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2010MCM Predictions of Locations of the Crimes Based on Geographical Profile

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- 1. Team # 7520 Page 1 of 37Predictions of Locations of the Crimes Based on Geographical ProfileSummary To make a prediction of the locations of the next crime, we have formulated andtested 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 bulls-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 RoadScheme for the generation of the geographical profile of a suspected serial criminalthat can help us to make predictions on the next crime site based on the past locationsof crimes and the distribution of the roads. A geographical profile describes anoptimal search process. To save the cost of search and improve the efficiency as muchas possible, a search should starts from the highest area and works down, which ismore likely to find the offender’s residence sooner. So the geographical profile is ascientific guide to plan the deployment of police force. In addition, we also give the measures of validation such as learning efficiencyand prediction accuracy to test different schemes, an analysis of the reliability ofdifferent algorithms in different circumstances and test of the sensitivity of parametersof our schemes. We also analyzed another case to test how well our schemesperformed on different serial criminals.
- 2. Team # 7520 Page 2 of 37Contents1 INTRODUCTION.................................................................................................................................4 1.1PROBLEM RESTATEMENT.........................................................................................................................4 1.2CONVENTIONS ......................................................................................................................................4 1.3OUR RESULTS.......................................................................................................................................52 ASSUMPTIONS AND ASSUMPTION JUSTIFICATIONS.............................................................63 LITERATURE REVIEW ....................................................................................................................74 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.....................................................................................................95 STATISTICAL CENTROID SCHEME............................................................................................10 5.1 MATHEMATICAL INTERPRETATION..........................................................................................................10 5.2 SIMULATE THE SCHEME.......................................................................................................................13 5.3 RESULT OF THE SCHEME......................................................................................................................136 PARETO TOTAL PROBABILITY SCHEME.................................................................................14 6.1 LOCATION OF THE ANCHOR POINT.........................................................................................................14 6.2 PREDICTION OF CRIME PROBABILITY.....................................................................................................167 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.............................................................198 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...............................................................................................229 MEASURES OF VALIDITY AND RELIABILITY.........................................................................23
- 3. Team # 7520 Page 3 of 37 9.1 MEASURES OF VALIDITY.....................................................................................................................23 9.2 ANALYSIS OF RELIABILITY...................................................................................................................2310 RESULTS FOR OTHER SERIAL CRIMINALS..........................................................................2411 SENSITIVITY TO PARAMETERS................................................................................................26 11.1 SAMPLE NUMBER .............................................................................................................................26 11.2 DISTANCE MEASUREMENT METHOD....................................................................................................2712 STRENGTHS AND WEAKNESSES..............................................................................................28 12.1 STRENGTHS......................................................................................................................................28 12.2 WEAKNESSES...................................................................................................................................2813 ALTERNATIVE APPROACHES AND FUTURE WORK...........................................................2914 CONCLUSIONS...............................................................................................................................29REFERENCES.......................................................................................................................................30APPENDIXES........................................................................................................................................33
- 4. Team # 7520 Page 4 of 371 Introduction1.1Problem RestatementWhat are the serial criminals? Like serial murder, there is a lack of consensus amongacademics and practitioners in the definition. In terms of serial murder, disagreementcenters on the number of victims, the presence/absence of a sexual element, and thecommon 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 serialmurder is used in the current research. In accordance with the crime classificationmanual developed by the FBI, serial murders are those that involve three or moreseparate events (Douglas et al., 1992), and most importantly, are repetitive sequentialhomicides of any nature. Frequently, serial murders involve a similarity of subject orpurpose (For example, the choice of victims, methods of killings, or the killer’smotivation; Aki, 2003: 6). Usually we can judge the law of committing the crime and establish theprediction scheme according to those intentions and other objective factors, such aslocation, population density distribution, transportation. Then using the predictionscheme we can predict the address of the criminal and the next crime site. However, itis difficult to determine the proportion of these factors in the process of judgment andprediction. There is no uniform standard. Previous studies and models mostly consideronly one factor (just as the distance) and get a prediction. The advantage is that themodel can be more widely applied to other serial criminals without many limitationsand is easy to operate. However, its disadvantages are also obvious. Because thefactor 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 geta satisfactory prediction. Different schemes will produce different probability distribution figures. We needto integrate these results and ultimately generate a geographical profile. Then wedevelop a technique to combine the results of the different schemes and generate auseful prediction. With this in mind we embark on our journey about criminalprofiling.1.2Conventions1.2.1 Terminology
