Team #26031

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Summary

     Faced with serial crimes, we usually estimate the possible location of next crime

by narrowing search area. We build three models to determine the geographical profile

of a suspected serial criminal based on the locations of the existing crimes. Model

One assumes that the crime site only depends on the average distance between the

anchor point and the crime site. To ground this model in reality, we incorporate the

geographic features G, the decay function D and a normalization factor N. Then we

can get the geographical profile by calculating the probability density. Model Two is

Based on the assumption that the choice of crime site depends on ten factors which is

specifically described in Table 5 in this paper. By using analytic hierarchy process

(AHP) to generate the geographical profile. Take into account these two geographical

profiles and the two most likely future crime sites. By using mathematical dynamic

programming method, we further estimate the possible location of next crime to

narrow the search area. To demonstrate how our model works, we apply it to Peter's

case and make a prediction about some uncertainties which will affect the sensitivity

of the program. Both Model One and Model Two have their own strengths and

weaknesses. The former is quite rigorous while it lacks considerations of practical

factors. The latter takes these into account while it is too subjective in application.

Combined these two models with further analysis and actual conditions, our last

method has both good precision and operability. We show that this strategy is not

optimal but can be improved by finding out more links between Model One and

Model Two to get a more comprehensive result with smaller deviation.

Key words:

expected utility

geographic profiling, the probability density, anchor point,

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Executive Summary

      Nowadays, in a serial crime, the spatial distribution of crime sites is arousing the

more and more attention of the criminologists and the geographers. The Serial

criminals, in a spirit of defiance, have endangered public security, gravely infringed

on the citizens' personal safety, lives and property, and are abhorred by the people

across the country. Since the offenders usually have no stable residence, it is very

difficult for the police to find out and arrest them. Therefore, in order to help the

police solve the crime as soon as possible to maintain social security and stability, a

more sophisticated technique is in urgent need to be developed to determine the

“geographical profile” of a suspected serial criminal based on the locations of the

crimes.

      This paper presents three methods, especially a new mathematical method

combining the advantages of the two previous models, which can generate a useful

prediction for law enforcement officers about possible locations of the next crime.

      Based on Bayesian statistical methods, Model One makes explicit connections

between assumptions on offender behavior and the components of the mathematical

model. It also takes into account local geographic features that either influence the

selection of a crime site or influence the selection of an offender’s anchor point.

What’s more, with rigorous inference formulas, this model has both precision and

operability.

      Model Two uses analytic hierarchy process (AHP). It takes full account of a

variety of factors relevant to the crime sites, assisting the police to take measures

adapted to local conditions so as to improve our work.

      The last method is composed from Model One and Model Two. Taken the

expected utility and other practical factors into consideration, it further estimate the

geographic profile generated by the previous two models.

      To conclude, we suggest the policemen put this mathematical method about

geographical profiling into practice for it will be of significant assistance to law

enforcement.

      The technical details are as follows:

      First of all, enhance the consciousness of the public social security and its

improvement in the potential criminal area and inform the local people that a series of

crimes have occurred recently, reminding them to keep vigilant and not to go into

remote areas alone.

      Second, the police departments should focus their activities, geographically

prioritize suspects, and concentrate saturation or directed patrolling efforts in those

zones where the criminal predator is most likely to be active.

      Third, after arresting the criminal, the police need to make experiential analysis

on all these kinds of serial crimes to prevent a similar case. What’s more, the police

departments set up an enhanced intelligence exchange network, especially in the area

which is the next crime site according to our prediction.

      Finally, search the suspect in the predicted criminal location or the likely

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residence of the offender.

     Generally speaking, our method has good maneuverability and practicability.

However, there are various uncertain factors in the situation which cannot be predicted

such as the weather condition and the traffic conditions. Additionally, the

determination and analysis of weight-coefficients on the various factors is very

subjective. As a result, the actual site scene of next crime may be outside of our

geographical profiling. Therefore, the police deployments must be based on the

analysis of local actual condition instead of applying our model blindly.

     Furthermore, our prerequisite that the offender has the only one stable anchor

point differs from the actual conditions. Since the offender is very likely to change his

residence, the police had better search the suspect according to the latest information.

     Maintaining social stability and ensuring the safety of residents will bring well-

being and peace to all mankind. Every one of us should make effort to make our

society become better. Hope that this paper will be of some help to prevent the

criminal behaviors.

