2010 mcm predictions of locations of the crimes based on geographical profile

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Predictions of Locations of the Crimes Based on Geographical Profile

Summary
To make a prediction of the locations of the next crime, we have formulated and
tested three schemes for geographical profiling, in terms of:
 The Statistical Centroid Scheme which is based on the assumption that the
anchor point is near the centroid of the crime sites ,using Manhattan distance
to metric space and Rossmo’ formula to describe the probability distribution
of the crime sites around the anchor point. It’s applicable for bull's-eye
spatial pattern of crime.
 The Pareto Total Probability Scheme is applicable for various patterns spatial
pattern of crime. The determination of the place of residence is a
comprehensive consideration of the probability of all crime sites, which
makes this scheme more accurate than the Statistical Centroid Scheme.
 The Geographical Road Scheme which is suitable for series of robbers and
offenders who flee every where extracts the data of roads in the map of the
region where the series crimes occurs with the help of Google Earth and
image processing algorithms.
We recommend the Pareto Total Probability Scheme and the Geographical Road
Scheme for the generation of the geographical profile of a suspected serial criminal
that can help us to make predictions on the next crime site based on the past locations
of crimes and the distribution of the roads. A geographical profile describes an
optimal search process. To save the cost of search and improve the efficiency as much
as possible, a search should starts from the highest area and works down, which is
more likely to find the offender’s residence sooner. So the geographical profile is a
scientific guide to plan the deployment of police force.
In addition, we also give the measures of validation such as learning efficiency
and prediction accuracy to test different schemes, an analysis of the reliability of
different algorithms in different circumstances and test of the sensitivity of parameters
of our schemes. We also analyzed another case to test how well our schemes
performed on different serial criminals.


Contents

1 INTRODUCTION.................................................................................................................................4

1.1PROBLEM RESTATEMENT.........................................................................................................................4
1.2CONVENTIONS ......................................................................................................................................4
1.3OUR RESULTS.......................................................................................................................................5

2 ASSUMPTIONS AND ASSUMPTION JUSTIFICATIONS.............................................................6

3 LITERATURE REVIEW ....................................................................................................................7

4 PRINCIPLES FOR SCHEME DESIGN............................................................................................8

4.1 CRIME SPATIAL PATTERN.......................................................................................................................8
4.2 PROBABILITY DISTRIBUTION OF CRIME SITE.............................................................................................8
4.3 BUFFER ZONE......................................................................................................................................9
4.4 DISTANCE MEASUREMENT STANDARD.....................................................................................................9

5 STATISTICAL CENTROID SCHEME............................................................................................10

5.1 MATHEMATICAL INTERPRETATION..........................................................................................................10
5.2 SIMULATE THE SCHEME.......................................................................................................................13
5.3 RESULT OF THE SCHEME......................................................................................................................13

6 PARETO TOTAL PROBABILITY SCHEME.................................................................................14

6.1 LOCATION OF THE ANCHOR POINT.........................................................................................................14
6.2 PREDICTION OF CRIME PROBABILITY.....................................................................................................16

7 TECHNIQUE OF COMBINING RESULTS TO GENERATE A USEFUL PREDICTION.......18

7.1 STATISTICAL CENTROID SCHEME AND PARETO TOTAL PROBABILITY SCHEME .............................................18
7.2 PREDICTION OF ATTACK SITE BASED ON GEOGRAPHICAL PROFILE.............................................................19

8 INCORPORATE GEOGRAPHIC ROAD DATA FOR PREDICTION........................................20

8.1 TAKE INTO ACCOUNT OTHER FACTORS TO OPTIMIZE GEOGRAPHICAL PROFILE.................................................20
8.2 GEOGRAPHIC ROAD SCHEME................................................................................................................21
8.3 INCORPORATION OF DIFFERENT SCHEMES...............................................................................................22

9 MEASURES OF VALIDITY AND RELIABILITY.........................................................................23


9.1 MEASURES OF VALIDITY.....................................................................................................................23
9.2 ANALYSIS OF RELIABILITY...................................................................................................................23

10 RESULTS FOR OTHER SERIAL CRIMINALS..........................................................................24

11 SENSITIVITY TO PARAMETERS................................................................................................26

11.1 SAMPLE NUMBER .............................................................................................................................26
11.2 DISTANCE MEASUREMENT METHOD....................................................................................................27

12 STRENGTHS AND WEAKNESSES..............................................................................................28

12.1 STRENGTHS......................................................................................................................................28
12.2 WEAKNESSES...................................................................................................................................28

13 ALTERNATIVE APPROACHES AND FUTURE WORK...........................................................29

14 CONCLUSIONS...............................................................................................................................29

REFERENCES.......................................................................................................................................30

APPENDIXES........................................................................................................................................33


1 Introduction

1.1Problem Restatement

What are the serial criminals? Like serial murder, there is a lack of consensus among
academics and practitioners in the definition. In terms of serial murder, disagreement
centers on the number of victims, the presence/absence of a sexual element, and the
common characteristics of victims (Egger, 1998; 1984; Holmes & DeBurger, 1998;
Dietz, Hazelwood & Warren, 1990; Myers et al., 1993; Cantor et al., 2000; Fox &
Levin, 2005). In order to include all types of serial killers, a broad definition of serial
murder is used in the current research. In accordance with the crime classification
manual developed by the FBI, serial murders are those that involve three or more
separate events (Douglas et al., 1992), and most importantly, are repetitive sequential
homicides of any nature. Frequently, serial murders involve a similarity of subject or
purpose (For example, the choice of victims, methods of killings, or the killer’s
motivation; Aki, 2003: 6).
Usually we can judge the law of committing the crime and establish the
prediction scheme according to those intentions and other objective factors, such as
location, population density distribution, transportation. Then using the prediction
scheme we can predict the address of the criminal and the next crime site. However, it
is difficult to determine the proportion of these factors in the process of judgment and
prediction. There is no uniform standard. Previous studies and models mostly consider
only one factor (just as the distance) and get a prediction. The advantage is that the
model can be more widely applied to other serial criminals without many limitations
and is easy to operate. However, its disadvantages are also obvious. Because the
factor the model considers is single, the predictable results are always not satisfactory.
But we can integrate the results of multiple predictions. Analysis those and we can get
a satisfactory prediction.
Different schemes will produce different probability distribution figures. We need
to integrate these results and ultimately generate a geographical profile. Then we
develop a technique to combine the results of the different schemes and generate a
useful prediction. With this in mind we embark on our journey about criminal
profiling.

