bf’ or a n As cii t ext file) a n d t h e ou t p u t
for m a t . As wit h t h e m a t h em a t ica l r ou t in e, t h e ou t pu t ca n be t o ArcView , M apIn fo,
Atlas*GIS , S u rfer for Win d ow s, S pat ial A n alyst, an d a s a n Ascii gr id file wh ich ca n be rea d
by ma n y ot h er GIS pa cka ges. All bu t S u rfer for Win d ow s r equ ire t h a t t h e r efer en ce grid be
cre a t ed by Crim eS tat.
The result is produced in t hr ee steps:
1.
Th e r out in e calcu la t es t h e dist a n ce bet ween ea ch r efer en ce cell of t h e gr id
a n d ea ch in cid en t loca t ion ;
2.
F or ea ch dis t a n ce m ea su r ed, t h e r ou t in e look s u p t h e ca lcu la t ed va lu e fr om
th e saved calibrat ion file; an d
3.
F or ea ch r efer en ce gr id cell, it su m s t h e va lu es of a ll t h e in cid en t s t o pr odu ce
a sin gle likelih ood est ima t e.
Ap pli ca ti on o f th e Ro u ti ne
To illus t r a t e t h e t ech n iques , th e r esu lts of t h e t wo m et h ods on a sin gle ca se a r e
com pa r ed. The case h a s been select ed beca u se t h e r ou t ines a ccu r a t ely est ima t e t h e
offen der ’s r es ide n ce. Th is w a s d one t o dem ons t r a t e h ow t h e t echn iqu es wor k . In t h e n ext
section, I’ll ask t h e qu est ion a bout h ow accu r a t e t h ese m et h ods a r e in gen er a l.
Th e ca se in volved a m a n wh o ha d comm it t ed 24 offen ses. Th ese in clu ded 13 t h eft s,
5 bu r gla r ies, 5 a ssa u lt s, a n d one r a pe. Th e spa t ia l d is t r ibu t ion wa s va r ied; m a n y of t h e
offen ses wer e clus t er ed bu t some wer e scat t er ed. Since th er e wer e m u ltiple t ypes of cr imes
com m itt ed by th is individu a l, a decision h a d t o be m a de over wh ich m odel t o u se t o
est ima t e t h e individu a l’s r esiden ce. In t h is ca se, th e t h eft (lar cen y) m odel wa s selected
sin ce th a t wa s t h e domin a n t t ype of crim e for t h is in divid u a l.
F or t h e m a t h em a t ica l fu n ct ion , t h e t r u n ca t ed n ega t ive exp on en t ia l wa s ch osen fr om
t a ble 10.3 wit h t h e pa r a m et er s bein g:
P ea k lik elih ood
P ea k dis t a n ce
E xp on en t
4.76%
0.38 miles
0.193015
F or t h e k er n el den sit y m odel, t h e calibr a t ed fun ction for la r ceny wa s s elect ed (figu r e
10.16).
10.61
F igur e 10.21 sh ows t h e r esu lt s of t h e es t im a t ion for t h e t wo met h ods. Th e out pu t is
fr om S u rfer for Win d ow s (Golden Soft wa r e, 1994). Th e left pa n e sh ows t h e r esu lt s of th e
m a t h em a t ical fun ction wh ile t h e r ight pa n e sh ows t h e r esu lt s for t h e k er n el de n sit y
fu n ct ion . Th e in cid en t loca t ion s a r e sh own a s cir cles wh ile t h e a ct u a l r esid en ce loca t ion of
t h e offen der is sh own a s a squ a r e. Sin ce t h is is a su r fa ce m odel, t h e h igh est loca t ion h a s
th e highest pr edicted likelihood.
In bot h ca ses, t h e m odels pr edict ed qu it e a ccu r a t ely. Th e dis cr epa n cy (er r or )
bet ween t h e pr edict ed pea k loca t ion a n d t h e a ct u a l r esid en ce loca t ion wa s 0.66 m iles for
t h e m a t h em a t ica l fu n ct ion a n d 0.36 m iles for t h e ker n el d en sit y fu n ct ion . F or t h e
m a t h em a t ica l m odel, t h e a ct u a l r esid en ce loca t ion (squ a r e) is seen a s sligh t ly off fr om t h e
pea k of t h e su r fa ce wh er ea s for t h e ker n el dens ity m odel t h e discrepa n cy fr om t h e pea k
ca n n ot be seen .
Never t h eless, th e differ en ces in t h e t wo su r fa ces sh ow dist inction s. The
m a t h em a t ica l model ha s a sm oot h decline from t h e pea k likelihood loca t ion , alm ost like a
con e. Th e ker n el d en sit y m odel, on t h e ot h er h a n d, s h ows a m or e ir r egu la r dis t r ibu t ion
with a pea k loca t ion followed by a s u r r ou n din g ‘t r ou gh’ followed a pea k ‘r im’. This is du e t o
t h e ir r egu la r dis t a n ce deca y fu n ct ion ca libr a t ed for la r cen y (s ee figu r e 10.16). Bu t , in bot h
ca ses, t h ey more or less iden t ify th e a ct u a l res iden ce loca t ion of t h e offen der .
Cho ic e of Cali bra tio n Sa m ple
Th e ca libra t ion sa m ple is cr itical for eith er m et h od. Ea ch m et h od a ssu m es t h a t t h e
dis t r ibu t ion of th e ser ia l offen der will be sim ila r t o a s a m ple of ‘lik e’ offen der s. Obviously,
dis t in ct ion s ca n be m a de t o m a k e t h e ca libr a t ion sa m ple m or e or less sim ila r t o t h e
pa r t icu lar ca se. For exa m ple, if a dist a n ce decay fu n ct ion of a ll cr imes is select ed, th en a
m odel (of eit h er t h e m a t h em a t ica l or ker n el d en sit y for m ) will h a ve less differ en t ia t ion
t h a n for a d ista n ce deca y fu n ct ion fr om a s pecific t ype of cr ime. Similar ly, brea kin g down
t h e t ype of cr ime by, sa y, m ode of oper a t ion or t ime of da y will pr odu ce bet t er
differ en t ia t ion t h a n by gr ou pin g a ll offen der s of t h e sa m e t yp e t oget h er . Th is pr ocess ca n
be t a k en on in defin it ely u n t il t h er e is t oo lit t le da t a t o m a k e a r elia ble est im a t e. An
a n a lyst sh ou ld t r y to find a s close a ca libra t ion sa m ple t o t h e a ct u a l as is possible, given
t h e lim it a t ions of t h e da t a .
F or exam ple, in ou r ca libra t ion da t a set , th er e wer e 4,694 bu r gla r y in ciden t s wh er e
bot h t h e offen der ’s hom e r es id en ce a n d t h e in cid en t loca t ion wer e k n own . Th e a pp roxim a t e
t ime of t h e offen se for 2,620 of t h e bur gla r ies wa s k n own a n d, of t h ese, 1,531 occu r r ed a t
n igh t bet ween 6 pm a n d 6 a m . Thu s, if a pa r t icu lar ser ial bu r gla r for wh om t h e police a r e
in t er es t ed in ca t ch in g t en d s t o com m it m os t of h is bu r gla r ies a t n igh t , t h en ch oos in g a
ca libra t ion sa m ple of n igh t t ime bu r gla r s will gener a lly pr odu ce a bet t er est ima t e t h a n by
groupin g all bur gla r s t ogeth er . Similar ly, of t h e 1,531 n igh t t ime bu r gla r ies, 409 were
com m it t ed by in divid u a ls wh o ha d a pr ior r ela t ion sh ip wit h t h e vict im . Aga in , if t h e
a n a lyst s su spect t h a t t h e bur gla r is r obbing h om es of people h e kn ows or is a cqu a int ed
wit h , t h en selectin g t h e su bset of n ight t im e bu r gla r ies com m it t ed a ga in st a kn own vict im
10.62
Figure 10.21:
Predicted and Actual Location of Serial Thief
Man Charged with 24 Offenses in Baltimore County
Predicted with Mathematical and Kernel Density Models for Larceny
Residence Location = Square
Crime Locations = Circles
Mathematical Model:
Truncated Negative Exponential
Kernel Density Model
wou ld p r odu ce even bet t er differ en t ia t ion in t h e m odel t h a n t a k in g a ll n igh t t im e bu r gla r s.
H owever, event u a lly, wit h fu r t h er su b-groupin gs t h er e will be insu fficient da t a .
