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Geographic Proﬁling: Nowhere To Hide

Summary

The study of serial crimes prediction has been a fruitful area of mathematical

research for decades. Here we attempt to analyze and model the serial crimes

prediction based on the time and locations of the past crime scenes.

Firstly, we divide the serial crimes prediction problem into two small prob-

lems: one is to develop different schemes to generate a geographical proﬁle and

then develop a technique to combine the results; the other is to generate a use-

ful prediction of the location of the next crime. In addition, we study the Peter

Sutcliffe’s murder case to illustrate every step of our solution.

For the ﬁrst problem, based on the least effort principle, we build the Cen-

ter of Minimum Distance Model and obtain the possible residence area of the

criminal. Then based on Bayes’ Theorem and Rayleigh distribution function,

we build the Bayesian Model and get another possible residence area. After

that, we design a technique of using a minimum circle to cover the two areas to

combine the two schemes.

For the second problem, using Rayleigh distribution function, we obtain a

preliminary probability distribution of the crime site based on the residence de-

termined in the ﬁrst problem . Taking geographical character and the offender’s

geographical preference into account, we utilize cluster analysis to divide all

the crime sites into 4 zones. In these 4 zones, we construct 4 two-dimensional

normal distributions around the 4 circle centers with the standard deviations

being the radii of the circles. In view of the inﬂuence of the crime time, we add a

time factor to the preliminary distribution. As a result, the preliminary distribu-

tion is modulated by geographical and temporal factors, producing an ultimate

prediction, which is rather satisfactory after validation.

Finally, an executive summary is presented for law enforcement ofﬁcers.

2010年美国国际大学生数学建模竞赛国际一等奖--张位春（07试点4），李臻（07试点2），邢照军（07国贸2）

Contents

Summary................................................................................... 1

1 Introduction ....................................................................... 3

1.1 Background ..................................................................... 3

1.2 Geographic Proﬁling.......................................................... 3

1.3 Objectives........................................................................ 3

2 Problem Analysis ............................................................... 4

2.1 Overall Strategy................................................................ 4

2.2 The Strategies for Problem I................................................. 4

2.3 The Strategy for Problem II ................................................. 4

3 Center of Minimum Distance Model.................................. 5

3.1 Notations ........................................................................ 5

3.2 Simplifying Assumptions.................................................... 6

3.3 Center of Minimum Distance Model ..................................... 6

3.4 Case Study....................................................................... 6

4 Bayesian Model.................................................................. 9

4.1 Notations and Deﬁnitions ................................................... 9

4.2 Simplifying Assumptions.................................................... 10

4.3 Bayesian Model ................................................................ 10

4.4 Case Study....................................................................... 12

4.5 The Discussion of α............................................................ 12

5 Combination of the Results of the Two Schemes................ 13

6 Prediction of the Next Crime.............................................. 14

6.1 Initial Prediction Method .................................................... 14

6.2 Developed Prediction Method ............................................. 15

7 Validation of Prediction...................................................... 20

8 Evaluating of Solutions ...................................................... 20

8.1 Strengths ......................................................................... 20

8.2 Weaknesses...................................................................... 21

8.3 Future Consideration ......................................................... 21

9 Executive Summary for the Chief of Police........................ 22

9.1 Preparatory Work.............................................................. 22

9.2 Overview of the Potential Issues........................................... 22

9.3 Suggestion....................................................................... 23

References ................................................................................. 24

Team #7120 page4 of 24

2 Problem Analysis

2.1 Overall Strategy

For simplicity, we divide the problem into two small problems

• Problem I: To develop at least two different schemes to generate a geo-

graphical proﬁle and then develop a technique to combine the results of

the different schemes.

• Problem II: To generate a useful prediction based on the geographical pro-

ﬁle.

Having ensured this, our strategy becomes

• Designing at least two different schemes to search out the residence of the

criminal ﬁrstly based on the time and locations of the past crime scenes

and designing a technique to combine the results of the different schemes.

• Utilizing the residence to predict the location of the next crime.

2.2 The Strategies for Problem I

After literature study, there are two categories of strategies to solve the Prob-

lem I: one is spatial distribution strategies and the other is probability distance

strategies[4].

Spatial distribution strategies include a number of different procedures, all of

which predict the home location of a serial offender by calculating a central point

from a distribution of crime site locations. Some common spatial distribution

strategies include the center of the circle, centroid, median, geometric mean,

harmonic mean, and center of minimum distance.

Probability distance strategies begin with the assumption that an offender’s

crime site locations deﬁne their activity space, and that this area contains the

offenders residence. Probability distance strategies differ from one another in

terms of the shape of the mathematical function applied around each crime site

and the assumptions regarding the relationship between where offenders reside

and where they commit their offences. Common probability distance functions

include the negative exponential, normal, lognormal, linear and truncated neg-

ative exponential.

The results of the schemes should be combined by certain technique to pro-

duce some useful geographical proﬁle.

2.3 The Strategy for Problem II

It is well known that the next crime is committed at a place where the proba-

bility of crime is the highest. So, for prediction, we try to obtain a planar proba-

bility distribution of being the next crime site. To accomplish this, several factors

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