2010年美国大学生数学建模竞赛B题一等奖
数学建模美赛翻译-关于2010年数学建模美国赛b题的翻译
2010年美国数学建模邀请赛试题2010-02-19 09:09PROBLEM A: The Sweet SpotExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe th at “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?PROBLEM B: CriminologyIn 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a “center of mass” of the locations of the attacks. In the end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the “geographical profile” of a suspected serial criminal ba sed on the locations of the crimes.Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for lawenforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings.In addition to the required one-page summary, your report should include an additional two-page executive summary. The executive summary should provide a broad overview of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool. The executive summary will be read by a chief of police and should include technical details appropriate to the intended audience.。
2010年数学建模B题—上海世博会影响力的定量评估
2010年上海世博会影响力的定量评估摘要上海世博会的举办对中国乃至世界的快速发展都产生了深远的影响。
本文着眼于上海世博会促进旅游业这一侧面,提出并解决三个重要的问题。
一:上海世博会对上海入境人数的贡献旅游业发展具有趋势性、周期性、随机性,根据这一规律建立入境人数本底趋势模型,使用内插法处理过的2007年7月至2010年3月数据拟合出本底趋势线,进而求出2010年4、5、6、7月的本底值,与实际人数相比得到世博会对入境人数平均贡献率为22.41%。
二:未来几个月上海入境人数走势利用第一问模型得出08-11月世博会给上海带来544815人的额外入境人数。
三:世博会给上海带来的直接利益通过对已知的世博会入园人数进行分析,建立每天入园人数的时间序列分析模型,由于人员的变动性,故采用时间序列分解法求解。
运用趋势外推法加权拟合出长期趋势直线,综合考虑影响参观人数的随机因素,预测出上海世博会最终入园人数为7010.82万人。
门票总收入达103.88亿元。
关键字:本底趋势线内插法定量评估时间序列分析模型趋势外推法一、问题重述题目背景:2010年上海世博会是首次在中国举办的世界博览会。
从1851年伦敦的“万国工业博览会”开始,世博会正日益成为各国人民交流历史文化、展示科技成果、体现合作精神、展望未来发展等的重要舞台。
请你们选择感兴趣的某个侧面,建立数学模型,利用互联网数据,定量评估2010年上海世博会的影响力。
问题提出:上海世博会促进旅游业这一侧面,提出了三个重要的问题。
问题一:上海世博会期间,上海的入境人数有什么变化,给出相应的数学模型,并计算世博会对入境人数的平均贡献率。
问题二:未来几个月上海入境人数走势。
问题三:从互联网获取每天入园参观人数,建立每天的参观人数的预测模型,并预测最终入园人数并估算世博会的门票总收入。
二、符号约定三、模型假设1、忽略国家政策、军事、节假日等方面对上海入境人数的影响。
2、将世博会期间的天气情况影响限制在一定波动范围内。
2010美国数学建模B题论文
Team#8254page1of16 Play your best hit:The mystery in baseball batMCM Contest Question ATeam#8254February22,2010Contents1Introduction (2)1.1Problem Restatement (2)1.2What is the sweet spot on earth? (2)1.3How to cork a bat? (2)1.4The material of baseball bat matters? (2)2Assumption (3)3A Simplified Model (3)3.1Model Description (3)3.2Finding the”sweet spot” (5)4An Augmented Model (6)4.1Model Description (6)4.2Why no corked bats and metal bats? (8)4.2.1Qualitative Analysis (8)4.2.2Quantitative Analysis (9)5Sensitivity (12)6Conclusion (13)6.1Strength (13)6.2Weakness (13)6.3Recommendation (13)7References (13)8Appendix (14)Team#8254page2of16 1Introduction1.1Problem RestatementEvery experienced hitter knows that there is a spot on the baseball bat that when hitting with this spot,the hand of the hitter feels no pain and the ball can be hit farther.This is the”sweet spot”.According to the theory of torque,this spot should be at the end of the bat,but the experience of the hitters proves it wrong.The location of”sweet spot”on a given baseball bat is approximately 6-1/2”from the end of the bat[1].Our purpose is to establish a model to give a scientific explanation.In addition,some players believe that drilling a cylinder in the end of the bat andfilling it with cork or rubber,namely,to cork the bat [2],enhances the”sweet spot”effect.We would like to found our model and use it to interpret it.What’s more,the question that whether the material of bats matters is also a part discussed in this paper.1.2What is the sweet spot on earth?There are many definitions of”sweet spot”[3].Here in this paper,we define it as the location where maximum energy is transferred to the ball.1.3How to cork a bat?”Corking”is to drill a1inch diameter hole6inches longitudinally into the bat’s barrel end.The structure of the real bat and the corked bat can be showed in Fig1.In our augmented model,we will analyze the effect of corking.1.4The material of baseball bat matters?Compared to the wood bats,aluminum bats is not easily be broke,and as the aluminum bats are hollow,the thickness of the shell can be manipulated so that the center of mass may be more closer to the handle,and consequently reducing the perceived weight while swinging.In this way,it increases the mobility of(a)Normal baseball bat(b)Corked baseball batFig1:Normal baseball bat and corked batTeam#8254page3of16 the hitter.What’s more,using aluminum bat can largely improve the energy of the ejecting ball,which may be too hard to catch or even dangerous for players. All those features lead to the prohibition of the using of aluminum bat[4].We will explain it in details later in this paper.Fig2:Cross section of aluminum bat2AssumptionAs the hitting process is too complex,we make the following assumptions in order to simplify the problems and establish our models:1.The better hitting effect means a larger exit speed of the ball and a shorteracceleration time of the bat.2.When the hitter swings,hitter-bat system rotates around a vertical axis that”penetrates”the hitter.3.The collision of the ball and the bat is one-dimensional.4.The hitter always hits the ball at the”sweet spot”.3A Simplified Model3.1Model DescriptionIn our simplified model,we omit the situations that the player might modify the baseball bat and concentrate on explaining why the”sweet spot”is not at the end of the bat but a few distances from the end.When you hit a ball,the bat vibrates in response.These vibrations travel in waves up and down the length of the bat.At one point,called”the node”, the waves always cancel each other out.If you hit the ball on the bat’s node, the vibrations from the impact will cancel out,and you won’t feel any stinging or shaking in your hand.Since little of the bat’s energy is lost to vibrations when this spot is hit,more can go to the ball[5].We will found a model based onTeam#8254page4of16 the characteristics showed as italics.In our model,we ignore the swing of the hitter’s arms when hit.We denote the spot where the hitter holds the bat as pivot, assuming that the pivot isfixed,and the bat is mounted on the pivot so that it can swing around the pivot freely.The parameters we use are given in the tabular Notation,and the force and motion diagram is showed in Fig3.NotationSymbols MeaningF ball the applied force in the collisionF px,F py the vertical and horizontal component forces of the force in the pivotcm the center of mass of the batp the impact pointpivot the spot where the hitter holds the batI pivot the moment of inertia of the bat with respect to pivotL x−y the distances between x and yαthe rotation angular acceleration of the batm bat the mass of the batL sum the whole length of the batFig3:The force and motion diagram of the batNext,we will analyze the model above with the knowledge of kinetics in order tofind the”sweet spot”.Team #8254page 5of 163.2Finding the ”sweet spot”According to the theorem of moment of momentum we have:F ball ·L pivot =I pivot ·α(1)and based on the theorem of motion of center of mass:∑F =ma c we have:F ball +F px =m bat ·L pivot −cm ·α(2)Therefore,with equation (1)and (2)we have:F hand =F ball m bat ·L pivot −cm ·L pivot −p I pivot−1 Thus we know that the horizontal component force is 0whenL pivot −p =I pivotm bat L pivot −cmIn