Th is poin t h a s been r a ised in a r ecen t deba t e. Va n Koppen a n d De Keijser (1997)
a r gu ed t h a t a dis t a n ce deca y fun ction t h a t com bin ed m u lt iple in ciden t s comm it t ed by t h e
sa m e individu a ls cou ld dist or t t h e est ima t ed r elat ion sh ip com pa r ed t o select ing incident s
comm itted by different individua ls.6 Ren ger t , P iqu er o a n d J on es (1999) a r gu ed t h a t su ch a
dist r ibut ion is never t h eless m ea n ingfu l. In ou r lan gua ge, t h ese a r e t wo differ en t su bgr ou ps - per son s com m it t in g m u lt ip le offen ses com pa r ed t o per son s com m it t in g on ly on e
offen se. Com binin g th ese t wo su b-groups int o a sin gle ca libra t ion da t a set will on ly m ea n
t h a t t h e r esu lt will ha ve less d iffer en t iat ion in pr edict ion t h a n if t h e su b-groups wer e
se pa r a t ed out .
Act u a lly, t h er e is n ot m u ch d iffer en ce, a t lea s t in Ba lt im or e Cou n t y. F r om t h e
41,426 ca ses, 18,174 were com m itt ed by per sons wh o wer e on ly list ed once in t h e da t a ba se
wh ile 23,251 offen ses wer e com m itt ed by per sons wh o wer e list ed t wo or m or e t imes (7,802
ind ividu a ls). Cat egor izin g th e 18,174 cr imes a s com m itt ed by ‘sin gle inciden t offen der s
a n d t h e 23,251 cr im es a s com m it t ed by ‘m u lt ip le in cid en t offen der s’, t h e den sit y d is t a n ce
deca ys fu n ct ion s wer e ca lcu la t ed u sin g t h e ker n el d en sit y m et h od (F igu r e 10.22).
Th e dist r ibut ion s a r e r em a r ka bly sim ilar . Ther e a r e some su bt le differ en ces. The
a ver a ge jou r n ey t o cr im e t r ip dis t a n ce m a de by a sin gle in cid en t offen der is lon ger t h a n for
m u lt iple in ciden t offend er s (4.6 m iles comp a r ed t o 4.0 miles, on a ver a ge); t h e differ en ce is
h igh ly sign ifica n t (p #.0001), pa r t ly beca u se of t h e ver y la r ge sa m ple sizes. H owever , a
visu a l insp ection of t h e dist a n ce decay fu n ctions s h ows t h ey a r e sim ila r . Th e sin gle
in cid en t offen der s t en d t o h a ve sligh t ly m or e t r ip s n ea r t h eir h om e, s ligh t ly fewer for
dis t a n ces be t ween a bout a m ile u p t o th r ee m iles , a n d s ligh t ly m ore longer t r ips . Bu t , t h e
differ en ces a r e n ot ver y la r ge.
Th er e a r e severa l rea sons for t h e sim ilar ity. Fir st , som e of t h e ‘sin gle inciden t
offen d er s ’ a r e a ct u a lly m u lt ip le in cid en t offen d er s wh o h a ve n ot been ch a r ged wit h ot h er
in ciden t s. Secon d, s ome of t h e sin gle in ciden t offend er s a r e in t h e pr ocess of becom in g
m u ltiple incident offen der s so th eir beh a vior is pr oba bly sim ilar . Thir d, th er e m a y not be a
m a jor differ en ce in t r a vel pa t t er n s by t h e n u m ber of offens es a n in dividu a l comm it s,
cer t a inly com pa r ed t o t h e m a jor differ en ces by type of cr ime (see gra ph s a bove). In oth er
words , t h e dist in ction bet ween a sin gle offend er crim e t r ip a n d a m u lt iple offen der crim e
t r ip is just a n oth er su b-group com pa r ison a n d, a pp a r en t ly, n ot t h a t im por t a n t .
N ever t h eles s, it is im p or t a n t t o ch oos e a n a pp rop ria t e s am p le fr om wh ich t o es t im a t e a
lik ely h om e ba se loca t ion for a ser ia l offen der . Th e m et h od depen ds on a sim ila r sa m ple of
offen der s for com pa r is on .
Sam ple Da ta Se ts for Jou rne y to Crime Rou tine s
Th r ee sa m ple da t a set s from Ba ltim or e Cou n t y ha ve been pr ovided for t h e jou r n ey
t o cr im e r out in e. Th e da t a set s a r e sim u la t ed a n d d o not r epr esen t r ea l da t a . Th e firs t file
- J t cTest 1.dbf, ar e 2000 simu lat ed r obber ies wh ile t h e secon d file - J t cTest 2.dbf, ar e 2500
10.64
Figure 10.22:
Journey to Crime Distances
Kernel Density Estimate of Single and Multiple Incident Offenders
8%
(by offender type)
Percent of all crimes
10%
6%
Multiple incident offender
4%
Single incident offender
2%
0%
0
10
5
20
15
Distance from offender's home (miles)
25
Hot Spot Verification in Auto Theft Recoveries
Bryan Hill
Glendale Police Department
Glendale, AZ
We use CrimeStat as a verification tool to help isolate clusters of activity
when one application or method does not appear to completely identify a problem.
The following example utilizes several CrimeStat statistical functions to verify a
recovery pattern for auto thefts in the City of Glendale (AZ). The recovery data
included recovery locations for the past 6 months in the City of Glendale which were
geocoded with a county-wide street centerline file using ArcView.
First, a spatial density “grid” was created using Spatial Analyst with a grid
cell size of 300 feet and a search radius of 0.75 miles for the 307 recovery locations.
We then created a graduated color legend, using standard deviation as the
classification type and the value for the legend being the CrimeStat “Z” field that is
calculated.
In the map, the K-means (red ellipses), Nnh (green ellipses) and Spatial
Analyst grid (red-yellow grid cells) all showed that the area was a high density or
clustering of stolen vehicle recoveries. Although this information was not new, it did
help verify our conclusion and aided in organizing a response
Constructing Geographic Profiles
Using the CrimeStat Journey-To-Crime Routine
Josh Kent,
Michael Leitner,
Louisiana State University
Baton Rouge, LA
The map below shows a geographic profile constructed from nine crime sites
associated with a Baton Rouge serial killer, Sean Vincent Gillis, who was
apprehended on April 29, 2004 at his residence in Baton Rouge. Eight of the nine
are body dump sites and the ninth is a point of fatal encounter. All crime sites were
located in the City of Baton Rouge and surrounding parishes. Gillis’s hunting style
can best be described as that of a typical ‘localized marauder’.
The Journey-to-crime routine, implemented in CrimeStat , was applied to
simulate the travel characteristics of Gillis to and from the known crime sites.
Gillis’s travel behavior was calibrated with different mathematical functions that
were derived from the known travel patterns of 301 homicide cases in Baton Rouge.
The profile was estimated using Euclidean distance and the negative
exponential distance decay function. It predicts the actual residence of Gillis
extremely accurately. The straight-line error distance between the predicted and
the actual residence is only 0.49 miles. The proportion of the entire study area that
must be searched in order to successfully identify the serial offender’s residence is
0.05% (approximately 0.98 square miles out of a 2094.75 square miles study area).
6
FEATURES
$
T
U
%
Crime Scenes
Interstates
Water
Study Area Gird
S7
#
6
8
OFFENDER RESIDENCE PREDICTION
Predicted Residence
Search Cost
S
#
0
Miles
Actual Residence
LOW
NUMBER OF CRIME SCENES:
CELL SIZE:
METRIC:
MODEL:
HIT SCORE:
SEARCH COST:
ERROR DISTANCE:
HIGH
9
0.49 miles2
Direct Path Euclidean Distance Metric
Negative Exponential Distance Decay Function
0.05 (99.95% accuracy)
0.98 miles2 (2 grid cells)
0.49 miles (distance from predicted to actual residence)
.
,
1 10
.
,
-
1$
TU
S
#
#8 %
S
10
S5
#
#6
S
.
,
12
3
#
S
S2
#
S4
#
.
,
10
S3
#
sim u la t ed bu r gla r ies. Bot h files h a ve coor din a t es for a n or igin loca t ion (H om eX, H om eY)
a n d a dest ina t ion loca t ion (In ciden t X, Incident Y). User s can u se t h e ca libra t ion r ou t ine t o
ca lcu la t e t h e t r a vel d is t an ces bet ween t h e or igin s a n d t h e d es t in a t ion s. A t h ir d d at a set Ser ial1.dbf, ar e sim u lat ed incident loca t ion s for a ser ial offen der . User s can u se t h e J t c
est im a t ion r out in e t o iden t ify t h e lik ely r esiden ce locat ion for t h is in dividu a l. In r u n n in g
t h is r out in e, a r efer en ce grid n eed s t o be overla id (see ch a pt er 3). For Ba lt im ore Coun t y,
a ppr opr iat e coor din a t es for t h e lower -left cor n er a r e -76.910 lon git u de a n d 39.19 0 la t it u d e
a n d for t h e u pper -r igh t cor n er a r e -76.320 lon git u de a n d 39.72 0 la t it u d e.
