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Pro8ect Participants
Towson University Applied MathematicsLaboratory
Undergrad(ate research pro8ects inapplied mathematics.
o(nded in 9,:/
1ational )nstit(te of *(stice$pecial thanks to $tanley ;rickson
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Geographic Profiling
The Question:
Given a series of linked crimes committed bythe same offender' can we make predictionsabo(t the anchor point of the offender
The anchor point can be a place ofresidence' a place of work' or some othercommonly visited location.
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Geographic Profiling
(r E(estion is operational.
This places limitations on available data.
;Fample
A series of , linked vehicle thefts in&altimore "o(nty
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;Fample
ADDRESS DATE_FROM TIME DATE_TO TIME REMARKS
918 M 01/18/2003 0800 01/18/2003 0810 VEHICLE IS 01 TOYT CAMRY,LEFT VEH RUNNING
1518 L 01/22/2003 0700 01/22/2003 072 VEHICLE IS 99 HOND ACCORDSTL!REC, """#/M
$AIR,DRIVING MAROON ACCORD"
731 CC 01/22/2003 07 01/22/2003 07% VEHICLE IS 02 CHEV MALI#U
STL!REC
1527 K 01/27/2003 110 01/27/2003 110 VEHICLE IS 97 MERC COUGAR,
LEFT VEH RUNNING
151 G 01/29/2003 0901 01/29/2003 0901 VEHICLE IS 99 MITS
DIAMONTE, LEFT VEH RUNNING
115 K 01/29/2003 1155 01/29/2003 115% VEHICLE IS 00 TOYT RUNNER
STL!REC, &' ARREST NFI
593 R 12/31/2003 0%32 12/31/2003 0%32 VEHICLE IS 92 #M( 525,
(ARMING U$ VEH
127 G 02/17/200 0820 02/17/200 0830 VEHICLE IS 00 HOND ACCORD,
(ARMING VEH
9 S 05/15/200 0210 05/15/200 0%00 VEHICLE IS 0 SU)I ENDORO
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Distance
;(clidean
Manhattan
$treet grid
d1x, y =x1y1x2y2
d2x, y =x1y12x2y22
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$patial Distrib(tion $trategies
"entroid%
Crime locations
Average
Average
Anchor Point
centroid=1n i=1n
x i
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$patial Distrib(tion $trategies
"enter of minim(m distance% is the val(eof that minimi?es
Crime locations
Distance sum = 10.63
Distance sum = 9.94
Smallest possible sum!
Anchor Point
cmdy
D y =i=1
n
dx i, y
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$patial Distrib(tion $trategies
"ircle Method%
Anchor point contained in the circle whose
diameter are the two crimes that arefarthest apart.Crime locations
Anchor Point
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Probability Distrib(tion $trategies
The anchor point is located in a region with ahigh hit scoreH.
The hit score has the form
where are the crime locations and is adecay f(nction and is a distance.
Sy=i=1
n
fdy,xi
Sy
= f dz,x1
fdz,x2
fdz,xn
xid
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Probability Distrib(tion $trategies
Linear%
f d=ABd
Hit Score
Crime Locations
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>ossmo
Manhattan distance metric.
Decay f(nction
The constants and are empirically
defined
f d={k
dh
if dB
k Bgh
2Bdg if dB
k , g , h B
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>ossmo
B=1h=2
g=3
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"anter' "offey' #(ntley I Missen
;(clidean distance
Decay f(nctions
fd=A ed
fd=
{ 0 if dA ,
B if AdBCe
dif dB .
,
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Dragnet
A=1=1
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Levine
;(clidean distance
Decay f(nctions
Linear1egativeeFponential
1ormal
Lognormal
fd=ABd
fd=A ed
f d= A
2 S2 exp [
dd2
2S2 ]
fd= A
d2S2
exp[ln dd2
2S2 ]
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"rime$tat
rom Levine
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"rime$tat
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$hortcomings
These techniE(es are all ad hoc.