1.2Conventions
1.2.1 Terminology


 Activity Space: Those places regularly visited by a person in which the
majority of their activities are carried out. It comprises an individual’s
activity sites and the routes used to travel between them, and is contained
within the awareness space.
 Anchor Point: The base from which an individual resides or regularly
operates; usually the single most important location in a person’s life.
 Buffer Zone: An area centre around the criminal’s residence within which
targets are viewed as less desirable because of the perceived risk associated
with operating too close to home.
 Centrograph: A form of spatial analysis that focuses on the central
tendency of a point pattern.
 Circle Hypothesis: The hypothesis that marauders reside within their
offence circle, while commuters reside without.
 CPA: Crime pattern analysis.
 Criminal Geographic Targeting (CGT): A computerized spatial profiling
model that determines the most probable area of offender residence through
the production of a jeopardy surface or geographical profile from a criminal
hunting algorithm. It is the primary methodology used in geographic
profiling.
 Distance Decay: The reduction in probability of spatial interaction with the
increase in distance. Most crime trips follow a distance-decay pattern as
measured from the offender’s residence.
 Manhattan Distance: Distance measured along an orthogonal (e.g.,
northing and easting) grid layout of street blocks.
 Mean Centre: See spatial mean.
 Spatial Mean: A univariate measure of the central tendency of a point
pattern, the geographic “centre of gravity.” Also known as the centroid or
mean centre.
1.2.2 Variables
We will define the following variables here as they are used widely throughout our
paper. Additional variables may be defined later, but will be confined to a particular
section.
 ( SM x , SM y ) refers to the spatial mean of crime sites.

 C refers to the total number of crime sites.
 xn , yn refer to the coordinates of the n th crime site.

 B refers to the radius of the buffer zone.

1.3 Our Results

To make a prediction of the locations of the next crime based on the time and
locations of the past series crime scenes, we have formulated and tested three schemes
for geographical profiling with the aid of computer. The basic characteristics and


scope of application of these three schemes are as follows:
 The Statistical Centroid Scheme is based on the assumption that the anchor
point is near the Centroid of the crime sites. It’s applicable in terms with
offenders whose spatial pattern of crime is bull's-eye pattern (George Regret ,
1996) .They have a fixed residence and are accustomed to committing the crime
where is near the region of which the center is their anchor point, rather than
itinerant criminal. As for this scheme, we only consider a single anchor point. We
use Manhattan distance to metric space and Rossmo’ formula describe the
probability distribution of the crime sites around the anchor point.
 The Pareto Total Probability Scheme is applicable in terms with offenders
whose spatial patterns of crime are various patterns such as bull’s-eye pattern,
bimodal pattern, tear drop pattern and so on. They also have a fixed residence, but
they would not choose crimes site around their house. They may tend to choose a
number of directions or a fixed residence between their workplace and residence.
The distribution of the probability of committing crime is anisotropic.
 The Geographical Road Scheme is suitable for series of robbers and offenders
who flee everywhere. With the ad of Google Earth and image processing
algorithm and Google Earth satellite map technology, we can to extract the roads
within the map of the region where the series crimes occurs. We consider the flee
route of the offenders and find that they prefer to choose the place which is not
far from the main roads. However, they would not choose to commit crimes on
the roads in order to prevent possible exposure of their criminal acts. Their
strategy of the selection of possible crime sites is within a fixed limit of the
narrow area to balance the risks of possible exposure of their criminal acts and
opportunities to escape form criminal scene.

2 Assumptions and Assumption Justifications

About crimes
 The crimes belong to a series of crimes that have occurred. That is to say, they
are likely to be committed by the same person. The so-called serial criminal refers
to the continuous criminal such as homicide, rape and theft and the process,
means, methods, objectives of which are same or similar. Only serial criminals
generally have the same identity of characteristics.
 Offenders are rational. Criminals committing the crime after they has been
carefully planned to avoid the risk of crime and increase the success rate of crime
as much as possible.
 The rule of distance decay is applicable to the cases. The distance between the
crimes site and the criminals’ home are always not too far for the majority of


cases. Because the offenders want to take action in a familiar environment
.However, they hate to happen to their acquaintance and exposure their own
identifications. As for the choice of crime site, they want to maximize opportunity
and minimize the risk.

About criminal time
 Criminals lurking in the normal population. They also have normal work and
life. They generally commit the crime at night or early hours of morning. Take
Peter Sutcliffe as an example, all the 13 murders occurred between 7:30 pm to
2:15 (Wikipedia, 2009).

3 Literature Review

Before introducing our own model, we first provide a very brief overview of
geographic profiling. The problem of geographic profiling is studied by some
scientists such as Kim Rossmo from the 20th century.
Geographical patterns in crime have been noted since the mid-19th century
pioneering work of Andre-Michel Gerry and Lambert-Adolph Quenelle who mapped
information about violent and property offences and examined their spatial
relationship to poverty (Brantingham&Brantingham, 1981c; Vold&Bernard, 1986).
The most famous spatial crime studies were conducted in the early 20th century, when
the city of Chicago served as an inspiration source and a field of experimentation for
University of Chicago sociologists (Warren, 1972; Williams&McShane, 1988). The
geographical focus of criminology had shifted from regional areas to city
neighborhoods.
Geographic profiling is a criminal investigative methodology that analyzes the
locations of a connected series of crimes to determine the most probable area of
offender residence. Typically used in cases of serial murder or rape (but also arson,
bombing, robbery, and other crimes), the technique helps police detectives prioritize
information in large-scale major crime investigations that often involve hundreds or
thousands of suspects and tips.
A sound mathematical algorithm for the geographic profiling should possess
( Mike O’Leary, 2009)
 The method should be logically rigorous.
 There should be explicit connections between assumptions on offender behavior
and the components of the model.
 The method should be able to take into account local geographic features; in
particular, it should be able to account for geographic features that influence the
selection of


 A crime site and geographic features that influence the potential anchor points of
offenders.
 The method should be based on data that are available to the jurisdiction(s) where
the offences occur.
 The method should return a prioritized search area for law enforcement officers.

4 Principles for Scheme Design

Before the scheme design, we need to determine a number of conditions, namely
crime spatial pattern, probability distribution of crime site, buffer zone and distance
measurement standard, so that we can operate on the schemes. Our analysis of these
conditions is as follows.

4.1 Crime Spatial Pattern

In the simplest case, offenders’ residences lie at the centre of their crime patterns and
can be found through the spatial mean (Brantingham and Brantingham, 1981). The
intricacy of most criminal activity spaces, however, indicates that more complex
patterns are the norm.
George Rengert (1996) proposes four hypothetical spatial patterns for the
geography of crime sites:
 uniform pattern, with no distance-decay influence;
 bull’s-eye pattern, exhibiting distance decay and spatial clustering around
the offender’s anchor point;
 bimodal pattern, with crimes clustered around two anchor points;
 teardrop pattern, centered around the offender’s primary anchor point, with
a directional bias towards a secondary anchor point.
We do not consider the first pattern. Because there is no law of a uniform pattern,
it is difficult to predict. For the last three patterns, we can promote the second pattern
and apply it to the third and fourth patterns. We can integrate two bull’s-eye patterns
and get a new pattern which equals to the bimodal pattern. Just like this, we can also
integrate two bull’s-eye patterns which have different weights and get a new pattern
which equals to the teardrop pattern.
Through the analysis above, we use the bull’s-eye pattern for our scheme design.