D ra w Cr im e T ri p s
Th e J ou r n ey to Crim e m odu le in clud es one u t ilit y th a t ca n h elp visu a lize t h e
pa t t er n befor e select in g a pa r t icula r est im a t ion m odel. Th is is a Dr a w Cr im e Tr ips r out in e
t h a t sim ply dra ws lines bet ween t h e or igin a n d des t ina t ion of ind ividu a l cr ime t r ips. The
X an d Y coordina tes of an origin and destina tion locat ion a re input an d th e rout ine draws a
lin e in ArcView ‘sh p’, M apIn fo ‘m if’, Atlas*GIS ‘bn a ’ or As cii for m a t .
F igu r e 10.23 illu st r a t es t h e dr a wing of t h e kn own t r a vel dist a n ces for 444 ra pe
ca ses for wh ich t h e r esid en ce loca t ion of t h e r a pis t wa s kn own . Of t h e 444 ca ses, 113 (or
25.5%) occur r ed in t h e r es ide n ce of th e r a pis t . H owever , for t h e r em a in in g 331 cas es , t h e
r a pe loca t ion wa s n ot t h e r esid en ce loca t ion . As seen , m a n y of t h e t r ip s a r e of qu it e lon g
dis t a n ces. Th is would su ggest t h e u se of a n jou r n ey t o cr im e fun ction t h a t h a s m a n y t r ips
a t zer o dis t a n ce bu t wit h a m or e gr a du a l d eca y fu n ct ion .
H o w Ac c u r a t e a re t h e Me t h o d s ?
A cr itical ques t ion is how accu r a t e a r e t h ese m et h ods? The jou r n ey to cr ime m odel
is ju st t h a t , a m odel. Wh et h er it in volves u sin g a m a t h em a t ical fu n ction or a n em pir icallyder ived on e, t h e a ss u m pt ion in t h e J t c rout in e is t h a t t h e dist r ibu t ion of in ciden t s will
pr ovide in for m a t ion a bou t t h e h om e ba se loca t ion of t h e offen der . In t h is sen se, it ’s n ot
u n like t h e wa y m ost crim e a n a lyst s will work wh en t h ey a r e t r ying t o fin d a ser ia l
offen d er . A t yp ica l a p pr oa ch will be t o p lot t h e d is t r ibu t ion of in cid en t s a n d r ou t in ely
se a r ch a geogra ph ic ar ea in a n d a r oun d a ser ia l crim e pa t t er n , n otin g offen der s w h o ha ve
a n a r r es t h ist ory m a t chin g cas e a t t r ibu t es (MO, t ype wea pon , su sp ect descr ipt ion, et c.).
Beca u se a h igh pr oport ion of offen ses a r e com m itt ed wit h in a sh or t dist a n ce of offen der
r esiden ce’s, t h e m et h od ca n freq u en t ly lea d t o th eir a pp r eh en sion . Bu t , in doing t h is
m et h od, th e a n a lyst s a r e n ot u sin g a soph ist icat ed st a t ist ical m odel.
Tes t S am ple of Se rial Offe n de rs
To explor e t h e a ccu r a cy of t h e a ppr oa ch , a sm a ll sa m ple of 50 ser ial offen der s wa s
is ola t ed fr om t h e da t a ba se a n d u sed a s a t a r get sa m ple t o t est t h e a ccu r a cy of t h e m et h ods.
Th e 50 offen der s a ccou n t ed for 520 individua l cr ime in ciden t s in t h e da t a ba se. To t est t h e
J t c met h od sys t em a t ically, t h e followin g dist r ibu t ion wa s s elect ed (ta ble 10.4). Th e sa m ple
wa s n ot r a n dom, but wa s selected t o pr odu ce a ba lan ce in t h e n u m ber of inciden t s
10.68
Figure 10.23:
Journey to Rape Trips
Distance from Residence to Rape Location for 444 Known Offenders
Baltimore County
City of Baltimore
Rape trips
Baltimore County
City of Baltimore
N
W
E
S
0
10
20 Miles
com m itt ed by each in dividu a l an d t o, rough ly, app r oxim a t e t h e dist r ibut ion of inciden t s by
ser ia l offen der s. E a ch of t h e 50 offen der s wa s is ola t ed a s a sepa r a t e file so th a t ea ch cou ld
be a n a lyzed in Crim eS tat.
Id e n t i fy i n g t h e Cr im e T y p e
E a ch of t h e 50 offen der s wa s cat egor ized by a crim e t ype. Only two of t h e offen der s
com m itt ed t h e sa m e cr ime for a ll t h eir offen ses; most com m itt ed t wo or m or e differ en t
t ypes of cr imes . Ar bitr a r ily, each offen der wa s t yped a ccor din g to th e cr ime t ype th a t
h e/sh e most frequ ent ly com m itt ed; in t h e two ca ses wh ere t h ere wa s a t ie between t wo
crim e t ypes , t h e m ost sever e wa s s elect ed (i.e., per son a l cr im e over pr oper t y cr im e). While
I r ecogn ize t h a t t h er e is a r bit r a r in ess in t h e a ppr oa ch , it seem ed a pr a ct ica l s olu t ion . An y
er r or in cat egor izin g an offen der wou ld be ap plica ble to all th e m et h ods. Th e cr ime t ypes
for t h e 50 offen der s a ppr oxim a t ely m ir r or ed t h e dis t r ibu t ion of in cid en t s: la r cen y (29);
veh icle t h eft (7); bu r gla r y (5); r obber y (5); assa u lt (2); ba n k r obber y (1); an d a r son (1).
Id e n ti fy in g th e Ho m e B as e an d In c id e n t Lo ca ti on s
In t h e da t a ba se, ea ch of t h e offen der s wa s list ed a s h a vin g a r esid en ce loca t ion . F or
t h e a n a lysis, t h is wa s t a ken a s t h e origin loca t ion of th e jour n ey t o cr im e t r ip. S im ila r ly,
t h e inciden t loca t ion wa s t a ken a s t h e d estina tion for t h e t r ip. Op er a t ion a lly, th e crim e t r ip
is t a k en a s t h e dist a n ce fr om t h e origin locat ion t o th e dest in a t ion locat ion . H owever , it is
ver y p ossible t h a t som e cr im e t r ip s a ct u a lly s t a r t ed fr om ot h er loca t ion s. F u r t h er , m a n y of
t h ese individu a ls ha ve moved th eir r esidences over t ime; we on ly ha ve th e last kn own
r esiden ce in t h e da t a ba se. Un for t u n a t ely, t h er e wa s n o ot h er in for m a t ion in t h e digita l
da t a ba se t o a llow m or e a ccu r a t e ident ifica t ion of t h e h om e loca t ion . In oth er wor ds, t h er e
m a y be, a n d p r obably a r e, n u m er ous er r ors in t h e es t im a t ion of th e jour n ey t o cr im e t r ip.
H owever , t h ese er r ors would be s im ila r a cross a ll m et h ods a n d s h ould n ot a ffect t h eir
r ela t ive a ccu r a cy.
E v a lu a t e d Me t h o d s
E leven m et h ods wer e compa r ed in es t im a t in g t h e lik ely r esid en ce locat ion of th e
offen d er s . F ou r of t h e m et h od s u s ed t h e J t c r ou t in es a n d s even wer e s im p le s pa t ia l
dis t r ibu t ion m et h ods (t a ble 10.5).
Th e m ea n cen t er a n d cent er of m inim u m dist a n ce a r e discuss ed in cha pt er 4. The
cen t er of m in im u m dis t an ce, in pa r t icu la r , is m or e or les s t h e geogr a ph ic cen t er of
dis t r ibu t ion in t h a t it ign or es t h e va lu es of pa r t icu la r loca t ion s; t h u s, locat ion s t h a t a r e fa r
a wa y fr om t h e clu st er (ext r em e va lu es) h a ve n o effect on t h e r esu lt . Wh en t h e cen t er of
m inim u m dist a n ce is ca lcu lat ed on a r oa d n et wor k in wh ich ea ch segm en t is weigh t ed by
t r a vel t im e or speed, t h e r esu lt is t h e cen t er of m in im u m t r a vel t im e, t h e poin t a t wh ich
t r a vel t ime t o ea ch of t h e inciden t s is m inim ized. The d irection a l mea n , tr ian gula t ed
m ea n , geom et r ic a n d h a r m on ic m ea n s a r e discuss ed in cha pt er 4.