other words,when the distance between the pivot and the p is I pivot m bat ·L pivot −cm ,the p is the sweet spot.In order to find the specific location of the ”sweet spot”on a certain bat,we quote the data from [6]and choose a C243wooden bat,thus we get the parame-ters L pivot −cm =0.42m ,I pivot =0.208kg ·m 2,L knob −pivot =0.15m ,m bat =0.905kg ,L sum =0.86m .Then we calculate L pivot −p =0.55m ,so that L knob −p =0.70m .From the re-sult we can obviously see that the ”sweet spot”is about 0.16m far from the end of the fat part of the bat.The forces on different location of the bat can be showed in Fig4.Fig 4:The rotation system in a hitting processIn this way,we have successfully demonstrated that the experience of the hitters is right.Team#8254page6of16 4An Augmented Model4.1Model DescriptionThe previous model fails to take the situation that some hitter may modify the bat into account.What’s more,in real baseball game it is impossible that the arms of the hitter are stationary when hit.So we augmented ourfirst model. In this augmented model,we assume that the hitter and the bat form a rotation system in which the hitter stays vertical and the bat stays horizontal.The hitting process can thus be modeled as:the system gets a torque T generated by the hitter and rotates around the axis which vertically”penetrates”the hitter and then hits a ball.After hitting process,the ball ejects out with a velocity of v f. The distance between the impact spot and the axis is r.The process is described in Fig5.Fig5:The rotation system in a hitting processWe notice that the ball gets maximum velocity after hitting means the effect of hitting is optimal,so we should focus on the exit speed of the ball.Two processes are displayed as follows:Process1:The hitter swings the bat and accelerates the bat to an angular velocity ofω.Process2:After the imperfect inelastic collision of bat and ball,the ball ejects out with a velocity of v f.In process1,the moment of inertia of bat and the air resistance will hamper the rotation of the system of hitter-bat.It is easy to calculate the liner velocity of any spot on the bat by the kinematics formula v=ωr,and according to the formulas given by Keith Koenig in reference[7],we know that the angular ve-Team#8254page7of16 locity and the linear velocity of the impact spot on the bat before the collision in process1are:ω=TK Dtanhcosh−1expK DI hitter+I batθv bat=rω(3) where•r is the rotation radius,equals the length from the impact spot to the axis.•T is the torque applied to the hitter-bat system which is generated by the hitter.•K D is an aerodynamic parameter,which is given byK D=12ρC D DL44(for more please read reference[7])•tanh is hyperbolic tangent function.•cosh−1is anti-hyperbolic cosine function.•exp is exponential function.•θis the angle that the bat rotates.•I hitter is the hitter’s moment of inertia with respect to the axis.•I bat is the bat’s moment of inertia with respect to the axis.In our case,we treat T,K D,I hitter,θas constant parameters while r and I bat are the only variables.In process2,we assume that the bat and ball have an one-dimensional col-lision and they both have some extent of deformation.The deformation trans-forms a small part of the kinetic energy(about25%,reference[8])into potential energy stored in bat and ball,while most part(about75%)of it dismisses due to the friction and the oscillation of the bat.And referring to reference[7]we havev f=COR−r M1+r Mv ball+COR+11+r Mv bat(4)where•COR is the coefficient of restitution,it will be discussed in the next part.•v ball is the velocity of the ball right before the collision with the bat.Team #8254page 8of 16•v bat is the velocity of the bat right before the collision with the bat.by equation (3)and (4),we getv f =COR −r M 1+r M v ball +COR +11+r M r T K D tanh cosh −1 exp K D I hitter +I batθ (5)The parameters:COR ,I bat ,r M are explained in Appendix .4.2Why no corked bats and metal bats?In order to reveal the influence of each parameter on the velocity of the ball thus to figure out the effect of the corking behavior and predict different be-havior for wood and metal bats,we make some analysis about the application procedure of our augmented model.4.2.1Qualitative AnalysisWe assume that the velocity of the ball is constant before the collision with the bat,and the impact spots are on the same position of the bat,namely,r is a constant.The corking has two aspects of influences:one is the decrease of mass,the other is the deviation of the center of mass.1.Corked bat•The influence on v fDue to the lower density of material corked in the barrel,the whole mass of bat inclines and the center of mass moves to the rotation axis a little.On one hand,it will cause the decrease of I bat as I bat = r 2dm ,and according to the expression of v f :v f =COR −r M 1+r M v ball +COR +11+r M r T K D tanh cosh −1 exp K D I hitter +I batθ we know that the value of r T K D tanh cosh −1 exp K D I hitter +I batθ will increase.On the other hand,as we haver M =m ball (z −z p )2I cm +m bat L 2pivot −cmTeam#8254page9of16 we know that if m bat decreases and the center of mass deviates toknob,I cm and L pivot−cm will both decline,and that will make r M in-crease and lead to the decline of both COR−r M1+r M v ball and COR+11+r M.How-ever,it is impossible for us to ensure whether the value of v f willincrease or decrease without specific data.We will explain it later inthe Quantitative Analysis section.•The influence onflexibilityWe regard the procedure that the hitter-bat system is accelerated fromstationary situation to rotating with an angular velocity ofωas auniformly accelerated motion procedure.By kinematic formulas weknow that the acceleration time is t=2θω,and from the expression ofωmentioned in page6,section4.1,we know that as the I bat decreasesdue to corking,the value ofωwill increase,so that the accelerationtime will be shorten and in this way theflexibility develops.2.Metal bats vs.wood batsUsing our augmented model we are able to analyze the behavior of bats that have different composition material.•The influence on v fMetal bats(usually aluminum bats)have trampoline effect due togood elasticity,so that it has larger COR than wood bats(usually ashbats).In addition,as the aluminum bats have less mass,the influenceof material is just like that in discussing corked bats.It is obvious thatin order tofind which kind of bat’s v f is larger we need the supportof data.