4.2 Probability Distribution of Crime Site

Usually there are some laws between the probability distribution of crime site and the


distance between crime site and anchor point. The offender’s search behavior follows
some form of distance-decay function (Brantingham & Brantingham, 1981, 1984;
Rhodes & Conly, 1981).
Levine (2009a) gives a number of choices for the decay function, including:
 Linear: f (d ) = A + Bd ;
−βd
 Negative exponential: f (d ) = Ae ;

Normal: f ( d ) = A(2π S ) exp[−(d − d ) 2 S ] ;
2 −1 2 2 2


Lognormal: f (d ) = A(2π d S ) exp[−(ln d − d ) 2S ] ; and
2 2 −1 2 2 2


Truncated negative exponential: f ( d ) = Bd if d < C and f ( d ) = Ae
−βd
 if
d ≥C.
Search pattern probabilities can also be modeled by a Pareto function, starting
from the sites and routes that compose the activity space and then decreasing as
distance away from the activity space increases. The Pareto function, named after the
Italian economist, is suitable for fitting data that have a disproportionate number of
cases close to the origin, making it appropriate for modeling distance-decay processes
(Brantingham & Brantingham, 1984). We will discuss the concrete form of
distribution function in Section 5.1.

4.3 Buffer Zone

There is usually a “buffer zone” that centers around the criminal’s residence,
comparable to what Newton and Swoope (1987) call the coal-sack effect. Within this
zone, targets are viewed as less desirable because of the perceived level of risk
associated with operating too close to home. For the offender, this area represents an
optimized balance between the maximization of opportunity and the minimization of
risk. The buffer zone is most applicable to predatory crimes; for affective-motivated
offences it takes on less importance, as can be seen by the fact that domestic
homicides usually occur within the residence. The radius of the buffer zone is
equivalent to the modal crime trip distance (Rossmo, 2000).
We discussed the determination of the modal crime trip distance in Section 5.1.

4.4 Distance Measurement Standard

Usually people use one of the two methods (ie, the Euclidean distance and the
Manhattan distance) to measure two points in a map. When the influence of the streets
is obvious, that is because the two sites in the map is close we can’t ignore the street
influence on the distance, we use the Manhattan distance. If the two sites are far from


each other we choice the Euclidean distance.
Based on the above considerations, we require: when the two sites are in the same
city we choice the Manhattan distance. Otherwise, we choice the Euclidean distance.

5 Statistical Centroid Scheme

We design the Statistical Centroid Scheme based on the statistical laws of the
predecessors. The crime sites are always around the anchor point. Therefore we treat
the centroid of the known crime sites as the anchor point. According to this we predict
the probability distribution of crime sites.

5.1 Mathematical Interpretation

In this subsection, we describe the mathematics of the centroid, the radius of the
buffer zone and the function of the probability distribution of crime sites.
 Centroid Formula
We use Centrography to calculate the centroid.
We calculate the centroid (sometimes referred to as the spatial mean or mean
centre) of the known crime sites. The centroid is defined as:
( SM x , SM y )

where :
 C 
SM x =  ∑ xn  C
 n =1 

 C 
SM y =  ∑ yn  C
 n =1 
and:
SM x is the x coordinate of the spatial mean;

SM y is the y coordinate of the spatial mean;

C is the total number of crime sites; and
xn , yn are the coordinates of the n th crime site.

 Radius of Buffer Zone B
So far, there is no uniform standard for dividing buffer zone. According to the view
that the radius of the buffer zone is equivalent to the modal crime trip distance, we


just need to get the modal crime trip distance if we want the radius of the buffer zone.
The process to get the modal crime trip distance is as follows:
According to the description of the definition of the centroid we calculate the the
standard distance. The standard distance is defined as:
C
Sd = (∑ rns 2 ) C
n =1

where:
Sd is the standard distance;
C is the total number of crime sites;
rns is the distance between the centroid and the n th crime site.
and:
rns 2 = ( xn − SM x ) 2 + ( yn − SM y ) 2

From the formula above we can get the average distance of the crime sites Sd .
This is just the modal crime trip distance.
The reason why we do it in this way is the definition of the modal crime trip
distance is the distance between a crime site and the offender’s residence. We can get
a better result in that way.
 Function of the Probability Distribution of Crime Sites
In section 4.2 we have already analyzed that the relationship between crime
probability and the distance between the crime site and anchor point can be modeled
by a Pareto function. It takes the general form:
y = k / xb

Most of the studies on this distribution are in the finance field. In the field of
probability distribution of crime sites, the Rossmo’s formula proposed by
mathematician Kim Rossmo (1995) is able to obtain a satisfactory result through the
test of practice.
Rossmo's formula is defined as:
C
φ (1 − φ )( B g − f )
pij = k ∑ [ + ]
n =1 (| xi − xn | + | y j − yn |) f (2 B − | xi − xn | − | y j − yn |) g

where:
| xi − xn | + | y j − yn |> B ⊃ φ = 1

| xi − xn | + | y j − yn |≤ B ⊃ φ = 0

and:
pij is the resultant probability for point ij ；

φ is a weighting factor;


k is an empirically determined constant;
B is the radius of the buffer zone;
C is the number of crime sites;
f is an empirically determined exponent;
g is an empirically determined exponent;
xi , y j are the coordinates of point ij

xn , yn are the coordinates of the n th crime site.
Rossmo's formula reflects that the Probability Distribution of Crime Sites
increases first and then decrease with the increasing of the distance between crime site
and the anchor point. Result is shown in Figure 1.

Figure 1.the Rossmo's formula figure

X-axis represents the distance between crime site and the anchor point;

Y-axis represents the Probability of Crime Sites

( k =1, g = 2 , f =1)

This is consistent with common sense, the offenders are unlikely to commit
crimes in the area closer to anchor point because they could be easily identified. As
the distance from anchor point increases, the likelihood of offenders selecting the
target increases. But when it up to a certain value the likelihood decreases because it
is too far from the anchor point.


5.2 Simulate the Scheme

In section 5.1 we have defined the formulas we will use in the scheme. Now we show
the implementation of the Statistical Centroid Scheme:
Step1.Get the map of the serial criminals and establish the coordinates;
Step2.Get the coordinate of the crime sites in this coordinates;
Step3.Use the (0.2) and (0.3) to calculate the coordinate of the centroid of the

crime site ( SM x , SM y ) ;

Step4.Use the ( SM x , SM y ) got in step 3 to calculate the radius of the buffer zone

according to (0.4) and (0.5);
Step5.Use (0.7) to calculate the probability of every point in the map as the crime
sites;
Step6.Get the probability of every point in the map and draw the contour map in
which the high-level value represents the probability of every point.

5.3 Result of the Scheme

Through the Statistical Centroid Scheme we get the figure of the Probability
Distribution of Crime Sites with the centroid of the crime sites as the anchor site.
Figure 2 and Figure 3 are the results of the Statistical Centroid Scheme by using the
data of the serial murders of Peter Sutcliffe in 1981.