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1. Introduction

     Clues derived from the locations connected to violent repeat criminal offenders,

such as serial murderers, arsonists, and rapists, can be of significant assistance to law

enforcement. Such information helps police departments to focus their activities,

geographically prioritize suspects, and to concentrate directed patrolling efforts in

those zones where the criminal offender is most likely to be active. By examining

spatial data connected to a series of crime sites, this methodological model generates a

probability map that indicates the area most likely to be the locations of the next crime.

     This paper presents two mathematical models to illustrate how geographical

analysis of serial crime conducted within a geographic information system can assist

crime investigation. Techniques are illustrated for determining the possible residence

of offenders and for predicting the location of the next crime based on the time and

locations of the existing crimes.

     First, we present a mathematical survey of some of the algorithms that have been

used to solve the geographic profiling problem. The geographic profiling problem is

the problem of constructing an estimate for the location of the anchor point of a serial

offender from the locations of the offender’s crime sites.

     The approach that we develop will make use of at these two different schemes to

generate a geographical profile. What’s more, we develop a third technique to

combine the results of the two previous schemes and generate a useful prediction for

law enforcement officers. The prediction provides some kind of estimate or guidance

about possible locations of the next crime based on the time and locations of the past

crime scenes. Our method will also provide some kind of estimate about how reliable

the estimate will be in a given situation, including appropriate warnings. The

executive summary will provide a broad overview of the potential issues. It will also

provide an overview of our approach and describe situations when it is an appropriate

tool and situations in which it is not an appropriate tool.

     The purpose is to apply geographical analysis to serial crime investigations to

predict the location of future targets and determine offender residence.

1.1 Restatement of the Problem1.1

     In order to indicate the origin of geographical profiling problems, the following

background is worth mentioning.

     In 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a

number of other people to vicious attacks. One of the methods used to narrow the

search for Mr. Sutcliffe was to find a “center of mass” of the locations of the attacks.

In the end, the suspect happened to live in the same town predicted by this technique.

Since that time, a number of more sophisticated techniques have been developed to

determine the “geographical profile” of a suspected serial criminal based on the

locations of the crimes.

     Our team has been asked by a local police agency to develop a method to aid in

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their investigations of serial criminals. The approach that we develop will make use of

at least two different schemes to generate a geographical profile. We also develop a

technique to combine the results of the different schemes and generate a useful

prediction for law enforcement officers. The prediction will provide some kind of

estimate or guidance about possible locations of the next crime based on the time and

locations of the past crime scenes. Our method will also provide some kind of estimate

about how reliable the estimate will be in a given situation, including appropriate

warnings.

1.2 Survey of Previous Research1.2

Existing Methods

     To understand how we might proceed let us begin by adopting some common

notation:

●

A point x will have two components x = ( x1 , x 2 , ) .

 These can be latitude and longitude

● These can be the distances from a pair of reference axes

●

●

The series consists of n crimes at the locations x1 , x2 , … , xn .

The offender’s anchor point will be denoted by z .

Distance between the points x and y will be d ( x, y) .

Existing algorithms begin by first making a choice of distance metric d ; they

●

●

then select a decay function f and construct a hit score function S ( y ) by computing

S ( y) = ∑ f ( d ( xi , y)) = f ( d ( x1, y)) + … + f ( d ( xn , y))

i =1

n

     Regions with a high hit score are considered to be more likely to contain the

offender’s anchor point than regions with a low hit score. In practice, the hit

score S ( y) is not evaluated everywhere, but simply on some rectangular array of

points y jk = ( y1j , yk2 ) for j ∈{1, 2,… , J } and k ∈ {1, 2, …, K } giving us the array of

values S jk = S ( y jk ) [4].

   Rossmo’s method, as described in (Rossmo, 2000, Chapter10) chooses the

Manhattan distance function for d and the decay function

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         ⎧k

         ⎪ dh⎪

f (d ) = ⎨

            kB g − h

         ⎪

         ⎪ (2 B − d ) g

         ⎩

if d < B

if d ≥ B

Other Studies

      In the study of Crime Analyst’s task, Bryan Hill considers that the use of the

“probability grid method” (PGM) can narrow the search as it pertains to tactical crime

analysis in the ArcView Geographic Information Systems (GIS) environment. The

main point of his theory will be that any current statistical method of predicting the

next hit location in a crime series is operationally ineffectual when the suspect covers

a large geographic area. When the analyst combines several statistical methods and

intuitive, logical thought processes into a combined “grid” score, the analysis product

can be made more operationally effective. This new grid surface allows the analyst to

make a better prediction of the next hit in a crime series and is useful in isolating

specific target locations for law enforcement deployment efforts. This easy to apply

“PGM method” allows the analyst to use sound statistical methods, as well as their

experience and knowledge of a crime series to narrow the focus and potential hit area.