•The influence onflexibilityThe aluminum bats are lighter than wood bats so that the influenceonflexibility is the same as corked bats.4.2.2Quantitative Analysis1.Data usingAs the qualitative analysis cannot show the exact differences between wood bats,corked wood bats(corked bats)and aluminum bats,we quote the data from reference[7][12][13][14]to verify our augmented model.Table1:Value of ConstantsTeam#8254page10of16Constants Valuem ball/(g)134.2v ball/(m/s)25(assumption)T316N·mK D/(N·m·s2)0.00544θ 2.36radr/(m)0.762R hitter−pivot/(m)0.305I hitter/(kg·m2)0.444Table2:Value of VariablesValues Wood bat Corked bat Aluminum batCOR0.500.500.58mass/(g)876834827I p/(kg·m2)0.2110.1950.17L cm−pivot/(cm)424039.5I bat/(kg·m2)0.5160.4760.446z−z p/(m)0.470.460.45r M0.14030.14540.1596 Using the data above we can calculate the three relative values of v f that are47.1m/s,47.6m/s and51.3m/s.The I bat is calculated based on parallel axis theorem.For a better compare of the hitting effects of the three bats,we use com-puter to make a hitting experiment simulation about theflying trajectories of the ball after hitting:Assuming that the ball dose a slanting parabolic motion and the launch angle of the ball after hitting with the bat isθ=30◦.Fig6shows the trajec-tories of balls that are hit by the three kinds of bats.2.The influence of three kinds of bats onflexibilityCombine the formula t=2θωwith the expression ofω,we have the expres-sion of the acceleration time t:t=2θTK Dtanhcosh−1expK DI hitter+I batθAfter using the values given in the table1and table2,we get:Kinds of bats t/(s)Wood Bats0.121Corked Bats0.118Aluminum Bats0.116Team#8254page11of16Fig6:The trajectories of the balls hit by the three kinds of bats It is easy to see that the acceleration time of corked bat is0.003s shorter than that of wood bat.During this time,the incoming ball moves0.075m farther,that is to say,theflying distance of the incoming ball from the ser-vice point to the collision point when using a corked bat is0.075m farther than that of wood bat,so that the hitter would feel more easy to deal with the incoming ball as he or she has more time to react and accelerate the bat when using a corked bat.In the same way we know that when usinga aluminum bat rather than a wood bat,the hitter has0.005s longer and0.125m farther to handle the bat.3.Summary of our analysisAt this point,we are able to answer the second and third questions about the corking behavior and the matters of different materials.•Why does Major League Baseball prohibit”corking”?From the analysis above we know that for a baseball bat used in ourmodel,if it is corked,the swinging velocity of bat right before hittingthe ball will increase0.5m/s and theflexibility of swinging can alsobe developed,as the mass of the bat decreases and the center of massof it deviates.•Why does Major League Baseball prohibit metal bats?According to our model,the hitting effect is closely linked with thematerials.We know from our qualitative verification that the hittingvelocity of using a aluminum bat is4.2m/s more than that of woodbat,and the deviation of the center of mass further gains0.005s reac-tion time for the hitter.As the ball hit by aluminum bat has a muchbigger velocity,it makes the catching of the ball difficult and evendangerous.Team#8254page12of16 5SensitivityAfter analyzing the application of our augmented model,wefind it neces-sary to analyze the sensitivity of our model,aiming at implementing it more effectively.We choose to research the exit speed of normal wood bat with differ-ent parameters.1.The influence of massThe change of mass will influence both r M and I bat,but as the value of them are too hard to obtain by direct calculating,we make some simplificationsbelow:r M=m ball (z−z p)2I p=m ball(z−z p)2I cm+m bat L2pivot−cmIn this expression,as I cm is very small(about0.04kg·m2),we treat it as a constant one.It is the same to z−z p(about0.47m)and L pivot−cm(about 0.42m).Hence,the value of r M only correlates to m bat,and we haver M=0.1340.4720.04+m bat×0.422then in I bat=I cm+m bat·L2hitter−cm,we also treat L hitter−cm(about0.725m) as a constant,so we haveI bat=0.04+m bat×0.7252Relating to equation(5),we denote COR=0.5and get:m bat(kg)0.830.850.870.89v f(m/s)46.846.7646.7146.672.The influence of materialThe major influence of different materials is they have different COR s.When the mass is constant,we have I bat=0.516kg·m2,and r M=0.1403, andfinally we get:COR0.40.450.50.550.6v f/(m/s)42.344.747.149.551.9From the results above,we can see that COR seems a more prominent influence.Team#8254page13of16 6Conclusion6.1StrengthFirst of all,this paper solves the problem of”sweet spot”,and we give an easy formula to calculate the position of”sweet spot”.Secondly,we analyze the hitting process and divide it into two stages,and discuss the factors that affect the exit speed in details while giving a formula that can describe this stage.Then we answer the questions through qualitative analysis and quantitative calculation.Finally,we make a analysis about the sensitivity and prove the rationality by comparing the results.6.2WeaknessFirst of all,as the real hitting process is too complex to analyze,we make several simplifications in order to facilitate the founding of model.In model 1,we just regard the bat as a pendulum rod with one endfixed,which is a little different from the real situation.And in model2,we simplify the complex process of hitting into two stages.Secondly,the data wefind are not precise,especially for the value of COR which we regard as constant.Additionally,the calculations in this paper are also simplified,thus the accu-racy of our results declines.6.3RecommendationThe biggest disadvantage of our model is lacking experiments,and if we have time and facilities to do some experiments,the result must be more reliable.For example,the equation(6)in Appendix can be used to measure COR,and in order to measure the value of I pivot,we could refer to[15]and use the method to obtain data.With this data we can verify our model in a better way.7References[1]/wiki/Sweet spot[2]/drussell/bats-new/corkedbat.html[3]/drussell/bats-new/sweetspot.html[4]/wiki/Aluminum Bats vs.Wood Bats[5]/baseball/sweetspot.htmlTeam#8254page14of16[6]/sysengr/slides/baseballBat.ppt[7]Keith Koenig,Nan Davis Mitchell,Thomas E.Hannigan,J.Keith Clutter.The influence of moment of inertia on baseball/softball bat swing speed.SportsEngineedng(2004)7,105-117.[8]Alan M.Nathana.Characterizing the performance of baseball bats.Am.J.Phys.,Vol.71,No.2,February2003134-143[9]/wiki/Coefficient of restitution[10]Lv ZhongjieHuang Fenglei.Coefficient of Restitution of a Circular PlateDuring Inelastic Collision.Transactions of Beijing Institute of Technology.Vol.28No.4.[11]P.J.Drane and J.A.Sherwood.Characterization of the effect of temperatureon baseball COR performance.[12]/sysengr/slides/baseballBat.ppt[13]/docs/621958/How-Does-a-Baseball-Bat-Work[14]/wiki/moment of inertia[15]/drussell/bats-new/bat-moi.html8AppendixParameters ExplanationIn expression(5),page8,section4.1,there are several parameters that will influence thefinal velocity of the ball:1.COR•What is COR?The coefficient of restitution(COR),or bounciness of an object is afractional value representing the ratio of velocities after and before animpact[9].Fig7:The one-dimensional collision processTeam #8254page 15of 16The coefficient of restitution is given byCOR =v 1f −v 2f v 1−v 2(6)where–v 1is the velocity of object 1before the collision.–v 2is the velocity of object 2before the collision.–v 1f is the velocity of object 1after the collision.–v 2f is the velocity of object 2after the collision.All the parameters above are scalars.In the ideal situations,we may have a so-called plastic collision when COR =0,namely the deformation of the material cannot re-store.And when COR =1,called perfectly elastic collision,is a situa-tion that the deformation can restore entirely.In general,the value of COR varies from (0,1).