Figure 2. the three-dimensional map of the probability distribution of crime sites

From Figure 2 we can see that the probability distribution of crime sites just like a
volcano. The center of the volcano is the anchor point. Probability distribution


increases first and then decreases. Figure 3 is the contour map representation of the
Figure 2.

Figure 3. the contour map of the probability distribution of crime sites

We mark the 13 crime sites of Peter Sutcliffe in this map. Half of the sites are
located in the relatively high probability of committing the crime. For this serial
murders the assumptions and reasoning of the Statistical Centroid Scheme is in line
with the actual situation.

6 Pareto Total Probability Scheme

Different from the Statistical Centroid Scheme the Pareto Total Probability Scheme
uses backstepping to predict the probability distribution of the anchor point according
to the probability distribution of crime sites. Relative to the assumption of Centroid in
the Statistical Centroid Scheme this method can determine the location of the anchor
point more reasonable and is more operational. This method is proposed by Kim
Rossmo and the accuracy rate is high by a large number of practical testing.

6.1 Location of the Anchor Point

In this method we will use the formulas definition in Section 5.1. The steps of the
method are as follows:


Step1.Get the map of the serial criminals and establish the coordinates;.
Step2.Get the coordinate of the crime sites in this coordinates;
Step3.Use the (0.2) and (0.3) to calculate the coordinate of the centroid of the

crime sites ( SM x , SM y ) ;

Step4.Use the ( SM x , SM y ) got in step 3 to calculate the radius of the buffer zone

according to (0.4) and (0.5);
Step5.Use (0.7) to calculate the probability of every point in the map by using the
coordinates of the known crime sites.
Step6.Get the probability of every point in the map and draw the three-dimensional
map and the contour map in which the high-level value represents the
probability of every point. These maps reflect the probability distribution of
the anchor point.
Step7.In the figures we can find some summits that represent a relatively high
probability. These areas are the most likely location of the anchor point.
Through the processes above, we can get the map that represents the probability
distribution of the anchor point. Figure 4 is the draw the three-dimensional map of the
probability distribution of the anchor point using the data of the serial murders of
Peter Sutcliffe in 1981. The red areas in the map are the most likely location of the
anchor point.

Figure 4. the three-dimensional map of the probability distribution of anchor point

Figure 5 is the contour map representation of the Figure 4. We mark the 13 crime
sites of Peter Sutcliffe and the centroid of the crime sites in this map. From Figure 5
we can see that the centroid is not located in the red areas in the map. This represents
that there are some differences on the prediction between the Statistical Centroid
Scheme and the Pareto Total Probability Scheme.


Figure 5. the contour map of the probability distribution of the anchor point

By using the map of the probability distribution of the anchor point, the police
can start searching from the high probability areas. This method can reduce the
police’s search areas effectively and accelerate the speed of detection.

6.2 Prediction of Crime Probability

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

offender, estimate the probability density that xnext will be the location of the next
offence. The Bayesian approach to this problem is to calculate the posterior predictive
distribution.
P ( xnext | x1 ,K , xn ) = ∫∫∫ P ( xnext | z, α ) P ( z , α | x1 ,K , xn )dz (1) dz (2) dα

where :
z is the location of the offender’s anchor point;
α is the average distance that the offender is willing to travel to offend;
p( x | z,α ) is the probability that an offender with a single stable anchor

point z and average offence distance α commits a crime at the location x .
John Wiley & Sons, Ltd. (2009) simplified (0.10) like this:
P ( z , α | x1 ,K , xn ) = P ( x1 | z, α )L P ( x1 | z, α ) H ( z )π (α )


where :
H ( z) is the prior probability density function for the distribution of anchor

points before any information from the crime series is included;
π (α ) is the probability density function for the prior distribution of the

offender’s average offence distance, again before any information from
the crime series is included.
According to the theory above, we design the prediction of the probability
distribution of crime sites. The steps of the scheme are as follows:
Step1.Use the probability we calculate in Section 6.1 as the weights ki of every

point;
Step2.Let every point in the map to be the anchor point (one at a time and every
point can only be used once). We calculate the probability distribution of the
anchor point in every points except the point used as the anchor point every
time. For example, when point a is used as the anchor point we calculate the
probability distribution pi − a of the anchor point in every other point and do

pi − a ∗ ka .

Step3.Calculate the sum of the probability of every point in the map

∑p i −a ∗ ka , (a = 1,K , n) . This is the probability the offender will commit the

crime at this point.
Through the Pareto Total Probability Scheme we get a figure of the Probability
Distribution of Crime Sites different from the figure get from the Statistical Centroid
Scheme. Figure 6 is based on the result of Figure 5. The high-level value is the
probability of every point.

Figure 6. the three-dimensional map of the probability distribution of anchor point


Figure 5 is the contour map representation of the Figure 4. We mark the 13 crime
sites of Peter Sutcliffe in this map. From the map we can see most sites are located in
the relatively high probability of committing the crime. For this serial murders the
prediction ability of the Pareto Total Probability Scheme is strong.

Figure 7. the contour map of the probability distribution of the anchor point

7 Technique of combining results to generate a useful

prediction

7.1 Statistical Centroid Scheme and Pareto Total Probability Scheme

Both Statistical Centroid Scheme and Pareto total probability Scheme are
geographical profiles for the criminal's offence probability distribution situation based
on the Rossmo formula and the distance variable of both are Manhattan distance,
thought they have different assumptions and solving methods. Statistical Centroid
Scheme assumed that the offender’s lodging is the center of mass which is determined
by all the known attack sites. Regard the lodging as the center, the possibility of
another attack for an arbitrary spot fits the Rossmo formula. Therefore, we get the
outline of the geographical profile of offender’s another attack, which fits the crime
space model of bull’s-eye.
Pareto total probability Scheme takes the assumption of ‘center of mass is the
lodging’ for a grand. It adopts Rossmo equation to describe the probability
distribution and the attack rate of every zone is the overlay of a single attack rate for
every possibility lodging of offender. Therefore, we get a more accurate geographical
profile about the offender’s lodging. Then, for every possibility of offender’s lodging,


take integration for the next attack possibility.

7.2 Prediction of Attack Site Based on Geographical Profile

The principles to combine the geographical profiles generated by Statistical Centroid
Scheme and Pareto total probability Scheme is:
 If a certain area lies in higher crime rate place in model one and model two, like
the red zone shown as figure 8, then the next attack rate in this zone is the largest;
 If a certain area lies in higher crime rate place in either model one or model two,
then the next attack rate in this zone is still larger;
 If a certain area lies in neither model one nor model two’s higher crime rate place,
like the blue zone shown as figure 8, then the next attack rate in this zone is the
small;
Therefore, we could overlay the attack rate in the same zone in order to get a
new distribution of attack rate. Figure in shows the new distribution of attack rate.

Figure 8. attack rate distribution after overlay

Figure 9. crime rate’s reliable ability
By integration, we got the attack rate’s reliable ability, shown as Figure 9. We


could see that the larger the search range is the larger possibility to hunt the murder.
However, it may be costly. Therefore, we should determine the hunting range for
different situation and purpose.