10.70
Ta ble 10 .4
S e r i a l Offe n d e r s U s e d i n Ac c u r a c y Ev a lu a t i o n
Num ber of
Offe n de rs
4
4
4
4
4
4
3
3
3
2
2
2
2
1
1
1
1
1
1
1
1
1
_________
50
N u m be r o f Cri m e s
Committed by
Each Pe rson
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
24
33
_________
520
10.71
Ta ble 10 .5
Co m p a ri s o n Me th o d s fo r E s ti m a ti n g th e Ho m e B a s e o f a S e ri a l Offe n d e r
J o u r n e y t o Cr im e M e t h o d s
Ma t h em a t ica l model for a ll cr imes
Math emat ical model for specific crime type
Ker n el dens ity m odel for a ll cr imes
Ker n el den sit y m odel for s pecific cr im e t ype
S p a t ia l D i s t ri b u t io n Me t h o d s
Mea n cen t er
Cen t er of m in im u m d is t a n ce
Cen t er of m in im u m t r a vel t im e
(ca lcu lat ed on r oa d n et wor k weigh t ed by tr a vel t ime)
Directiona l mea n (weight ed) ca lcu lat ed wit h ‘lower left cor n er ’
a s or igin
Tr ia n gu la t ed m ea n
Geom et r ic m ea n
H a r m on ic m ea n
Th e T e s t
E a ch of t h ese eleven m et h ods wer e r u n a ga in st ea ch of t h e files cr ea t ed for t h e
ser ial offen der s. For t h e seven ‘m ea n s’ (m ea n cen t er , geom et r ic m ea n , ha r m on ic m ea n ,
d ir ect ion a l m ea n , t r ia n gu la t ed m ea n , cen t er of m in im u m d is t a n ce, cen t er of m in im u m
t r a vel t im e), t h e m ea n wa s it self t h e be st gu ess for t h e lik ely r esid en ce locat ion of th e
offen der . F or t h e fou r jou r n ey t o cr im e fu n ct ion s, t h e gr id cell wit h t h e h igh est lik elih ood
est ima t e wa s t h e best guess for t h e likely res iden ce loca t ion of t h e offen der .
Me a s u r e m e n t o f E rr o r
F or ea ch of t h e 50 offen der s, err or wa s defined a s t h e dist a n ce in m iles between t h e
‘best guess ’ a n d t h e a ct u a l loca t ion . For ea ch offen der , th e dist a n ce bet ween t h e est ima t ed
10.72
h om e ba se (t h e ‘best gu ess’) a n d t h e a ct u a l r esid en ce loca t ion wa s ca lcu la t ed u sin g d ir ect
dis t a n ces. Ta ble 10.6 pr esen t s t h e r esu lt s. Th e da t a sh ow t h e er r or by m et h od for ea ch of
t h e 50 offen der s. Th e t h r ee r igh t colu m n s sh ow t h e a ver a ge er r or of a ll m et h ods a n d t h e
m in im u m er r or a n d m a xim u m er r ors obta in ed by a m et h od. Th e m et h od wit h t h e
m inim u m er r or is boldfa ced; for some cases , two met h ods a r e t ied for t h e m inim u m . The
bot t om t h r ee r ows sh ow t h e m ed ia n er r or , t h e a ver a ge er r or a n d t h e s t a n da r d devia t ion of
t h e er r ors for ea ch m et h od a cross a ll 50 offen der s.
Th er e a r e a n u m ber of conclu sion s fr om t h e r esu lt s. F ir st , t h e degr ee of pr ecis ion
for a n y of t h ese m et h ods va r ies con sid er a bly. Th e pr ecis ion of t h e est im a t es va r y fr om a
low of 0.0466 m iles (a bou t 246 feet ) t o a h igh of 75.7 m iles. Th e over a ll pr ecis ion of th e
m et h ods is n ot ver y h igh a n d is h igh ly va r ia ble. Th er e a r e a n u m ber of possible r ea son s for
t h is, some of wh ich h a ve been dis cus sed a bove. Ea ch of t h e m et h ods p r odu ces a sin gle
pa r a m et er fr om wh a t is , essen t ia lly, a pr oba bilit y d is t r ibu t ion wh er ea s t h e dis t r ibu t ion of
m a n y of t h ese in cid en t s a r e wid ely dis per sed. F ew of t h e offen der s h a d su ch a
con cen t r a t ed pa t t er n t h a t on ly a sin gle loca t ion wa s p os sible. Sin ce t h es e a r e p roba bilit y
dist r ibut ion s, not ever yon e follows t h e ‘cen t r a l ten den cy’. Also, som e of t h ese offen der s
m a y ha ve m oved du r in g t h e per iod in dica t ed by t h e in ciden t s, t h er eby sh iftin g t h e spa t ia l
pa t t er n of inciden t s a n d m a kin g it difficu lt t o ident ify th e las t r esiden ce.
A s econ d con clu s ion is t h a t , for a n y on e offen d er , t h e m et h od s p r od u ce s im ila r
r esu lt s. F or m a n y of th e offend er s t h e differ en ce bet ween t h e bes t est im a t e (th e m in im u m
error) an d th e worst estimat e (th e maximu m err or) is not great . Thus, the simple meth ods
a r e gen er a lly a s good (or ba d) a s t h e m or e soph is t ica t ed m et h ods.
Th ir d, a cr oss a ll m et h ods, t h e cen t er of m in im u m t r a vel t im e, wh ich is ca lcu la t ed on
a r oa d n et wor k (see ch a pt er s 3 a n d 16), a n d it ’s dis t a n ce-ba sed ‘cou sin ’ - t h e cen t er of
m in im u m t r a vel t im e, h a d t h e lowest a ver a ge er r or. Th u s, t h e a pp r oxim a t e geogra ph ic
cen t er of t h e d is t r ibu t ion wh er e t r a vel t im e t o ea ch of t h e in cid en t s wa s m in im a l p r od u ced
a s good a n est im a t e a s t h e m ore s oph ist icat ed m et h ods. H owever , it wa sn ’t pa r t icula r ly
close (3.8415 m iles, on a ver a ge). Th e wor st m et h od wa s t h e t r ia n gu la t ed m ea n ; it h a d a n
a ver a ge er r or of 7.6472 m iles . Th e t r ia n gu la t ed m ea n is p r odu ced by vector geomet r y a n d
will n ot n eces sa r ily ca p t u r e t h e cen t er of t h e d is t r ibu t ion . Ot h er t h a n t h is , t h er e wer e n ot
grea t differ en ces. This r einfor ces t h e poin t a bove th a t t h e m et h ods a r e a ll, m or e or less,
descr ibin g t h e cent r a l t en den cy of t h e dist r ibu t ion . For offend er s t h a t don ’t live in t h e
cent er of t h eir dis t r ibu t ion, t h e er r or of a m et h od will n ecessa r y be h igh .
Lookin g a t ea ch of t h e 50 offen der s, t h e m et h ods va r y in t h eir effica cy. F or
exa m ple, t h e J t c ker n el fu n ct ion for a ll cr im es wa s t h e be st or t ied for be st for 17 of th e
offen der s, bu t wa s a ls o t h e wor st or t ied for wor st for 9. Sim ila r ly, t h e J t c ker n el fu n ct ion
for t h e specific cr imes wa s best or t ied for best for 8 of t h e offen der s, but wor se for 4. Even
t h e m ost con sist en t wa s best for 4 offen der s, but a lso wor st for on e. On t h e ot h er h a n d,
t h e t r ia n gu la t ed m ea n , wh ich h a d t h e wor st over a ll er r or , p r odu ced t h e best est im a t e for 9
of t h e in divid u a ls w h ile it pr odu ced t h e wor st est im a t e for 25 of th e in divid u a ls. Th u s, t h e
t r ian gula t ed m ea n t en ds t o be very accu r a t e or very ina ccu r a t e; it h a d t h e h igh est
va r ia n ce, by fa r .