- 5. Team # 7520 Page 5 of 37 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 VariablesWe will define the following variables here as they are used widely throughout ourpaper. Additional variables may be defined later, but will be confined to a particularsection. ( 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 ResultsTo make a prediction of the locations of the next crime based on the time andlocations of the past series crime scenes, we have formulated and tested three schemesfor geographical profiling with the aid of computer. The basic characteristics and
- 6. Team # 7520 Page 6 of 37scope 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 bulls-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 JustificationsAbout 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
- 7. Team # 7520 Page 7 of 37 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 ReviewBefore introducing our own model, we first provide a very brief overview ofgeographic profiling. The problem of geographic profiling is studied by somescientists such as Kim Rossmo from the 20th century. Geographical patterns in crime have been noted since the mid-19th centurypioneering work of Andre-Michel Gerry and Lambert-Adolph Quenelle who mappedinformation about violent and property offences and examined their spatialrelationship to poverty (Brantingham&Brantingham, 1981c; Vold&Bernard, 1986).The most famous spatial crime studies were conducted in the early 20th century, whenthe city of Chicago served as an inspiration source and a field of experimentation forUniversity of Chicago sociologists (Warren, 1972; Williams&McShane, 1988). Thegeographical focus of criminology had shifted from regional areas to cityneighborhoods. Geographic profiling is a criminal investigative methodology that analyzes thelocations of a connected series of crimes to determine the most probable area ofoffender residence. Typically used in cases of serial murder or rape (but also arson,bombing, robbery, and other crimes), the technique helps police detectives prioritizeinformation in large-scale major crime investigations that often involve hundreds orthousands 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
- 8. Team # 7520 Page 8 of 37 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 DesignBefore the scheme design, we need to determine a number of conditions, namelycrime spatial pattern, probability distribution of crime site, buffer zone and distancemeasurement standard, so that we can operate on the schemes. Our analysis of theseconditions is as follows.4.1 Crime Spatial PatternIn the simplest case, offenders’ residences lie at the centre of their crime patterns andcan be found through the spatial mean (Brantingham and Brantingham, 1981). Theintricacy of most criminal activity spaces, however, indicates that more complexpatterns are the norm. George Rengert (1996) proposes four hypothetical spatial patterns for thegeography 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 patternand apply it to the third and fourth patterns. We can integrate two bull’s-eye patternsand get a new pattern which equals to the bimodal pattern. Just like this, we can alsointegrate two bull’s-eye patterns which have different weights and get a new patternwhich 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 SiteUsually there are some laws between the probability distribution of crime site and the
- 9. Team # 7520 Page 9 of 37distance between crime site and anchor point. The offender’s search behavior followssome 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, startingfrom the sites and routes that compose the activity space and then decreasing asdistance away from the activity space increases. The Pareto function, named after theItalian economist, is suitable for fitting data that have a disproportionate number ofcases close to the origin, making it appropriate for modeling distance-decay processes(Brantingham & Brantingham, 1984). We will discuss the concrete form ofdistribution function in Section 5.1.4.3 Buffer ZoneThere 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 thiszone, targets are viewed as less desirable because of the perceived level of riskassociated with operating too close to home. For the offender, this area represents anoptimized balance between the maximization of opportunity and the minimization ofrisk. The buffer zone is most applicable to predatory crimes; for affective-motivatedoffences it takes on less importance, as can be seen by the fact that domestichomicides usually occur within the residence. The radius of the buffer zone isequivalent 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 StandardUsually people use one of the two methods (ie, the Euclidean distance and theManhattan distance) to measure two points in a map. When the influence of the streetsis obvious, that is because the two sites in the map is close we can’t ignore the streetinfluence on the distance, we use the Manhattan distance. If the two sites are far from
- 10. Team # 7520 Page 10 of 37each other we choice the Euclidean distance. Based on the above considerations, we require: when the two sites are in the samecity we choice the Manhattan distance. Otherwise, we choice the Euclidean distance.5 Statistical Centroid SchemeWe design the Statistical Centroid Scheme based on the statistical laws of thepredecessors. The crime sites are always around the anchor point. Therefore we treatthe centroid of the known crime sites as the anchor point. According to this we predictthe probability distribution of crime sites.5.1 Mathematical InterpretationIn this subsection, we describe the mathematics of the centroid, the radius of thebuffer 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 meancentre) 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 BSo far, there is no uniform standard for dividing buffer zone. According to the viewthat the radius of the buffer zone is equivalent to the modal crime trip distance, we
- 11. Team # 7520 Page 11 of 37just 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 thestandard 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 tripdistance is the distance between a crime site and the offender’s residence. We can geta better result in that way. Function of the Probability Distribution of Crime SitesIn section 4.2 we have already analyzed that the relationship between crimeprobability and the distance between the crime site and anchor point can be modeledby 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 ofprobability distribution of crime sites, the Rossmo’s formula proposed bymathematician Kim Rossmo (1995) is able to obtain a satisfactory result through thetest of practice. Rossmos 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;
- 12. Team # 7520 Page 12 of 37 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. Rossmos formula reflects that the Probability Distribution of Crime Sitesincreases first and then decrease with the increasing of the distance between crime siteand the anchor point. Result is shown in Figure 1. Figure 1.the Rossmos 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 commitcrimes in the area closer to anchor point because they could be easily identified. Asthe distance from anchor point increases, the likelihood of offenders selecting thetarget increases. But when it up to a certain value the likelihood decreases because itis too far from the anchor point.
