7071（一等奖）

Team Control Number 7071 Problem chosen B MLE & DRMA Geographical Profiling Model Abstract Being familiar with the unsolved mystery of Jack the Ripper, serial crime has proven to be a difficult puzzle for both criminal investigators and the police. However, more emphases have been put on a suspect’s geographic patterns with the development of Geographical Profiling Technique. Motivated by the Page-Rank Algorithm used by Google search engine, we develop a Danger-Rank Matrix Algorithm (DRMA) to rank the dangerous degree of the potential serial criminal locations and predict the high-risk areas. The technique of Maximum Likelihood Estimation (MLE) is used to examine the probability of crime in certain situation. The model also can predict the location of next crime directly without considering where the suspect lives. As a real world study, we successfully apply our model to the notorious “black ghosts” case recently occurred in China. From the view of implementation, our model is easy to be operated by the local police agency by offering a criminal information database system. Team # 7071 Page 1 of 17 Contents 1 Introduction ................................................................................................................. 2 1.1 Serial Crimes (Murders) ................................................................................... 2 1.2 Geographical Profiling Technique and Serial Crimes ...................................... 3 1.3 Objectives ......................................................................................................... 3 2 Simplifying Assumptions ............................................................................................ 3 About the serial criminals: ...................................................................................... 3 About the model:..................................................................................................... 4 3 The Model ................................................................................................................... 4 3.1 The Maximum Likelihood Estimation .............................................................. 4 3.2 The Danger-Rank Matrix Model ....................................................................... 6 4 Case Study .................................................................................................................. 8 4.1 The “Black Ghosts” Case in China ................................................................... 8 4.2 Peter Sutcliffe’s Case ...................................................................................... 11 5 Evaluating the Model ................................................................................................ 12 6 Conclusions ............................................................................................................... 13 7 Executive Summary for a Chief of Police ................................................................ 14 7.1 Overview of the Approach .............................................................................. 14 7.2 Situations Where the Approach is Appropriate ............................................... 15 7.3 Specific Recommendations for Implement ..................................................... 15 References .................................................................................................................... 16 Team # 7071 Page 2 of 17 1 Introduction In 1888, the throats of five women prostitutes from the slums were cut prior to abdominal mutilations in London by an unidentified killer lately named “Jack the Ripper”. While not the first serial killer, Jack the Ripper's case was the first to create a worldwide fear and frenzy of the people. The unsolved mystery of Jack the Ripper still symbolizes the great difficulty to understand these dangerous predators. Serial crime is a frightening and perplexing phenomenon which has proven to be a difficult puzzle for both criminal investigators and the police. However, the locations where the crimes happen are not completely random, but instead have a degree of underlying structure. In order to provide guidance for the police to deduce the likely whereabouts of the perpetrator of a series of linked criminal offences, it is our task to develop a mathematical model to geographically locate the suspect based on the past crimes. 1.1 Serial Crimes (Murders) Serial murder is the intentional killing of two or more victims, at separate times and is primarily predatory in nature. Before each murder, there is a building of tension through fantasy, which is then relieved by the murder and the killer have a “cooling-off period”. With an ultimate goal of murdering a victim, each victim represents an intrinsic motive that fulfills a psychological need of the offender. (Bryan Nelson, 2004) Team # 7071 Page 3 of 17 1.2 Geographical Profiling Technique and Serial Crimes A relatively recent development in the investigation is the emphasis on a suspect’s geographic patterns: where a victim is selected, where the crime was committed or where the bodies are dumped. This information can help the police to position the potential area of the suspect’s residence. Take “the Yorkshire Ripper” Peter Sutcliffe for example, the suspect was finally found living in the same town as the “center of mass” technique suggested. Geographic Profiling is defined as a methodology that uses the locations of a connected series of crimes to determine the most probable area that an offender lives in (Keith Harries, 1999). By referring to previous literatures, we can conclude that crimes tend to occur at locations where suitable, in terms of profit and risk, offenders encountered victims. 