10.73
Table 10.6
Accuracy of Methods for Estimating Serial Offender Residences
(N= 50 Serial Offenders)
*
Center of
Center of
Number
Primary
* Mean
Minimum
Minimum
Triangulated
Geometric
Harmonic
Jtc Kernel:
Jtc Kernel:
Jtc Math:
Jtc Math:
of
Crime
* Center
Distance
Travel Time Mean
Mean
Mean
All Crimes
Crime Type
All Crimes Crime Type
Dataset
Crimes
Type
* Error (miles)
Error (miles)
Error (miles)
Error (miles)
Error (miles)
Error (miles)
Error (miles)
Error (miles)
Error (miles)
Error (miles)
------------------- -------------------- --------------------- * ---------------------------- ----------------------------------------------------------------------------------- -------------------------- ----------------------------------------------------- --------------------------- ----------------------- ------------------------3A
3
Larceny
*
31.5991
32.4477
32.2975
32.4109
31.5995
31.6000
32.7824
32.7880
32.7824
32.7880
3B
3
Larceny
*
13.2303
12.1683
12.1207
24.1531
13.2311
13.2319
10.7526
14.4929
10.7526
11.2501
0.6775
5.8416
0.6775
6.0946
3C
3
Bank robbery
*
2.8348
0.9137
0.9588
2.7767
2.8335
2.8322
3D
3
Burglary
*
2.9733
3.2603
2.6907
6.1013
2.9728
2.9724
4.6038
3.3883
3.3882
3.7931
4.3341
3.8217
4.2436
4.2436
4.2527
4.2364
4.2527
4.2590
*
*
3.6144
4.2154
2.6907
3.8217
6.1013
4.3341
0.1158
2.0563
4.6789
0.0466
17.8985
1.9621
4.4733
0.2925
17.3292
1.9623
4.4733
0.2926
17.3276
0.3125
4.9681
0.0703
0.2018
4.3563
0.0703
0.3125
4.2637
0.0703
0.2784
4.3563
0.4560
*
*
*
0.9473
4.5026
0.1790
0.1158
4.2637
0.0084
2.0563
4.9681
0.4560
15.9738
1.7733
16.4518
1.3586
2.2450
1.3564
0.2068
17.8655
0.6974
15.9739
0.5140
16.4526
0.6974
*
*
16.9881
0.8269
15.9738
0.0565
17.8985
1.7733
2.2442
2.7886
2.7886
0.2371
7.9621
0.9171
5.1837
0.9174
5.1837
0.1577
2.4205
0.4267
5.1271
4.8554
4.9393
3.0922
0.4267
5.2256
*
*
*
3.9704
0.5184
5.3933
2.2442
0.1577
4.8554
16.4518
0.9174
7.9621
0.9625
1.3710
1.3192
3.2431
1.3700
1.3184
3.2405
3.1126
0.2580
2.3800
0.5272
1.3566
0.2580
2.0831
0.5272
*
*
1.6348
0.7641
0.9625
0.0051
3.1126
1.7928
1.2506
3.9022
12.4107
5.0481
3.9021
12.4115
1.9718
3.1364
14.8293
10.8567
8.5357
2.6253
3.0532
8.6148
8.5357
1.9718
3.0532
8.8275
*
*
*
2.9686
3.2627
10.5776
1.2506
2.3176
8.5357
6.5209
3.9023
14.8293
5.0460
7.9975
0.3223
75.7424
2.2684
2.2682
0.0892
7.9975
0.7191
0.0892
7.6274
0.7191
*
*
7.1204
8.5260
5.0460
0.0892
10.8567
75.7424
6.3167
1.1458
1.6203
4.0475
5.4565
6.2653
2.1776
1.3684
5.5915
6.0264
1.0042
1.3043
3.5809
6.0229
1.0042
1.3027
3.5825
8.4210
1.7475
2.1513
6.1166
1.3510
1.8707
1.3340
*
*
*
*
6.3714
1.3459
1.5971
2.7931
6.0165
1.0041
1.2020
0.5900
8.4210
2.1776
2.1513
5.5915
4.8574
6.9916
5.2532
8.1850
3.7738
0.9353
4A
4B
4
4
Vehicle theft
Larceny
*
*
4.2436
1.9618
4.2670
0.3100
4C
4D
5A
4
4
5
Larceny
Assault
Larceny
*
*
*
4.4733
0.2925
17.3308
4.4733
0.1905
16.6459
5B
5C
5
5
Larceny
Larceny
*
*
1.3609
2.2458
0.2481
2.6832
0.0565
5D
6A
6B
5
6
6
Larceny
Larceny
Vehicle theft
*
*
*
0.9169
5.1837
1.3720
0.2250
5.2081
1.1869
0.8021
5.0644
1.1535
6C
6D
6
6
Larceny
Larceny
*
*
1.3199
3.2458
0.3157
2.3324
0.0051
7A
7B
7C
7
7
7
Larceny
Larceny
Burglary
*
*
*
3.9023
12.4100
5.0501
3.4185
9.2973
7.1477
3.1998
9.9031
6.4354
2.3176
7D
8A
7
8
Larceny
Larceny
*
*
2.2686
6.0298
0.7733
6.0165
8B
8C
8D
9A
8
8
8
9
Larceny
Larceny
Vehicle theft
Robbery
*
*
*
*
1.0041
9B
9C
10A
10B
10C
9
9
10
10
10
Larceny
Robbery
Larceny
Larceny
Larceny
*
*
*
*
*
8.1923
3.7778
0.9358
2.8581
0.8052
11A
11B
11C
12A
12B
11
11
11
12
12
Vehicle theft
Robbery
Vehicle theft
Larceny
Larceny
*
*
*
*
*
2.9127
0.3250
1.2689
3.3881
0.5562
3.2715
0.3250
1.7157
4.2334
0.5361
13A
13B
14A
14B
15A
13
13
14
14
15
Larceny
Assault
Larceny
Arson
Vehicle theft
*
*
*
*
*
6.3282
7.2857
1.4943
1.4943
1.9363
0.6898
0.7282
0.8706
0.3727
0.7189
15B
16A
17A
18A
19A
15
16
17
18
19
Robbery
Vehicle theft
Burglary
Larceny
Larceny
*
*
*
*
*
0.4914
2.1107
1.6484
0.6308
8.6462
20A
21A
22A
24A
33A
20
21
22
24
33
Burglary
Burglary
Larceny
Larceny
Robbery
*
*
*
*
*
6.3520
1.2396
3.6828
1.7959
3.9901
1.3059
3.5794
5.2527
1.1437
1.6944
2.3780
5.7156
10.6555
3.8454
0.5159
3.4940
0.7251
*
*
All Methods
* Average
Minimum
Maximum
* Error
Error
Error
* -------------------- ------------------------------ --------------------*
32.3095
31.5991
32.7880
*
13.5384
10.7526
24.1531
*
2.6441
0.6775
6.0946
4.5096
0.0084
17.0832
2.7443
3.2838
1.7928
6.5209
2.7419
6.2962
1.3510
0.5900
1.3340
7.8257
12.4578
4.9015
7.1961
10.3957
5.1862
0.3720
6.5709
6.2520
12.4578
4.6206
0.2601
10.3095
5.9265
12.0514
4.3445
0.7172
6.4758
*
*
*
*
*
5.8989
9.9554
4.8797
0.6315
6.0856
4.8574
6.9916
3.5670
0.0606
2.8491
7.8257
12.4578
11.0042
1.1003
14.2219
0.8404
3.4335
0.9060
3.4282
0.4235
2.8945
5.5843
1.2786
3.2087
0.7011
2.2049
5.2132
*
*
*
*
*
1.3410
3.2343
0.3596
1.7107
5.0048
0.7251
2.9127
0.2263
0.6984
3.2639
5.5938
3.6936
0.7011
2.8945
10.9241
0.7897
7.6438
1.6501
0.3359
0.9631
7.9915
2.0824
0.7631
0.6213
*
*
*
*
*
0.8716
6.9831
1.5940
1.0770
0.4905
0.4973
6.0244
1.4572
0.2596
0.0251
2.8003
7.9915
2.0824
1.9368
0.8086
0.8155
0.6468
2.5911
0.2879
0.2132
1.5128
0.6546
2.4033
0.5268
0.6985
*
*
*
*
*
0.7741
0.5651
2.6949
0.7761
0.4911
0.3362
0.3422
1.5957
0.1000
0.0417
1.5128
0.8254
8.2311
1.6484
1.0876
9.7022
9.5548
0.7945
*
*
*
*
*
9.2224
6.3426
0.9509
2.8895
1.0921
8.6462
0.5934
0.4965
2.0949
0.2658
10.2869
28.3094
1.2776
3.6828
2.3033
1.2020
0.4822
4.8179
11.0042
1.1003
14.2219
2.8491
6.4051
0.7451
3.4493
0.2709
1.4115
4.2640
5.5938
3.1192
0.2513
1.4750
10.9241
0.8050
2.9130
0.3250
1.2709
3.3867
0.8049
2.9134
0.3250
1.2729
3.3852
0.9059
3.6936
0.4235
2.8945
6.4050
0.2263
0.6984
3.2639
0.4973
2.8003
6.8066
1.4572
0.6681
6.0244
0.5562
6.3248
1.4944
1.9365
0.6899
0.5562
6.3213
1.4944
1.9368
0.6900
0.7897
7.6438
1.6501
0.3434
0.3359
0.6709
7.4607
1.5954
0.6058
0.3359
0.7277
0.4914
2.1107
1.6461
0.6329
0.7271
0.4914
2.1107
1.6438
0.6349
0.8155
0.6468
0.4855
0.5693
1.6404
0.2879
0.3383
8.6486
6.3486
1.2393
3.6803
1.7975
8.6511
6.3452
1.2390
3.6777
1.7991
3.5670
0.0251
1.5279
1.4498
0.8086
0.8741
0.3362
0.4914
2.0995
0.3093
0.4196
0.3422
0.8254
8.2311
1.0227
1.0876
9.4195
5.7969
0.8861
2.6232
0.5892
5.0481
9.3665
7.4256
1.0564
2.3597
0.9322
2.0555
0.1000
0.0417
3.4056
8.6772
28.3094
1.2776
2.0949
2.3033
7.9975
6.2022
1.5298
2.1513
1.9133
5.2529
8.1886
3.7758
0.9355
2.8536
9.9787
0.1577
0.0606
1.5957
0.2879
0.2132
10.2869
0.5934
0.5243
2.4937
0.2658
9.2708
0.8673
0.5243
2.8944
0.3574
0.2596
0.5934
1.0253
2.4937
0.4222
0.4965
2.8944
0.6587
7.2505
3.9940
3.9979
7.9485
7.6939
8.1907
7.9439 *
5.9463
3.4056
8.1907
------------------- -------------------- --------------------- * ---------------------------- ------------------------- ------------------------ -------------------------------- -------------------------- ----------------------------------------------------- --------------------------- ----------------------- ------------------------- * -------------------- ------------------------------ --------------------Median Error = *
2.5517
2.2159
2.2076
3.4704
2.5509
2.5502
1.9494
2.0102
2.0615
2.1440
Mean Error = *
4.0434
3.8441
3.8415
7.6472
4.0429
4.0424
4.0395
4.0305
4.0163
4.1467 *
SD Error =
*
5.2696
5.4392
5.4619
12.1867
5.2696
5.2695
5.6244
5.6807
5.5961
5.4727 *
F our t h , t h e m edia n er r or is sm a ller t h a n t h e a ver a ge er r or. Th a t is, t h e m edia n is
t h e point a t wh ich 50% of th e case s h a d a sm a ller er r or a n d 50% ha d a la r ger er r or.