- 13. Team # 7520 Page 13 of 375.2 Simulate the SchemeIn section 5.1 we have defined the formulas we will use in the scheme. Now we showthe 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 SchemeThrough the Statistical Centroid Scheme we get the figure of the ProbabilityDistribution 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 thedata 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 avolcano. The center of the volcano is the anchor point. Probability distribution
- 14. Team # 7520 Page 14 of 37increases first and then decreases. Figure 3 is the contour map representation of theFigure 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 arelocated in the relatively high probability of committing the crime. For this serialmurders the assumptions and reasoning of the Statistical Centroid Scheme is in linewith the actual situation.6 Pareto Total Probability SchemeDifferent from the Statistical Centroid Scheme the Pareto Total Probability Schemeuses backstepping to predict the probability distribution of the anchor point accordingto the probability distribution of crime sites. Relative to the assumption of Centroid inthe Statistical Centroid Scheme this method can determine the location of the anchorpoint more reasonable and is more operational. This method is proposed by KimRossmo and the accuracy rate is high by a large number of practical testing.6.1 Location of the Anchor PointIn this method we will use the formulas definition in Section 5.1. The steps of themethod are as follows:
- 15. Team # 7520 Page 15 of 37 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 probabilitydistribution of the anchor point. Figure 4 is the draw the three-dimensional map of theprobability distribution of the anchor point using the data of the serial murders ofPeter Sutcliffe in 1981. The red areas in the map are the most likely location of theanchor 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 crimesites of Peter Sutcliffe and the centroid of the crime sites in this map. From Figure 5we can see that the centroid is not located in the red areas in the map. This representsthat there are some differences on the prediction between the Statistical CentroidScheme and the Pareto Total Probability Scheme.
- 16. Team # 7520 Page 16 of 37 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 policecan start searching from the high probability areas. This method can reduce thepolice’s search areas effectively and accelerate the speed of detection.6.2 Prediction of Crime ProbabilityGiven a series of crimes at the locations x1 ,..., xn committed by a single serialoffender, estimate the probability density that xnext will be the location of the nextoffence. The Bayesian approach to this problem is to calculate the posterior predictivedistribution. 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 )
- 17. Team # 7520 Page 17 of 37 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 probabilitydistribution 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 ProbabilityDistribution of Crime Sites different from the figure get from the Statistical CentroidScheme. Figure 6 is based on the result of Figure 5. The high-level value is theprobability of every point. Figure 6. the three-dimensional map of the probability distribution of anchor point
- 18. Team # 7520 Page 18 of 37 Figure 5 is the contour map representation of the Figure 4. We mark the 13 crimesites of Peter Sutcliffe in this map. From the map we can see most sites are located inthe relatively high probability of committing the crime. For this serial murders theprediction ability of the Pareto Total Probability Scheme is strong. Figure 7. the contour map of the probability distribution of the anchor point7 Technique of combining results to generate a usefulprediction7.1 Statistical Centroid Scheme and Pareto Total Probability Scheme Both Statistical Centroid Scheme and Pareto total probability Scheme aregeographical profiles for the criminals offence probability distribution situation basedon the Rossmo formula and the distance variable of both are Manhattan distance,thought they have different assumptions and solving methods. Statistical CentroidScheme assumed that the offender’s lodging is the center of mass which is determinedby all the known attack sites. Regard the lodging as the center, the possibility ofanother attack for an arbitrary spot fits the Rossmo formula. Therefore, we get theoutline of the geographical profile of offender’s another attack, which fits the crimespace model of bull’s-eye. Pareto total probability Scheme takes the assumption of ‘center of mass is thelodging’ for a grand. It adopts Rossmo equation to describe the probabilitydistribution and the attack rate of every zone is the overlay of a single attack rate forevery possibility lodging of offender. Therefore, we get a more accurate geographicalprofile about the offender’s lodging. Then, for every possibility of offender’s lodging,
