美赛数学建模优秀论文

1、Yet Another Mathematical Approach to Geographic ProlingControl #7502February 22, 20101Control #7502Page 2of 21AbstractGeographic proling is a mathematical technique to derive infor-mation about a serial crime spree given the locations and times of previous crimes in a given crime series. We have cre

2、ated a crime prediction model by using the anchor-distance-decay and hot-spotting geographic proling techniques. Based on the observation that the anchor-distance-decay model tends to predict a large area while hot-spotting provides a narrow, but not necessarily compherensive area, we devised a heur

3、istic scheme to combine the results of the two tech-niques. We then tested the combined model relative to both the anchor-distance-decay and hot-spotting proling techniques on a data set generated for this purpose. We found that using a specic combina-tion of the two results in an eective search are

4、a for law enforcement. The anchor-distance-decay geographic proling assumes that an oender chooses future crime targets dependent on distance from an “anchor point” (suchas a home or workplace and the targets desir-ability based on data from other crimes. The geographic prole yields a probability di

5、stribution indicating the likelihood of an anchor point at a particular location accounting for geographical factors and his-torical crime data. From a generated anchor point distribution, we compute the predictive distribution of possible future locations of the next crime. The other prediction sch

6、eme is hot-spotting , in which we generate a probability distribution of likely locations of future crimes by assuming that their geographic location is close to that of the of-fenders prior crimes, especially crimes commited most recently. Our combined model is a combination of the two that optimiz

7、es their rel-ative performance.Due to the diculty of obtaining sets of crime data, we generated a data set on which to test all three models. We believe that the generated data is representative of data that would be used by law enforcement agencies using these models.2Control #7502Page 3of 21 Conte

8、nts1Introduction 5 2Background 5 2.1Key Terms and Denition . . . . . . . . . . . . . . . . . . . . 5 2.2Cluster analysis and hotspot prediction . . . . . . . . . . . . . 6 2.3Rossmo and Geographic Proling . . . . . . . . . . . . . . . . 63.2Estimation based on prior oenses . . . . . . . . . . . . .

9、. . . 124Combined Model 13 5Example 14 5.1Anchor-Distance Decay . . . . . . . . . . . . . . . . . . . . . . 14 5.2Hot-spotting model . . . . . . . . . . . . . . . . . . . . . . . . 14 5.3Combined Model . . . . . . . . . . . . . . . . . . . . . . . . . 156Conclusion 15 A Executive Summary 173Control

10、#7502Page 4of 21 List of Figures1Generated historical crime data . . . . . . . . . . . . . . . . . 19 2Population density based on U.S. Census data . . . . . . . . . 19 3Prediction of home base location, demonstrating OLearys model of geographic proling . . . . . . . . . . . . . . . . . . . 20 4Pred

11、iction of next crime location using anchor-distance-decay 20 5Prediction of next crime location using hot-spotting . . . . . . 21 6Prediction of next crime location using the combined model . 21 4Control #7502Page 5of 21 1IntroductionGeographic proling is the process of aggregating and generalizing

12、geographic data about a series of crimes. This process has many uses for law-enforcement as it allows predictins such things as the criminals home or workplace, or the location of the next crime.Geographic proling of criminals is a relatively new eld, started by K. Rossmo around 1987, but only very

13、recently made mathematically rigorous by M. OLeary. However, geographic proling of crime locations has ex-isted for a long time in the form of hot-spotting, whereby police presense is increased in areas of high crime frequency.In our model, most geographic proling techniques are a result of work by

14、M. OLeary due to his careful and thorough work on the eld. M. OLearys analysis can be found in 4.2Background2.1Key Terms and DenitionFor our paper we clarify the following denitions A serial oender an individual who has committed a serial crime. A serial crime a crime in a series of 3or more crimes

15、committed at dierent times and locations. An anchor-point is the location from where a serial criminal is op-erating such as a home or workplace. It is assumed that the oender operates from one location and does not move his starting location in between the crimes. Geographic proling is the process

16、by which geographic information about an event or individual can be determined. Geographic proling of the oender establishes the oenders home base. Geographic proling of the crime series establishes the areas where the crimes are likely to occur, and thus provides a prediction for the location of th

17、e next crime. 5Control #7502Page 6of 212.2Cluster analysis and hotspot predictionA class of attempts to use software to aid in crime prevention rely on fore-casting crime “hotspots”, that is, areas where crimes are more likely to occur. The hotspots allow police to redirect resources to those areas.