8 Incorporate Geographic Road Data for prediction

8.1 Take into account other factors to optimize geographical profile

Many different crime factors and environmental elements are considered in the
construction and interpretation of a geographic profile. (Kim Rossmo, 2000)The
schemes established above are mainly from the perspective of the spatial distribution
of crime site, we can also consider other factors. The most relevant ones include:
 Offender type — The type and number of offender(s) affect crime geography. If
multiple criminals living apart are involved, the geographic file will focus on the
dominant one’s residence. Large, amorphous gangs may not be suitable for
geographic profiling because of changing group composition. Psychological
profiling assists in interpreting offender behaviors by providing information on
personality, background, and level of organization.
 Target backcloth — Constrained or patchy target backcloths limit the degree of
offender choice, affecting the importance of certain crime site types for the
profile.
 Bus stops and rapid transit stations — Offenders without vehicles may use
public transit or travel along bicycle and jogging paths. The locations and routes
of these should be taken into consideration.
 Physical and psychological boundaries — People are constrained by physical
boundaries such as rivers, ocean, lakes, ravines, and highways. Psychological
boundaries also influence movement. For example, a criminal of low
socioeconomic status may avoid an upper class area, or a black offender might
not wish to go into a white neighborhood.
 Zoning and land use — Zoning (e.g., residential, commercial, industrial) and
land use (e.g., stores, bars, businesses, transportation canters, major facilities,
government buildings, military institutions) provide keys as to why someone may
be in a particular area. Police in.
 Neighborhood demographics — Some sex offenders prefer victims of a certain
racial or ethnic group. These groups may be more common in certain
neighborhoods than in others, affecting spatial crime patterns.
 Victim routine activities — The pattern of routine victim movements provides
insight to how the offender is searching for targets.


8.2 Geographic Road Scheme

Our comprehensive consideration about the impact of geographical distribution of
the traffic road on the probability of crime bring out the Geographic Road Scheme
based on the geographical distribution of the traffic road. The main assumptions are as
followings:
 People, including criminals, do not travel as the crow flies. Not only must they
follow street layouts, but they are most likely to travel along major arterial routes,
freeways, or highways.
 Experienced criminals usually plan well escape route before offenses, so most of
them would rather choose the crime site that is not far from road. They can
complete the rapid departure at the same time.
 Generally, a lot of people and traffic are on the road. In order to reduce the risk of
crime, criminals will not commit a crime on the road.
This new method allows us a simple way to incorporate geographic features into
the model. Indeed, let us suppose that offender target selection depends on more than
just the distance from the anchor point to the crime site locations, but that it depends
on some features in the local geography. One way to account for this is to suppose that
the offence probability density is proportional to both a distance decay term and to a
function that measures the attractiveness of a particular target location. ( Mike
O’lerary，2009) .Doing so, we obtain the following expression.

P ( x / z, α ) = D(d ( x, z ), α )G ( x ) N ( z )

The factor G ( x) is used to account for the local geographic features that influence

the selection of a crime site. High values for G ( x) indicate that x is a likely target for
typical offenders; low values indicate x is a less likely target. The remaining factor
N is a normalization required to ensure that P is a probability distribution. Its value
is completely determined by the choices of D and G and has the form.
We use the difference between two normal distribution functions to describe
distribution of crime probability where is centered with the point in the roads .The
steps of the implementation are as follows:
Step1.Use Google Earth to obtain the geographical coordinates of 13 crime sites of
Peter Sutcliffe in the satellite map which shows roads at the same time.
Step2.Use MATLAB to generate a two-dimensional Laplacian of Gaussian function
f ( x, y ) .

Step3.Take the G and B components in the RGB map, then minus them respectively.
Add the results as Im Road .
Step4.Make convolution formula operation on Im Road .


Step5.Remove the part where the value is less than zero in Im Road to get the image
we want.

The geographical profile based on the Geographic Road Scheme is shown in
Figure 10.The deeper the color is the larger the probability of next serial criminal.

Figure 10.The geographical profile based on the Geographic Road Scheme

8.3 Incorporation of Different Schemes

To obtain a more precise geographical profile, we combine the geographical
profile base on the Geographical Road Scheme And the Pareto Total Probability
Scheme .Then we get a figure which shows both the impacts of roads and The spatial
distribution of previous crime site on the Probability distribution of next crime site.
For the road, the deeper the color is, the larger the probability of next serial criminal
is. For contour lines, each line represents the level of probability values marked on
the figure.

Figure 11.The geographical profile based on the Pareto- total probability Scheme and the
Geographic Road Scheme


9 Measures of Validity and Reliability

9.1 Measures of Validity

The methods of the evaluation of the validity of geographical profile include the test
of real probability, hit score percentage, learning rate and so on.
 Test of Real Probability
For the sample data of serial criminals, we use data of crime occurred early to
build models and generate a geographical profile. Then make use of data of crime
occurred late to test the accuracy of forecasts of the geographical profile. As for real
crime site, the greater the predicted values of crime probability, the better the model
is. The test of our models adopted this method (see Section 13).
 Hit Score Percentage
Dr. Kim, Rossmo came up with this method to test the validity of geographical
profile. The success of the model is measured by the hit score percentage --- the ratio
of the total number of points with scores equal or higher to the hit Score, to the total
number of points within the hunting area. This is equivalent to the percentage of the
total area that must be searched before the offender’s residence is found, assuming an
optimal search process (i.e., one that started in the locations with the highest scores
and then worked down). The extent of the search area --- the territory police have to
search in order to find the offender --- is equal to the size of the hunting area
multiplied by the hit score percentage. The smaller the hit score percentage is, the
better the model is.
 Learning Rate
Learning rate describes the speed that the model make use of limited number
of sample data to generate a good Forecast Accuracy .The faster learning rate
means that to achieve the same Forecast Accuracy the model just need fewer
sample data. Even if the police just know a few crime sites, they can get
reasonable accurate forecast about the next crime site.

9.2 Analysis of Reliability

Our model has its own reasonable applications occasions .It’s necessary to analysis
the reliability of our models in different occasions . There is an element of subjectivity
in determining which crime sites in a given case are useful predictors.
 It is usually assumed that the offender has not moved or been displaced during the
time period of these crimes, but if this has occurred, then more locations are
required.