1.3 Objectives Our main goal is to develop a mathematical model to provide guidance about possible locations of the next crime (high-risk areas) based on the time and locations of the past crimes scenes for the police agency. In order to achieve this goal, a “geographical profile” of a suspected serial criminal is established. Other specific evidences such as transportation and population density will be considered. 2 Simplifying Assumptions About the serial criminals: ●That there is indeed a series of crimes that have been committed by the same offender. Team # 7071 Page 4 of 17 ●That the offender is a 'local hunter' and not a 'poacher' who comes into the area to commit crimes from elsewhere. ●That the same kind of people are the victims of the crime, since different behavioral patterns are likely to be adopted by offenders depending on different targets. ● That the series of linked crimes are relatively complete and that any missing crimes should not be "spatially biased". Police from different jurisdictions share information completely with each other. ●That the perpetrator has a "single stable anchor point over the time period of the crimes", which is to say that they haven't moved home recently or otherwise changed their locations of crimes. About the model: ●We assume that different kinds of serial crimes (usually including serial murders, serial rapes and serial robberies) share the same characteristics. And we use the features of serial murders to represent the characteristics. ●We assume that there is no gender difference between serial killers. ●We assume that the location where the crime is committed and where the bodies are dumped overlaps. 3 The Model 3.1 The Maximum Likelihood Estimation The probability of the commitment of serial crimes varies due to different features of different locations. In order to estimate the possibility for a serial criminal to commit crimes in block with certain feature(s), we divide the whole potential criminal locations into different blocks. Team # 7071 Page 5 of 17 Given that the event of the commitment of serial crime is A and the probability of the commitment of serial crimes of any block lA is )( lAP . It should be noticed that the value of )( lAP can usually be inferred by experience. Take three features: the density of population, the transportation and the function of community into consideration, we have: 1B : The event that the density of population is high. 1B : The event that the density of population is low 2B : The event that the transportation is well developed 2B : The event that the transportation is poorly developed 3B : The event that the function of community is perfect 3B : The event that the function of community is substandard Figure 1 Block ●Suppose that Block 3 has high density of population, then the probability of the commitment of serial crime can be estimated as )( )( )|( 1 13 13 BP BAP BAP  . ●Suppose that Block 4 has high density of population and well developed transportation system as well, then the probability of the commitment of serial crime Team # 7071 Page 6 of 17 can be estimated as )( )( )|( 21 214 214 BBP BBAP BBAP  . ●If the block has three or more than three features, the estimation method above still works. 3.2 The Danger-Rank Matrix Algorithm In order to narrow down the range of high-risk areas, we develop the Danger-Rank Matrix Algorithm to rank the dangerous degree of potential criminal locations based on their different features. Thus we need to label each location with “importance score” based on the following hypotheses: ●We suppose that the offender commits crime at any location, for the other locations, the more they are equipped with features which are convenient for the offender to commit crime, the higher importance score the location will get. Let us use Figure 2 to illustrate: Figure 2 Team # 7071 Page 7 of 17 Suppose the offender may commit crime at any one of the four locations. 1. The arks in Figure 2 represent the travel direction of the offender, which means the targeted location has features convenient for his crime and attract him to go there. 2. We use jn to stand for the number of “attractive places”. For location 1, jn =2. 3. If one location has no attractive features for the serial criminal, meaning that the location suffers no danger, we score 0 for that location. ●The more distinctive the feature of the area is, the higher the score will be. When considering this standard, we can look for some kind of “signature behaviors”, which can be seen as a reflection of the underlying traits of a specific offender. Because of the outstanding distinctiveness of “signature behaviors”, we give high importance score for areas fit them. Given that nnijaA  ][ , where       0 1 jij na , if j links to i , then the formula is mSAmM  )1( where ]1,0(m M : A weighted average of A and S S : A nn matrix with all entries n 1 A : Column-stochastic matrix m : Constant To rank the degree of danger of the four locations in Figure 2, We have                1 312 32 3 3 4 421 3 41 2 4 1 x x xxx x xx x x x , and then                    0100 3 1 01 2 1 3 1 00 2 1 3 1 000 A We set 12.0m and the importance scores are 1288.01 x , 1855.02 x , 3488.03 x , Team # 7071 Page 8 of 17 3369.04 x . Therefore, we can rank the locations by 2143 xxxx  and location 1 is the location with the highest risk. Our scheme is based on Google’s Page-Rank algorithm, which ranks the importance of web pages according to an eigenvector of a weighted link matrix. We analogize looking for the high-risk area as searching information through search engine. Thus, the features of certain areas such as well developed transportation system or high density of rich people can be regarded as the “key words” when searching information. 