      In addition, when the crime series has sufficient suspect information, journey to

crime analysis using the CrimeStat software can be used to provide investigators with

a list of probable offenders from law enforcement records available to the analyst [4].

2. Model Overview

1、Model OneModel

    In Model One, we assume that the crime sites depend on the distance between the

anchor point and the crime site. Generate a hot zone model according to probability

formula, then combine the geographic features G with the decay function D and add a

normalization factor N into the model .At last, we can get the “geographical profile”.

 Model2、Model Two

     In Model Two, we take account of other factors relevant to crime site location, we

have studied various literature to summary two aspects denoted by U( Utility from

crimes) and P(the probability of success) specifically including the following ten

factors: the responding speed of the police, public security situation, resistances’

diathesis, density of registered inhabitants, the advantageous position, the number of

offenses, the distance from the anchor point of the offender, the time required for

committing the crime, number of target persons, offender’s mental satisfaction from

the crime.

      Then, we use Analytic Hierarchy Process (AHP) to get weighted factors. Finally,

according to the formula we can draw E of some areas. A wide range of criminal area

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can then be divided into several small areas. Then, we give the proper scores for each

small area according to the actual condition. When the area is small enough, we can

get the geographical profile. Sites with higher scores have high probability to be crime

areas.

 The3、The two models can be synthesized as one method

     For their different emphasis (The theory of Model One is quite rigorous, but it

ignores the factors such as the geographic features and criminal motivation while

these factors is important for selecting criminal crime sites. On the other hand, Model

Two takes these into account, while it is too subjective in practical application which

may easily cause the deviation), the geographical profiles we get from two models

must be different. By using mathematical dynamic programming method to establish

the model, this method explains the common rules of the offender’s choice on the

crime sites. Therefore, when we get several geographical profiles from the previous

two models, we can use this method to predict the next offence site.

     To sum up，we can generate the geographical profile in the investigations of serial

criminals, providing some kind of estimate or guidance about possible locations of the

next crime and narrow the search.

3. Model One

Symbols:

symbols

   P ( x)

     x

     z

     a

     D

     G

     N

meaning

probability density function

the crime sites

the location of the offender’s anchor point

The average distance that the offender is willing to travel to offend.

the effect of distance decay

the geographic features

a normalization factor

Hypothesis：

1、 We assume that our offender chooses potential locations to commit crimes

  randomly according to some unknown probability density function P .

2、 We assume that P depends upon z and a.

3、 We suppose that that the values of the anchor point z and average offense

  distance a are unknown, but the form of the distribution P is known.

4、 The offender has only one residence which is unchanged.

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3.1 A Mathematical Approach

      In order to looking for an appropriate model, we start with the simplest possible

situation. Since we know nothing about the offender, we assume that our offender

chooses potential locations to commit crimes randomly according to some unknown

probability density function P . For any geographic region R , the probability that our

offender will choose a crime location in R can be found by adding up the values of

 P in R , giving us the probability

∫∫

R

P(x) d ( x)1 d ( x)

2

Upon what sorts of variables should the probability density P( x) depend?

     The fundamental assumption of geographic profiling is that the choice of an

offender’s target locations is influenced by the location of the offender’s anchor point

z. Therefore, we first assume that P depends upon z. (Provided that the offender has

a single anchor point and that it is stable during the crime series.) A second important

factor is the distance the offender is willing to travel to commit a crime from their

anchor point.(Different offender’s have different levels of mobility- an offender will

need to travel farther to commit some types of crimes than others ) Let a denote the

average distance that the offender is willing to travel to offend. So a varies between

offenders and crime types [4].

     Let us suppose that that the values of the anchor point z and average offense

distance a are unknown, but the form of the distribution P is known. Then the

problem can be stated mathematically as:

Given a sample x1 ,… , xn (the crime sites) from the distribution P( x z, a) with

parameters z and a to determine the best way to estimate the parameter z (the anchor

point).

      One approach of this mathematical problem is the theory of maximum likelihood.

 First, construct the likelihood function [4]:

n

L(y, a) = ∏ P( xi y , a ) = P( x1 y, a) … P( xn y, a)

i =1

    In order to get the best choice of z, make the likelihood as large as possible by

maximizing the log-likelihood:

λ (y, a) = ∑ lnP( xi y, a) = lnP( x1 y, a) + … + ln P( xn y, a)

i =1

n

     This approach is rigorous; however, it is unsuitable as simple point estimates for

the offender’s anchor point are not operationally useful. Therefore, we continue our

analysis by using Bayes Theorem[4].