- 19. Team # 7520 Page 19 of 37take integration for the next attack possibility.7.2 Prediction of Attack Site Based on Geographical ProfileThe principles to combine the geographical profiles generated by Statistical CentroidScheme 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
- 20. Team # 7520 Page 20 of 37could 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 fordifferent situation and purpose.8 Incorporate Geographic Road Data for prediction8.1 Take into account other factors to optimize geographical profile Many different crime factors and environmental elements are considered in theconstruction and interpretation of a geographic profile. (Kim Rossmo, 2000)Theschemes established above are mainly from the perspective of the spatial distributionof 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.
- 21. Team # 7520 Page 21 of 378.2 Geographic Road Scheme Our comprehensive consideration about the impact of geographical distribution ofthe traffic road on the probability of crime bring out the Geographic Road Schemebased on the geographical distribution of the traffic road. The main assumptions are asfollowings: 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 intothe model. Indeed, let us suppose that offender target selection depends on more thanjust the distance from the anchor point to the crime site locations, but that it dependson some features in the local geography. One way to account for this is to suppose thatthe offence probability density is proportional to both a distance decay term and to afunction that measures the attractiveness of a particular target location. ( MikeO’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 influencethe selection of a crime site. High values for G ( x) indicate that x is a likely target fortypical 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 valueis completely determined by the choices of D and G and has the form. We use the difference between two normal distribution functions to describedistribution of crime probability where is centered with the point in the roads .Thesteps 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 .
- 22. Team # 7520 Page 22 of 37Step5.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 inFigure 10.The deeper the color is the larger the probability of next serial criminal. Figure 10.The geographical profile based on the Geographic Road Scheme8.3 Incorporation of Different Schemes To obtain a more precise geographical profile, we combine the geographicalprofile base on the Geographical Road Scheme And the Pareto Total ProbabilityScheme .Then we get a figure which shows both the impacts of roads and The spatialdistribution 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 criminalis. For contour lines, each line represents the level of probability values marked onthe figure. Figure 11.The geographical profile based on the Pareto- total probability Scheme and the Geographic Road Scheme
- 23. Team # 7520 Page 23 of 379 Measures of Validity and Reliability9.1 Measures of ValidityThe methods of the evaluation of the validity of geographical profile include the testof 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 tobuild models and generate a geographical profile. Then make use of data of crimeoccurred late to test the accuracy of forecasts of the geographical profile. As for realcrime site, the greater the predicted values of crime probability, the better the modelis. 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 geographicalprofile. The success of the model is measured by the hit score percentage --- the ratioof the total number of points with scores equal or higher to the hit Score, to the totalnumber of points within the hunting area. This is equivalent to the percentage of thetotal area that must be searched before the offender’s residence is found, assuming anoptimal search process (i.e., one that started in the locations with the highest scoresand then worked down). The extent of the search area --- the territory police have tosearch in order to find the offender --- is equal to the size of the hunting areamultiplied by the hit score percentage. The smaller the hit score percentage is, thebetter 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 ReliabilityOur model has its own reasonable applications occasions .It’s necessary to analysisthe reliability of our models in different occasions . There is an element of subjectivityin 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.