•What factors affect COR ?MaterialCOR represents the deformation recovery ability of the material.Gen-erally speaking,the more elastic the material is,the higher the value of COR will be.Impact velocityCOR decreases when the impact velocity increases.[10]The Temperature and Relative Humility of The EnvironmentCompared to the factors above,another two factors,the tempera-ture and relative humility have a relatively smaller influence.COR decreases when the temperature decreases and it decreases when the relative humility increases.[11]2.I bat•MOI (moment of inertia)[14]Moment of inertia is a measure of an object’s resistance to changes in its rotation rate.It is the rotational analog of mass,the inertia of a rigid rotating body with respect to its rotation.The moment of iner-tia plays much the same role in rotational dynamics as mass does in linear dynamics,determining the relationship between angular mo-mentum and angular velocity,torque and angular acceleration,and several other quantities.It is denoted asI = r 2dmwhere m is mass and r is the perpendicular distance to the axis of rotation.Team#8254page16of16•Parallel Axis TheoremI z=I cm+mL2where I cm is the moment of inertia of the rotor with respect to thecenter of mass,and the L is the distance from the center of mass toaxis z.•The factors affect MOIFigure:It influences the location of the center of mass,thereby affectsthe distance from the center of mass to rotation axis.Mass:Its increase is proportional to the increase of MOI.3.r MIn reference[8],Alan M.Nathan develops a formula relating v f to the initial speed of the ball v ball and the initial speed of the bat at the impact pointv bat as:r M=m ball (z−z p)2I pwhere•m ball and m bat are the ball and the bat’s mass respectively.•Z is the location of the impact point.•Z p is the location of the pivot point.•I p is the moment of inertia of the bat with respect to the pivot point. From the expression above we can see that reducing the mass of the bat m bat while keeping the other parameters constant will lead to a augment of r m.。
2010高教社杯全国大学生数学建模竞赛一等奖论文B题
上海世博会的Байду номын сангаас响
——长三角地区经济互动式发展与产业结构升级 摘要
本文是关于定量评估上海世博会影响力的问题。通过建立“经济互动式发展模型” 得到了上海与周边城市的经济相互影响的形式, 进而建立 “产业结构升级规律模型” 得 到长三角区域产业结构升级规律。 首先, 依据国际大都市产业结构特点将长三角地区产业分为两类, 并以上海为中心, 根据上海世博会的影响力的辐射情况(按地理位置) ,将长三角地区城市分为三层;通 过搜索和处理长三角区域各城市往年两类产业值数据,经过回归分析,求得了一元线性 回归方程,作为描述上海与各层城市之间泛一和泛三两类产业值的函数。再根据回归系 数分析得到上海与各层次城市之间的经济的互动发展方式,得到结论如下: i. 对于泛一类产业: 外层对中心层的影响大小 (用回归系数表示) 依次为 2.8748、 3.2435、5.6573(趋势:变大) ,中心层对外层的影响大小依次为 0.038757、0.001001、 0.001145(趋势:减小) 。则外层城市的发展会带动中心层城市的发展,而且随着世博影 响层的向外扩展,影响力变大;而中心层城市的发展对外层城市的影响随着世博影响层 的向外扩展而减小,整体上对外层城市依赖性强。 ii. 对于泛三类产业:外层对中心层的影响大小依次为 3.19、6.0243、10.1221,中 心层对外层的影响大小依次为 0.001173、0.002842、0.005502,变化趋势都是增大的。 说明中心层与外层城市的发展互动性强。 iii. 通过对方程的回归系数的分析,得出“长三角地区经济逐渐趋于一体化 ”、“ 2010 年上海世博会的影响力不会因为世博会的结束减弱,而是更加深远长久”等结论。 其次,本文根据上海与各层城市之间泛一和泛三两类产业值的函数,推导出描述产 业结构升级规律的“单城市产业结构关系式”和“相关城市产业结构关系式” ,将城市 的产业结构与其年度生产总值建立函数关系。若产业结构分类多于两类,也可以根据该 模型求解多个产业元素与相关城市的年度生产总值的关系式。该关系式在根据宏观数据 分析成分变量比重关系,对城市产业结构升级的研究有一定的借鉴作用。 最后,根据建立的模型对世博会影响力进行分析,可以得出,世博对各世博影响层 城市生产总值的拉动量分别为:6.91%、7.68%、5.81%、7.52%,进一步说明了世博会 对整个长三角地区的经济影响巨大。本文还对上海建设国际大都市的进程进行预测,得 出“世博会将加快上海国际大都市化步伐 ”的结论。
2010年数学建模赛区B题一等奖论文4
上海世博会影响力的定量评估上海世博会影响力的定量评估摘 要2010上海世博会备受世界瞩目,也影响我国各个区域,尤其对上海。
本文首先根据上海市1978年至上海申报世博会成功前期的GDP 数据,运用时间序列预测模型,预测出后期GDP 数据,并与实际GDP 作对比,接着分析与经济增长具有密切关系的九项指标数据,讨论各项经济指标的关联度,最后利用主成分分析法找出这九项指标中的主要影响经济增长的因子,即可反映上海世博对经济增长的影响力。
对于模型一,由于经济数据具有较强的自相关性,先采用ADF 检验时间序列的平衡性,接着通过对时间序列取对数、二次差分进行平衡化,然后运用SPSS 得到了拟合函数2(10.231)(1)ln (10.981)0.02t t B B x B ε−−=++,并通过拟合统计值检验了时间序列模型的可靠性,从而对后期数据作出较准确预测,最后通过比较预测值与实际时,得到了世博对上海的经济起到了明显的作用。
对于模型二,为了分析世博对经济增长产生的影响,是如何通过影响经济增长的各项指标来实现,找出引起经济增长的具有代表性受世博影响的九项指标。
利用SPSS 软件九项指标的数据做回归分析,并结合MATLAB 软件拟合得到单个指标随时间变化的函数。
再综合对九项指标数据进行分析,利用灰色系统理论对这九项指标进行关联度分析,并MATLAB 编程得到九项指标的关联系数如下图:1r2r3r4r5r6r7r8r9r0.8822 0.7667 0.8981 0.7124 0.6897 0.7417 0.6482 0.6663 0.6436 对于模型三,采用主成分分析法,通过SPSS 处理,到得影响经济的三个主成分,其方差贡献率之和达到95.517%,主成分与各因子的关系如下234567891010.9890.9850.9930.9440.9870.9930.1860.5330.647F x x x x x x x x x =+++++−−−234567891020.0120.0180.0120.0610.0510.0140.9160.7420.286F x x x x x x x x x =−−+−+−−−−234567891030.0970.0200.0700.2380.0470.0630.0780.1540.698F x x x x x x x x x =+++++−−− 从其表达式分析可以得到,世博导致了外来总投资,社会固定资产投资总额,消费受额,外贸总额,交通,旅游总收入等有较大幅度的增加。
2010数学建模竞赛B题分析
选取近3次综合世博会作为上海世博会纵向影响力的参考,它们分 别是1992年西班牙塞维利亚世博会,2000德国汉诺威世博会以及2005 年日本爱知世博会,将从上述城市竞争力的各个方面进行讨论,表现上 海世博会对城市竞争力影响的大小。其数据如下表:
年份
主办方
世博投资 (亿元)
世博收益 (亿元)
举办市第三产 业产值增长率
C
I f In If
100%
If
In
正常举行世博会上海净投资额 假设不举办世博会上海净投资额
通过查找相关资料我们得出本次上海市由亍世博会而进行的基础配 套设施投资(丌包括园区投资)总额预测值为3200亿元左右[1]。而丌丼 办世博会的投资额预测值为2346.8亿元(见表2)。利用上述公式计算得 到世博会的相关净投资率约为0.363。 带动投资在上海丌丼办世博会情况下和丼办世博会情况下的比较值 为 0.637:1。
灰色系统(Grey System)理论是我国著名学者邓聚龙教授2O世纪8O年代初 创立的一种兼备软硬科学特性的新理论.该理论将信息完全明确的系统定义为 白色系统,将信息完全丌明确的系统定义为黑色系统,将信息部分明确、部分 丌明确的系统定义为灰色系统.由亍客观世界中,诸如工程技术、社会、经济 、农业、环境、军事等许多领域,大量存在着信息丌完全的情况.要么系统因 素戒参数丌完全明确,因素关系丌完全清楚;要么系统结构丌完全知道,系统 的作用原理丌完全明了等,从而使得客观实际问题需要用灰色系统理论来解决 。
现在来比较世博会的海外影响力,首先注意到在世博会开幕乊前, 关键字Shanghai expo的点击指数较为稳定,保持在7左右。为了比 较世博会前后点击率的差别,我们对世博会期间点击率的积分结果取 平均数,得到世博会当年,关键字Shanghai expo的平均点击指数 24.5657。 上述结果表明:上海世博对上海的知名度影响力贡献比达 1:3=0.3333:1。
数学建模竞赛b题
学期四的平 均成绩:
74.9
10.29
学期一
学期二
学期三
学期四
各个学期的成绩波动由大到小的顺序是:第二学期>第四学期>第一学期>第三学期。 而且有表格二也证实了这个波动的顺序。这说明第三学期不同学生的成绩之间的差距最小, 第二学期不同学生的成绩之间的差距最大。 1.可以看到各个学生之间的差异是不同的,说明不同的学生的基础条件是有所不 同。基础较好的学生在取得更加优异的成绩上是本身具有优势的。 2.样本标准差呈折线变化,但整体来说,是有逐渐减小的趋势的。这说明虽然基 础条件不同,但是,学生成绩之间的差距是逐渐变小的。即基础条件不太好的学 生取得了进步的。
通过对学生的各学期的成绩的统计处理分析,并针对题目的三个 问题建立了数学模型,在模型的求解过程中,利用统计学软件 Excel计算工具,编写相应的函数,对建立的模型进行求解,得出 相应的结论。 问题一:我们假设第i个学期学生的平均成绩为Xi,则可利用excel 软件来求得Xi的平均值,用所得的平均值来分析说明所给出的学 生的整体情况 问题二:由于现在需要采用适当的方法来评价学生的学习情况, 为此我们把这目的转化为求学生的各学期Xi的变化规律及所能反 映学生成绩波动的方差变化。利用Excel软件编写相应函数关系式, 进行绘图,结合数据和图形,评价学生的学习情况。 问题三:其模型问题二相似,不同之处在于要收集数据,对数据 走向进行预测。为此我们将四个学期的成绩当成一个总体进行分 析。当总体数据n越大,则总体近似服从正态分布,故后两个学期 也近似服从正态分布。因此,可以选择不同的置信区间来估算得 出后两个学期学生的成绩的大致范围。
2010AMC10美国数学竞赛B卷
2010 AMC10美国数学竞赛B卷1. What is 10010031001003()()?---(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 02. Makarla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?(A) 15 (B) 20 (C) 25 (D) 30 (E) 353. A drawer contains red, green, blue, and white socks with at least 2 of each color, What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?(A) 3 (B) 4 (C) 5 (D) 8 (E) 94. For a real number x, define ♡(x) to be the average of x and x2. What is♡(1)+ ♡(2)+ ♡(3)?(A) 3 (B) 6 (C) 10 (D) 12 (E) 205. A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?(A) 2 (B) 3 (C) 4 (D) 5 (E) 66. A circle is centered at O, AB is a diameter and C is a point on the circle with∠COB=50°. What is the degree measure of ∠CAB?(A) 20 (B) 25 (C) 45 (D) 50 (E) 657. A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?(A) 16 (B) 24 (C) 28 (D) 32 (E) 368. A ticket to a school play cost x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for x are possible?(A) 1 (B) 2 (C) 3 (D) 4 (E) 59. Lucky Larry’s teacher asked him to substitute numbers for a, b, c, d, and e in the expression ((()))---+and evaluate the result. Larry ignored the parent thesea b c d ebut added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for a, b, c, and d were 1, 2, 3, and 4, respectively. What number did Larry substitute for e?