 Only crime locations that are accurately known should be used. For example,
encounter sites may be imprecise if they have to be inferred from last known
victim sighting. In some investigations, the locations of certain sites may be
completely unknown.
 Analyses of the crime site type with the most locations results in lower expected
hit score percentages. Multiple offences in the same immediate area should not be
double counted. The degree of spatial temporal clustering must be assessed as
crime sites too close in time and space are probably non-independent events.
The different criminal spatial patterns also affected the reliability of scheme. The
Statistical Centroid Scheme is based on the assumption that the anchor point is near
the Centroid of the crime sites. It’s applicable in terms with offenders whose spatial
pattern of crime is bull's-eye pattern (George Regret ,1996) .They have a fixed
residence and are accustomed to committing the crime where is near the region of
which the center is their anchor point, rather than itinerant criminal. As for this
scheme, we only consider a single anchor point. We use Manhattan distance to metric
space and Rossmo’ formula describe the probability distribution of the crime sites
around the anchor point. The Pareto Total Probability Scheme is applicable in terms
with offenders whose spatial patterns of crime are various patterns such as bull’s-eye
pattern, bimodal pattern, tear drop pattern and so on. They also have a fixed residence,
but they would not choose crimes site around their house. They may tend to choose a
number of directions or a fixed residence between their workplace and residence. The
distribution of the probability of committing crime is anisotropic. The Geographical
Road Scheme is suitable for series of robbers and offenders who flee everywhere. We
consider the flee route of the offenders and find that they prefer to choose the place
which is not far from the main roads. However, they would not choose to commit
crimes on the roads in order to prevent possible exposure of their criminal acts. Their
strategy of the selection of possible crime sites is within a fixed limit of the narrow
area to balance the risks of possible exposure of their criminal acts and opportunities
to escape form criminal scene.

10 Results for Other Serial Criminals

In order to test how well our schemes performed on reliability with different attacks,
we also analyzed another case. Figure12 (a) gives the geographical profile of this case
and the triangle shows the place of committing a crime.
Different from Peter Sutcliffe case, the traffic is extremely convenient around the
offence place in this case, and the majority places of committing a crime are
concentrated on hub of communications. Compared to Sutcliffe, the murderer in this
case has larger escaping distance within the same time. From Statistical Centroid
Scheme, we know that the larger distance from murder’s lodging, the lower possibility


he commits a crime. Therefore, the murder may choose commit a crime remotely and
in this case, we choose the Pareto Total Probability Scheme, which turn out to
performer well enough.

Figure 12(a). Geographical Profile

Figure 12(b).
As Figure 12(b) shown, the probability of committing a crime in two areas of A,
B is the greatest, while other areas have a low possibility. The murderer's real offence
place has landed on two areas of A, B mostly, which had fully proved the offence
probability distribution that model two predicts has stronger practicability. In most
instances, the police can fixes up police strength according to the offence probability
distribution depicted by scheme2, which improved the hunting probability to a certain
extent.


11 Sensitivity to Parameters

11.1 Sample number

In this section, we use different sample number to test the accuracy of crime
rate’s prediction of Pareto Total Probability Scheme and Statistical Centroid Scheme.
Take Peter Sutcliffe for example, the sample number is selected according to the
murder time’s order. In this case, we assumed that 5 crime sites are known. Then
calculate the 6th crime site’s rate using Pareto Total Probability Scheme and Statistical
Centroid Scheme separately. After that, calculate the 7th crime site’s rate if 6 crime
sites are known. Do this until 12 crime sites are known. The result is shown as
Figure8.

Figure 13. sensitivity of Sample number.

From Figure13 we know that, both of the two schemes’ prediction rate presents
the law of rising first then reducing and rising again at last, along with the sample
number’s increase, which fits the common sense appropriately.
For the Pareto Total Probability Scheme, since the 6th and 9th crime site are far
from the rest of the crime site, which is not fit the normal principle. Therefore, when
we adopt the former 6 crime sites as the sample number to calculate the 7th crime
site’s rate, the result turn out to be some big deviation. As a result, the 8th calculate
result seem to be influenced to some extent. However, the 9th result rises again due to
the closed distance from 6th crime site. After that, the prediction rate seems to be more
stable because there is no more abnormal crime site again.
Statistical Centroid Scheme’s changing pattern similar the Pareto Total


Probability Scheme, but has some certain of lag, which means the Statistical Centroid
Scheme is no so sensitive as the Pareto Total Probability Scheme on the changing of
the sample number.
For vertical comparison, we could see that the Pareto Total Probability Scheme’s
prediction results are superior to the Statistical Centroid Scheme’s. Therefore, the
Pareto Total Probability Scheme has a higher predictive accuracy.

11.2 Distance Measurement Method

In this section, we want to test whether the Euclidean distance or the Manhattan
distance has higher prediction accuracy in the Pareto Total Probability Scheme. The
detach method is similar to that described in section13.1. The difference is just to
replace the Manhattan distance with the Euclidean distance. Figure9 shows the results
by using different distance type.

Figure 15. sensitivity analysis for distance measurement

From horizontal view, both of the two curves’ change pattern is similar, which


shows that under the same scheme, if we use different distance measurement formula,
we will get the same prediction on condition of adopting the same sample number.
However, the prediction results are somewhat server if we using Manhattan distance.
From vertical view, the Manhattan distance’s prediction result is superior to the
Euclidean distance’s and has higher prediction accuracy.

12 Strengths and Weaknesses

12.1 Strengths

The two schemes of our approach take the murder’s lodging into consideration.
Scheme 1 adopts the bull’s-eye assumption, which fits most serial crime’s character.
Scheme 2 uses CGT to reach the target of making a prediction for criminal’s lodging.
Both schemes had made a crime site prediction and got a good prediction result.
Then we make a integration for these two schemes and get a geographical profile.
Above all, we work out the crime site’s reliability: 92.23%, 71.82%, 18.71%, which
are extremely important for Investigators.
What’s more, we also analyze the traffic conditions around the crime site.
Through Peter Sutcliffe’s case, we learn that most of the crime site are close to the
main transport routes. Then we make a integration among the former schemes and the
situation, in this way, we could get a more accurate prediction.

12.2 Weaknesses

There are so many influence factors to take into consideration and it is hard to
construct a model concluding all of them.
Besides, our model is based on the happened cases, so there is inevitable to
have some conditions which are not suitable for the case. For example, the CGT
model works on the assumption that a relationship, modeled on some form of
distance-decay function, exists between crime location and offender residence. Since
the model cannot locate the residence of a criminal that lies outside of the boundaries
of the hunting area map, it is necessary to limit the process to non-commuting
offenders.


13 Alternative Approaches and Future Work

There were several extensions to our approach that we could not pursue due to the
time constraint. We have also considered the time limit for offender activity space.
Through the mastery of the time of the committing crimes we can know how long the
offender spent to commit a crime every time. Then we combine the time and the speed
of the offender to calculate the radius of the offender activity space. By drawing the
circles with the known crime sites as the center one at a time we can get a map with
many overlaps on it. These overlaps are most probably the location of the anchor
point. After integrate the result of this scheme and the scheme we design in above
sections we can predict the probability distribution of the anchor point more
accurately.
In addition, we need further analysis for the influence of the traffic in the
Geographical Road Scheme. Quantify it and get the weights of different traffic
conditions. Thus we can combine the Geographical Road Scheme with the other two
schemes. We can also consider the population density just like the traffic and the data
of it is not difficult to obtain.
At last, to promote the approach, we have already discussed it in Section 4.1. Our
approach takes into account the second pattern. For different patterns (except for
uniform pattern) we can promote our approach and apply it to them. For the bimodal
pattern, the distribution of the known crime sites mainly focuses on two areas. We can
divide all the crime sites into two parts with one cluster in each. We use our approach
on each part and integrate the results at last. For the teardrop pattern, like the bimodal
pattern we also divide all the crime sites into two parts with one cluster in each. But
we should add the different weights on the two parts. Then use our approach on each
part and integrate the results. According to this idea our approach can be applied to
more kinds of serial criminals.