- 24. Team # 7520 Page 24 of 37 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. TheStatistical Centroid Scheme is based on the assumption that the anchor point is nearthe Centroid of the crime sites. It’s applicable in terms with offenders whose spatialpattern of crime is bulls-eye pattern (George Regret ,1996) .They have a fixedresidence and are accustomed to committing the crime where is near the region ofwhich the center is their anchor point, rather than itinerant criminal. As for thisscheme, we only consider a single anchor point. We use Manhattan distance to metricspace and Rossmo’ formula describe the probability distribution of the crime sitesaround the anchor point. The Pareto Total Probability Scheme is applicable in termswith offenders whose spatial patterns of crime are various patterns such as bull’s-eyepattern, 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 anumber of directions or a fixed residence between their workplace and residence. Thedistribution of the probability of committing crime is anisotropic. The GeographicalRoad Scheme is suitable for series of robbers and offenders who flee everywhere. Weconsider the flee route of the offenders and find that they prefer to choose the placewhich is not far from the main roads. However, they would not choose to commitcrimes on the roads in order to prevent possible exposure of their criminal acts. Theirstrategy of the selection of possible crime sites is within a fixed limit of the narrowarea to balance the risks of possible exposure of their criminal acts and opportunitiesto escape form criminal scene.10 Results for Other Serial CriminalsIn 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 caseand the triangle shows the place of committing a crime. Different from Peter Sutcliffe case, the traffic is extremely convenient around theoffence place in this case, and the majority places of committing a crime areconcentrated on hub of communications. Compared to Sutcliffe, the murderer in thiscase has larger escaping distance within the same time. From Statistical CentroidScheme, we know that the larger distance from murder’s lodging, the lower possibility
- 25. Team # 7520 Page 25 of 37he commits a crime. Therefore, the murder may choose commit a crime remotely andin this case, we choose the Pareto Total Probability Scheme, which turn out toperformer 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 murderers real offenceplace has landed on two areas of A, B mostly, which had fully proved the offenceprobability distribution that model two predicts has stronger practicability. In mostinstances, the police can fixes up police strength according to the offence probabilitydistribution depicted by scheme2, which improved the hunting probability to a certainextent.
- 26. Team # 7520 Page 26 of 3711 Sensitivity to Parameters11.1 Sample number In this section, we use different sample number to test the accuracy of crimerate’s prediction of Pareto Total Probability Scheme and Statistical Centroid Scheme.Take Peter Sutcliffe for example, the sample number is selected according to themurder time’s order. In this case, we assumed that 5 crime sites are known. Thencalculate the 6th crime site’s rate using Pareto Total Probability Scheme and StatisticalCentroid Scheme separately. After that, calculate the 7th crime site’s rate if 6 crimesites are known. Do this until 12 crime sites are known. The result is shown asFigure8. Figure 13. sensitivity of Sample number. From Figure13 we know that, both of the two schemes’ prediction rate presentsthe law of rising first then reducing and rising again at last, along with the samplenumber’s increase, which fits the common sense appropriately. For the Pareto Total Probability Scheme, since the 6th and 9th crime site are farfrom the rest of the crime site, which is not fit the normal principle. Therefore, whenwe adopt the former 6 crime sites as the sample number to calculate the 7th crimesite’s rate, the result turn out to be some big deviation. As a result, the 8th calculateresult seem to be influenced to some extent. However, the 9th result rises again due tothe closed distance from 6th crime site. After that, the prediction rate seems to be morestable because there is no more abnormal crime site again. Statistical Centroid Scheme’s changing pattern similar the Pareto Total
- 27. Team # 7520 Page 27 of 37Probability Scheme, but has some certain of lag, which means the Statistical CentroidScheme is no so sensitive as the Pareto Total Probability Scheme on the changing ofthe sample number. For vertical comparison, we could see that the Pareto Total Probability Scheme’sprediction results are superior to the Statistical Centroid Scheme’s. Therefore, thePareto Total Probability Scheme has a higher predictive accuracy.11.2 Distance Measurement MethodIn this section, we want to test whether the Euclidean distance or the Manhattandistance has higher prediction accuracy in the Pareto Total Probability Scheme. Thedetach method is similar to that described in section13.1. The difference is just toreplace the Manhattan distance with the Euclidean distance. Figure9 shows the resultsby using different distance type. Figure 15. sensitivity analysis for distance measurement From horizontal view, both of the two curves’ change pattern is similar, which
- 28. Team # 7520 Page 28 of 37shows 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 theEuclidean distance’s and has higher prediction accuracy.12 Strengths and Weaknesses12.1 StrengthsThe 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%, whichare 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 themain transport routes. Then we make a integration among the former schemes and thesituation, in this way, we could get a more accurate prediction.12.2 WeaknessesThere are so many influence factors to take into consideration and it is hard toconstruct a model concluding all of them. Besides, our model is based on the happened cases, so there is inevitable tohave some conditions which are not suitable for the case. For example, the CGTmodel works on the assumption that a relationship, modeled on some form ofdistance-decay function, exists between crime location and offender residence. Sincethe model cannot locate the residence of a criminal that lies outside of the boundariesof the hunting area map, it is necessary to limit the process to non-commutingoffenders.