(A) -5 (B) -3 (C) 0 (D) 3 (E) 510. Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rainin the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?(A) 18 (B) 21 (C) 24 (D) 27 (E) 3011. A shopper plans to purchase an item that has a listed price greater than $100 and can use any one of the three coupons. Coupon A gives 15% off the listed price, Coupon B gives $30 off the listed price, and Coupon C gives 25% off the amount by which the listed price exceeds $100.Let x and y be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is y-x?(A) 50 (B) 60 (C) 75 (D) 80 (E) 10012. At the beginning of the school year, 50% of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and 50% answered “No”. At the end of the school year, 70% answered “Yes” and 30% answered “No”. Altogether, x% of the students gave a difference between the maximum and the minimum possible values of x?(A) 0 (B) 20 (C) 40 (D) 60 (E) 8013. What is the sum of all the solutions of 2602=--?x x x(A) 32 (B)60 (C)92 (D) 120 (E) 12414. The average of the numbers 1, 2, 3, …, 98, 99, and x is 100x. What is x? (A)49101 (B) 50101 (C) 12 (D) 51101 (E) 509915. On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse ’s total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?(A) 25(B) 27 (C) 29 (D) 31 (E) 3316. A square of side length 1 and a circle of radiusshare the same center. What is the area inside the circle, but outside the square?(A)13π- (B) 29π- (C) 18π (D) 14(E) 29π17. Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea ’s score was the median among all students, and hers was the highest score on here team. Andrea ’s teammates Beth and Carla placed 37th and 64th , respectively. How many schools are in the city?(A) 22(B) 23 (C) 24 (D) 25 (E) 2618. Positive integers a, b, and c are randomly and independently selected withreplacement from the set {1, 2, 3, …, 2010}. What is the probability that abc ab a++ is divisible by 3?(A) 13(B) 2981(C) 3181(D) 1127(E) 132719. A circle with center O has area 156π. Triangle ABC is equilateral, BC is a chordon the circle, OA=, and point O is outside △ABC. What is the side length of △ABC?(A)(B) 64 (C) (D) 12 (E) 1820. Two circles lie outside regular hexagon ABCDEF. The first is tangent to AB, and the second is tangent to DE, Both are tangent to lines BC and FA. What is the ratio of the area of the second circle to that of the first circle?(A) 18 (B) 27 (C) 36 (D) 81 (E) 10821. A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?(A) 1/10 (B) 1/9 (C) 1/7 (D) 1/6 (E) 1/522. Seven distinct pieces of candy are to be distributed among three bags. The red bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?(A) 1930 (B) 1931 (C) 1932 (D) 1933 (E) 193423. The entries in a 3×3 array include all the digits from 1 through 9, arranged so thatthe entries in every row and column are in increasing order. How many such arrays are there?(A) 18(B) 24 (C) 36 (D) 42 (E) 6024. A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?(A) 30(B) 31 (C) 32 (D) 33 (E) 3425. Let a>0, and let P(x) be a polynomial with integer coefficients such that(1)(3)(5)(7),(2)(4)(6)(8).P P P P a and P P P P a ========- What is the smallest possible value of a?(A) 105(B) 315 (C) 945 (D) 7! (E) 8!。
2010年全国数学建模B题答案
2010年上海世博会对居民消费结构影响力的定量评估摘要本文从世博会的筹备期间(2003年---2009年)对上海居民消费结构的影响进行定量评估研究。
消费结构是一项反映居民消费水平的重要指标,包含居民的收入水平、消费支出、消费分类三部分[1]。
为了全面反映和研究居民的生活消费状况,我们采取了一系列相互联系的统计指标对上海居民的消费结构进行定量研究。
在对大量的数据分析基础上,研究了上海市居民的收入水平的变化;并且从上海市的几个主要消费群体来分析上海市居民的收入与支出的变化情况;对消费分类的研究,我们选取了食品、衣着、居住、家庭设备用品及服务、交通和通信、文教娱乐用品及服务、医疗保健、商品和服务作为消费分类的八项指标,利用主成分分析的方法对各个主成分进行了详细的定量分析,并运用matlab编程利用曲线拟合的方法做了假设不存在世博会时的预测,再将所搜集到的实际值与预测值作差,我们定义该差值为影响力指数,通过影响力指数的大小来说明上海世博会对上海市居民消费分类的影响,影响力指数越大,说明世博会对上海居民消费结构的影响越深,进而定量评估了上海世博会对上海市居民的消费结构的影响情况。
消费结构的升级产生的经济势力是持久强大的,了解了上海世博会的对上海居民消费结构的影响后,若能顺势调控,则能充分带动经济的发展,为支撑我国国民经济的稳定快速发展提供动力。
关键词:消费结构主成分分析定量评估预测曲线拟合 matlab一 问题的提出2010年上海世博会是首次在中国举办的世界博览会.从1851年伦敦的“万国工业博览会”开始,世博会正日益成为各国人民交流历史文化、展示科技成果、体现合作精神、展望未来发展等的重要舞台.请你们选择感兴趣的某个侧面,建立数学模型,利用互联网数据,定量评估2010年上海世博会的影响力.二 符号说明np x 第n 个样品的第p 个指标 X标准化数据矩阵R 变量的关系矩阵p λ 关系矩阵的特征值p μ p λ所对应的单位特征向量i y 第i 个主成分y 1995年到2002年dy 影响力指数三 模型的假设1、本文所作的影响力评估是针对上海市居民的消费结构.2、本文所作的影响力评估仅限于世博会筹备期间及召开期间的居民消费结构.3、消费结构是一项反映居民消费水平的重要指标,要全面反映和研究居民的生活消费状况,包含居民的收入水平、消费支出、消费分类三部分.4、上海居民消费由食品、衣着、居住、家庭设备用品及服务、交通和通信、文教娱乐用品及服务、医疗保健、商品和服务八部分组成.5、居民的收入是决定居民的消费水平和消费结构的主要因素,收入水平的高低直接决定消费水平的高低.四 模型的建立及求解上海是我国最大的经济中心城市,随着2010年上海世博会的日益临近,将对上海经济发展发挥巨大的作用.投资、消费和出口被称为经济发展的三架马车,2010年的世博会为上海经济发展提供了会前的投资拉动和会后的需求拉动两个方面的刺激,消费是需求的基础,有效地投资必须准确的把握需求的变化.消费是人们为了满足生活需要而消耗产品和服务的行为和过程, 是满足人们生存、发展和生活享受所必需的行为.人们基本的消费状况, 既能反映需求规律, 又成为其他需求的基础,因此, 评估消费状况和需求趋向便成为政府和企业了解市场的起点.据中国社科院的研究, 2001年投资、消费和出口对国内GDP 增长的贡献分别是77%、34%和-11%. 从2002 年上海的统计数据来看, 同样是外需下降、出口下滑, 依靠增幅达31.7%的社会固定资产投资和9.8%的社会消费品零售总额的增长, 才保证了上海经济10.4%的高速增长.由此也可以看到投资和消费是推动上海经济发展的两个最基本因素。
2010数学建模竞赛A B题 高教社杯全国大学生数学建模竞赛题目
2010高教社杯全国大学生数学建模竞赛题目(请先阅读“全国大学生数学建模竞赛论文格式规范”)A 题 储油罐的变位识别与罐容表标定通常加油站都有若干个储存燃油的地下储油罐,并且一般都有与之配套的“油位计量管理系统”,采用流量计和油位计来测量进/出油量与罐内油位高度等数据,通过预先标定的罐容表(即罐内油位高度与储油量的对应关系)进行实时计算,以得到罐内油位高度和储油量的变化情况。
许多储油罐在使用一段时间后,由于地基变形等原因,使罐体的位置会发生纵向倾斜和横向偏转等变化(以下称为变位),从而导致罐容表发生改变。
按照有关规定,需要定期对罐容表进行重新标定。
图1是一种典型的储油罐尺寸及形状示意图,其主体为圆柱体,两端为球冠体。
图2是其罐体纵向倾斜变位的示意图,图3是罐体横向偏转变位的截面示意图。
请你们用数学建模方法研究解决储油罐的变位识别与罐容表标定的问题。
(1)为了掌握罐体变位后对罐容表的影响,利用如图4的小椭圆型储油罐(两端平头的椭圆柱体),分别对罐体无变位和倾斜角为α=4.10的纵向变位两种情况做了实验,实验数据如附件1所示。
请建立数学模型研究罐体变位后对罐容表的影响,并给出罐体变位后油位高度间隔为1cm 的罐容表标定值。
(2)对于图1所示的实际储油罐,试建立罐体变位后标定罐容表的数学模型,即罐内储油量与油位高度及变位参数(纵向倾斜角度α和横向偏转角度β )之间的一般关系。
请利用罐体变位后在进/出油过程中的实际检测数据(附件2),根据你们所建立的数学模型确定变位参数,并给出罐体变位后油位高度间隔为10cm 的罐容表标定值。
进一步利用附件2中的实际检测数据来分析检验你们模型的正确性与方法的可靠性。
草根分享论坛 图吧 分享吧 海之南 分享斑竹 share8 好学吧附件1:小椭圆储油罐的实验数据 附件2:实际储油罐的检测数据油油浮子 出油管油位探测装置 注油口 检查口地平线 2m 6m 1m 1m3 m油位高度 图1 储油罐正面示意图 油位探针油位探针α地平线 图2 储油罐纵向倾斜变位后示意图油油浮子出油管油位探测装置注油口 检查口水平线α油油浮子出油管油位探针注油口水平线1.2m1.2m1.78m图3 储油罐截面示意图(b )横向偏转倾斜后正截面图地平线β地平线垂直线油位探针(a )无偏转倾斜的正截面图油位探针油位探测装置地平线 油3m油2010高教社杯全国大学生数学建模竞赛题目(请先阅读“全国大学生数学建模竞赛论文格式规范”)B题2010年上海世博会影响力的定量评估2010年上海世博会是首次在中国举办的世界博览会。
2010年数学建模竞赛B题上海世博会经济综合评估
2010年上海世博会经济综合评估(杨建林、熊明会、李艳梅)摘 要本文综合评估了2010年上海世博会对经济的影响,文中着重从科技进步、旅游发展、吸引外资三方面进行研究、讨论。
第一方面:科技进步对经济的影响。
在基于索洛进步模型(科技进步对经济增长的作用模型)的基础上,利用索洛进步模型函数关系式),,(L K t Y Y =对相关数据进行微分,找出科技进步率与GDP 的增长速度的关系,最后得到科技影响经济的一个关系E=%100⨯ya(a 为科技进步率、y 为GDP 增长速度、E 为科技对经济的作用),在此基础上,得到V=E ⨯m=25%⨯9%=2.25%>0(V 为世博对经济的影响、m 为工业增加值比去年增加值的增长值),说明世博会带来的科技对经济有极大的促进作用。
第二方面:世博旅游对经济的影响。
通过收集数据,用Matlab 绘图拟合得出游客人数与门票收入之间的函数关系y=18400L+N(L 为旅游人数、N 为我们假定的游客消费与支出间的一个恒量),其次,运用近几年上海旅游收入与总的财政收入数据之间进行拟合得到一个函数,其关系式为y=3.19x-816.85,通过计算得相关统计量R=∑∑∑==--=------ki ki iiki i iy x xy y x xy 111)()())((,有|R|﹥c(c 为临界值,此时相关系数r=0.01),则线性相关显著。
再运用相关实际数据计算出两个收入之间的比例,得出财政收入增长的大部分依靠旅游收入,所以旅游是提高经济收入的一个重要因素。
第三方面:世博外商投资对经济的影响。
通过1986—2006年外商投资与上海GDP 的数据,运用SPSS 对数据进行拟合,得到符合Logistic 模型的曲线,曲线方程为Y=1/((1/u)+b0*(b1^t)) (其中u=2000为函数的上限),由已知的2010年1月到7月的数据,预测出2010年部分月份的外商投资对经济增长的影响,分析2010年外商投资的趋势,假设非正常增长的外商投资是世博会带来的,从而推算出世博促进了上海经济的发展。
2010年数学建模试题(全部)
2010高教社杯全国大学生数学建模竞赛题目(请先阅读“全国大学生数学建模竞赛论文格式规范”)A 题 储油罐的变位识别与罐容表标定通常加油站都有若干个储存燃油的地下储油罐,并且一般都有与之配套的“油位计量管理系统”,采用流量计和油位计来测量进/出油量与罐内油位高度等数据,通过预先标定的罐容表(即罐内油位高度与储油量的对应关系)进行实时计算,以得到罐内油位高度和储油量的变化情况。
许多储油罐在使用一段时间后,由于地基变形等原因,使罐体的位置会发生纵向倾斜和横向偏转等变化(以下称为变位),从而导致罐容表发生改变。
按照有关规定,需要定期对罐容表进行重新标定。