     Bayes Theorem then implies:

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P(z, a x) =

P(z, a x)π ( z, a)

      p( x)

（1）

Here π ( z , a) is the prior distribution.

     It represents our knowledge of the probability density for the anchor point z and

the average offense distance a before we incorporate information about the crime.

     If we assume that the choice of anchor point is independent of the average

offense distance, we can write:

π ( z , a) = H ( z )π ( a)

(2)

H ( z ) is the prior distribution of anchor points, and π (a) is the prior distribution

of average offense distances.

     We will assume that the offender’s choices of crime sites are mutually

independent, so that

P( x1 , …, xn z, a) = P( x1 z,a) … P( xn z,a)

Suppose that an unknown offender has committed crimes at x1 ,… , xn , and that

(3)

The offender has a unique stable anchor point z.

The offender chooses targets to offend according to the probability density

P( x z, a) where a is the average distance the offender is willing to travel.

The target locations in the series are chosen independently.

The prior distribution of anchor points is H ( z ) , the prior distribution of the

average offense distance is π ( a ) and these are independent of one another.

      Then the probability density that the offender has anchor point at the location z

satisfies

P( z x1,…, xn ) ∝ ∫ P x1 z, a） P xn z, a）H ( z)π ( a) da（…（

(4)

3.2 Simple Models for Offender Behavior

     What’s more, we need to be able to construct reasonable choices for our model

of offender behavior, if our fundamental mathematical result is to have any practical

or investigative value.

     One simple model is to assume that the offender chooses a target location based

only on the Euclidean distance from the offender’s anchor point to the offense

location and that this distribution is normal. In this case we obtain

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P( x z,a ) =

 1⎛π⎞

    exp ⎜ − 2 ( x − z) 2 ⎟

4a2⎝ 4a⎠

(5)

    If we make the prior assumptions that the average offense distance and the

anchor point are unchanged, and the offender commits n crimes at the crime site

locations x1 , … , xn , then

                    1⎛π

P( z x1 ,… , xn ) =( 2 ) n exp ⎜ − 2

                   4a⎝ 4a

n

∑

i =1

            ⎞

( xi − z )2 ⎟

            ⎠

      We see that the anchor point probability distribution is just a product of normal

distributions; the maximum likelihood estimate for the anchor point is simply the

mean center of the crime site locations. We also mention that in this model of

offender behavior, this is also the mode of the posterior anchor point probability

distribution [4].

      Another reasonable model for offender behavior is to assume that the offender

still chooses a target location based only on the Euclidean distance from the offense

location to the offender’s anchor point, but now the distribution is a negative

exponential so that

P( x z,a ) =

2⎛2⎞

    exp ⎜ − x − z ⎟

π a2⎝a⎠

(6)

     Once again, because of our prior assumptions that the average offense distance

and the anchor point are unchanged, and the offender commits n crimes at the crime

site locations

x1 ,… , xn

, then we have

                    2n⎛ 2n⎞

P( z x1 ,… , xn ) =( 2 ) exp ⎜ − ∑ xi − z ⎟

                   πa⎝ a i =1⎠

      We see that this is just a product of negative exponentials centered at each crime

site. Further, the corresponding maximum likelihood estimate for the offender’s

anchor point is simply the center of minimum distance for the crime series locations [4].

      This preceding analysis was predicated on the prior assumptions that the average

offense distance and it is known in advance. Similarly, the existing methods

mentioned in this paper all rely on decay functions f with one or more parameters

which also need to be determined in advance. Unlike those methods, our method does

not require that we make a choice for the parameter in advance.

3.3 Realistic Models for Offender Behaviorffender

What would a more realistic model for offender behavior look like?

Consider a model in the form：

P( x z, a) = D( d( x, z), a) G( x) N ( z)

(7)

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▲ D models the effect of distance decay using the distance metric d(x, z)

1、We can specify a normal decay, so that

D(d , a ) =

 1⎛π⎞

    exp ⎜ − 2 d 2 ⎟

4a2⎝ 4a⎠

2、We can specify a negative exponential decay, so that

D( d , a ) =

2⎛2 ⎞

    exp ⎜ − d ⎟

π a2⎝a⎠

 Any choice can be made for the distance metric (Euclidean, Manhattan, et.al)

      ▲ G models the geographic features that influence crime site selection

 High values for G(x) indicate that x is a likely target for typical offenders;

 Low values for G(x) indicate that x is a less likely target

      ▲ N is a normalization factor, required to ensure that P is a probability

distribution

N ( z) =

1

∫∫ D(d ( y, z ), a )G(y) d ( y) d ( y)

1

2

      N is completely determined by the choices for D and G.

