美国大学生数学建模竞赛B题一等奖.pdf

Team#6539Page 1 of 26SummarySummarySummarySummaryFaced with serial crimes,we usually estimate the possible location of next crimeby narrowing search area.Webuild three models to determine the geographical profileof a suspected serial criminal based on the locations of the existing crimes.ModelOne assumes that the crime site only depends on the average distance between theanchor point and the crime site.Toground this model in reality,we incorporate thegeographic features G,the decay function D and a normalization factor N.Then wecan get the geographical profile by calculating the probability density.Model Two isBased on the assumption that the choice of crime site depends on ten factors which isspecifically described in Table 5 in this paper.By using analytic hierarchy process(AHP)to generate the geographical profile.Take into account these two geographicalprofiles and the two most likely future crime sites.By using mathematical dynamicprogramming method,we further estimate the possible location of next crime tonarrow the search area.To demonstrate how our model works,we apply it to Peterscase and make a prediction about some uncertainties which will affect the sensitivityof the program.Both Model One and Model Two have their own strengths andweaknesses.The former is quite rigorous while it lacks considerations of practicalfactors.The latter takes these into account while it is too subjective in application.Combined these two models with further analysis and actual conditions,our lastmethod has both good precision and operability.Weshow that this strategy is notoptimal but can be improved by finding out more links between Model One andModel Two to get a more comprehensive result with smaller deviation.KeyKeyKeyKey words:words:words:words:geographic profiling,the probability density,anchor point,expected utilityTeam#6539Page 2 of 26E E E Executivexecutivexecutivexecutive S S S SummaryummaryummaryummaryNowadays,in a serial crime,the spatial distribution of crime sites is arousing themore and more attention of the criminologists and the geographers.The Serialcriminals,in a spirit of defiance,have endangered public security,gravely infringedon the citizens personal safety,lives and property,and are abhorred by the peopleacross the country.Since the offenders usually have no stable residence,it is verydifficult for the police to find out and arrest them.Therefore,in order to help thepolice solve the crimeassoon as possible to maintain social security and stability,amore sophisticated technique is in urgent need to be developed to determine the“geographical profile”of a suspected serial criminal based on the locations of thecrimes.This paper presents three methods,especially a new mathematical methodcombining the advantages of the two previous models,which can generate a usefulprediction for law enforcement officers about possible locations of the next crime.Based on Bayesian statistical methods,Model One makes explicit connectionsbetween assumptions on offender behavior and the components of the mathematicalmodel.It also takes into account local geographic features that either influence theselection of a crime site or influence the selection of an offendersanchor point.Whatsmore,with rigorous inference formulas,this model has both precision andoperability.Model Two uses analytic hierarchy process(AHP).It takes full account of avariety of factors relevant to the crime sites,assisting the police to take measuresadapted to local conditions so as to improve our work.The last method is composed from Model One and Model Two.Taken theexpected utility and other practical factors into consideration,it further estimate thegeographic profile generated by the previous two models.Toconclude,we suggest the policemen put this mathematical method aboutgeographical profiling into practice for it will be of significant assistance to lawenforcement.The technical details are as follows:First of all,enhance the consciousness of the public social security and itsimprovement in the potential criminal area and inform the local people that a series ofcrimes have occurred recently,reminding them to keep vigilant and not to go intoremote areas alone.Second,the police departments should focus their activities,geographicallyprioritize suspects,and concentrate saturation or directed patrolling efforts in thosezones where the criminal predator is most likely to be active.Third,after arresting the criminal,the police need to make experiential analysison all these kinds of serial crimes to prevent a similar case.Whatsmore,the policedepartments set up an enhanced intelligence exchange network,especially in the areawhich is the next crime site according to our prediction.Finally,search the suspect in the predicted criminal location or the likelyTeam#6539Page 3 of 26residence of the offender.Generally speaking,our method has good maneuverability and practicability.However,there are various uncertain factors in the situation which