国际数学建模大赛一等奖论文Research on the Location and Time of Serial Criminals

1、for office use only t1 _ t2 _ t3 _ t4 _ team control number problem chosen b for office use only f1 _ f2 _ f3 _ f4 _ research on the location and time of serial criminals february 22,2010 abstract twisted souls and antisocial spirits have resulted in serial murders. in order to prevent such crimes w

2、hich cause great harm to public security, the paper has established effective early warning systems and efficient detection programs by using mathematical models so that we can reduce the number of potential victims significantly. warning preparations: first, the paper makes a statistical analysis o

3、f the victims personality characterictics (such as physical characteristics, clothing, occupations, etc.), locations and time of the criminal offenses, then the paper has a principal factor analysis of those factors so that we can rule out the obvious or non-intrinsic factors. the principal factors

4、obtained above are the characteristics of the crime targets in the serial murders. according to the conclusion, early warnings are given to the vulnerable groups and residents with the same situation, residence and daily schedule so as to strengthen their vigilance. scheme one: the establishment of

5、geographical profile based on time-series analysis model according to the statistics of the locations of serial crimes, the paper predicts the distance between adjacent cases via gps (global positioning system) and establishes a time series analysis model. the paper uses spss software to predict the

6、 distance between the possible locations of the next crime and the past crime scenes. the data of the serial murders committed by peter sutcliffe and of the serial robberies in dallas are used to verify the model: at 95% confidence level, the data obtained tally with the real circumstances. scheme t

7、wo: the establishment of geographical profile based on the adaptive network-based fuzzy inference system (anfis) and fuzzy matter-element analysis based on a large number of crime statistics, the paper has established the adaptive network-based fuzzy inference system (anfis) model and the subtractiv

8、e clustering initializing fuzzy inference system (scifis) model. the crime data in dallas from 1992 to 2007 are taken as the sample to verify the accuracy of the anfis model for the prediction of the distribution of the citys high crime areas and the areas where similar criminal cases take place. th

9、en, based on results of neural network training, the crime areas obtained are made further screening. the complex element matrix is established by using fuzzy matter-element analysis. thus the associated vector is generated through the relationship among the three elements of matter-element. the pos

10、sible locations of the next crime are screened through the comparison of correlation. the paper predicts the serial robberies in dallas and compares the data with the facts using the model reaching 95% confidence level. therefore, the model is also reasonable and reliable. the prediction of the time

11、 of crime: based on the time series formula, the paper predicts the time of the next crime by commensurability of ternary, quaternion and quintuple and the existing time of criminal offenses. the criminal is predicted over time and space based on the geographical profile generated by the two schemes

12、 mentioned and time prediction method mentioned above. in this way, the case solving rate can be significantly increased, which is also helpful to build a harmonious society. keywords: principal factor analysis; time-series analysis; adaptive network- based fuzzy inference system; fuzzy matter-eleme

13、nt analysis; commensurability; executive summary in view of the increasing criminal rate in the world, the polices final goal is to increase their case solving rate. based on this, a model is established under the premise of serial criminal cases, which is used to predict the time and location of th

14、e criminal cases. there are two schemes in the model, which can be used to predict the geographical profile and one method to predict the time of the crime. the first scheme can predict the circular region of the next crime by using the time series analysis model and further reduces the predicted re

15、gion by using the principal factor analysis method. the second scheme can predict the next possible location of similar cases by using the adaptive network-based fuzzy inference system. then a fuzzy matter- element analysis is made of the common characteristics of the victims in the serial crimes so

16、 that the prediction region can be further reduced. while predicting the geographical files, the commensurability of ternary, quaternion and quintuple is used to determine the time of the next criminal offense. in this way, the suspected serial criminal can be determined from the two aspects: time a

17、nd space. this model has some advantages as follows: 1. the simplicity of its applying method scheme one： this paper primarily confirms each accurate location where the crimes have emerged, following that step, it calculates the distance between neighboring places, then puts the figures into statist

18、ical software of spss, by adopting its predicted function of time series to predict the distance between the address of this crime and next one. meanwhile, it can be identified that the predicted distance will move within a domain, predicting it as an annular region on the map, which can be defined

19、as the potential place where next crime may happen. by analyzing the environment of the criminal places and each accurately characters of criminals to ensure the mainly attacking targets of the criminals. furthermore, by carrying out the experienced analysis to the annular region, it can reduce the

20、criminal regions where the next crime may happen. scheme two： firstly, the researchers compile the anfis fuzzy systematic procedure into the matlab software, and input the times of criminals of its completely criminal category which belong to the studied city, subsequently, it can produce the list o

21、f possible happened criminal regions of all kinds of crimes in each cities. moreover, the researchers compile the subtractive clustering initialization fuzzy deducing anfis model into the matlab software, and input the data of serial crimes and the potential criminal regions got from the above proce