18、 The hotspots are based on the compilation of geographic data from recent crimes. For ex-ample, Bowers, Johnson, &Pease 1attempt to decrease the size of clusters to make the prediction more eective at the expense of failing to cover outly-ing crimes. Other studies, such as Liu &Brown 3evalua

19、te the likelihood of crimes occurring at certain locations based on favorable factors, such as an-nual income and homeowner age. Most of these studies are observation-based and not mathematically rigorous.2.3Rossmo and Geographic ProlingIn 1991, while travelling on a bullet train in Japan, Dr. Kim R

20、ossmo began to realize a new approach to using software to help solve crimes. Rossmo was interested in using past crime data to generate a geographical prole of the criminal, that is, the likely area from whree the oender operates. Rossmos model was the following: The oender is more likely to target

21、 areas that are closer to his home or workplace than areas which are farther away. The oender does not target areas that are too close for psychological fear of being detected.Based on this model, he came up with the formulap x =N n(d (x, x n f+(1 B g f (2B d (x, x n gWhere N is the normalization co

22、nstant and f, g are empirically determined. d (x 1, x 2 is the metric-dependent distance function. Rossmo used the Man-hattan metric,d (x 1, x 2 , (y 1, y 2 =|x 2 x 1|+|y 2 y 1|To produce what is known today as Rossmos Formula.6Control #7502Page 7of 21 3Known Prediction Schemes3.1OLearys New Mathema

23、tical Approach to Crimi-nal ProlingOLeary 4approaches the geographic proling problem by developing a rigorous and explicit mathematical framework. OLearys framework takes into account geographic features for both anchor point prediction and crime site selection, and returns a distribution of oenders

24、 home base location. This location distribution and the same model can then be used to forecast future crimes.For the simplest case, suppose the oender has committed only one crime at location x . Then we compute an estimate for the probability distribution for the anchor point z using Bayes Theorem

25、,P (z , |x = P (x |z , (z , P (x where P (z , |x is the posterior distribution ; the probability density that the oender anchor point z and average oense distance given oender has committed a crime at the location x . P (x is the marginal distribution , which must be independent of z and to retain e

26、quality (ratherthan proportionality. (z , is the prior distribution ; our knowledge of the probability den-sity that oender has anchor point z and average oense distance before considering crime series.With the assumption that z is independent of the average oense distance we can factor to obtain(z

27、, =H (z (7Control #7502Page 8of 21where H (z is the prior probability density function for the distribution of anchor points before incorporating crime series. ( is the probability density function for the prior distribution of the oenders average oense distance before incorporating crime series. He

28、nce we haveP (z , |x P (x |z , (z , ,For the typical case, to estimate probability density for the anchor point z given crime series x 1,. , x n , Bayes Theorem impliesP (z , |x 1,. , x n = P (x 1,. , x n |z , (z , P (x 1,. , x n .Again P (z , |x 1,. , x n is the posterior distribution giving the pr

29、obability density that the oender has anchor point z and average oense dis-tance given oender has committed oenses at x 1,. , x n . P (x 1,. , x n is the marginal distribution , since it is independent of z and x it can be ignored. (z , x can be factored again into (z , x =H (z ( with same def-initi

30、ons for H and made above.Assuming that the oense sites are independent we can reduceP (x 1,. , x n |z , =P (x 1|z , ···P (x n |z , .Substituting we get thatP (z , |x 1,. , x n P (x 1|z , ···P (x n |z , H (z ( .8Control #7502Page 9of 21 Finally we can since we are only i

31、nterested in z we take the conditional distribution to obtainP (z |x 1,. , x n P (x 1|z , ···P (x n |z , H (z ( d.where P (z |x 1,. , x n gives the probability density that the oender has anchor point z given oenses at locations x 1,. , x n . This is a general framework for geographic

32、 proling that allowing for many possible choices of P (x |z , . This also allows us to easily add additional parameters or remove parameter without signicantly altering the mathematics.In construction of this framework two fundamental assumptions were made z is independent of ; average distance the

33、oender is willing travel is independent of oenders anchor point. oenders choice of crime sites are pairwise independent.Both assumptions are reasonable (othersources? maybe he uses in paper pg. 19 and necessary for some factorization. What remains is deciding a model for Oender Behavior P (x |z , .T

34、he simple model for oender behavior is the encompasses most models in the current literature. The following models make the assumption that all oenders have the same average oense distance known in advance. If we assume that an oender chooses a target location based only on the Euclidean distance fr

35、om oense location to oenders anchor point, then we result in a (normalbivariate distributionP (x |z , =142exp42|x z |2 .Assuming that all oenders have the same average oense distance obtain a product of normal distributionsP (z |x 1,. , x n =142nexp42nn =i|x i z |2 . 9Control #7502Page 10of 21 Anoth

36、er model of oender behavior is the maximum likelihood estimate for the anchor point as the mean center of the crime site locations. This is the centrography mentioned in Rossmos Thesis 5and basis of Rossmos Formula; this is also the mode of the posterior anchor point probability distribution P (z |x

37、 1,. , x n .Another behavior model is that the oender choose a target location based only on the Euclidean distance from the oense location to the oenders anchor point, but as a (bivariatenegative exponential so thatP (x |z , =22exp2|x z | .and by making the assumption that all oenders have same ave

38、rage oense distances and all anchor points are equally likely, thenP (z |x 1,. , x n =22nexp2ni =1|x i z | .And the maximum likelihood estimate for the oenders anchor point is sim-ply the center of minimum distance for the crime series locations.And nally we have the hit-score method popular in curr