      G models the geographic features that influence crime site selection, with high

values indicating the location was more likely to be targeted by an offender.

      Then, how can we calculate G?

      Use available geographic and demographic data and the correlations between

crime rates and these variables that have already been published to construct an

appropriate choice for G(x).

      Different crime types have different etiologies; in particular their relationship to

the local geographic and demographic backcloth depends strongly on the particular

type of crime. This would limit the method to only those crimes where this

relationship has been well studied.

      Some crimes can only occur at certain, well-known locations, which are known

to law enforcement For example, gas station robberies, ATM robberies, bank

robberies; liquor store robberies .This does not apply to all crime types- e.g. street

robberies, vehicle thefts.

      We can assume that historical crime patterns are good predictors of the

likelihood that a particular location will be the site of a crime.

      Suppose that historical crimes have occurred at the locations c1, c2. . . cn.

Choose a kernel density function

K y λ）（

     λ is the bandwidth of the kernel

density function.

Figure 1

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Calculate

N

G (x) = ∑ K (x- ci λ )

i =1

(8)

     The bandwidth λ can be e.g. the mean nearest neighbor distance.

     We have assumed:

     Each offender has a unique, well-defined anchor point that is stable throughout

the crime series

     The function H(z) represents our prior knowledge of the distribution of anchor

points before we incorporate information about the crime series.

     Suppose that anchor points are residences- can we estimate H(z)?

     ●Population density information is available from the U.S. Census at the block

level, sorted by age, sex, and race/ethnic group.

        1、We can use available demographic information about the offender

Nblocks

2、Set

H (z) =

∑ = p K (z − q

i

i =1

i

Ai )

3、Here block i has population pi, center qi, and area Ai.

●Distribution of residences of past offenders can be used.

Calculate H ( z ) using the same techniques used to calculate G(x).

3.4 Future Offense Predictionffense

Given a series of crimes at the locations x1 , … , xn committed by a single serial

offender, estimate the probability density P( xnext x1 , …, xn ) , that Xnext will be the

location of the next offense [4]. The Bayesian approach to this problem is to calculate

the posterior predictive distribution:

P( xnext x1 , …, xn ) = ∫∫∫ P( xnext z , a) P( z, a x1 , …, xn ) d ( z)1 d ( z) 2 d( a)

We can use the method above to simplify, and so obtain the expression:

P( xnext x1 , …, xn ) ∝ ∫∫∫ P( xnext z , a) P( z, a x1 , …, xn ) H ( z)π ( a) d( z)1 d( z) 2 d( a)

    This approach makes the same independence assumptions about offender

behavior as our fundamental result.

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4. Model Two

Symbols:

symbols

w

E

rB1 ，…rB10

mB1 ,…mB10

P

U

Meaning

Weight

Total score

Each score

Various factors

The probability of success

The utility from crimes

Assumption：

The selection for crime sites depends on ten factors that demonstrated in Table 5 in the

paper.

4.1 Analysis of models

      The selection for crime sites depends on various factors instead of only one factor

that the distance from the anchor point to the crime site. Therefore, Combine a large

number of previous studies and related papers with our own thinking, we have

summed up 10 key factors and a grading system about how to make the decisions on

where the criminal locations will be (see Table 5), which are listed as follows: the

responding speed of the police, public security situation, resistances’ diathesis, density

of registered inhabitants, the advantageous position, the number of offenses, the

distance from the anchor point of the offender, the time required for committing the

crime, number of target persons, offender’s mental satisfaction from the crime.

Classify these 10 index factors into two categories P and U, based on the actual

situation. Let the former six factors belong to P and the latter four belong to U.

      Next, we use Analytic hierarchy process (AHP) to get weighted factors (Table 4).

Then we give the proper scores for each small area according to the actual condition.