- 29. Team # 7520 Page 29 of 3713 Alternative Approaches and Future WorkThere were several extensions to our approach that we could not pursue due to thetime 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 theoffender spent to commit a crime every time. Then we combine the time and the speedof the offender to calculate the radius of the offender activity space. By drawing thecircles with the known crime sites as the center one at a time we can get a map withmany overlaps on it. These overlaps are most probably the location of the anchorpoint. After integrate the result of this scheme and the scheme we design in abovesections we can predict the probability distribution of the anchor point moreaccurately. In addition, we need further analysis for the influence of the traffic in theGeographical Road Scheme. Quantify it and get the weights of different trafficconditions. Thus we can combine the Geographical Road Scheme with the other twoschemes. We can also consider the population density just like the traffic and the dataof it is not difficult to obtain. At last, to promote the approach, we have already discussed it in Section 4.1. Ourapproach takes into account the second pattern. For different patterns (except foruniform pattern) we can promote our approach and apply it to them. For the bimodalpattern, the distribution of the known crime sites mainly focuses on two areas. We candivide all the crime sites into two parts with one cluster in each. We use our approachon each part and integrate the results at last. For the teardrop pattern, like the bimodalpattern we also divide all the crime sites into two parts with one cluster in each. Butwe should add the different weights on the two parts. Then use our approach on eachpart and integrate the results. According to this idea our approach can be applied tomore kinds of serial criminals.14 ConclusionsWe have formulated and tested three schemes for geographical profiling with the aidof computer. The Statistical Centroid Scheme is based on the assumption that theanchor point is near the Centroid of the crime sites. It’s applicable in terms withoffenders whose spatial pattern of crime is bulls-eye pattern (George Regret ,1996) .The Pareto Total Probability Scheme is applicable in terms with offenders whosespatial patterns of crime are various patterns such as bull’s-eye pattern, bimodalpattern, tear drop pattern and so on. The Geographical Road Scheme is suitable for
- 30. Team # 7520 Page 30 of 37series of robbers and offenders who flee everywhere. We find that the learning efficiency and prediction accuracy of the Pareto TotalProbability Scheme is higher than the Statistical Centroid Scheme. Because the ParetoTotal 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 ofresidence is a comprehensive consideration of the probability of all known crime sites,this scheme is more accurate than the Statistical Centroid Scheme. The GeographicalRoad Scheme combines image processing algorithm and satellite map to extract thecontours of the road within the region which includes some crime sites. It build amodel separately on the impact of the traffic road on the probability distribution of thecrime sites and is combined with movement patterns and crime site selectionpsychology of offenders, which is a certain novelty and uniqueness. We recommend the Pareto Total Probability Scheme and the Geographical RoadScheme for the generation of the geographical profile of a suspected serial criminalthat can help us to make predictions on the next crime site based on the past locationsof the crimes and the distribution of the traffic road. In fact, a geographical profiledescribes an optimal search process. To save the cost of search and improve theefficiency as much as possible, a search should starts from the highest area and worksdown , which is more likely to find the offender’s residence sooner than a randomprocess would. So the geographical profile is a scientific guide to plan the deploymentof police force. The real search efficiency is therefore an indicator of the performanceof geographical profile and the scheme to generate the geographical profile. In addition, we also give the measures of validation to test different schemes andan analysis of the reliability of different algorithms in different circumstances. We testthe sensitivity of parameters of our schemes and make some recommendations baseon the test results .ReferencesEgger SA 1998. The killers among us. New Jersey: Prentice HallEgger 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 HolmesRM & Holmes ST (eds), Contemporary perspectives on serial murder. ThousandOaks: Sage: 5–16Dietz PE, Hazelwood RR & Warren J 1990. The sexually sadistic criminal and hisoffenses. Bulletin of the American Academy of Psychiatry and the Law 18(2): 163–
- 31. Team # 7520 Page 31 of 37178Myers WC et al. 1993. Malignant sex and aggression: an overview of serial sexualhomicide. Bulletin of the American Academy of Psychiatry and the Law 21(4): 435–451Cantor DV et al. 2000. Predicting serial killers’ home base using a decision supportsystem. Journal of quantitative criminology 16(4): 457–478Fox JA & Levin J 2005. Extreme killing: understanding serial and mass murder.Thousand Oaks CA: Sage publicationsDouglas JE et al. 1992. Crime classification manual: a standard system forinvestigating and classifying violent crimes. San Francisco, CA: Jossey-BassAki K 2003. Serial killers: a cross-cultural study between Japan and the United States.Graduate thesis: California State UniversityBrantingham, 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 AnchorPoints of Routine Activities." Paper presented at the meeting of the American Societyof Criminology, San Francisco, CA.Brantingham, P. L., & Brantingham, P. J. (1981). Notes on the geometry on crime. InP. 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 incrime. 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.