Chat is their theoretical 8(stification
Chat ass(mptions are being made abo(tcriminal behavior
Chat mathematical ass(mptions are being
made#ow do yo( choose one method overanother
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$hortcomings
The conveF h(ll effect%
The anchor point always occ(rs inside the
conveF h(ll of the crime locations.Crime locations
Convex Hull
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$hortcomings
#ow do yo( add in local information
#ow co(ld yo( incorporate socio-
economic variables into the model$nook' Individual differences in distance travelled by
serial burglarsMalc?ewski' Poet? I )ann(??i' Spatial analysis of
residential burglaries in London, Ontario&ernasco I 1ie(wbeerta' How do residential burglarsselect target areas?
sborn I Tseloni, The distribution of householdproperty crimes
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A 1ew Approach
)n previo(s methods' the (nknown E(antitywas%
The anchor point
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A 1ew Approach
Let be the density f(nction for theprobability that an offender with anchor point
commits a crime at location .
This distrib(tion is o(r new (nknown.
This has criminological significance.
)n partic(lar' ass(mptions abo(t theform of are eE(ivalent toass(mptions abo(t the offender!sbehavior.
Px;z
z x
Px;z
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The Mathematics
Given crimes located at themaximum lielihood estimatefor the anchorpoint is the val(e of that maFimi?es
or eE(ivalently' the val(e that maFimi?es
x1,x2,,xn
mle yLy =
i=1
n
Px i, y
=Px1,yPx2,yPxn,y
y=i=1
n
lnPx i, y
=lnPx1,ylnPx2,y lnPxn, y
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>elation to
$patial Distrib(tion $trategies)f we make the ass(mption that offenderschoose target locations based only on adistance decay f(nction in normal form' then
The maFim(m likelihood estimate for theanchor point is the centroid.
Px;z= 1
22exp[xz
2
22 ]
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>elation to
$patial Distrib(tion $trategies)f we make the ass(mption that offenderschoose target locations based only on adistance decay f(nction in eFponentiallydecaying form' then
The maFim(m likelihood estimate for theanchor point is the center of minim(mdistance.
Px;z= 1
22exp [xz2]
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>elation to
Probability Distance $trategiesChat is the log likelihood f(nction
This is the hit score provided we (se;(clidean distance and the linear decay
for
y =
i=1
n
[ln 2
2
x iy
]Sy f d=ABd
A=ln 22B=1 /
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Parameters
The maFim(m likelihood techniE(e does notreE(ire a prioriestimates for parametersother than the anchor point.
The same process that determines the bestchoice of also determines the best choiceof .
Px;z,= 1
22exp [xz
2
22 ]z
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&etter Models
Ce have recapt(red the res(lts of eFistingtechniE(es by choosingappropriately.
These choices of are not veryrealistic.
$pace is homogeneo(s and crimes are
eE(i-distrib(ted.
$pace is infinite.
Decay f(nctions were chosen arbitrarily.
Px;z
Px;z
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&etter Models
(r framework allows for better choices of.
"onsider
Px;z
Px;z=D dx,zG x Nz
Geographic
factors
NormalizationDistance Decay
(Dispersion Kernel)
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The $implest "ase
$(ppose we have information abo(t crimescommitted by the offender only for a portionof the region.
W
E
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The $implest "ase
>egions
% *(risdiction
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The $implest "ase
Ce set
Ce choose an appropriate decay f(nction
The reE(ired normali?ation f(nction is
G x={1 x0 x
D xz=exp [xz2
22 ]
Nx ;z =
[
exp
yz
2
2
2
dy
1dy
2
]
1
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The $implest "ase
(r estimate of the anchor point is thechoice of that maFimi?es
expi=1n x iy
2
22
[exp
y222
d1
d2
]n
mley
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The $implest "ase
(r st(dents wrote code to implement thismethod last year' and tested it on real crimedata from &altimore "o(nty.
Ce (sed Green!s theorem to convert thedo(ble integral to a line integral.
