Geographical Profiling

8/13/2019 Geographical Profiling

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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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"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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"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(

Geographical Profiling

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