7070（一等奖）

Team control number 7070 Problem chosen B _______________________________________________________________ 2010 Mathematical Contest in Modeling (MCM) Summary Sheet Summary Since the mystery case of Jack the Ripper occurred, the study of serial crime has been a fruitful area in mathematical research for decades. Here we attempt to build a model to generate a geographical profile and predict the next location of crime based on the past information to aid the local investigation. The model contains two schemes and a prediction based on results of the two schemes. In scheme one, according to the theory developed by Newton and Swoope, the geographic mean of past crime sites could probably be the offender’s residence. Then a “best-fit” function to describe the relationship between the crime frequency and the distance from the residence to each crime site could be estimated, whose error term could be relatively small because of the use of a variety of fitting methods. Next, we can generate a geographic profile based on the relationship between crime frequency and the distance from each crime site to residence. After that, the relationship between time interval of each crime and location of crime is estimated by linear regression to adjust the geographic profile we have obtained. In Scheme Two, different scores, which are calculated by weighting the factors, are assigned to each unit of the divided map. The area with higher scores has a higher crime frequency. Finally, we combine the results of two schemes by weighting summation. An estimator called crime index is introduced to rank the level of crime possibility based on the combined results. According to crime index, the final geographic profile illustrated by different colors could be generated. The areas with highest crime possibility measured by crime index are the prediction of the next location of crime, which can be obtained from the geographic profile directly by the local policy. Furthermore, some specific examples are provided to explain and test the model. An executive summary is also attached to outline when and how to use our model to predict location of next crime. Team # 7070 1 / 35 Contents Introduction.......................................................................................................................................2 Assumptions......................................................................................................................................2 Model ................................................................................................................................................3 Scheme One ..............................................................................................................................3 Scheme Two............................................................................................................................15 Combination and Prediction....................................................................................................17 Example ..........................................................................................................................................19 Scheme One ............................................................................................................................20 Scheme Two............................................................................................................................24 Combination and Prediction....................................................................................................27 Discussion.......................................................................................................................................29 Weakness.................................................................................................................................29 Strengths..................................................................................................................................30 Further consideration ......................................................................................................................31 Executive summary.........................................................................................................................32 References.......................................................................................................................................34 Team # 7070 2 / 35 Introduction Since 1888, serial crime became part of our cultural lexicon (Rumbelow, 1988). The famous mystery case, Jack the Ripper, was certainly not the first nor last of this type. This kind of crime could have a large negative effect on the society and also cause the shock, fear, anger, and panic among the community. Therefore, increasing concerns about this crime has leaded to many researches. This paper aims to generate a geographical profile to predict the next location of crime according to the past crime information. Two different schemes, based on the geographic profiling theory (D.Kim Rossmo, 2000) and the weights method are used. In addition, some examples are given to explain and text the model as well. Assumptions 1. Criminal offenders are rational: 1) Offenders make decisions that benefit themselves by least effects. 2) The crime is planned before it happens. 2. Serial crime is defined as crime involves at least five separate events with an emotional cooling-off period between homicides. The offender plans his crime during the cool off period and when the time is right, he select the location and proceeds with his plan. 3. The locations of the each crime have some relevance with the offenders’ residence. 4. All the crimes are happened in the local place. 5. The location of next crime is in connection with all the time and locations of the past crimes. 6. The locations of crime selected by the offender are affected by some common factors, for example traffic convenience, population density and Team # 7070 3 / 35 crime rate in the local place. Model In order to aid the local police agency in their investigations of serial criminals, a model contains three main parts have been developed: Scheme I, scheme II to generate a geographic profile and then a prediction method about the next crime based on the previous profile. In the mean time, an executive summary will be attached as an overview and guidance of how to use the model. Scheme One This scheme aims to construct a geographic profile, which presents the different level of crime frequency, based on the locations and time of past serial crimes. According to the theory of D. Kim Rossmo (2000)---- “Crimes often occur in relatively close proximity to the home of the offender”, we assume that all of the locations of crime are distributed around one “center of mass”, which can be regarded as the residence of the offender. Variables C i ( ix , iy ) the i th location of the crime measured by latitude ix ,longitude iy . R( x , y ) the residence of the offender measured by latitude x ,longitude y . d i the distance between the i th location and the offender’s residence f the frequency of crime in a certain area Step one---find the most possible