&altimore co(nty was simply a polygonwith +,/: vertices.
exp
y
2
22
d1 d2=
!
2
yexp
y
2
e
rn ds{ z0 z
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The $implest "ase
To calc(late the maFim(m' we (sed the&G$ method.
$earch in the direction where
or the 9-D optimi?ation we (sed thebisection method.
Dn " fynDn1=D n1g
TD ng
dTg dd
T
dTg
Dn gdTgd TDnd
Tg
d=yn 1y
ng=" fyn 1" fyn
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$ample >es(ltsBaltimore CountyVehicle Theft
Predicted Anchor PointOffender's Home
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&etter Models
This is 8(st a modification of the centroidmethod that acco(nts for possibly missingcrimes o(tside the 8(risdiction.
"learly' better models are needed.
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&etter Models
>ecall o(r ansat?
Chat wo(ld be a better choice of
Chat wo(ld be a better choice of
Px;z=D dx,zG x Nz
D
G
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Distance Decay
rom Levine
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Distance Decay
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Distance Decay
$(ppose that each offender has a decayf(nction where variesamong offenders according to the distrib(tion
.
Then if we look at the decay f(nction for alloffenders' we obtain the aggregate
distrib(tion
fd ; 0,#
$
Fd=%0
#
fd ; $ d
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Distance Decay
f d= A
d2S2
exp [ ln dd2
2 S2 ]
A=d=0.1
$caling Parameters$hape Parameters
0.51
2
3
4&=2 S2
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Distance Decay
1 2 3 4
0.2
0.4
0.6
0.8
;ach offender has a lognormal decay f(nctionThe offender!s shape parameter has a lognormal decay
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1 2 3 4
0.2
0.4
0.6
0.8
Distance Decay
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Distance Decay
)s this real' or an artifact
#ow do we determine the bestH choice of
decay f(nctionThis needs to be determined in advance.
Cill it vary depending on
crime typelocal geography
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Geography
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Geography
1o calibration is reE(ired if is calc(latedin this fashion.
An analyst can determine what historicaldata sho(ld be (sed to generate thegeographic target density f(nction.
Different crime types will necessarily
generate different f(nctions .
G x
G x
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$trengths of this ramework
All of the ass(mptions on criminal behaviorare made in the open.
They can be challenged' tested' disc(ssedand compared.
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$trengths
The framework is eFtensible.
Bastly different sit(ations can be modelled
by making different choices for the formand str(ct(re of .
e!g!ang(lar dependence' barriers.
The framework is otherwise agnostic abo(tthe crime seriesK all of the relevantinformation m(st be encoded in .
Px;z
Px;z
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$trengths
This framework is mathematically rigoro(s.
There are mathematical and criminological
meanings to the maFim(m likelihoodestimate .mle
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Ceaknesses of this ramework
G)G
The method is only as acc(rate as the
acc(racy of the choice of .)t is (nclear what the right choice is for
;ven with the simplifying ass(mption that
this is diffic(lt.
Px;zPx;z
Px;z=D dx,zG x Nz
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Ceaknesses
There is no simple closed mathematical formfor .
>elatively compleF techniE(es arereE(ired to estimate even for simplechoices of .
The error analysis for maFim(m likelihood
estimators is delicate when the n(mber ofdata points is small.
mle
mlePx;z
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Ceaknesses
The framework ass(mes that crime sites areindependent' identically distrib(ted randomvariables.
This is probably false in general
This sho(ld be a solvable problem tho(gh...
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Ceaknesses
Ce only prod(ce the point estimate of .
Law enforcement agencies do not want
4 Marks the $potH.A search area' rather than a point estimateis far preferable.
This sho(ld be possible with some &ayesiananalysis
mle
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(estions
"ontact information%
Dr. Mike !Leary
Director' Applied Mathematics LaboratoryTowson University
Towson' MD +9+2+
J9/-0/J-0J20
molearyNtowson.ed(
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