数学建模美赛一等奖论文

1、Investigations of Geographical Profile to Predict Serial CriminalSummaryFirst, we analyze the characteristics of serial criminal. Based on the analysis, we propose a method combining two schemes to generate a geographical profile to aid in the investigations of serial criminal.In scheme I, we propos

2、e a Spatial-time Sequence Model. In this model, we define attraction factor determined by crime rate and repulsion factor by chronological order. Comparing gravitation law, we take former crime locations into consideration as well, and put forward a formula expressing the effect of these factors.In

3、scheme II, we propose an Anchor Point Model. In this model, we define an anchor point which is the center of the criminals activity. Then we use two methods, average distance and linear regression, to describe the mobility of criminal. Based on the mobility we profile the anchor point location. The

4、scheme then generates geographical profile of next crime location with the assistance of anchor point. Moreover, we outline two advanced models which consider Buffer Zone and the crime time of a day.Applying both schemes to the famous serial crime committed by Peter Sutcliffe in West Yorkshire, Engl

5、and separately, we get two different Geographical Profile of next crime location.Afterwards, we combine the results of the two schemes and generate an integrated geographical profile. So the advantages of each scheme can be shared. Then, we are able to achieve a more accurate prediction.In order to

6、help law enforcement officers make the best decision, we analyze the variation of reliability and cost-effective while the searching area changes. To achieve the balance between reliability and cost-effective, we define a decision index to assist us to make the best choice. Using decision index to g

7、et the optimal decision in Peter Sutcliffes, we get the searching area accounting for 15% total area as the optimal one.At last, we analyze the strength and weakness of each model, point out the aspects which can be improved, and draw a conclusion to our investigation.ContentsExecutive Summary . . 3

8、1 Introduction . 52 Analysis of the problem . 55 Combination . . 19 5.1 How to combine . . 19 5.2 Prediction . 205.3 Results analysis . 206 Decision Making . . 21 6.1 Reliability estimation and cost-effective . . 216.2 Decision index . 227 Strength and weakness . 238 Further study . . 249 Conclusion

9、 . . 24 10 References . . 24Executive SummaryIn this paper, we discuss typical serial criminal, which is defined to be patterns of crime committed by one person, in large numbers with no apparent rhyme, reason, or motivation. Due to the characteristics of the serial criminal above, it is not proper

10、to apply the conventional investigation methods in this circumstance.To assist the investigation, multiple methods have been proposed in history. The most significant method is generating a Geographical Profile for assistance. There are many approaches to generate the profile. Based on the existing

11、achievement in this field, we propose an integrated method to do the work.To execute our method, you should: First , collect the time and locations of the happened cases of this serial crime;gain access to the local crime rate and map out it, the crime rate at point j denoted as AAjj. Second , subst

12、itute the crime rate map and the locations of happened cases intothe formula below, you can get the a probability distribution map1;PPPPPPPPjjDDDDDDDDjj = DDjjCCDD=1 Third , substitute the time order and the locations of happened cases into the formula below; you can get the repulsion force distribu

13、tion map2;PPPPPPPPjjDDDDDDDDjj = DDjjCCDD=1 Forth , use the locations of happened cases to determine the mobility (denotedas N of criminal and outline the anchor point location of the criminal. Here we define anchor as a place similar to the criminals residence.PPPPPPPPjj= eeRRDDjjNN CCDD=1If you go

14、t the exact crime time of a day, you can use it to improve the accuracyof the result. Fifth , if the happened cases indicate that the criminal avoid committing thecrime around his anchor point, you should take a Buffer Zone into consideration. Sixth , with the assist of anchor point location, determ

15、ine the probabilitydistribution map of next crime location, denoted as map3, by applying the formula:PPPPPPNNkk=eeRRDDjjNNSeventh , stack up map1, map2 and map3; we can generate the Geographical Profile of next crime location to assist the investigation.Eighth , in order to decide the scale of searc

16、hing area, you should draw out the reliability curve from the Geographical Profile to reflect the tendency of it, with the formula:RReeRRRR= CCPPCCPPddDDPPAfterwards, calculate the cost-effective (CF:CCCC=Finally , you can get the decision index, which reflect the balance between reliability and cos

17、t-effective. Decision index expressed as:DDDD=1RReeRRRR+2CCCC; 1+2=1Your decision optimizes the balance when the index reaches the maximum value.Overview of our approach:Our method can only be applied to typical serial crime investigation; in our paper we only calculate the case with criminals behav

18、ior obeys negative-exponential function. The characteristic vary from criminal to criminal, therefore, the negative-exponential function cannot fit all the situations well, so we should modify the function to meet the special situation.Moreover, searching the anchor point is also a promising scheme