22、dure into this software, then it can produce the possible happened regions of the criminal rate of the definitely criminals. therefore, it is necessary to do the checking in the regions where there is highly criminal rates. assurance of time prediction sorting out the cases happened time according t

23、o the months sequence, taking “the yorkshire ripper” for example, for the first time, it happens on july 5, 1975, and the number is 7. adopting it as a standard, it can deduce a figure of each happened region. once again, the researchers compile the procedures of commensurability of ternary, quatern

24、ion and quintuple in the matlab software, and then to input these date, it can produce the predicted criminal time. (limited to the month accurately, take january, 1975 as the standard of the model). 2. the highly reliability the predictive domain in the geographic files of scheme one is made under

25、the circumstance of its significant test level to be 5%, thats to say, the reliability of scheme one is 95%. the scheme two carries out the error testament by using the data of dallas, thus ,it can obtain the permitting errors difference to be 0.0683, in another word, the reliability is 93.2%. becau

26、se it is impossible to enlarge the errors difference by combining of the two methods, therefore, the least value of reliability is 93.2%. the time prediction adopts the method of commensurability of ternary, quaternion and quintuple; the acquired predictive time is the largest figure among all of po

27、ssible criminal time, which proves to be the possible criminal time. 3. the wide range of its application scheme one testifies the possibility of the model by adopting two kinds of psychological criminal types (the murder is identified as emotional crime, while, the robber is identified as profitabl

28、e crime). it proves that scheme one is accurately right to most of the cases. scheme two primarily calculates the regions where the similar crimes happened, it can be applied into the prediction of non-serial criminals instead of selection which does not take advantage of serial crimes, and therefor

29、e, its application turns to be more widely. because the two methods take applications into consideration from difference aspects and the repeating factors can be neglected, then it is impossible to add the errors difference by combining of the two methods, on the contrary, it may be reduce the diffe

30、rence. when the researchers do the predictions, it can take advantage of the commensurability method to increase the probability of arresting criminals, and decrease the losing of property and the labor. this method effectively reduces the waiting time of policemen in the predictive regions. content

31、s introduction.6 analysis.6 model preparation:.7 assumption.8 modeling and solution.8 1.appropriate warning .8 (1) theory of the principle factor analysis.8 (2) analysis of yorkshire ripper case.9 2. building a model to generate the geographic profile.10 scheme one: to establish a time series model.

32、10 (1) introduction to time series analysis.10 (2) random time series theory 3 .11 (3) solving the cases of the yorkshire ripper .15 (4) the analysis of the continuous robbery in dallas.18 scheme two: constructing the predicting model of anfis .19 (1)t-s (takagi-sugeno) fuzzy model.19 (2) systematic

33、 structure of anfis.20 (3) the systematic algorithm of anfis.21 (4) analysis of the series criminals cases in dallas .22 (5) fuzzy matter-element analysis scheme .25 (6)analysis of the series criminals cases in dallas .27 3. commensurability prediction of real time crime .29 (1) commensurability pre

34、diction.29 (2) model checking .30 the merit and demerit of the model.33 the direction of improvement of the model.34 references.35 introduction nowadays, because many people have certain distorted spirits in their subconscious minds, thus, it causes many serial crimes in the world. taking the notori

35、ous “yorkshire ripper” case for example, it is necessary to construct a kind of quickly and effective scheme to help the policemen to arrest the criminals. according to the requirements of the local police agency, a scheme should be developed to aid their investigations of those suspected serial cri

36、minals. the solution should make use of at least two different schemes to generate a geographical profile. it also should develop a technique to combine the results of the different schemes and generate a useful prediction for law enforcement officers. the prediction must provide some kind of estima

37、te or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. if there are any other evidence in the estimate, the researchers must provide specific details about how the researchers incorporate the extra information. finally, the researchers wil

38、l provide an additional executive summary to increase the efficiency of capturing the criminals indeed. analysis currently, the crimes rate is increasing in the world, under the social pressure, family violence and unequal treatment and so on, lead to a phenomenon that some people suffer mental defo

39、rmities, and take a revenge on others. so they pursue their inner satisfaction by killing, robbery and other unlawful criminal acts to release their hatred. in view of this social phenomenon, in order to arrest criminal suspects as soon as possible, the regular criminal psychology analysis only can

40、assist to investigate the criminal means, the crime characteristics and the psychological changes of the criminal suspect. the scheme of, however, generating geographic file can effectively predict the next crime location of the suspect and the crime cycle can be predicted based on time series, this

41、 can help the local police arrest the criminals quickly. two schemes to generate geographic profile: scheme one: (1) based on the respective characteristics of various victims, using the principle factor analysis scheme to extract the principle factor. (2) based on the principal factor analysis sche