4 Case Study 4.1 The “Black Ghosts” Case in China To examine the effectiveness and stability of our model, we apply our model to the “black ghost” serial criminal happened in Anhui and Henan Province from April, 2002 to July, 2005. We first estimate average serial criminal probability 018.0)( lAP according to previous experience. In this case, we find that the transportation is more important as the “black ghosts” commit crimes along the highway, so we reasonably decide that 12.0)|( 1 BAP l , 20.0)|( 2 BAP l , 17.0)|( 3 BAP l after several estimations. According to the information above, we simply divide our potential locations into four districts as follows: Team # 7071 Page 9 of 17 Figure 3 Then we have                2 12 2 12 1 4 42 3 1 2 32 1 x x xx x x x xx x according to our Danger-Rank Matrix Model. Next we assigned several values of m and choose the best one: 16.0m . At last we get the ranking of the four locations and location 1 is the most dangerous one. In order to ensure the stability of our model, we choose anther 165.0m and get the same rank. m 1x 2x 3x 4x 16.0m 0.3502 0.1871 0.2757 0.1871 165.0m 0.3497 0.1873 0.2758 0.1873 4231 xxxx  Team # 7071 Page 10 of 17 Figure 4 the testing of the case in China Figure 5 the proving of the case in China 1 2 4 3 Team # 7071 Page 11 of 17 4.2 Peter Sutcliffe’s Case We consider the time of committing the crime in Sutcliffe’s case from two aspects: ●The cycle of the criminal time: we use sin-function to imitate the cycle of the criminal time by analyzing the “cooling-off” period between crimes. Figure 6 ●The day of week of serial crime: we use a radar diagram to analyze the day of week of serial crime. Team # 7071 Page 12 of 17 0 1 2 3 4 5 6 Monday Tuesday Wednsday ThursdayFriday Saturday Sunday Figure 7 Serial murder by day of week It can be seen from the diagrams that the crime mainly happened on Saturday and a longer “cooling-off” period comes after three short ones. When predicting the time of next crime, the cycle and the day of week may be considered. 5 Evaluating the Model The Danger-Rank Matrix Model has some obvious advantages. Firstly, it is convenient for the police to operate. It works just like a database. Once the police type in the key features, they can get the high-risk areas directly. Secondly, compared to traditional technique, our Danger-Rank Matrix Model can predict the location of next crime directly without considering where the suspect lives. This advantage guarantees the “first-mover advantage” of the police. Thirdly, as can be seen from the case study of the “black ghosts”, our model is relatively stable. Team # 7071 Page 13 of 17 6 Conclusions With the goal to generate a geographical profile and help the police to give prediction of the location of next crime, we combine the Maximum Likelihood Estimation and the Danger-Rank Matrix Algorithm (MLE & DRMM). Our model can help the police to rank the degree of danger and narrow down the locations with high risk. We also applied our model to the real case of the “black ghosts” in China to text the stability. Team # 7071 Page 14 of 17 7 Executive Summary for a Chief of Police 7.1 Overview of the Approach In order to foreclose further occurrence of serial crimes timely and effectively, we develop a Danger-Rank Model to forecast the next potential criminal location (“high-risk area). The procedural diagram below shows the steps of implementing the Model: Past serial criminal Record Key features Database Potential crime map Importance scores of Location Prediction of possible locations of next crime Warnings to some high-risk areas New serial criminal record Analyze Update Information exchange Generate Police Input Rank Output Geographical Profile Model Aid Team # 7071 Page 15 of 17 7.2 Situations Where the Approach is Appropriate Our approach is appropriate when the serial crimes fit the features as follows: 1) The offender is a “local hunter” and has a stable anchor point over the time period of the crimes. 2) The victims of the crime are the same kind of people. 3) The information about the crime is abundant. 4) When the exact address of the suspect is unknown or hard to get, our model can still predict the location of next crime. 7.3 Specific Recommendations for Implement ●In order to save the cost and police force, the police should concentrate on the locations of the highest risk area. ●Always remember that geographical profiling only can offer some kind of support. It should be combined with other detective techniques to enhance its efficiency. Other factors such as the influence of the press and psychological profile should not be ignored either. Team # 7071 Page 16 of 17 References [1] K．Bryan and T. Leise www.rose-hulman.edu/~bryan [2] http://zmdgaj.zhumadian.gov.cn/Article/ShowArticle.asp?ArticleID=50 [3] http://en.wikipedia.org/wiki/Peter_Sutcliffe [4] Bryan Nelson, 2004 http://www.deviantcrimes.com/defineserialmurder.htm [5] Keith Harries, 1999 http://everything2.com/title/Geographic+profiling