19、to execute.1 IntroductionSerial crime is a typical category of criminology. This type of crime usually cannot be solved by conventional methods, due to lacking of discipline, organization and schedule. Moreover, the targets of crime have some degree of randomness. Peter Sutcliffe has committed thirt

20、een murders without being caught, finally arrested by using the “center of mass” method to locate his residence. Here the problem arises: how to locate the suspicion of a serial crime accurately?In history, several developed techniques have been put forward to solve this problem. Early methods 1 inc

21、lude: Standard Deviation Rectangles, Jennrich/Turner Ellipse and Modified Correlated Walk Analysis. In 1993, D. Kim Rossmo2 has adapted location theory to serial offenders and outlined a mathematical approach to identifying the offenders home base location, which is still widely used.In order to pre

22、dict next crime location of serial criminal asked by the problem, we analyzed the characteristics of existing methods. And then we proposed an improved method combined with two different schemes to generate a geographical profile. Finally, with the assistance of geographical profile, we are able to

23、make a useful prediction.2 Analysis of the problemFirst, we should gain a clear idea of the characteristics of serial criminal. Generally, the serial murder is defined to be patterns of murder committed by one person, in large numbers with no apparent rhyme, reason, or motivation 3. Then, we can get

24、 the key characteristic of serial criminal.Taking the essential criminal behavior of serial criminal into consideration, so that we can outline pertinent schemes.According to the serial crime characteristics, offenders always choose crime location with uncertainty, but somehow related to multiple fa

25、ctors involving their personality, social statue, etc. We are not able to give a certain answer but come up with a Geographical Profile associated with probability.With the assistance of Geographical Profile, the searching decision can be made. And we can estimate the reliability of the decision as

26、well.The cost-effective to search the criminal should be taken into consideration, which enables us to make a optimal strategy to balance the confidence level and cost-effective.3 Scheme I3.1 AnalysisAlthough the serial criminals choose crime location and time randomly, the decision making still fol

27、lows some general rules:Routine activity theory 2 suggests that offender tends to occur in locations which have considerable profit and relatively less risks.A serial criminal activity space is restricted to a limited area. Since the criminal is familiar with the surrounding circumstance, the locati

28、on of next crime tends to be chosen in criminals active area.3.2 Definition and symbolsCrime hot spots: geographical areas with the highest crime rate.High-crime-area : areas have high incidents of crime. 3.3 AssumptionCrime hot spots and high-crime-area are considered suitable to commit crime. So t

29、hey are more attractive to serial criminals.The farther location is away from former crime locations, the less probable for criminals to choose it.Serial offenders avoid to committing crime at same location successively.3.4 ModelingAs weve already analyzed, the farther the place is from the former c

30、rime locations,the less probability for offenders to choose the place. That is to say, the probability of one place to be the next crime location decays while its distance from former crime locations increases. Comparing to the idea of gravitation law, we get:PPPPPPPPjjDDDDDDDDjj = DDDDDDDDjjCCDD=1P

31、PPPPPDD represents the probability determined by the distance; C is the number of committed crimes;1. Attraction effectThe formula above should include a parameter AAjj, which associated with the circumstance of location j. The parameter is defined to measure at what degree the location appeals to t

32、he criminal. According to given assumption this parameter can be expressed as:AAjj=crime rate at point jj (2 2. Time sequence effectIf a serial offender commits crimes at the same location successively, he will be caught easily. So offenders tend to change location to avoid being arrested. The longe

33、r the time interval between two crimes is, the less location interaction the two crimes have. In another word, if the interval is short, the criminal tends to choose location far away from the former one. The relationship can be expressed as:RReeRRCC= DDDDDDDDjjRReeRRCC represents the repulsion forc

34、e between the ith crime location and next one. DD represents the sequence interval between the i th crime and next crime. Negative sign means this factor reduces the probability of a certain place to be thenext crime location.3. Parameter KCompare equation (2 and(3 to (1, we can get the expression o

35、f K :KK=AAjj eeDDSubstituting K into equation (1, the model is re-proposed as follows:PPPPPPPPjjDDDDDDDDjj = DDjjCCDD=1Figure 1 Scheme Is Idea3.5 Generating Geographical ProfileWe will demonstrate how to apply our model to practical serial crime case. The procedure is shown as below:Step1:Collect th

36、e information of the serial crime: former case sequence and locations.Step2: Get a crime rate map of the region where the serial crime happened.Step3: Divide the region into a set of discrete small rectangle segments.Step4: Use crime rate of each area segment and former crime locations to calculate