Over a ll, m os t of t h e ca s es wer e fou n d wit h in a sh or t er d is t a n ce t h a n t h e a ver a ge wou ld
ind ica t e. This ind ica t es t h a t severa l ca ses h a d very la r ge err or s wh er ea s m ost h a d sm a ller
er r or s; th a t is, th ey were ou tliers. Over a ll m et h ods, t h e J t c ker n el ap pr oa ch for a ll cr imes
h a d t h e lowest m edia n er r or (1.95 m iles). In fact , all four J t c met h ods h a d s m a ller m edia n
er r or s t h a n t h e s im p le cen t r ogr a ph ic m et h od s. In ot h er wor ds , t h ey a r e m or e a ccu r a t e
t h a n t h e cent r ogr a ph ic m et h ods m ost of t h e t im e. Th e pr oblem in a pp lying t h is logic in
pr a ct ice, h owever , is t h a t on e wou ld n ot kn ow if t h e ca se bein g s t u died is t yp ica l of m ost
cas es (in wh ich ca se , t h e er r or w ould be r ela t ively sm a ll) or wh et h er it wa s a n out lier . In
ot h er wor ds, t h e m edia n wou ld defin e a sea r ch a r ea t h a t ca pt u r ed a bou t 50% of t h e ca ses,
bu t wou ld be ver y wr on g in t h e ot h er 50%. If we cou ld som eh ow develop a m et h od for
id en t ifyin g wh en a ca se is ‘t yp ica l’ a n d wh en it is n ’t , in cr ea sed a ccu r a cy will em er ge fr om
t h e J t c m et h ods. Bu t , un t il t h en , th e sim ple cen t er of m inim u m t r a vel t ime will be th e
most accur at e meth od.
F ift h , t h e a m ou n t of er r or va r ies by t h e n u m ber of in cid en t s. Ta ble 10.7 below
sh ows t h e a vera ge err or for ea ch m et h od a s a fu n ct ion of t h r ee size clas ses: 1-5 inciden t s;
6-9 in ciden t s; a n d 10 or m ore in ciden t s. As ca n be s een , for ea ch of th e t en m et h ods, t h e
er r or decr ea ses wit h in cr ea sin g n u m ber of in cid en t s. In t h is sen se, t h e m ea su r ed er r or is
r es pon sive t o th e sa m ple size from wh ich it is ba se d. I t is, p er h a ps , n ot s u r pr isin g t h a t
with on ly a h a n dful of inciden t s n o m et h od can be very pr ecise.
Sixth , th e r elat ive a ccu r a cy of ea ch of t h ese m et h ods va r ies by sam ple size. The
m et h od or m et h ods wit h t h e m in im u m er r or a r e bold fa ced. F or a lim it ed n u m ber of
in ciden t s (1-5), t h e J t c ma t h em a t ical fun ction for a ll cr im es (i.e., th e n ega t ive expon en t ia l
with t h e pa r a m et er s from t a ble 10.5) pr odu ced t h e est ima t e with t h e leas t er r or , followed
by t h e J t c k er n el fu n ct ion for a ll cr im es ; t h e wa s t h e t h ir d bes t . Th e d iffer en ces in er r or
bet ween t h ese wer e n ot ver y gr ea t . F or t h e m id dle ca t egor y (6-9 in cid en t s), t h e cen t er of
m in im u m dis t a n ce pr odu ced t h e lea st er r or followed by t h e J t c m a t h em a t ica l fu n ct ion for
t h e specific cr ime t ype. For t h ose offen der s wh o h a d com m itt ed t en or m or e cr imes , th e J t c
k er n el fun ction for t h e specific cr im e t ype pr odu ced t h e best est im a t e, followed by t h e
cen t er of m in im u m d is t a n ce. Th e t wo m a t h em a t ica l fu n ct ion s pr od u ced t h e lea s t a ccu r a cy
for t h is su b-gr ou p, t h ou gh a ga in t h e differ en ces in er r or a r e n ot ver y big (2.2 m iles for t h e
best com pa r ed t o 2.7 m iles for t h e wor st ). In ot h er wor ds, on ly wit h a sizea ble n u m ber of
in ciden t s d oes t h e J t c ker n el den sit y a pp r oach for sp ecific crim es pr odu ce a good est im a t e.
It is bet t er t h a n t h e ot h er a ppr oa ch es, bu t on ly sligh t ly bet t er t h a n t h e sim ple m ea su r e of
t h e cen t er of m inim u m dist a n ce.
S e a rc h Are a
A n u m ber of r es ea r ch er s h a ve been in t er es t ed in t h e con cep t of a s ea r ch a r ea for t h e
police (Ros sm o, 2000; Ca n t er , 2003). Th e con cept is t h a t t h e jou r n ey t o cr im e m et h od ca n
define a sm a ll sea r ch a r ea wit h in wh ich t h er e is a h igh er pr obabilit y of find in g t h e
offen der . Th e a ver a ge or m ed ia n er r or dis cu ss ed above ca n be u sed to d efin e s uch a sea r ch
a r ea if t r ea t ed a s a r a diu s of a circle. While in t u itive, I’m n ot su r e wh et h er t h is r epr esen t
10.75
Table 10.7
Method Estimation Error and Sample Size:
Average Error of Method by Number of Incidents (miles)
*
Number of * Mean
Incidents * Center
-------------- * ----------3-5 * 6.9553
Center of Center of
Minimum Minimum
Triangulated Geometric Harmonic
Distance Travel Time Mean
Mean
Mean
------------------------------------------------------ ---------------- --------------6.4861
6.4768
9.3672
6.9160
6.9545
Jtc
Kernel:
All
----------6.4622
*
6-9 *
4.2596
4.0753
4.1000
10.6160
4.3331
4.2576
4.4805
*
10+ *
2.3832
2.3149
2.2980
4.8136
2.4575
2.3827
2.4880
Jtc
Kernel:
Crime types
------------------7.2321
Jtc
Math:
All
-----------
Jtc
Math:
Crime types
------------------6.3278
6.9954
*
*
*
*
*
All Methods
Average
Minimum
Error
Error
----------------- -------------7.0173
6.3278
4.2274
*
4.2020 *
4.8800
4.0753
2.2176 2.6725
*
2.6243 *
2.6652
2.2176
4.2489
-------------- * ----------- -------------- -------------------------------------- ---------------- --------------- ----------- ------------------- ----------- ------------------- * ----------------- --------------
a m ea n in gfu l st a t ist ic. F or exa m ple, t a k in g t h e a ver a ge er r or of t h e cent er of m in im u m
dis t a n ce (3.84 m iles) would p r odu ce a s ea r ch a r ea of 46.4 s qu a r e m iles, n ot exa ctly a sm a ll
a r ea in wh ich t o fin d a ser ia l offen der . E ven if we t a k e t h e m edia n er r or of 1.94 m iles fr om
t h e J t c ker n el a pp r oach for a ll crim es (1.94 m iles ) will st ill pr odu ce a sea r ch a r ea of 11.9
squ a r e m iles, a n d it would be cor r ect only h a lf t h e t im e
In oth er words , t h ese m et h ods a r e st ill ver y im pr ecise. F u r t h er , t h e er r or is liable
t o in crea se over t im e, r a t h er t h a n decrea se. With a bout 50% of th e U .S. popu la t ion living
in s u bu r bia (Demogr a ph ia, 1998) a n d wit h 90% of Am er ica n h ou seh olds owning a t leas t
on e m ot or vehicle (U.S. Cens u s Bu r ea u , 2000), th e a vera ge dista n ces t r a veled by offen der s
h a s pr oba bly been in cr ea sin g over t ime s ince most t ypes of t r ips h a ve also shown in cr ea ses
in t r a vel over t im e. Th is m ea n s t h a t u n les s p olice ca n fin d a wa y t o n a r r ow d own t h e
se a r ch a r ea con sid er a bly, t h e m et h ods don ’t r ea lly h elp beyon d w h a t police int u it ively do
a n yway, na m ely look n ea r t h e dist r ibut ion of t h e inciden t s com m itt ed by ser ial offen der s.