- 32. Team # 7520 Page 32 of 37Holmes RM & DeBurger JE 1998. Profiles in terror: the serial murderer, in HolmesRM & Holmes ST (eds), Contemporary perspectives on serial murder. ThousandOaks: Sage: 5–16Dietz PE, Hazelwood RR & Warren J 1990. The sexually sadistic criminal and hisoffenses. Bulletin of the American Academy of Psychiatry and the Law 18(2): 163–178Myers WC et al. 1993. Malignant sex and aggression: an overview of serial sexualhomicide. Bulletin of the American Academy of Psychiatry and the Law 21(4): 435–451Cantor DV et al. 2000. Predicting serial killers’ home base using a decision supportsystem. Journal of quantitative criminology 16(4): 457–478Fox JA & Levin J 2005. Extreme killing: understanding serial and mass murder.Thousand Oaks CA: Sage publicationsDouglas JE et al. 1992. Crime classification manual: a standard system forinvestigating and classifying violent crimes. San Francisco, CA: Jossey-BassAki K 2003. Serial killers: a cross-cultural study between Japan and the United States.Graduate thesis: California State UniversityBrantingham, 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 AnchorPoints of Routine Activities." Paper presented at the meeting of the American Societyof Criminology, San Francisco, CA.Brantingham, P. L., & Brantingham, P. J. (1981). Notes on the geometry on crime. InP. 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 incrime. New York: Macmillan.Newton, Jr., M. B., & Swoope, E. A. (1987). Geoforensic analysis of localized serial
- 33. Team # 7520 Page 33 of 37murder: 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 crimeincident locations. Ned Levine & Associates, Annandale, VA and the NationalInstitute of Justice, Washington, DC. Retreived July 20 2009 fromhttp://www.icpsr.umich.edu/crimestatMike O’Leary N. (2009). The Mathematics of Geographic Profiling. Journal ofInvestigative Psychology and Offender Profi ling. J. Investig. Psych. Offender Profi l.6: 253–265Wikipedia 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:OxfordUniversity Press.Williams, F. P., & McShane, M. D. (1988). Criminological theory. Englewood Cliffs,NJ: Prentice Hall.Appendixesfunction p = model1killrate()x = [ 835.9544 934.2431 972.8121 726.4682 936.7314 274.8376 720.2473 710.2941298.4766 629.4236 727.7123 808.5828 896.9183];y = [38.1518 102.8482 53.0817 96.6274 86.6741 677.6507 111.5573 340.4830625.3960 248.4151 130.2197 100.3599 77.9650];sum = 0;sumx = 0;sumy = 0;for i=1:13
- 34. Team # 7520 Page 34 of 37 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 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 349.3666200.6537 222.1098 267.9814 228.0287];y = [ 59.3193 71.8970 81.5152 97.7922 98.5321 88.9139 215.4307 94.0929 128.8666255.3834 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
- 35. Team # 7520 Page 35 of 37 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 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 710.2941298.4766 629.4236 727.7123 808.5828 896.9183];y = [38.1518 102.8482 53.0817 96.6274 86.6741 677.6507 111.5573 340.4830625.3960 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;
- 36. Team # 7520 Page 36 of 37endtemp = 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)+1.652/8.307*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);% end
- 37. Team # 7520 Page 37 of 37endfigure,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);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

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