37、the probability of each area effected by the two factors, the calculation to generate a probability map bases on the following formula:PPPPPPPPjjDDDDDDDDjj = DDjjCCDD=1Step5:Measure probability of each area effected by former crime sequence; generate another probability map with the help of:RReeRRCC

38、= DDjjStep6: Stack up the two maps generated by Step4 and Step5, get the final map of probability.To illustrate the usage of scheme I, wed like to demonstrate the processes to generate Geographical Profile in a real circumstance. Therefore we choose the typical serial crimes committed by Peter Sutcl

39、iffe mentioned in the problem. We will apply our first scheme to generate Geographical Profile of it. 1. First we make our effort to process every crime in Peter Sutcliffes case6, and pointed out every critical location in Google Map. Those locations include: every crime scene, location of his resid

40、ence and the place he was arrested. The processed map is shown as below: Figure 2 Yorkshire Ripper Locations2. To determine the attractive effect, we collected the crime data. Then we divide the real map into discrete rectangle pieces and generate a visual map of the crime rate: Figure 3 Crime Rate

41、in This AreaThe point with higher altitude has a higher crime rate in the left figure.In the right one, each circle represents the points with same crime rate.We can see that there three are hot-spots in this area, and the area can be roughly divided into two high crime areas.3. Consider the former

42、crime locations and the crime rate of each area, we can generate a map of probability for each point to be the next crime location. This map does not consider the former crime locations time order, we name the map as unmodified ProS : Figure 4 Unmodified ProSThe figure expresses the information as f

43、ollows; The height describes the probability distribution of the site, where theoffender is going to perform the next attack. The main peaks occur around Leeds, Bradford, Greater Manchester.4. Take the time order of the crime sequence into consideration; we draw out themap of repulsion force of each

44、 former crime scene lay on the other points. Figure 5 Repulsion ForceFrom the map, we can find out that: Only latest two crime locations will have a great effect on the choice of nextcrime location, the earlier ones effects can be ignored. Although there is repulsion force, the force decreases sharp

45、ly while thedistance from the force peak point increases.5. Apply the repulsion force map to adjust the unmodified ProS map, we get a newmap, named modified ProSmap:Figure 6 Modified ProSCompare the modified map and the unmodified one carefully, we can discover that in the repulsion area the altitud

46、e of ProS has been pulled down. However, the general shape of the map does not change much at the big picture.Reconsider the unmodified map and the modified map, which acted as the Geographical Profiles in Peter Sutcliffes case. Wed like to highlight the practical significance behind the mathematica

47、l constructions:Compare the Geographical Profiles to the crime rate map, we find they are in an amazing accordance with each other. Therefore the comparison result support the first assumption that: crime hot spots and high-crime-area are considered suitable to commit crime, so they are more attract

48、ive to serial criminals.Analyze the reason why all the crime locations lie in one of the high-crime-area not in both two. We come to the conclusion that: the crime tends to choose next crime location near the former ones, the fact of Peter Sutcliffes case verifies our second assumption.In both unmod

49、ified map and the modified map, we only use the information of the earlier 12 crime locations. The 13th one is predicted through that information. The level lines in unmodified map and the modified map suggest the probability to be the 13th one. Compare the Geographical Profile of the 13th one and t

50、he actually point of 13th location, we find that the 13th location lies in the high altitude level line. The first scheme is reliable.4 Scheme II4.1 AnalysisScheme I offers a way to generate the Geographical Profile and predict next crime location directly, however, the scheme inhere some shortcomin

51、gs: if the serial criminal never commits the next crime, the method will not work well to assist catchthe criminal; if the prediction is not precise enough, more people will be attacked. In order to compensate the flaws of scheme I, we analyze the criminal psychology and propose a second scheme from

52、 another approach.The paper Mapping Mayhem The Geography of Crime 4 provides an evidence that serial criminals usually operate inside a personal “comfort zone”, where they feel familiar, safe and powerful.The “comfort zone” center is always some place sepcial to the criminal, such as home, workplace

53、 or some other similar place. Since the place is likely to be the most important place in the criminals daily life, we can first generate a Geographical Profile of the place acting as an anchor point to the criminal.Using this Geographical Profile, we can narrow the searching area for criminal and c

54、ontinue our approach to predict next crime location.4.2 AssumptionThe longer the distance between crime location and a certain point, the less probable for this point to be anchor point.The longer the distance between a location and anchor point, the less probable for this place to be the next crime location.The serial offenders avoid to committing crime at same location successively. 4.3 Definition and symbolsAnchor point: The “comfort