location of offender’s residence Based on the theory of the geographic profiling, the residence could be re regarded as the geographic mean of each location. It can be easily caculated by using n series of data which are availiable from past crimes as: Team # 7070 4 / 35 ( ) ( ) ),(,yx, n y n x RyxRR ii ∑∑== n the total number of the serial crime at present Step two---find the distance between crime location and the residence After estimated the residence of the offender, the distance between each crime site and residence id could be calculated as ( ) ( )22 iii yybxxad −+−= a the distance per longitude, which is 111km b the distance per latitude , which is also approximately 111km Step three---find the frequency of crime in a certain area First, we find ),...,,max( 21 ndddl = , then we divide l into k intervals, k is the number of the interval and decided by the specific features of the serial crime and the local place. For example, we have =l 6km and then we make k=6, then we get interval (0,1),(1,2) (2,3), (3,4) (4,5), (5, 6). Second, k concentric circles could be drawn using the radius r= k l , k l2 ,…, l . If k=6 and l=6, we have 6 concentric circle and the radius of them are 1km, 2km,…,6km. Third, we could get a number of cyclic areas constructed by the concentric circles, denoted as area A, A 2 ,…, A n . Now the number of crime in each area could be counted, denoted as a 1 , a 2 ,…, a n . Final, according to the date we get from the above, the frequency of crime in each area could be calculated as n i i Af = . Step four---study the relationship between the frequency of crime in each area if and the distance between crime location and the Team # 7070 5 / 35 residence id Based on the shape of the data, the previous experience and the distance decay theory(Hoshua David Kent, 1994), different fitting methods are used to describe the relationship between the two variables, including the logarithmic, negative exponential, truncated negative exponential and polynomial. This is the example of a Euclidean Distance decay Model: The first three regression lines are shown as follow. Therefore, a better function could be chosen with the largest correlated correlation R 2 to describe the relationship between the two variables. Then we get the “best-fit” line on the graph. Step Five---generate the geographic profile by the different levels of crime frequency After getting the “best-fit” line, two horizontal lines could be plotted on the graph to divide the frequency into three levels, denoted by low frequency, medium frequency and high frequency. Then based on the different frequency of crime and the distance between crime site and residence, a geographic Team # 7070 6 / 35 profile could be got. For example, the “best fit” line is the negative exponential line, the graph is The geographical will look like: R(x, y) high low medium Team # 7070 7 / 35 Step six---take the time factor into consideration First, the time interval between each past crime is denoted as it ( .1,...,2,1 −= ni ) Second, explore a relationship between the time interval it and the distance between crime location and residence id ( ni ,...,3,2= ). By taking advantage of the data we get about it and id , do linear regression using different methods. If no obvious relationship comes out, ignore the effect of time factor on crime location. If obvious relationship is found, adjust the geographical profile we got from step five and get a new profile about the crime location. For example, if it said the time interval and the distance is positive correlated, it means after a certain period of time, the area with high frequency should be put a little far away the residence. Example A real serial crime case in Texas，USA is introduced to explain and text the above model. During 1959 to 1983, Henry Lee Lucas created a serial murder case mainly in Texas, USA. Although he had admitted to over 1000 murders, there were 11 of the murders that he had been convicted of. In those 11 of the murders, except one in Michigan and one in West Virginia, the rest of nine were located in Texas, which are listed below. Date Place August 1970 Kauffman County, Texas November 1977 Harrison County, Texas October 1979 Willimson County, Texas September 1981 Brownfield, Texas August 1982 Denton, Texas Team # 7070 8 / 35 September 1982 Ringgold, Texas December 1982 Hale County, Texas Match 1983 Montgomery County, Texas April 1984 Montgomery County, Texas By the help of the software, GoogleTM Earth , the location of crimes can be found in graph below. The latitude and longitude of each location of crime are listed in table below. N(°) W(°) 32.633 96.317 32.550 94.300 30.750 97.683 33.167 102.267 Team # 7070 9 / 35 33.200 97.933 33.817 97.933 34.100 101.867 30.317 95.467 30.317 95.467 In order to text scheme one, the data from the first murder to the eighth could be used to form the scheme. Then, estimate the location of the ninth murder by the method. Step one---find the residence of the offender For ( ) ),(, n y n x RyxR ii ∑∑= In the example: n=8; n ∑ ix =32.319; n y∑ i =97.587; ( )yxR , =(32.319,97.587 ) The residence in the map can be pointed out. Team # 7070 10 / 35 Step two--- calculate the distance between crime location and the offender’s residence According to the formula ( ) ( )22 iii yybxxad −+−= (a the distance per longitude, which is 111km b the distance per latitude, which is also approximately 111km ) Then we get crime time d(km) 1 145.28 2 365.76 3 174.43 4 527.90 5 110.90 6 170.68 7 514.55 Team # 7070 11 / 35 8 323.68 Step three---find the frequency of crime in a certain area First, find kmdddl n 55.514),...,,max( 21 ＝= And in order to easy calculation, we make kml 600= . Then we divide l into 6 intervals as (0,100), (100,200), (200,300), (300,400), (400,500), (500,600). Second, six concentric circles could be drawn using the radius r=100, 200,…, 600. Third, we could get six cyclic areas constructed by the concentric circles, denoted as area A, A 2 ,…, A 6 . Now the number of crime in each area could be counted, which are 0, 4, 0, 2, 0, 2. Finally, according to the date we get from the above, the frequency of crime in each area could be calculated as n Af ii = . Then we get if Interval Frequency (0,100) 0 (100,200) 50% (200,300) 0 (300,400) 25% (400,500) 0 (500,600) 25% Step four---explore the relationship between the frequency of crime in each area if and the distance between crime location and the residence id . Team # 7070 12 / 35 In this case, polynomial fitting method with degree of five is regarded to get the “best fit” line. (A hypothesis) Step Five---generate the geographic profile by the different levels of crime frequency. After getting the “best-fit” line through the polynomial fitting method, two horizontal lines with f=1.25 and f=0.3 are plotted on the graph to divide the frequency into three levels, denoted by low frequency, medium frequency and high frequency. Team # 7070 13 / 35 Then we can get Boundary point of each interval Distance r1 133 r2 166 r3 250 r4 310 r5 500 r6 625 Then based on the different frequency and the distance between crime location and residence, a geographic profile could be got as Team # 7070 14 / 35 Red region 0.3