Ca u ti on a ry No te s
Of cour se, t h is is a lim it ed t est . It wa s a sm a ll sa m ple (on ly 50 cas es) fr om a sin gle
jur isdict ion (Ba ltim or e Cou n t y). The sa m ple wa sn ’t even r a n domly select ed, but ch osen t o
exa m in e t h e a ccu r a cy by a r a n ge of s am p le s izes . Th u s, t h e con clu sion s a r e on ly t en t a t ive
an d mu st be seen as h ypoth eses for fur th er work . Clearly, more resear ch is needed.
Never t h eless, t h er e a r e cer t a in ca u t ion s t h a t m u st be con sid er ed in u sin g eit h er of
t h ese jou r n ey t o cr im e m et h ods (t h e m a t h em a t ical or t h e em pir ical). Fir st , a sim ple
t ech n ique, su ch a s t h e cen t er of m inim u m dist a n ce, ma y be as good a s a m or e sophist ica t ed
t ech n iqu e. It doesn ’t a lwa ys follow t h a t a soph is t ica t ed m et h od will p r odu ce a n y m or e
a ccu r a cy t h a n a sim ple on e. F or t h e t im e bein g, I wou ld a dvise cr im e a n a lyst s wh o a r e
t r yin g t o det ect a pa t t er n in t h e dis t r ibu t ion of th e in cid en t s of a ser ia l offen der t o do
exa ctly w h a t t h ey h a ve been doin g, ba sica lly look in g a t t h e da t a a n d m a k in g a su bjective
guess about where th e offender m ay be residing. The kernel density J tc rout ine needs an
a dequ a t e a m oun t of in for m a t ion (i.e, a t lea st 10 in ciden t s) t o pr odu ce some wh a t pr ecise
est ima t es. Thes e t ech n iques sh ou ld be seen for n ow a s r esea r ch t ools r a t h er t h a n a s
dia gnost ics for ident ifyin g th e wh er ea bou t s of a n offen der . They ar e ju st t oo imp r ecise a n d
un reliable to depend on, at least u nt il more definitive results ar e obtained.
Secon d, th er e a r e ot h er lim ita t ion s t o t h e t ech n ique. The m odel m u st be ca libra t ed
for ea ch in divid u a l ju r is dict ion . F u r t h er , it m u st be per iod ica lly r e-ca libr a t ed t o a ccou n t
for ch a n ges in cr im e p a t t er n s . F or exa m p le, in u s in g t h e m a t h em a t ica l m od el, on e ca n n ot
t a ke t h e pa r a m et er s est ima t ed for Balt imore Coun t y (Ta ble 10.3) a n d a pply th em t o
a n ot h er cit y or if u sin g t h e ker n el d en sit y m et h od t a k e t h e r esu lt s fou n d a t on e t im e per iod
a n d a ssu m e t h a t t h ey will r em a in in defin it ely. Th e m odel is a pr oba bilit y m odel, n ot a
gu a r a n t ee of cer t a in t y. It p r ovid es gu es ses ba s ed on t h e s im ila r it y t o ot h er offen d er s of t h e
sa m e t ype of cr im e. In t h is s en se, a pa r t icula r ser ia l offen der m a y not be t ypical a n d t h e
m odel cou ld a ct u a lly or ien t police wr on gly if t h e offen der is differ en t fr om t h e ca libr a t ion
sa m ple. It will t a k e in sigh t by t h e in vest iga t in g officers t o kn ow wh et h er t h e pa t t er n is
t yp ica l or n ot .
10.77
Th ir d, a s a t h eore t ical m odel, t h e jou r n ey t o cr im e a pp r oach is qu it e sim ple. It is
ba sed on a dis t r ibu t ion of in cid en t s a n d a n a ssu m ed t r a vel d is t a n ce deca y fu n ct ion . F r om
t h e per spective of m odeling t h e t r a vel beha vior of offen der s, it is limit ed. As m en t ion ed
a bove, t h e m et h od does not u t ilize in for m a t ion on t h e dist r ibut ion of t a r get oppor t u n ities
n or d oes it u t ilize infor m a t ion on t h e t r a vel m ode a n d r out e t h a t a n offend er t a k es. It is
pu r ely a st a t is t ica l m odel. Th e r esea r ch a r ea of geogr a ph ic p r ofilin g a t t em pt s t o go beyon d
st a t is t ica l d escr ip t ion a n d u n der st a n d t h e cogn it ive m a ps t h a t offen der s u se a s well a s h ow
t h es e in t er a ct w it h t h eir m otives. Th is is good a n d s h ould clear ly gu ide fut u r e r es ea r ch.
Bu t it h a s t o be un der st ood t h a t t h e t h eory of offend er t r a vel beh a vior is n ot ver y well
develop ed, cer t a in ly com pa r ed t o ot h er t yp es of t r a vel beh a vior . F u r t h er , s om e t yp es of
cr im e t r ip s m a y n ot even st a r t fr om a n offen der ’s r esid en ce, bu t m a y be r efer en ced fr om
a n oth er locat ion , su ch a s veh icle t h efts occur r in g nea r dis posa l locat ion s. Rou t in e a ctivit y
t h eor y wou ld su ggest m u lt ip le or igin s for cr im es (Coh en a n d F els on , 1979).
Th e exis t in g m odels of t r a vel dem a n d u se d by t r a n sp ort a t ion p la n n er s (wh ich h a ve
t h em selves been crit icized for be in g t oo sim ple) mea su r e a va r iet y of fa ctor s t h a t h a ve only
been m a r gin a lly inclu ded in t h e cr ime t r a vel lit er a t u r e - t h e a vailability of opport u n ities,
t h e con cen t r a t ion of offen der t ypes in cert a in a r ea s, t h e m ode of t r a vel (i.e., a u t o, bus,
wa lk), t h e specific r ou t es t h a t a r e t a ken , th e int er a ct ion bet ween t r a vel t ime a n d t r a vel
r ou t e, a n d ot h er fa ct or s. It will be im por t a n t t o in cor por a t e t h ese elem en t s in t o t h e
u n der st a n din g of jou r n ey t o cr im e t r ip s t o bu ild a m u ch m or e com pr eh en sive t h eor y of h ow
offen d er s op er a t e. Tr a vel beh a vior is ver y com p lica t ed a n d we n eed m or e t h a n a st a t is t ica l
dis t a n ce m odel t o a dequ a t ely u n der st a n d it . Th e n ext seven ch a pt er s look a t a n a pplica t ion
of t r a vel dem a n d t h eory t o cr im e t r a vel.
Also, it’s n ot clea r wh et h er kn owing a n offend er ’s ‘cogn it ive m a p’ will h elp in
pr edict ion . Th er e h a ve been n o eva lu a t ion s t h a t h a ve com pa r ed a st r ict ly st a t is t ica l
a ppr oa ch wit h a n a ppr oa ch t h a t u t ilizes in for m a t ion a bou t t h e offen der a s h e or sh e
u n der st a n ds t h e en vir on m en t . It ca n n ot be a ssu m ed t h a t in t egr a t in g in for m a t ion a bou t
t h e per cept ion of t h e en vir on m en t will a id pr edict ion . In m ost t r a vel d em a n d for eca st s
t h a t t r a n spor t a t ion en gin eer s a n d pla n n er s m a k e, cogn it ive in for m a t ion a bou t t h e
en vir on m en t is n ot u t ilized except in t h e defin it ion of t r ip pu r pose (i.e., wh a t t h e pu r pose of
t h e t r ip wa s). Th e m odels u se t h e a ct u a l t r ip s by or igin a n d dest in a t ion a s t h e ba sis for
for m u la t in g p r ed ict ion s , n ot t h e u n d er s t a n din g of t h e t r ip by t h e in d ivid u a l.
Un der st a n din g is im por t a n t from t h e viewpoin t of developin g t h eor y or for w a ys t o
com m u n ica t e wit h peop le. Bu t , it is not n eces sa r ily u sefu l for pr ed ict ion . In sh or t ,
u n der st a n din g a n d p r ediction a r e n ot t h e sa m e t h in g.