42、me, to the extracted principle factor, using time series analysis scheme to get the distance between the locations of this crime and the next one. scheme two: based on the subtractive clustering fuzzy inference system initialization model and the anfis-based model of adaptive neural-fuzzy inference

43、system, according to the 18 years of crime data in dallas, texas, we can make a reasonable and effective prediction to its high incidence of crime. further screening the region received through the previous neural network training and using fuzzy matter-element analysis to sift out the crime locatio

44、n of the next time. prediction of the crime time: based on the determination of the geography profile, we can predict the crime time of the criminal in order to improve the case solving rate. according to the prediction function of the surability of ternary to the time series, this model predicts th

45、e crime time of peter sutcliffe. model preparation: peter sucliffe peter sutcliffe is the son of john and kathleen sutcliffe. he was born in the small town of bingley, just six miles north of bradford. he is inward-looking, and performed poor at school. at the age of 15, he left school. he once made

46、 a living as a grave-digger. his personality and work had a great effect on him as one of the worlds top ten serial killers. in the 5 years from october 30, 1975 to november 17, 1980 , peter sutcliffe killed a total of 13 people, and 10 of them subjected to his malicious attacks. he was arrested unt

47、il may 22, 1981. peter sutcliffe was very cruel when killing. he first pounded the victim in the head with a hammer, and then stabbed a sharp screwdriver into the victims head and abdomen, then conducted fierce beating and kicking. the victim and the persons suffered his malicious attacks are all ov

48、er the five central united kingdom cities: leeds, bradford, huddersfield, manchester, and keighley, which involves in 23 cases. 1 victims most victims lived in yorkshire in northern england, near the residence of peter sutcliffe. the locations of the crimes are relatively near to each other. all the

49、 victims were women, and most of them were prostitutes, rangeing in age from 16 to 47. status quo of dallas: according to the data published by the u.s. police authorities, in the cities which the population of more than 1 million lived in, dallas enjoys the highest crime rate in 2004. it made dalla

50、s the first malicious city in a row of seven years. in various crimes, the murder rate in dallas ranks second in 2004, and there were 38 criminal cases occurred per thousand. 2 assumption 1.in scheme one, there is no specific direction for the determined crime region. 2.the psychology of each type o

51、f criminals are similar. 3.there is no particular decisive evidence in the cases. 4.each victim is independent of each other , that is, there is no close relationship among them. modeling and solution 1.appropriate warning according to the analyses of various criminal cases, we find that victims kil

52、led in a chain of serial crimes are impacted by various factors, such as the victims physical characteristics and clothing, location of crime and so on. so appropriate warning is required to extract important factors. we can achieve this goal by adopting principal factor analysis scheme. (1) theory

53、of the principle factor analysis the basic idea of principal factor analysis is to find out few random variables which can control the original variables ,through the analysis of ), 1(mifi internal structure of the correlation coefficient matrix of the variable. the princople of selecting the common

54、 factor is to make it contain information the original variables as much as possible, building a model , ignoring , using toreplaceefaxef , with this model to reproduce the relevant ralationship among various sub-)(pmx variables of the original variable , thus to simplify the variables and ), 1(pixi

55、x reduce the dimension. the mathematical model is: 111 nmmnn efax that is, nnnmnn m m n e e e f f f aaa aaa aaa x x x 2 1 2 1 21 22221 11211 2 1 among them, is called factor loading, indicating that the lading of the th variable ij ai in the common factor th, which reflects the correlation between t

56、he variables and j the main factors; is called the common factor, which is common in ), 2 , 1(nifi all the variable factors; is a special factor , including the random error. they meet: i e and is not relevanfe ；，that is , the jiffcov ji , 0),(mjiffcovfvar jii , 2 , 1,1),()(。 covariance matrix of th

57、e variable is order unit matrix.fm ；，that is, the covariance matrix of jieecov ji , 0),(njievar ii , 2 , 1,)( 2 。 the variable is order diagonal matrix.en (2) analysis of yorkshire ripper case in this case, the main characteristics of victims include: physical appearance, height, sex, occupation, dr

58、ess and so on. the main characteristics of the victim lication include: the distance from the road, the nearby population density and occupational distributions. by principal factor analysis of these factors and similarity comparison of serial cases, characteristics of victims can be drawn as follow

59、s: tall long-haired women, gorgrously-dressed prostitutes, and the victim locations are generally those streets or parks where prostitutes frequently stayed with small populations passing, while mostly ,the crime time was between 11 pm and 2am. therefore, we present warnings against residents who li

60、ve in the places with these geographical characteristics. 2. building a model to generate the geographic profile scheme one: to establish a time series model (1) introduction to time series analysis recognizing the changeable characteristics which time series possess in order to choose a different a