(Table 5). (There are some data which is difficult for us to achieve but they may be

easily achieved by the local police .）Finally, we can work out the total score of E:

E =w

P1

× rP1 + … + w P 6 × rP 6 + wU 1 × rU 1 + … + wU 4 × rU 4

4.2 Use AHP to get the weighted factors

    Then we can get a comparison matrix as follows:

    The probability of success (P) is as important as the expected utility (U), so

EPU= 1，EUP=1

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  ⎡1 1⎤

E=⎢⎥

  ⎣1 1⎦

Ui and the judgment matrix of U :

    ⎡1

    ⎢1

    ⎢

    ⎢7

U = ⎢1

    ⎢

    ⎢5

    ⎢1

    ⎢

    ⎣2

7

1

3

7

5

1

3

1

5

2⎤

1⎥

 ⎥

7⎥

1⎥

 ⎥

5⎥

 ⎥

1⎥

 ⎦

5

4

1

2

1

6

5

4

2

1

7

1

6

1

1

3

6⎤

 ⎥

3⎥

 ⎥

1⎥

 ⎥

5⎥

1⎥

 ⎥

5⎥

 ⎥

3⎥

 ⎥

1⎥

 ⎥

 ⎦

Pi and the judgment matrix of P:

    ⎡1

    ⎢1

    ⎢

    ⎢3

    ⎢1

    ⎢

    ⎢7

P = ⎢1

    ⎢

    ⎢5

    ⎢1

    ⎢4

    ⎢

    ⎢1

    ⎢6⎣

3

1

1

6

1

4

1

2

1

3

7

6

1

2

7

5

     According to all levels of comparison matrix, let’s obtain its maximum of

eigenvalue, coincidence indicator and the weight corresponding to all factors by using

Matlab. The results are shown in the table below:

comparisonMaximumcoincidencethe weight of evaluation indexes

matrixw

                  eigenvalue λ maxindicator C I

E

λ

λ

max

=2

0

[0.5000; 0.5000]

U

max

=4.1341

0.0447

0.1127

[0.4971;0.0495;0.1016;0.3519]

[0.4319;0.2105;0.0298;

0.0448;0.1798;0.1033]

P

λ max = 6.5633

                                    Table 1

The coincidence indicator can be computed by the formula

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CI=

λ

max

−n

n −1

The results are shown in the form below. Find the average random coincidence

indicator R I . The average random coincidence indicator (exponent number is within

15) is as follows:

exponent 1

number

             0

2

0

10

1.49

3

0.52

11

1.52

4

0.89

12

1.54

Table 2

Based on

   = C I , we can work out coincidence ratios:

CR

5

1.12

13

1.56

6

1.26

14

1.58

7

1.36

15

1.59

8

1.41

R

I

exponent 9

number

         1.46

R

I

R

I

C

Rw

C

RU

C

RP

0

0.0502

Table 3

0.0894

C

Rw

、 C RU 、 C RP are all less than 0.1 within the scope of consistency, which shows

that the index factors we set meet the requirements, that is, they have some credibility

within the error range.

      Combination weight for all levels. Calculate the weight of the index factors Ui, Pi.

For example,

w

U1

=0.4971 × 0.5000=0.24855

( i=1, 2, …6) ,and

In the same way, we can work out w Pi

See Table 4:

w

Ui

( i=1, 2, …4).

w

w

P1

w

w

P2

w

w

P3

w

w

P4

w

P5

w

P6

0.21595

U1

0.10525

U2

0.0149

U3

0.0224

U4

0.0899

0.05165

0.24855

0.02475

0.0508

Table 4

0.17595

Team # 6539

Page 16 of 26

4.3 Scoring system for factors

The

responding

speed of the

police

      ⎧1

      ⎪m⎪

rP1 = ⎨1 − P1

      ⎪ 10

      ⎪0

      ⎩

mP1 = 0

mP1 stands for the

0< mP1 < 10 responding speed of the

            police. 10 represents very

mP1 =10

            fast, 6 represents fast,3

            represents slow, 0 represents

            very slow.

public

security

situation

       ⎧1

       ⎪m⎪

rP 2 = ⎨1 − P 2

           50

       ⎪

       ⎪0

       ⎩

mP 2 ≥ 50

mP 2 stands for the region's

0< mP 2 < 50 monthly criminal records. If

             criminal record is 0, then the

 mP 2 = 0

             public security situation is

             good; if Criminal record is

             larger than 50, then the

             public security situation is

             bad.