On t h e oth er h a n d, t h e jou r n ey t o cr im e r out in e, pa r t icula r ly th e k er n el de n sit y
appr oach, can be useful for police depar tm ents if u sed car efu lly. If t h er e a r e su fficient
ca ses t o bu ild a n est im a t e (i.e., 10 or m or e in cid en t s), it ca n pr ovide a ddit ion a l in for m a t ion
t o officer s in vest iga t in g a ser ia l offen der by r edu cin g t h e n u m ber of possible su spect s t h a t
m igh t be lin k ed t o a s er ies of crim es . It can a lso pr ovide s ome dir ect ion in orien t in g t h e
deploym en t of officer s a n d d et ect ives in vest iga t in g wh a t a pp ea r t o be ser ia l offen se s. I t
pr ovides gu esses a bout wh er e t h e offend er m ight be livin g, bu t ba sed on sim ila r it ies wit h
pr eviou s offen der s for t h e sa m e t yp e of cr im e. It ’s n ot goin g t o give a n exa ct est im a t e of
10.78
wh er e a n offend er is livin g, bu t will pr ovide som e in sigh t s in t o which a r ea s t h e in dividu a l
m igh t be loca t ed. Th e J t c m odel s h ou ld be seen a s a su pplem en t t o ot h er t ech n iqu es, n ot a
com plet e solu t ion . Like a ll t h e st a t ist ical t ools in Crim eS tat, it m u st be u se d ca r efu lly a n d
in t elligen t ly. The ph ilosoph y of cr im e a n a lysis m u st a lwa ys be t o us e a t echn iqu e wit h
t h ou ght a n d wit h a syst em a t ic pr ocedu r e.
10.79
En d n ot e s fo r Ch ap te r 10
1.
It sh ould a lso be poin t ed out t h a t t h e u se of dir ect dis t a n ces will u n der est im a t e
t r a vel dis t a n ces pa r t icula r ly if t h e st r eet n et work follows a gr id.
2.
Th er e a r e, of cou r se, m a n y ot h er t ypes of m a t h em a t ica l fu n ct ion s t h a t ca n be us ed
t o descr ibe a declin in g likelih ood wit h dis t a n ce. In fact , t h er e a r e a n in finit e
n u m ber of su ch fu n ct ion s. H owever, t h e five t ypes of fu n ct ion s pr esen t ed h er e a r e
com m only u sed. We a voided t h e in ver se dis t a n ce fu n ction becau se of it s p oten t ia l t o
dis t ort t h e lik elih ood r ela t ions h ip.
1
f(d) = ------d ijk
wh er e k is a power (e.g., 1, 2, 2.5). F or la r ge dis t a n ces , t h is fu n ct ion ca n be a u sefu l
a ppr oxim a t ion of t h e lessen in g t r a vel in t er a ct ion wit h dis t a n ce. H owever , for sh or t
dis t a n ces, it doesn ’t wor k . As t h e dis t a n ce bet ween t h e r efer en ce cell loca t ion a n d
a n in cid en t loca t ion becom es ver y s m a ll, a ppr oa ch in g zer o, t h en t h e lik elih ood
est im a t e becom es ver y la r ge, a ppr oa ch in g in fin it y. In fa ct , for d ij = 0, t h e fun ction is
u n solvable. Since ma n y dista n ces bet ween r efer en ce cells an d in ciden t s will be zero
or close t o zero, t h e fu n ct ion becom es u n u sa ble.
3.
It is a ct u a lly t h e in ver se of t h e in ver se dis t a n ce fu n ct ion . If a dis t a n ce deca y
fu n ct ion d r op s off p r op or t ion a l t o t h e in ver s e of t h e d is t a n ce,
Yij = A* 1/d ij
wh er e Yij is t h e t r a vel lik elih ood, A is coefficien t , a n d d ij is t h e dis t a n ce fr om t h e
h ome ba se, t h en t h e opposit e - a d ist a n ce incr ea se is ju st t h e in ver se of t h is fu n ction
1
d ij
Zij = ------------------ = ------------ =
A* 1/d ij
A
4.
B* d ij
Th er e a r e sever a l s ou r ces of er r or a ssocia t ed wit h t h e da t a set . F ir st , t h ese r ecor ds
wer e a r r est r ecor ds p r ior t o a t r ial. Un doubt edly, some of t h e individu a ls wer e
in cor r ect ly a r r es t ed . S econ d , t h er e a r e m u lt ip le offen s es . In fa ct , m or e t h a n h a lf
t h e r ecor ds wer e for in divid u a ls w h o wer e list ed t wo or m ore t im es in t h e da t a ba se .
Th e t r a vel pa t t er n of r epea t offen der s m a y be slight ly differ en t t h a n for a ppa r en t
firs t -t ime offen der s (see figu r e 10.19). Thir d, m a n y of t h ese ind ividu a ls h a ve lived
in m u lt ip le loca t ion s . Con s id er in g t h a t m a n y a r e you n g a n d t h a t m os t a r e s ocia lly
n ot well a dju st ed, it would be exp ected t h a t t h ese in dividu a ls would h a ve m u lt iple
h om es. Thu s, t h e dist r ibut ion of inciden t s cou ld r eflect m u ltiple h om e bas es, ra t h er
t h a n on e. U n for t u n a t ely, t h e da t a we h a ve on ly gives a sin gle r esid en t ia l loca t ion ,
th e place at wh ich t hey were living when ar rested.
10.80
5.
If th e coordina te system is projected with th e dista nce units in feet, meters or m iles,
t h en t h e dist a n ce bet ween t wo point s is t h e h ypot en u se of a r ight t r ia n gle u sin g
E u clidea n geomet r y.
____________________
d AB = % (XA - XB )2 + (YA - YB )2
(3.1)
r ep ea t
wh er e ea ch locat ion is define d by a n X a n d Y coor din a t e in feet , m et er s, or m iles .
If t h e coor din a t e syst em is sph er ica l with u n its in la t itu des a n d lon git u des, t h en t h e
dista nce between t wo points is the Great Circle dista nce. All lat itudes an d
lon git u des a r e con ver t ed in t o ra dia n s u sin g
2B N
Radians (N) = ---------------360
(3.2)
r ep ea t
2B 8
Radians (8) = ---------------360
(3.3)
r ep ea t
Th en , t h e dis t a n ce bet ween t h e t wo poin t s is det er m in ed fr om
d AB = 2* Ar csin { Sin 2 [(NB - NA )/2] + Cos NA *Cos NB *Sin 2 [(8B - 8A )/2]1 /2 }
(3.4)
r ep ea t
wit h a ll a n gles be in g defin ed in r a dia n s (Sn yd er , 1987, p. 30, 5-3a ).
6.
Th ey a ls o a r gu ed t h a t t h e com bin a t ion of in cid en t s - wh ich t h ey ca lled ‘a ggr ega t ion ’,
wou ld dis t or t t h e r ela t ion sh ip bet ween dis t a n ce a n d in cid en ce lik elih ood beca u se of
t h e ecologica l fa lla cy. To m y m in d, t h ey a r e in cor r ect on t h is poin t . Da t a on a
dis t r ibu t ion of in ciden t s by dist a n ce tr a veled is a n in dividu a l ch a r a cter ist ic a n d is
n ot ‘ecologica l’ in a n y way. An ecologica l in fer en ce occu r s wh en da t a a r e a ggr egat ed
wit h a grou pin g va r ia ble (e.g., st a t e, cou n t y, cit y, cen su s t r a ct; see La n gbein a n d
Licht m a n , 1978). A freq u en cy dis t r ibu t ion of in dividu a l cr im e t r ip d ist a n ces is a n
in divid u a l p r oba bilit y d is t r ibu t ion , s im ila r , for exa m ple, t o a dis t r ibu t ion of
ind ividu a ls by height , weight , in com e or a n y ot h er ch a r a ct er ist ic. Of cou r se, th er e
a r e su b-set s of th e da t a t h a t h a ve been a ggr ega t ed (sim ila r t o heigh t s of men v.
h eigh t s of wom en , for exam ple). Clea r ly, iden t ifyin g sub-groups ca n m a ke bet t er
dis t in ction s in a dis t r ibu t ion. Bu t , it is s t ill a n in divid u a l pr obabilit y dis t r ibu t ion.
Th is d oesn ’t pr odu ce bias in est im a t in g a pa r a m et er , on ly var ia bilit y. For exa m ple
if a pa r t icula r dis t a n ce decay fu n ction im plies t h a t 70% of th e offend er s live wit h in ,
say, 5 miles of th eir comm itted incident s, then 30% don’t live with in 5 miles. In
ot h er wor ds, beca u se t h e da t a a r e in divid u a l level, t h en a dis t a n ce deca y fu n ct ion ,
wheth er estimat ed by a m at hema tical or a kern el density model, is an individua l
pr oba bilit y m odel (i.e., a n a t t em pt t o descr ibe t h e u n der lyin g d is t r ibu t ion of
in divid u a l t r a vel d is t a n ces for jou r n ey t o crim e t r ip s).
10.81