2010年美国大学生数学建模竞赛B题一等奖.pdf

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1、Team#6539Page 1 of 26SummaryFaced with serial crimes,we usually estimate the possible location of next crimeby narrowing searcharea.We build three models to determine the geographical profileof a suspected serial criminal based on the locations of the existing crimes.ModelOne assumes that the crime

2、site only depends on the average distance between theanchor point and the crime site.To ground 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 probabilitydensity.Model Two isBas

3、ed on theassumption that the choice of crime site depends on ten factors which isspecificallydescribed in Table 5 in this paper.By using analytic hierarchy process(AHP)to generatethe geographical profile.Take into account these two geographicalprofiles and the two most likely future crime sites.By u

4、sing mathematical dynamicprogrammingmethod,we further estimate the possible location of next crime tonarrow the search area.To demonstrate how our model works,we apply it to Peterscaseand make a prediction about some uncertainties which will affect the sensitivityof the program.Both Model One and Mo

5、delTwo 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 furtheranalysis and actual conditions,our lastmethod has both good

6、 precision and operability.We show that this strategy is notoptimal but can be improvedby findingout more links between Model One andModel Two to get a more comprehensive result with smaller deviation.Key words:geographicprofiling,the probabilitydensity,anchorpoint,expected utilityTeam#6539Page 2 of

7、 26Executive SummaryNowadays,in a serial crime,the spatial distribution of crime sites is arousing themore and more attention of the criminologistsand the geographers.The Serialcriminals,in a spirit of defiance,have endangered public security,gravely infringedon the citizens personal safety,lives an

8、d property,and are abhorred by the peopleacross the country.Since the offenders usually have no stable residence,it is verydifficultfor the police to find out and arrest them.Therefore,in order to help thepolice solve the crime as soon as possible to maintain social security and stability,amore soph

9、isticated technique is in urgent need to be developed to determine the“geographicalprofile”of a suspected serial criminal based on the locations of thecrimes.This paper presents three methods,especially a new mathematicalmethodcombining the advantages of the two previous models,which can generate a

10、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

11、 that either influence theselection of a crime site or influence the selection of an offendersanchor point.Whats more,with rigorous inference formulas,this model has both precision andoperability.Model Two uses analytic hierarchy process(AHP).It takes fullaccount of avariety of factors relevant to t

12、he crime sites,assisting the police to take measuresadapted to local conditions soasto improve our work.The last method is composed from ModelOne and ModelTwo.Taken theexpected utilityand other practical factors into consideration,it further estimate thegeographic profile generated by the previous t

13、wo models.To conclude,we suggest the policemen put this mathematical method aboutgeographical profilinginto practice for it willbe of significantassistance to lawenforcement.The technical details are asfollows:First of all,enhance the consciousness of the public social security and itsimprovement in

14、 the potential criminal areaand inform the local people that a seriesofcrimes have occurred recently,reminding them to keep vigilantand not to go intoremote areasalone.Second,the police departments should focus their activities,geographicallyprioritizesuspects,and concentrate saturation or directed

15、patrolling efforts in thosezoneswhere 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 exchan

16、ge network,especially in the areawhich is the next crime site according to our prediction.Finally,search the suspect in the predicted criminallocation or the likelyTeam#6539Page 3 of 26residence of the offender.Generally speaking,our method has good maneuverabilityand practicability.However,there ar

17、e various uncertain factors in the situation which cannot be predictedsuchas theweatherconditionandthe trafficconditions.Additionally,thedeterminationand analysis ofweight-coefficientson the various factors is verysubjective.As a result,the actual site scene of next crime may be outside of ourgeogra

18、phicalprofiling.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 cha

19、nge hisresidence,the police had better searchthe suspect according to the latest information.Maintainingsocial 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 willbe

20、 of some help to prevent thecriminal behaviors.Team#6539Page 4 of 261.IntroductionClues derived from the locations connected to violent repeat criminal offenders,such as serial murderers,arsonists,and rapists,can be of significant assistanceto lawenforcement.Such informationhelps police departments

21、to focus their activities,geographicallyprioritizesuspects,and to concentrate directed patrollingefforts inthose zones where the criminaloffender is most likelyto be active.By examiningspatial data connected to a seriesof crime sites,this methodological model generatesaprobability map that indicates

22、 the area most likely to be the locations of the next crime.This paper presents two mathematical models to illustratehow geographicalanalysis of serial crime conducted within a geographic informationsystem can assistcrime investigation.Techniques are illustrated for determining the possible residenc

23、eof 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 profilingproblem.The geographic profilingproblem isthe problem of construc

24、ting 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 schemestogenerate a geographical profile.Whats more,we develop a thirdtechnique tocombine the results of the t

25、wo previous schemesand generate a useful prediction forlaw enforcement officers.The prediction provides some 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 rel

26、iablethe estimate willbe ina givensituation,includingappropriatewarnings.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 t