Resistances’

diathesis

The

probability

of success

density of

registered

inhabitants

       ⎧1

       ⎪m⎪

rP 3 = ⎨1 − P 3

          10

       ⎪

       ⎪0

       ⎩

       ⎧1

       ⎪m⎪

rP 4 = ⎨ P 4

       ⎪ 10

       ⎪0

       ⎩

mP 3 = 0

mP 3 stands for

resistances’

0<mP 3 < 10 diathesis,10 stands for very

            good, 0 stands for very bad.

mP 3 = 10

mP 4 = 10

0<mP 4 < 10

mP 4 stands for density of

registered inhabitants.10 for

the highest, 6 for the

general, 3 for few people in

that area and 0 for no one

there.

mP 4 = 0

the

advantageo

us position

       ⎧1

       ⎪m⎪

rP 5 = ⎨ P 5

       ⎪ 10

       ⎪0

       ⎩

       ⎧1

       ⎪m⎪

rP 6 = ⎨ P 6

       ⎪3

       ⎪0

       ⎩

mP 5 = 10

0<mP 5 < 10

mP 5 stands for the extent of

favorableness. 10 for the

quite favorable, 6 for general,

 3 for not going well, 0 for

not executable.

mP 5 = 0

The number

of crimes

mP 6 ≥ 3

0<mP 6 < 3

mP 6 stands for the number of

crimes, greater than 3 times,

for the very skilled and

unskilled times for 0

mP 6 = 0

Team # 6539

Page 17 of 26

the distance

from the

anchor

point of the

offender

      ⎧1

      ⎪m⎪

ru1 = ⎨ u1

      ⎪ 10

      ⎪0

      ⎩

mu1 = 10

0< mu1 < 10

mu1 stands for the distance

from the anchor point of the

offender. Advantageous

distance for 10,

disadvantageous distance for

0.

mu1 = 0

The time

required for

committing

the crimes

mu 2 stands for the time

       ⎧1

       ⎪m⎪

ru 2 = ⎨ u 2

       ⎪ 10

       ⎪0

       ⎩

       ⎧1

       ⎪m⎪

ru 3 = ⎨ u 3

       ⎪ 10

       ⎪0

       ⎩

       ⎧1

       ⎪m⎪

ru 4 = ⎨ u 4

       ⎪ 10

       ⎪0

       ⎩

mu 2 =10

0< mu 2 < 10

mu 2 = 0

required for committing the

crimes. Long time for 0, less

longer time for 3, normal

long for 6, short time for 8,

very short time for 10

Utility from

crimes

Number of

target

persons

mu 3 =10

0<mu 3 < 10

mu 3 stands for the intensity

of the target population. Very

intensive for 10, intensive

for 8, generally for 6, less-

intensive 3, no goals for 0

mu 3 = 0

Offender’s

mental

satisfaction

mu 4 =10

0< mu 4 < 10

mu 4 stands for the

offender’s mental

satisfaction in that criminal

location. Very satisfied for

10, satisfied for 8, general

for 6, not very satisfied for

3,not satisfied at all for 0

mu 4 = 0

Table 5

4.4 Results

    We can get a wide range of criminal area and divide it into several small regions.

Next, we give the proper scores for each small region according to the actual condition.

Finally, using the formula

E =w

P1

× rP1 + … + w P 6 × rP 6 + wU 1 × rU 1 + … + wU 4 × rU 4

to calculate the E of each area.

     Sites with higher scores have high probability to be crime areas. The police

should enhance the patrols in these areas, and people should strengthen the awareness

of safety.

Team # 6539

Page 18 of 26

5. The synthesized method

     Up until now, we have already given two different models to show the geographic

profile of next possible crime site. But both these two models have advantages and

disadvantages. Model One allows us to find possible locations of the next crime as

well as the offender’s resistance based on only the time and locations of the past crime

scenes, but it ignores some critical factors related to the crime, such as the

geographical environment and the motive for the crime and other factors. Model Two

takes 10 factors which affects the selection of crime sites into consideration and

qualify them to work out the geographic profile, but it is too subjective and easily leads

to deflection when put into practice. Because of their different emphases, the two

models will not give the same geographical profile. Assume we work out n

geographical profiles through Model One and m geographical profiles through Model

Two. Apparently these geographical profiles will not be the same. Therefore, we come

up with the following new method which can combine the two models together.