图1是一种典型的储油罐尺寸及形状示意图,其主体为圆柱体,两端为球冠体。
图2是其罐体纵向倾斜变位的示意图,图3是罐体横向偏转变位的截面示意图。
请你们用数学建模方法研究解决储油罐的变位识别与罐容表标定的问题。
(1)为了掌握罐体变位后对罐容表的影响,利用如图4的小椭圆型储油罐(两端平头的椭圆柱体),分别对罐体无变位和倾斜角为α=4.10的纵向变位两种情况做了实验,实验数据如附件1所示。
请建立数学模型研究罐体变位后对罐容表的影响,并给出罐体变位后油位高度间隔为1cm 的罐容表标定值。
(2)对于图1所示的实际储油罐,试建立罐体变位后标定罐容表的数学模型,即罐内储油量与油位高度及变位参数(纵向倾斜角度α和横向偏转角度β )之间的一般关系。
请利用罐体变位后在进/出油过程中的实际检测数据(附件2),根据你们所建立的数学的罐地平线 图1 储油罐正面示意图 油位探针2010高教社杯全国大学生数学建模竞赛题目 (请先阅读“全国大学生数学建模竞赛论文格式规范”)B 题 2010年上海世博会影响力的定量评估 20101851年伦互联网数据,定量评估2010年上海世博会的影响力。
2010高教社杯全国大学生数学建模竞赛题目(请先阅读“全国大学生数学建模竞赛论文格式规范”)C 题 输油管的布置某油田计划在铁路线一侧建造两家炼油厂,同时在铁路线上增建一个车站,用来运送成品油。
数学建模美国赛历年试题.
历年美国大学生数学建模赛题目录MCM85问题-A 动物群体的管理 (3)MCM85问题-B 战购物资储备的管理 (3)MCM86问题-A 水道测量数据 (4)MCM86问题-B 应急设施的位置 (4)MCM87问题-A 盐的存贮 (4)MCM87问题-B 停车场 (5)MCM88问题-A 确定毒品走私船的位置 (5)MCM88问题-B 两辆铁路平板车的装货问题 (5)MCM89问题-A 蠓的分类 (5)MCM89问题-B 飞机排队 (6)MCM90-A 药物在脑内的分布 (6)MCM90问题-B 扫雪问题 (6)MCM91问题-B 通讯网络的极小生成树 (6)MCM 91问题-A 估计水塔的水流量 (7)MCM92问题-A 空中交通控制雷达的功率问题 (7)MCM 92问题-B 应急电力修复系统的修复计划 (7)MCM93问题-A 加速餐厅剩菜堆肥的生成 (7)MCM93问题-B 倒煤台的操作方案 (8)MCM94问题-A 住宅的保温 (8)MCM 94问题-B 计算机网络的最短传输时间 (9)MCM-95问题-A 单一螺旋线 (9)MCM95题-B A1uacha Balaclava学院 (10)MCM96问题-A 噪音场中潜艇的探测 (10)MCM96问题-B 竞赛评判问题 (10)MCM97问题-A Velociraptor(疾走龙属)问题 (11)MCM97问题-B为取得富有成果的讨论怎样搭配与会成员 (11)MCM98问题-A 磁共振成像扫描仪 (12)MCM98问题-B 成绩给分的通胀 (13)MCM99问题-A 大碰撞 (13)MCM99问题-B “非法”聚会 (13)MCM2000问题-A空间交通管制 (13)MCM2000问题-B: 无线电信道分配 (14)MCM2001问题- A: 选择自行车车轮 (14)MCM2001问题-B 逃避飓风怒吼(一场恶风...) .. (15)MCM2001问题-C我们的水系-不确定的前景 (15)MCM2002问题-A风和喷水池 (15)MCM2002问题-B航空公司超员订票 (16)MCM2002问题-C (16)MCM2003问题-A: 特技演员 (17)MCM2003问题-B: Gamma刀治疗方案 (18)MCM2003问题-C航空行李的扫描对策 (18)MCM2004问题-A:指纹是独一无二的吗? (18)MCM2004问题-B:更快的快通系统 (18)MCM2004问题-C安全与否? (19)MCM2005问题A.水灾计划 (19)MCM2005B.Tollbooths (19)MCM2005问题C:不可再生的资源 (20)MCM2006问题A: 用于灌溉的自动洒水器的安置和移动调度 (20)MCM2006问题B: 通过机场的轮椅 (20)MCM2006问题C : 抗击艾滋病的协调 (21)MCM2008问题A:给大陆洗个澡 (23)MCM2008问题B:建立数独拼图游戏 (23)MCM85问题-A 动物群体的管理在一个资源有限,即有限的食物、空间、水等等的环境里发现天然存在的动物群体。
2010数学建模B题
2010年上海世博会科技影响力的定量评估模型摘要:2010年上海世博会是首次在中国举办的世界博览会。
从1851年伦敦的“万国工业博览会”开始,世博会正日益成为各国人民交流历史文化、展示科技成果、体现合作精神、展望未来发展等的重要舞台。
“科学技术是生产力”是马克思主义的基本原理。
于是,我们放弃了大众化的旅游或经济问题,而是选择了有突出成效的科技成果,并从科技成果这一侧面,对世博会的影响力进行定量评价。
反应科技成果影响力的指标很多,本文选择了其中最重要的5个指标作为第一层评价因素,分别是实用性,创新性,严谨性,可探索性,可继承性。
又从世博会中选取了9个项目作为评价这5个指标的第二层因素,分别是世博轴,LED光源,立升直饮水,准4G技术,手机付款,物联网首秀,电子导游仪,数控水幕,农民机器人,从而建立了层次结构模型。
本文利用层次分析法,分别对两层评价因素,建立两两成对比较矩阵并进行一致性检验,进一步计算出特征向量及特征根,得出第一层及第二层个因素的权重向量。
然后,利用互联网调查得到的数据,分别对9个项目从5个方面进行评分,得到5项指标的得分向量,最后利用前面求得的权系数向量加权求和,得到上海世博会科技影响力的综合评价得分。
经过计算,综合评价得分为85.589,可见这次世博会的科技影响力让世人叹为观止,各媒体也争先报道此届世博的新科技,在百度上搜索“上海世博会科技”可得到21,800,000篇相关文章,可见其影响力之大。
关键字:层次分析法,一致性检验,权重,评分,加权求和一、 问题重述:2010年上海世界博览会(Expo 2010)是第41届世界博览会。
此次世博会是由中国举办的首届世界博览会。
上海世博会以“城市,总投资达450亿人民币,创造了世界博览会史上最大规模记录。
此次世博会以“城市,让生活更美好”为主题,展现了人们对未来生活的期盼。
其主要表现在它所包含的五个分主题:城市与经济发展关系、城市与可持续发展关系、城乡互动关系、城市与高科技发展关系、城市与多元文化发展关系。
2010数学建模国家一等奖论文(B题)
承诺书
我们仔细阅读了中国大学生数学建模竞赛的竞赛规则. 我们完全明白,在竞赛开始后参赛队员不能以任何方式(包括电话、电子邮件、网 上咨询等)与队外的任何人(包括指导教师)研究、讨论与赛题有关的问题。 我们知道,抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的 资料(包括网上查到的资料),必须按照规定的参考文献的表述方式在正文引用处和参 考文献中明确列出。 我们郑重承诺,严格遵守竞赛规则,以保证竞赛的公正、公平性。如有违反竞赛规 则的行为,我们将受到严肃处理。
其次,在研究上海世博会对其他相关产业影响时: 一、对于研究世博会对旅游业的影响时,本文采用了������������(1,1)灰色预测模型,得到 不受世博影响下的旅游业人数增长模型,再与上海市实际旅游人数增长进行比较得到了 受到世博影响所增加的旅游人数,并根据人均消费额得到了对旅游业������������������的拉动作用。 二、研究世博会对就业影响时,本文利用本底趋势线模型和最小二乘法模拟自然条 件下上海的就业人口数,再与实际数据进行对比得到了因世博而新增的岗位数。再利用 上海市人均������������������得到了这些岗位产生的经济效益。 然后,建立世博会经济影响力评价指标体系,以往届世博会各项指标为参考依据, 采用三层 BP 神经网络模型,经由������������������������������������������运算,预测出了上海世博会所拉动的������������������。 并因此反映上海世博会的经济影响力。 最后,本文以挤出效应为研究重点,分析了因上海世博会的外资引入对其他方面外 资引入的影响,量化反映出上海世博会的负面经济效应。
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系数为-0.093 和-0.107,呈不相关关系。 求解所得统计结果如下表所示:
美赛优秀论文2010MCM的B特等奖翻译
摘要:该模型最大的挑战是如何建立连环杀手犯罪行为的模型。
因为找出受害者之间的联系是非常困难的;因此,我们预测罪犯的下一个目标地点,而不是具体目标是谁。
这种预测一个罪犯的犯罪的空间格局叫做犯罪空间情报分析。
研究表明:最暴力的连环杀手的犯罪范围一般在一个径向带中央点附近:例如家庭,办公室,以及其他一些犯罪行为高发区(例如城镇妓女集中区)。
这些‘锚点’为我们的模型提供了基础。
我们假设整个分析域是一个潜在的犯罪现场。
罪犯的活动不受到任何条件约束;并且该区域足够大包括所有的打击点。
我们考虑的是一个可度量的空间,为预测算法创建了空间可能性。
此外;我们假设罪犯为一个暴力的系列犯罪者,因为研究表明窃贼和纵火犯不可能遵循某一空间模式。
一个锚点与多个锚点有着实质性的不同,首先讨论单个锚点的案例,建立坐标系并把罪犯最后犯罪地点与犯罪序列表示出来,并估计以前案件发生地地点,评估模型的可靠性,并且我们得到未来可能发生犯罪行为的锚点。
对于多个锚点的案例,我们通过聚类与排序的方法:将所给数据划分为几组。
在每组中找一个最重要的锚点,每一个分区都给定一个权值。
我们进行单点测试,利用以前的锚点预测最近的一个锚点,并且与其实际位置相比较。
我们从文献中摘录七个数据集,并且用其中四个改善我们的模型,检测其序列变化,地理集中位置和总锚点的数目。
然后通过其他三点来评估我们的模型。
结果显示多个锚点的模型的结果比较优。
引言:通过研究文献中以得出的连环案件罪犯地理空间往往是围绕罪犯日常活动的几个锚点附近的区域。
我们建立的预测模型就是在其规律下建立的,并且预测出一个表面的可能值和度量值。
第一个方案是通过重心法寻找出单个可能的锚点。
第二个方案是假设2到4个锚点,并且利用聚类算法的排序与分组的方法。
两种方案都是利用统计方法来缩小预测未来犯罪的地点区域背景:1981年peter sutcliffe的逮捕是法医生物学家stuart kind 通过利用数理原理成功预测出约克郡开膛手的住处的一个标志目前,信息密集型模型是通过热图技术建立确定特殊犯罪类型的热点或者是找出犯罪活动与某一地区之间的联系比率。
连环犯罪预测-2010美国大学生数学建模B题(MCM)
For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number7370Problem ChosenBFor office use onlyF1________________F2________________F3________________F4________________2010 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to each copy of your solution paper.)Type a summary of your results on this page. Do not includethe name of your school, advisor, or team members on this page.SummaryIn our paper, we define two types of serial murder, the single-hunting area type and the multi-hunting area type, and we mainly do research on serial murder of these two types.We work out a method to determine if a given serial murder belongs to the single-hunting area type or the multi-hunting area type, or neither of them.To do research on the cases of the single-hunting area type, we developed two sub-models, the Concentration Driven Model (CDM) and the Centro-based Model (CM). We use the “center of mass” and the search-radius calculated by an algorithm to figure out the scope of hunting area.We discretize the hunting area by divided it into small action spaces and all of our models depend on the discretized one.The CDM is based a truth that the distribution of preys will affect the criminal’s moving to a certain extent. We develop a discrete algorithm to calculate variation of the prey’s distribution, and an algorithm to search for the next possible crime locations.The CM is based on the fact that the criminal tends to move to some places far away from those latest crime locations to avoid be arrested, and CM also includes the random factors. We develop a partially-random algorithm to determine the next possible directions the criminal may choose based on the time and locations of the past crimes. We determine the next possible distance the criminal will go basing on a reference.We then develop an algorithm to solve cases of multi-hunting area based on the sub-models in our model development.For the sub-Models have take into consideration some direct driven factors, in our Combined Model, we consider some indirect factors that may affect the moving of criminals. We develop a method to figure out the main common properties among the locations of past crimes, and then use an matching-algorithm to figure out the superiority for each result that figured out by the sub-models, basing on the assumption that if a place have much common property with the past crime locations, there’s more chances that it will be the next crime location. And the police can select some places with high superiority to investigate.In our paper, we