cannot be predictedsuchas theweatherconditionandthe trafficconditions.Additionally,thedetermination and analysis of weight-coefficients on the various factors is verysubjective.As a result,the actual site scene of next crime may be outside of ourgeographical profiling.Therefore,the police deployments must be based on theanalysis of local actual condition instead of applying our model blindly.Furthermore,our prerequisite that the offender has the only one stable anchorpoint differs from the actual conditions.Since the offender is very likely to change hisresidence,the police had better search the suspect according to the latest information.Maintaining social stability and ensuring the safety of residents will bring well-being and peace to all mankind.Every one of us should make effort to make oursociety become better.Hope that this paper will be of some help to prevent thecriminal behaviors.Team#6539Page 4 of 261 1 1 1.IntroductionIntroductionIntroductionIntroductionClues derived from the locations connected to violent repeat criminal offenders,such as serial murderers,arsonists,and rapists,can be of significant assistance to lawenforcement.Such information helpspolice departments to focus their activities,geographically prioritize suspects,and to concentrate directed patrolling efforts inthose zones where the criminal offender is most likely to be active.By examiningspatial data connected to a series of crime sites,this methodological model generates aprobability map that indicates the area most likely to be the locations of the next crime.This paper presents two mathematical models to illustrate how geographicalanalysis of serial crime conducted within a geographic information system can assistcrime investigation.Techniques are illustrated for determining the possible residenceof offenders and for predicting the location of the next crime based on the time andlocations of the existing crimes.First,we present a mathematical survey of some of the algorithms that have beenused to solve the geographic profiling problem.The geographic profiling problem isthe problem of constructing an estimate for the location of the anchor point of a serialoffender from the locations of the offenderscrime sites.The approach that we develop will make use of at these two different schemes togenerate a geographical profile.Whatsmore,we develop a third technique tocombine the results of the two previous schemes and generate a useful prediction forlaw enforcement officers.The prediction providessome kind of estimate or guidanceabout possible locations of the next crime based on the time and locations of the pastcrime scenes.Our method will also provide some kind of estimate about how reliablethe estimate will be in a given situation,including appropriate warnings.Theexecutive summary will provide a broad overview of the potential issues.It will alsoprovide an overview of our approach and describe situations when it is an appropriatetool and situations in which it is not an appropriate tool.The purpose is to apply geographical analysis to serial crime investigations topredict the location of future targets and determine offender residence.1.1.1.1.1 1 1 1 RestatementRestatementRestatementRestatement ofofofof thethethethe ProblemProblemProblemProblemIn order to indicate the origin of geographical profiling problems,the followingbackground is worth mentioning.In 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting anumber of other people to vicious attacks.One of the methods used to narrow thesearch for Mr.Sutcliffe was to find a“center of mass”of the locations of the attacks.In the end,the suspect happened to live in the same town predicted by this technique.Since that time,a number of more sophisticated techniques have been developed todetermine the“geographical profile”of a suspected serial criminal based on thelocations of the crimes.Our team has been asked by a local police agency to develop a method to aid inTeam#6539Page 5 of 26their investigations of serial criminals.The approach that we develop will make use ofat least two different schemes to generate a geographical profile.Wealso develop atechnique to combine the results of the different schemes and generate a usefulprediction for law enforcement officers.The prediction will provide some kind ofestimate or guidance about possible locations of the next crime based on the time andlocations of the past crime scenes.Our method will also provide some kind of estimateabout how reliable the estimate will be in a given situation,including appropriatewarnings.1.1.1.1.2 2 2 2 SurveySurveySurveySurvey ofofofof PreviousPreviousPreviousPrevious ResearchResearchResearchResearchExistingExistingExistingExisting MethodsMethodsMethodsMethodsTo understand how we might proceed let us begin by adopting some commonnotation:A pointwill have two components.x12(,)xx x=These can be latitude and longitudeThese can be the distances from a pair of reference axesThe series consists of n crimes at the locations.12,nx xxThe offendersanchor point will be denoted by.zDistance between the pointsandwill be.xy(,)d x yExisting algorithms begin by first making a choice of distance metric;theydthen select a decay functionand construct a a a a hit score functionby computingf()S y11()(,)(,)(,)niniS yf d x yf d x yf d xy=+Regions with a high hit score are considered to be more likely to contain theoffendersanchor point than regions with a low hit score.In practice,the hitscoreis not evaluated everywhere,but simply on some rectangular array of()S ypointsforandgiving us the array of12(,)jkjkyyy=1,2,jJ1,2,kKvalues4.