39、ent methods by constructing hit-score functionS (y =ni =1f (d (x i , y The general framework of OLearys model allows for a realistic geographic proling by allowing a simple way to incorporate geographical features into the model. We suppose the probability density distribution is in the form P (x |z

40、 , =D (d (x , z , G (x N (z , where10Control #7502Page 11of 21 D (d, is a factor accounting for distance decay eect . d (x , z is a suitable distance metric used in D . G (x accounts for local geographic features that inuence the selection of oense site. N (z , is a normalization factor independent

41、of x .Like we have seen before, the distance decay function D can take many forms and any desired distance metric d . But, the main factor of interest is the geographical inuence on target weighting, G (x .A simple variant on G (x is a characteristic function approach, that isG (x =J (x =1if x J 0if

42、 x / J .This accounts for a region J corresponding to one or more regions of jurisdic-tion, i.e. area where crimes are known. Crimes can occur outside of J , but these are presumed to be unknown to proler. Also the anchor point may lie outside of J . Fortunately the model treats areas outside of J a

43、s having no information about crimes.G (x =Ni =1K (x c i | .As in the software code, a typical choice for is the mean nearest neighbor distance between historical crimesites c i and for K a simple normal/bivariate distribution.11Control #7502Page 12of 21 Now the prior probability density for oender

44、anchor points H (z incor-porates knowledge of oenders anchor points before using information from the crime series. If nothing is known H (z =1. Otherwise, like software, assuming that the anchor pint is the oenders home and the distribution of anchor points follows population density we can incorpo

45、rate U.S. census data, which is given at the block level. Using more kernel density parameter estimation, we calculate H (z byH (z = N blocksi =1=p i K (z q i | A i .where a block has population p i , center q i , and a chosen bandwidth equal to the side lenth of a square with area A i of the block.

46、 With available data, H (z can also be calculated by starting with similar anchor points of similar crime sprees and calculate H like G .The nal element is the estimation of the prior distribution of average distance to crime ( which is distribution of the average oense distances across oenders. The

47、 software uses a complicated approach of Tikhonov regularization and L-curve method to compute .Posed mathematically, to estimate the location of the serial oenders next target, given a series of crimes at x 1,. , x n committed by a single serial oender, we wish to estimate P (x next |x 1,. , x n wh

48、ere x x is the location of the next oense. The simple Bayesian approach is to calculate the posteriorpredictive distribution P (x next |x 1,. , x n =P (x next |z , P (z , |x 1,. , x n dz (1dz (2dP (x next |z , P (z , |x 1 ···P (z , |x n H (z ( dz (1dz (2d.Where we used the factoring m

49、ethods used earlier by making the same inde-pendence assumptions.3.2Estimation based on prior oensesOur second scheme to predict future crime locations is a time-dependent geographic proximity model. The fundamental assumptions of this model 12Control #7502Page 13of 21are: The oender prefers to targ

50、et areas that he has experience with, that is, areas that he has previously targeted The oender prefers to reoend in the same areaBased on these assumptions, we directly determine a probability distribution of the next oense as follows:1. For each past crime scene (x , t , we dene a kernel density f

51、unction K (x , =N (x , as a bivariate normal distribution centered at x and with variance where is the mean nearest neighbor distance over all crime scenes using the Euclidean metric. The bivariate normal distribution has the formN (x , =142exp 42|x z |2 . 2. We dene a temporal decay function (t whe

52、re t is the time elapsed since the crime. For simplicity, we dene to be linear in time with (t 0 =0. 5for the rst crime and (t n =1for the last crime.3. The search area is the sumP (x next |x 1,. , x n =ni =1K (x i , (t i This procedure is a variation of the one used in OLearys model 4to gener-ate a

53、 possible target density function provided a list of historical crime data. Our model instead generates a target density function specic to the oender and includes time-dependse whereas OLearys model does not.4Combined ModelLet P 1be the result from using OLearys method, and P 2be the result from th

54、e second proling scheme. Our heuristic combined estimate isP =P1+(1 P 213Control #7502Page 14of 21 To nd a “good enough” value for , we consider the eect of the two models on generating a useful prediction. While an anchor-distance-decay model can provide a very good estimate of the oenders home loc

55、ation, it is not very useful in predicting the location of the next crime. Practically, predicting based on the home base location yields a large watch area “centered” at the home base and encompassing all prior crimes. This area is too big to search for all practical purposes. However, it provides

56、us with a practical upper bound on the area where the next crime could occur.On the contrary, the second scheme eectively nds clusters in crimes and generates a much smaller watch area. Doing so increases the possibility of missing the next crime, but the area is large enough so that it catches most

57、 crimes. Thus the proximity-based model is more important to law enforcement than the anchor-distance-decay model.Lacking more precise comparative evaluation of the two models, we as-sume=0. 255ExampleTo test our combined model relative to the anchor-distance-decay and hot-spotting models, we generated data to represent a series of residential bur-glaries. We also generated a set of prior crime and home base location data. Figure 1shows the generated historical data from two crimes, plotted in Google Earth. We belie