study three cases of single-hunting area type and one case of multi-hunting area type to determine the best value of the key coefficients; we also use them to do the Sensitivity and Reliability Analysis, and we find that after the sample size approximately increases to 7, our model will give out relatively accurate results.We write a report to the police whether our model can be used to solve a given case, and if yes, when the model will give accurate results in terms of the sample size. We also give some suggestions that should be taken care of to the executive officer.Key words: serial-crime; prey-concentration; single-hunting area; multi-hunting areaTo Seize the Devils!Content1. Introduction (2)2. Terminology (2)3. Assumptions and Symbol (3)4. Sub-Models (3)4.1 Preparation (3)4.1.1 Determine the Search Radius and the Center of Mass (4)4.1.2 Hunting Area Division (5)4.1.3 Normalized Distance (ND) (6)4.2 Concentration Driven Model (CDM) (6)4.2.1 Overview (6)4.2.2 Frame of CDM (6)4.2.3 Model Definition (7)4.2.3.1 Concentration Calculator (7)4.2.3.2 Points Generator (8)4.3 Centro-based Model (CM) (10)4.3.1 Overview (10)4.3.2 Model Definition (10)4.3.2.1 Random Direction Algorithm (10)4.3.2.2 Random Distance Algorithm (12)4.4 Residence of the criminal (13)4.5 Parameter Determination (13)5. Model Comparison (13)6. Sub-Model Development (14)6.1 Problem Addressing (14)6.2 Identify the type of a case (14)6.3 Use the Basic Model (16)7. Model Combining (16)7.1 Overview (16)7.2 The Property Vector (16)7.3 Model Definition (17)8. Strengths and Weaknesses of the Model (18)8.1 Strengths (18)8.2 Weaknesses (18)9. Case Study (19)10. Sensitivity Analysis (20)11. Model Reliability (21)12. Executive Summary (22)13. References (24)Appendix (25)Appendix I (25)Appendix II (25)Appendix III (26)Appendix IV (26)1.I ntroductionSerial crime is generally defined as crimes in a serial or repetitive nature. [1]Serial murder, serial rape and serial arson are the crimes regarded as serial crimes. Serial crime is a frightening and perplexing phenomenon that has proven to be a difficult puzzle for both criminal investigators and criminological researchers. Despite being a rare event, this crime has a broad-based impact on the larger community (Jenkins, 1992a; Silverman & Kennedy, 1993). [2] In our paper, we try to develop a method to aid in a local police agency’s investigations of serial criminals.The objectives in our study are:z Make use of at least two different schemes to generate a geographical profile of serial criminals; z Develop a technique to combine the results of the different schemes and generate a useful prediction;z Provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings;z Provide an additional two-page executive summary, which provides a broad overview of the potential issues and an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool;Firstly, we determine what the meaning of the geographical profiling is. Geographic profiling is a criminal investigative methodology that analyzes the locations of a connected series of crimes to determine the most probable area of the murderer’s residence or his next crime location. [3] To generate a geographical profile of serial criminal, we should provide the murderer’s sphere of activities; predict the murderer’s next crime location.In our paper, firstly we work out a Concentration Driven Model (CDM) to make a prediction providing some kind of guidance about possible locations of the next crime from the microscopic point of view. Secondly, we add the concept of the center of hunting area and build a Centro-based Model (CM) to macroscopically predict the possible locations of the next crime. Thirdly, we combine both models and come up with a combined model aiming at generate a more reliable prediction of the possible locations of the next crime. Finally, we analyze the sensitivity and the reliability of our models. At the end of this paper, we provide an executive summary for a chief of police on the strategy for investigations of serial criminals.2.T erminologyz Crime location: a place where has just experienced a crime actionz Hunting area: a concentrated area that has experience a series of crimes. An area can be assumed to be hunting area only when the distance between two successive crime locations doesn’t vary much as the number of the crimes increases. In our model, the hunting area is a square.z Action space: we divided the hunting area into several small parts, and each part is an action space. The action space represents a district, a city, a block, or someplace else. It depends on the scale of a certain serial crime.z Single-hunting area and multi-hunting area: in our paper, we just investigate two types of serial crime. One is single-hunting area and the other is multi-hunting area.¾The type of single-hunting area contains only one hunting area.¾The type of multi-hunting area contains multiple hunting areas, and the distance of each pair of hunting areas is much larger than the distance between two successive crime locations within one hunting area.z The search-radius and the core-radius:¾The search-radius limits the search scale of the criminal, and the criminal cannot reachbeyond it. In our paper, the search-radius is half the side-length of the hunting area.¾The core-radius is an abstract valuable; it varies as the number of crimes increases. We will define it later in our paper.Here is a graph to illustrate the terms above:1 Graph 1z Latest crime location: the location of the latest crimez Prey: certain kind of people that the murderer tends to kill3.A ssumptions and SymbolAssumptions:z We treat the action space as a unit. A murderer’s basic step is to stay in a action space, or to move to the very neighboring action space.z The prey concentration of each action space is the natural distribution through the hunting area before the very first crime of the serial crime happened. Also, we consume that the natural distribution remains constant.Symbol:z K: the number of parts we divided the length of the hunting area intoz , … , … :the action space located in i row and the j column in the hunting areaz , :the concentration or the number of prey in Sq ,z … … :the array of concentration through the hunting areaz : the longitude of the location of i crimez : the latitude of the location of i crimez : the longitude of the “mass of center”z : the latitude of the “mass of center”4.S ub-Models4.1PreparationWe develop two sub-models in this part, both of which are based on a single-hunting area. In this part, we introduce several important concepts, such as the search radius, the center of mass, the ND, etc. Now, we are going to definite the scope of the hunting area. And the search radius and thecenter of mass are both used to describe the scope of the hunting area. One thing that we must bear in mind is that the center of mass is used to determine the scope of hunting area, rather than a place that the murderer will come back to after a crime action.4.1.1 Determine the Search Radius and the Center of MassThe search radiusThe search radius of the murderer’s hunting area is measured as of the maximum