27、ool.The purpose is to apply geographical analysis to serial crime investigations topredict the location of future targetsand determine offender residence.1.1.1 1 Restatement of the ProblemIn order to indicate the origin of geographical profiling problems,the followingbackground is worth mentioning.I

28、n 1981 Peter Sutcliffewas 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 predict

29、ed by this technique.Since that time,a number of more sophisticated techniques have been developed todetermine the“geographicalprofile”of a suspected serial criminalbased 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 26

30、their investigations of serial criminals.The approach that we develop will make use ofat least two different schemesto generate a geographical profile.We also develop atechnique to combine the results of the differentschemes and generate a usefulprediction for law enforcement officers.The prediction

31、 willprovide some kind ofestimate or guidance about possible locations of the next crime basedon the time andlocations of the past crime scenes.Our method will also provide some kind of estimateabout how reliable the estimate willbe in a given situation,includingappropriatewarnings.1.1.2 2 Survey of

32、 Previous ResearchExisting MethodsTo understand how we might proceed let us begin by adopting some commonnotation:A pointwill have two components.x12(,)xx x=Thesecan be latitude and longitudeThesecan be the distancesfrom a pair of reference axesThe series consists of n crimes at the locations.12,nx

33、xx?The offendersanchor point will bedenoted 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 hit score functionby computingf()S y11()(,)(,)(,)niniS yf d x yf d x yf d xy=+?Region

34、s with a high hit score are considered to be more likelyto contain theoffendersanchor point than regions witha lowhit score.In practice,the hitscoreis not evaluated everywhere,but simply on some rectangular array of()S ypointsforandgivingus the array of12(,)jkjkyyy=1,2,jJ?1,2,kK?values4.()jkjkSS y=R

35、ossmos method,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-?=?-?Other StudiesIn the study of Crime Analyststask,Bryan Hill considers that the use of the“probabilitygrid method”(PGM)can narr

36、ow the searchas it pertains to tactical crimeanalysis in the ArcViewGeographic InformationSystems(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 g

37、eographic area.When the analyst combines several statistical methods andintuitive,logical thought processesinto a combined“grid”score,the analysis productcan bemade more operationally effective.This new grid surface allows the analyst tomake a better prediction of the next hit in a crime series and

38、is useful in isolatingspecific target locations for law enforcement deployment efforts.This easy to apply“PGMmethod”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 ser

39、ies has sufficient suspect information,journey tocrime analysis using the CrimeStat software can be usedto provide investigators witha list of probable offenders from law enforcement records available to the analyst4.2.Model Overview1、Model OneIn Model One,we assumethat the crime sites depend on the

40、 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、Model TwoIn Model Two,we take

41、account of other factors relevant to crime site location,wehave studied various literature to summary two aspects denoted by U(Utilityfromcrimes)and P(the probabilityof success)specificallyincludingthe followingtenfactors:the responding speed of the police,public security situation,resistancesdiathe

42、sis,density of registered inhabitants,the advantageousposition,the number ofoffenses,the distance from the anchor point of the offender,the time required forcommittingthe crime,number of target persons,offendersmental satisfaction fromthecrime.Then,we useAnalytic Hierarchy Process(AHP)to get weighte

43、d 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 scoresfor eachsmall area according to the actual condition.When the area is small enough,we canget the geograp

44、hical profile.Sites with higher scoreshave high probability to be crimeareas.3、The two models can be synthesized asone methodFor their different emphasis(The theory of Model One is quite rigorous,but itignores the factors such as the geographic features and criminalmotivationwhilethese factors is im

45、portant 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 profiles we get from two modelsmust be different.By using mathematical dynamic programming method t

46、o establishthe model,this method explains the common rules of the offenderschoice on thecrime sites.Therefore,when we get several geographical profiles from the previoustwo models,we can usethis method to predict the next offence site.To sum up，we can generatethe geographical profile in the investig

47、ations of serialcriminals,providing some kind of estimate or guidance about possible locations of thenext crime and narrow the search.3.Model OneSymbols:Hypothesis：1、We assume that our offender chooses potentiallocations to commitcrimesrandomly according to some unknown probability density function.

48、P2、We assumethatdependsupon 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 known.aP4、The offender hasonly one residence which is unchanged.symbolsmeaning(x)Pprobability density functionxthe crime siteszthe lo

49、cation of the offendersanchor pointaThe averagedistance that the offender is willingto travel to offend.Dthe effect of distance decayGthe geographic featuresNa normalization factorTeam#6539Page 8 of 263.1 A MathematicalApproachIn order to looking for an appropriate model,we start with the simplest p

50、ossiblesituation.Since we know nothing about the offender,we assume that our offenderchooses potential locations to commit crimes randomly according to some unknownprobability density function.For any geographic region,the probabilitythat ourPRoffender will choose a crime location incan be found by