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

Team # 7520 of 36

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.

Team # 7520 of 36

Contents

1 INTRODUCTION..................................................................................................................................3

1.1 PROBLEM RESTATEMENT...........................................................................................................3

1.2 CONVENTIONS...........................................................................................................................4

1.3 OUR RESULTS............................................................................................................................6

2 ASSUMPTIONS AND ASSUMPTION JUSTIFICATIONS.............................................................7

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

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

4.1 CRIME SPATIAL PATTERN.................................................................................................................9

4.2 PROBABILITY DISTRIBUTION OF CRIME SITE...................................................................................9

4.3 BUFFER ZONE.................................................................................................................................10

4.4 DISTANCE MEASUREMENT STANDARD...........................................................................................10

5 STATISTICAL CENTROID SCHEME............................................................................................11

5.1 MATHEMATICAL INTERPRETATION..................................................................................................11

5.2 SIMULATE THE SCHEME.................................................................................................................14

5.3 RESULT OF THE SCHEME................................................................................................................14

6 PARETO TOTAL PROBABILITY SCHEME................................................................................16

6.1 LOCATION OF THE ANCHOR POINT.................................................................................................16

6.2 PREDICTION OF CRIME PROBABILITY.............................................................................................18

7 TECHNIQUE OF COMBINING RESULTS TO GENERATE A USEFUL PREDICTION.....20

7.1 STATISTICAL CENTROID SCHEME AND PARETO TOTAL PROBABILITY SCHEME.............................20

7.2 PREDICTION OF ATTACK SITE BASED ON GEOGRAPHICAL PROFILE..............................................21

8 INCORPORATE GEOGRAPHIC ROAD DATA FOR PREDICTION.......................................22

8.1 TAKE INTO ACCOUNT OTHER FACTORS TO OPTIMIZE GEOGRAPHICAL PROFILE..............................22

8.2 GEOGRAPHIC ROAD SCHEME.........................................................................................................23

8.3 INCORPORATION OF DIFFERENT SCHEMES.....................................................................................25

9 MEASURES OF VALIDITY AND RELIABILITY.........................................................................26

9.1 MEASURES OF VALIDITY................................................................................................................26

9.2 ANALYSIS OF RELIABILITY.............................................................................................................27

10 RESULTS FOR OTHER SERIAL CRIMINALS.........................................................................28

11 SENSITIVITY TO PARAMETERS...............................................................................................30

11.1 SAMPLE NUMBER..........................................................................................................................30

11.2 DISTANCE MEASUREMENT METHOD............................................................................................31

12 STRENGTHS AND WEAKNESSES..............................................................................................32

12.1 STRENGTHS..................................................................................................................................32

Team # 7520 of 36

12.2 WEAKNESSES................................................................................................................................32

13 ALTERNATIVE APPROACHES AND FUTURE WORK.........................................................33

14 CONCLUSIONS...............................................................................................................................34

REFERENCES.......................................................................................................................................35

Appendixes..............................................................................................................................................40

1 Introduction

1.1 Problem 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

Team # 7520 of 36

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.

1.2 Conventions 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.

Team # 7520 of 36

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 VariablesWe 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.

refers to the spatial mean of crime sites.

refers to the total number of crime sites.

, refer to the coordinates of the th crime site.

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

Team # 7520 of 36

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

Team # 7520 of 36

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:

http://en.wikipedia.org/wiki/Serial_murder

Team # 7520 of 36

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: ;

Negative exponential: ;

Normal: ;

Lognormal: ; and

Truncated negative exponential: if and if

.

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

Team # 7520 of 36

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 FormulaWe 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:

where :

Team # 7520 of 36

and:

is the coordinate of the spatial mean;

is the coordinate of the spatial mean;

is the total number of crime sites; and

, are the coordinates of the th crime site.

Radius of Buffer Zone 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:

where: is the standard distance; is the total number of crime sites;

is the distance between the centroid and the th crime site.

and:

From the formula above we can get the average distance of the crime sites . 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 SitesIn 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:

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.

Team # 7520 of 36

Rossmo's formula is defined as:

where:

and:

is the resultant probability for point ；

is a weighting factor;

is an empirically determined constant;is the radius of the buffer zone;is the number of crime sites;

is an empirically determined exponent;

is an empirically determined exponent;

are the coordinates of point

are the coordinates of the 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;

Team # 7520 of 36

Y-axis represents the Probability of Crime Sites

( , , )

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 ;

Step4. Use the 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.

Team # 7520 of 36

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.

Team # 7520 of 36

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 ;

Step4. Use the 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.

Team # 7520 of 36

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 committed by a single serial

offender, estimate the probability density that will be the location of the next

offence. The Bayesian approach to this problem is to calculate the posterior predictive distribution.

where :is the location of the offender’s anchor point;

is the average distance that the offender is willing to travel to offend;

is the probability that an offender with a single stable anchor

point and average offence distance commits a crime at the location .

John Wiley & Sons, Ltd. (2009) simplified (0.10) like this:

where :

is the prior probability density function for the distribution of anchor

points before any information from the crime series is included;

Team # 7520 of 36

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 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 is used as the

anchor point we calculate the probability distribution of the anchor

point in every other point and do .

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

. 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.

Team # 7520 of 36

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.

Team # 7520 of 36

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 abilityBy 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.

Team # 7520 of 36

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:

Team # 7520 of 36

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.

The factor is used to account for the local geographic features that influence

the selection of a crime site. High values for indicate that is a likely target

for typical offenders; low values indicate is a less likely target. The remaining factor is a normalization required to ensure that is a probability distribution. Its value is completely determined by the choices of and 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

.

Step3. Take the G and B components in the RGB map, then minus them respectively. Add the results as .

Step4. Make convolution formula operation on .Step5. Remove the part where the value is less than zero in 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.

Team # 7520 of 36

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.

Team # 7520 of 36

Test of Real ProbabilityFor 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

Team # 7520 of 36

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.

Team # 7520 of 36

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.

Team # 7520 of 36

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

Team # 7520 of 36

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.

Team # 7520 of 36

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

Team # 7520 of 36

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

Team # 7520 of 36

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

Team # 7520 of 36

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 serialmurder: 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

Team # 7520 of 36

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 serialmurder: 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/

Team # 7520 of 36

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:OxfordUniversity 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);endcenterx = sumx/13;centery = sumy/13;temp = 0;for j=1:13 temp = temp+(x(j)-centerx)^2+(y(j)-centery)^2;endtemp = 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

Team # 7520 of 36

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);endcenterx = sumx/12;centery = sumy/12;temp = 0;for j=1:12 temp = temp+(x(j)-centerx)^2+(y(j)-centery)^2;endtemp = 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

Team # 7520 of 36

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);endcenterx = sumx/13;centery = sumy/13;temp = 0;for j=1:13 temp = temp+(x(j)-centerx)^2+(y(j)-centery)^2;endtemp = 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

Team # 7520 of 36

p(m,n)=1.784/8.307*temp(1)+1.617/8.307*temp(2)+1.671/8.307*temp(3)+1.652/8.307*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);% endendfigure,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 imagen=13;coordinates=zeros(n,2);

Team # 7520 of 36

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

Technology