14 Conclusions

We have formulated and tested three schemes for geographical profiling with the aid
of computer. The Statistical Centroid Scheme is based on the assumption that the
anchor point is near the Centroid of the crime sites. It’s applicable in terms with
offenders whose spatial pattern of crime is bull's-eye pattern (George Regret ,1996) .
The Pareto Total Probability Scheme is applicable in terms with offenders whose
spatial patterns of crime are various patterns such as bull’s-eye pattern, bimodal
pattern, tear drop pattern and so on. The Geographical Road Scheme is suitable for


series of robbers and offenders who flee everywhere.
We find that the learning efficiency and prediction accuracy of the Pareto Total
Probability Scheme is higher than the Statistical Centroid Scheme. Because the Pareto
Total Probability Scheme did not assume that the place of residence is in the centroid.
It can response to various types of offenders. Since the determination of the place of
residence is a comprehensive consideration of the probability of all known crime sites,
this scheme is more accurate than the Statistical Centroid Scheme. The Geographical
Road Scheme combines image processing algorithm and satellite map to extract the
contours of the road within the region which includes some crime sites. It build a
model separately on the impact of the traffic road on the probability distribution of the
crime sites and is combined with movement patterns and crime site selection
psychology of offenders, which is a certain novelty and uniqueness.
We recommend the Pareto Total Probability Scheme and the Geographical Road
Scheme for the generation of the geographical profile of a suspected serial criminal
that can help us to make predictions on the next crime site based on the past locations
of the crimes and the distribution of the traffic road. In fact, a geographical profile
describes an optimal search process. To save the cost of search and improve the
efficiency as much as possible, a search should starts from the highest area and works
down , which is more likely to find the offender’s residence sooner than a random
process would. So the geographical profile is a scientific guide to plan the deployment
of police force. The real search efficiency is therefore an indicator of the performance
of geographical profile and the scheme to generate the geographical profile.
In addition, we also give the measures of validation to test different schemes and
an analysis of the reliability of different algorithms in different circumstances. We test
the sensitivity of parameters of our schemes and make some recommendations base
on the test results .

References

Egger SA 1998. The killers among us. New Jersey: Prentice Hall

Egger SA 1984. A working definition of serial murder and the reduction blindness.
Journal of police science and administration 12: 348–387.

Holmes RM & DeBurger JE 1998. Profiles in terror: the serial murderer, in Holmes
RM & Holmes ST (eds), Contemporary perspectives on serial murder. Thousand
Oaks: Sage: 5–16

Dietz PE, Hazelwood RR & Warren J 1990. The sexually sadistic criminal and his
offenses. Bulletin of the American Academy of Psychiatry and the Law 18(2): 163–


178

Myers WC et al. 1993. Malignant sex and aggression: an overview of serial sexual
homicide. Bulletin of the American Academy of Psychiatry and the Law 21(4): 435–
451

Cantor DV et al. 2000. Predicting serial killers’ home base using a decision support
system. Journal of quantitative criminology 16(4): 457–478

Fox JA & Levin J 2005. Extreme killing: understanding serial and mass murder.
Thousand Oaks CA: Sage publications

Douglas JE et al. 1992. Crime classification manual: a standard system for
investigating and classifying violent crimes. San Francisco, CA: Jossey-Bass

Aki K 2003. Serial killers: a cross-cultural study between Japan and the United States.
Graduate thesis: California State University

Brantingham, P.J. and P.L. Brantingham (eds.) (1981). Environmental Criminology.
Beverly Hills, CA: Sage.

Rengert, G.F. (1991). The Spatial Clustering of Residential Burglaries About Anchor
Points of Routine Activities." Paper presented at the meeting of the American Society
of Criminology, San Francisco, CA.

Brantingham, P. L., & Brantingham, P. J. (1981). Notes on the geometry on crime. In
P. J. Brantingham & P. L. Brantingham (Eds.), Environmental criminology (pp.
27-54). Beverly Hills: Sage.

Rhodes, W. M., & Conly, C. (1981). Crime and mobility: An empirical study. In P. J.
Brantingham & P. L. Brantingham (Eds.), Environmental criminology (pp. 167-188).
Beverly Hills: Sage.

Beverly Hills: Sage.Brantingham, P. J., & Brantingham, P. L. (1984). Patterns in
crime. New York: Macmillan.

Newton, Jr., M. B., & Swoope, E. A. (1987). Geoforensic analysis of localized serial
murder: The Hillside Stranglers located. Unpublished manuscript.

Rossmo, Kim D. (1995). Geographic profiling: target patterns of serial murderers.
Simon Fraser University. p. 225.

Egger SA 1984. A working definition of serial murder and the reduction blindness.
Journal of police science and administration 12: 348–387.


Holmes RM & DeBurger JE 1998. Profiles in terror: the serial murderer, in Holmes
RM & Holmes ST (eds), Contemporary perspectives on serial murder. Thousand
Oaks: Sage: 5–16

Dietz PE, Hazelwood RR & Warren J 1990. The sexually sadistic criminal and his
offenses. Bulletin of the American Academy of Psychiatry and the Law 18(2): 163–
178

Myers WC et al. 1993. Malignant sex and aggression: an overview of serial sexual
homicide. Bulletin of the American Academy of Psychiatry and the Law 21(4): 435–
451

Cantor DV et al. 2000. Predicting serial killers’ home base using a decision support
system. Journal of quantitative criminology 16(4): 457–478

Fox JA & Levin J 2005. Extreme killing: understanding serial and mass murder.
Thousand Oaks CA: Sage publications

Douglas JE et al. 1992. Crime classification manual: a standard system for
investigating and classifying violent crimes. San Francisco, CA: Jossey-Bass

Aki K 2003. Serial killers: a cross-cultural study between Japan and the United States.
Graduate thesis: California State University

Brantingham, P.J. and P.L. Brantingham (eds.) (1981). Environmental Criminology.
Beverly Hills, CA: Sage.

Rengert, G.F. (1991). The Spatial Clustering of Residential Burglaries About Anchor
Points of Routine Activities." Paper presented at the meeting of the American Society
of Criminology, San Francisco, CA.

Brantingham, P. L., & Brantingham, P. J. (1981). Notes on the geometry on crime. In
P. J. Brantingham & P. L. Brantingham (Eds.), Environmental criminology (pp.
27-54). Beverly Hills: Sage.

Rhodes, W. M., & Conly, C. (1981). Crime and mobility: An empirical study. In P. J.
Brantingham & P. L. Brantingham (Eds.), Environmental criminology (pp. 167-188).
Beverly Hills: Sage.