Symbols:

  symbols

  U

  s

  I

P （x）

  E

     meaning

Total income

the number of crimes

income for the offender

the probability of success for each crime

expected utility

Hypothesis：

1、we assume that the net income in of each crime occurred in the same region is

 unchanged, that is, for any i, j,

2、we assume in the same area, each time the probability of successfully committing

a crime is also unchanged, i.e. Pi1 = Pj1 , Pi 2 = Pj 2

      With the method of dynamic programming, two spatial variables, the expected

utility and the probability of success for each offense, are used to model the criminal’s

location choices. A criminal usually commits his first offence in the district which has

the highest probability of success but a lowest expected utility. If an area has both

higher expected utility and a higher probability of success, the criminal will commit

all his offences in this place. The model also suggests that crime prevention measures

should be adopted in the local conditions. “Covering” measures, such as patrolling,

should be taken in the poor residential districts or delinquency districts, while more

sophisticated and advanced measures should be introduced in the richer districts or the

districts where career criminals haunt.

Team # 6539

Page 19 of 26

5.1 The Foundation of Model

First, let the utility function for a certain criminal be given:

U = U 1 ( I1 ) + U 2 ( I 2 ) + … + U S ( I S )

(8)

      In this function, s indicates the number of crimes; Ii (i=1,2,…s) indicates the net

income for the ith time offender.

      Then, denote Pi as the probability of success for each crime (i.e., the probability

of not being caught). If the criminal is caught when he commits the hth crime, he will

lose Ih. Then we can get the total net income

I1 + I 2 + … + I h −1

     Meanwhile, we get the expected utility when the cumulative number of crime

reaches s.

E = U1 P (1 − P2 ) + (U1 + U 2 ) P P2 (1 − P ) + … + (U1 + U 2 + … + U s ) P P2 … Ps1131

s

h =1

h

i =1

h

= ∑ ( ∑ U i )∏ Pi (1 − Ph +1 )

i =1

(9)

For here Ps +1 ≡ 0 so we can change Function (9) into

E = U1 ∑ ∏ Pi (1 − Ph +1 ) + U 2 ∑ ∏ P (1 − P +1 ) + … + U s ∏ Pihi

h =1 i =1

h =2 i =1

i =1

s

h

s

h

s

(10)

For:

s

h

1 − P + ∑∏ Pi (1 − Ph +1 ) ≡ 11

h =1 i =1

So in Equation (10), the coefficient of U 1 can be abbreviated as P , and the1

coefficient of U 2 can be simplified as P P2 , the coefficient of U h can be simplified as1

h

∏P

i

i =1

In this way, Equation (9) can be further abbreviated as:

E = U1 P + U 2 P P2 + … + U s P P2 … Ps ≡ ∑ (U h )∏ Pi111

h =1

i =1

s

h

(11)

     Now assume that there are two potential criminal regions in a city, which are

labeled by superscripts l and 2. For the sake of simplicity, further assume that the net

income in of each crime occurred in the same region is unchanged, that is, for any i, j,

Ii1 = I j1 , Ii 2 = I j 2

Team # 6539

Page 20 of 26

But I 1 does not necessarily equal I 2

Meanwhile, assume in the same area, each time the probability of successfully

committing a crime is also unchanged, i.e. Pi1 = Pj1 , Pi 2 = Pj 2

Therefore, if the offender intends to commit s crimes, he needs to make location

choices for 2 s times.

      The following dynamic programming method reveals the optimal solution of the

offender’s location choice of committing the crime. Here assume that during the

planning period, the offender implement all his crimes. What’s more, he determines all

the optimal locations one by one from the back to the front. The planning period can

be any time that a crime will happen.

      Suppose the offender has already determined the locations of the crimes for the

first s − 1 times and attempts to optimize the location of sth crime.

      Denote the expected utility for committing sth crime in district 1 and district 2 as

E1 = U 1P1 + U 1P1P2 + … + U s1P1P2 … Ps1

E1 = U 1P1 + U 2 P1P2 + … + U s2 P1P2 … Ps2

(12)

(13)

     The first s-1 terms on the right side in Equation (12) and (12) has no region label,

because of the assumption of they are the same in the two equations mentioned above

and the only difference exits in the sth term. Therefore, the difference between the

expecting utilities of committing sth crime in district1 and district 2 is

s −1

∆E = E − E = (U P − U P )∏ Pi

i =1

1

2

1

s1

2

s2

(14)

      Equation (14) indicates that whatever the probability of success P or the

probability of being caught (1-P) is, the offender will choose the place which has the

highest expected profit to commit his last crime. When the utility function is monotone,

 and the offender does not care about his location of crime activity, the results

mentioned above also suggest that the offender will commit sth crime in the location

where the expected utility is highest[3].

      Generally, if the offender has already determined the crime locations of the first

time, the second time, the (k-1)th time, the (k+1)th time, …the sth time, the expected

utility of committing the kth crime in district1 and district2 are as follows:

k −1

1