()jkjkSS y=Rossmosmethod,as described in(Rossmo,2000,Chapter10)chooses theManhattan distance function for d and the decay functionTeam#6539Page 6 of 26 if()if (2)hg hgkdBdf dkBdBBd=OtherOtherOtherOther StudiesStudiesStudiesStudiesIn the study of Crime Analyststask,Bryan Hill considers that the use of the“probability grid method”(PGM)can narrow the search as it pertains to tactical crimeanalysis in the ArcView Geographic Information Systems(GIS)environment.Themain point of his theory will be that any current statistical method of predicting thenext hit location in a crime series is operationally ineffectual when the suspect coversa large geographic area.When the analyst combines several statistical methods andintuitive,logical thought processes into a combined“grid”score,the analysis productcan be made more operationally effective.This new grid surface allows the analyst tomake a better prediction of the next hit in a crime series and is useful in isolatingspecific target locations for law enforcement deployment efforts.This easy to apply“PGM method”allows the analyst to use sound statistical methods,as well as theirexperience and knowledge of a crime series to narrow the focus and potential hit area.In addition,when the crime series has sufficient suspect information,journey tocrime analysis using the CrimeStat software can be used to provide investigators witha list of probable offenders from law enforcement records available to the analyst4.2.2.2.2.ModelModelModelModel OverviewOverviewOverviewOverview1 1 1 1、ModelModelModelModel OneOneOneOneIn Model One,we assume that the crime sites depend on the distance between theanchor point and the crime site.Generate a hot zone model according to probabilityformula,then combine the geographic features G with the decay function D and add anormalization factor N into the model.At last,we can get the“geographical profile”.2 2 2 2、ModelModelModelModel TwoTwoTwoTwoIn ModelTwo,we take account of other factors relevant to crime site location,wehave studied various literature to summary two aspects denoted by U(Utility fromcrimes)and P(the probability of success)specifically including the following tenfactors:the responding speed of the police,public security situation,resistancesdiathesis,density of registered inhabitants,the advantageous position,the number ofoffenses,the distance from the anchor point of the offender,the time required forcommitting the crime,number of target persons,offendersmental satisfaction fromthe crime.Then,we use Analytic Hierarchy Process(AHP)to get weighted factors.Finally,according to the formula we can draw E of some areas.A wide range of criminal areaTeam#6539Page 7 of 26can then be divided into several small areas.Then,we give the proper scores for eachsmall area according to the actual condition.When the area is small enough,we canget the geographical profile.Sites with higher scores have high probability to be crimeareas.3 3 3 3、TheTheTheThe twotwotwotwo modelsmodelsmodelsmodels cancancancan bebebebesynthesizedsynthesizedsynthesizedsynthesized asasasas oneoneoneone methodmethodmethodmethodFor their different emphasis(The theory of Model One is quite rigorous,but itignoresthe factors such as the geographic features and criminal motivation whilethese factors is important for selecting criminal crime sites.On the other hand,ModelTwo takes these into account,while it is too subjective in practical application whichmay easily cause the deviation),the geographical profileswe get from two modelsmust be different.By using mathematical dynamic programming method to establishthe model,this method explainsthe common rules of the offenderschoice on thecrime sites.Therefore,when we get several geographical profiles from the previoustwo models,we can use this method to predict the next offence site.Tosum up，we can generate the geographical profile in the investigations of serialcriminals,providing some kind of estimate or guidance about possible locations of thenext crime and narrow the search.3 3 3 3.ModelModelModelModel OneOneOneOneSymbols:Hypothesis：1、We assume that our offender chooses potential locations to commit crimesrandomly according to some unknown probability density function.P2、We assume thatdepends upon z and a.P3、We suppose that that the values of the anchor point z and average offensedistanceare unknown,but the form of the distributionis kno