Beverly Hills: Sage.Brantingham, P. J., & Brantingham, P. L. (1984). Patterns in
crime. New York: Macmillan.

Newton, Jr., M. B., & Swoope, E. A. (1987). Geoforensic analysis of localized serial


murder: The Hillside Stranglers located. Unpublished manuscript.

Rossmo, Kim D. (1995). Geographic profiling: target patterns of serial murderers.
Simon Fraser University. p. 225.

Levine, N. (2009a). CrimeStat: A spatial statistics program for the analysis of crime
incident locations. Ned Levine & Associates, Annandale, VA and the National
Institute of Justice, Washington, DC. Retreived July 20 2009 from
http://www.icpsr.umich.edu/crimestat

Mike O’Leary N. (2009). The Mathematics of Geographic Profiling. Journal of
Investigative Psychology and Offender Profi ling. J. Investig. Psych. Offender Profi l.
6: 253–265

Wikipedia 2009. List of geographical profiling http:// en.wikipedia.org/wiki/
Geographic_profiling.

Brantingham,P.L.,&Brantingham,P.J.(1981).Notes on the geometry on crime.
In P.J.Brantingham&P.L.Brantingham(Eds.),Environmental criminology(pp.
27-54).Beverly Hills:Sage.

Vold,G.B.,&Bernard,T.J.(1986).Theoretical criminology.New York:Oxford
University Press.

Williams, F. P., & McShane, M. D. (1988). Criminological theory. Englewood Cliffs,
NJ: Prentice Hall.

Appendixes

function p = model1killrate()
x = [ 835.9544 934.2431 972.8121 726.4682 936.7314 274.8376 720.2473 710.2941
298.4766 629.4236 727.7123 808.5828 896.9183];
y = [38.1518 102.8482 53.0817 96.6274 86.6741 677.6507 111.5573 340.4830
625.3960 248.4151 130.2197 100.3599 77.9650];
sum = 0;
sumx = 0;
sumy = 0;
for i=1:13


sumx = sumx+x(i);
sumy = sumy+y(i);
end
centerx = sumx/13;
centery = sumy/13;
temp = 0;
for j=1:13
temp = temp+(x(j)-centerx)^2+(y(j)-centery)^2;
end
temp = temp/13;
rfang = sqrt(temp);
for m = 1:730
for n = 1:1200
temp1 = n-centerx;
temp2 = m-centery;
oshi(m,n) = sqrt(temp1^2+temp2^2);
if oshi(m,n) > rfang
p(m,n) = 1/(oshi(m,n)*1.828889e+003);
else
p(m,n) = (rfang)/((2*rfang-oshi(m,n))^2*1.828889e+003);
end
end
end
[X1,Y1] = meshgrid(1:1:1200,1:1:730);
mesh(X1,Y1,p)

function fun = model2homelocal()
x = [265.0220 253.9240 321.9916 259.8429 310.1537 297.5760 224.3294 349.3666
200.6537 222.1098 267.9814 228.0287];
y = [ 59.3193 71.8970 81.5152 97.7922 98.5321 88.9139 215.4307 94.0929 128.8666
255.3834 119.9882 327.1503];
sum = 0 ;
k=1;
sumx=0;sumy=0;
upmax = 0;max =0;downmax = 0;
for i=1:12
sumx = sumx+x(i);
sumy = sumy+y(i);
end
centerx = sumx/12;
centery = sumy/12;
temp = 0;
for j=1:12


end
temp = temp/12;
rfang = sqrt(temp);
for m = 1:450
for n = 1:680
temp4 =0;
for i=1:12
temp1 = n-x(i);
temp2 = m-y(i);
manhaten = abs(temp1)+abs(temp2);
if manhaten > rfang
temp3 = 1/(manhaten*k);
else
temp3 = rfang/((2*rfang-manhaten)^2*k);
end
temp4 = temp4+temp3;
end
p(m,n)=temp4;
end
end
[X1,Y1] = meshgrid(1:1:680,1:1:450);
meshc(X1,Y1,p)

function fun = model2killratepredict()
x = [ 835.9544 934.2431 972.8121 726.4682 936.7314 274.8376 720.2473 710.2941
298.4766 629.4236 727.7123 808.5828 896.9183];
y = [38.1518 102.8482 53.0817 96.6274 86.6741 677.6507 111.5573 340.4830
625.3960 248.4151 130.2197 100.3599 77.9650];
sum = 0 ;
p1=0;p2=0;p3=0;p4=0;p5=0;p6=0;
sumx = 0;
sumy = 0;
for i=1:13
sumx = sumx+x(i);
sumy = sumy+y(i);
end
centerx = sumx/13;
centery = sumy/13;
temp = 0;
for j=1:13


end
temp = temp/13;
rfang = sqrt(temp);
homex = [857 547 817 892 468];
homey = [315 10 347 263 83];
temp = [0 0 0 0 0 ];
for m = 1:730
for n = 1:1200
for i=1:5
temp1 = n-homex(i);
temp2 = m-homey(i);
manhaten = abs(temp1) + abs(temp2);
if manhaten > rfang
temp(i) = 1/(manhaten*1.569547e+003);
else
temp(i) = 1*rfang/((2*rfang-manhaten)^2*1.569547e+003);
end
end
p(m,n)=1.784/8.307*temp(1)+1.617/8.307*temp(2)+1.671/8.307*temp(3)+1.65
2/8.307*temp(4)+1.583/8.307*temp(5);
end
end
[X1,Y1] = meshgrid(1:1:1200,1:1:730);
meshc(X1,Y1,p)
figure;
contour3(X1,Y1,p);

Model3:
map1=imread('GISdiejia1.jpg');
map2=imread('GISdiejia2.jpg');
n=2;
hh=zeros(11*n,11*n);
for i=1:1/n:10
for j=1:1/n:10
x=i-5;
y=j-5;
hh(i*n,j*n)=-2/9*exp(-1/9*(x.^2+y.^2)^2)+4/81*(x.^2+y.^2)^2*exp(-1/9*(x.^2+y.^
2)^2);%
end


end
figure,mesh(hh)
imsub=double(map2(:,:,2))-double(map1(:,:,2))+double(map2(:,:,3))-
double(map1(:,:,3));
figure,imshow(map2)
imfiltered=imfilter(imsub,hh);
imfiltered=imfiltered.*(imfiltered>0);
figure,imshow(imfiltered,[]);
figure,imshow(-imfiltered,[]);

im = imread('PSbig.png');
image(im); axis image
n=13;
coordinates=zeros(n,2);
xy = [];
n = 0;
disp('Left mouse button picks points.')
disp('Right mouse button picks last point.')
but = 1;
while but == 1
[xi,yi,but] = ginput(1);
hold on;
plot(xi,yi,'w*')
n = n+1;
xy(n,:) = [xi,yi]
end

2010 mcm predictions of locations of the crimes based on geographical profile

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2010 mcm predictions of locations of the crimes based on geographical profile