distance from “center of mass” to crime location. The following regression equation reflects the maximum distance from “center of mass” to crime site against maximum distance between crime sites y a x 0.61(Equation 4.1.1)Where:y is the maximum distance in miles from the “center of mass” to crime site; and x is the maximum distance in miles between crime sites.The gradient of a in Equation 4.1.1 indicates an eccentric placement of the “center of mass” vis-à-vis the crime sites (a perfect centric placement would yield a gradient of 0.5). Regressions for U.S. and British serial murder crime location data yield values of 0.81 and 0.79, respectively, and an Australian study found gradients of 0.77 for rape, 0.60 for arson, and 0.65 for burglary. [2] The center of massCalculate the “center of the mass” by using Centroid-Formula: Longitude∑ ,Latitude ∑ VerificationWe review the serial murder of Peter Sutcliffe. Graph 2 shows Yorkshire Ripper locations within West Yorkshire (Victims 6 & 9 are off this map to the south west). In Graph 2 the labels of 1-13 were the victims’ locations where the bodies were found. The labels of 14-17 were the survivors and the label of 18 was the place where he was arrested.2 Graph 2Figure 1shows the quantified original image of the crimes, and Figure 2 shows the image without singularities.3 Figure 14 Figure 2Take the order of occurrence of the murder into consideration; we do the following works as the serial number n increases:1) Calculate the “center of the mass” by using Centroid-Formula: Longitude∑ ,Latitude ∑ ; 2) Calculate the maximum distance from the “center of mass” to crime site by using Equation4.1.1.Figure 3 shows all the positions of the “center of mass”, from which we can find that the “center of mass” tend to be lumped as we take more crimes into consideration.Figure 4 shows the variation of y as the serial number n increases. As is shown in 4.2, y is the maximum distance from the “center of mass” to crime location. Y tends to be stable as n increases. It means that the murder’s hunting area is limited to an area that centered with the “center of mass” and that has a radius of y.5 Figure 36 Figure 4All the above proves that it is reasonable to describe the scope of a hunting area by the center of mass and the search radius.4.1.2 Hunting Area DivisionIn order to apply our model to different scales of serial crime, we should normalize the hunting area, and treat it as a square 1 on a side. The procedure is as follows: We define the normalizationparameter as 2 times the length of search radius, and then get the scope of the hunting area normalized by dividing the distance by the normalization parameter. We divide the hunting area into k k action spaces. Each of the action spaces is a square on a side.4.1.3 Normalized Distance (ND)It is difficult to deal with problems of different order of magnitude. As the scope of the murderer’s hunting area differs at a wide range, we introduce the concept of Normalized Distance (ND) to make our model more tractable.We get the ND of a crime location by dividing the actual distance by 2 times the length of search radius.4.2 Concentration Driven Model (CDM)4.2.1 OverviewThe Concentration Driven Model (CDM) is a discrete model. In this model, we suppose that the murderer’s moving from one action space to another depends on preys’ distribution through the hunting area. Meanwhile, as the crimes being committed sequentially, the distribution will change.We develop an algorithm (we call it concentration calculator) to calculate the current preys’ distribution using the happening time and the location of the past crimes scenes. Then, we develop an algorithm (we call it points-generator) to search the next possible crime locations. The points-generator is a ‘discrete ‘steepest descent method.4.2.2 Frame of CDM7 Frame of CDM4.2.3Model Definition4.2.3.1 Concentration Calculatorz Change of the Prey’s DistributionAccording to our assumption above, each action space own a property of prey concentration. So we investigate the transfer of preys among the many action spaces to estimate the change of the prey distribution.z Causes for Transfer¾ A crime has just been committedAccording to our assumptions, when a crime has just been committed in an action space, all the preys in the action space will soon run away to other action spaces. This assumption is reasonable. In fact, people who will emerge in a certain place always for the reason that he will pass through the place on the way to work or some places else in his daily life, and he will go round if an crime has just happened in the place. This is always true when people choose the way to work or people choose a place for purchase, etc.Suppose the current crime location is represented by C , . We let the preys transfer to the eight neighboring action spaces according to a certain principle. We limit the transfer just among the crime location .We divided the eight spaces into two groups of A and B, Figure 5 can illustrate it:8 Figure 5We assume that each action space of group A will receives a% of the preys in the crime location, and each action space of group B receives b%, where a ,a b 25.This is for the reason that action space of group A connects with the crime location basing on edge, while group B basing on angle, and we think that the larger the contact area is, the more preys will transfer.Therefore, after a crime is committed, the instant transfer follows the principle below:1)When the coordinates of an action space i′,j′ is one of i 1,j , i 1,j , i,j 1 ,i,j 1 , C ′, ′ a% C , ;2)When the coordinates of an action space i′,j′ is one of (i 1,j 1), i 1,j 1 ,i 1,j 1 , i 1,j 1 , C ′, ′ b% C , ;3)C ′, ′ 0.¾As time goes by, preys that run away to other action spaces will come back After a crime being committed, the number of the preys in the crime location will reduce, but as time goes by, the influence will gradually reduce, preys will gradually come back because the way contain the crime location or the supermarket they choose in their daily life before the crime’s happening is their usual choice and it must be the best one. Thus the distribution will recover. We put forward assumptions below:In every time unit, there are a preys come back from each action space of group A, whileb preys from each action space of group B. We assume that the preys don’t transfer on largescale any more until the distribution recover to the normal level.Apart from the ending situation above, there is another ending situation, that is, when the concentration or number of preys in an action space of A or B group being affected by another crime later, that means it also exchange preys with other action spaces later, it may not return all the preys who come from the crime location .Under this circumstances, the return or the transfer between the certain action space and the crime location will terminate.¾Exception on the borderFor two kinds of points on the border shown in Figure 6 and Figure 7, we also divided the action spaces around the crime location into group of A and B, and then figure deal with it in the similar way:9Figure 6 10Figure 7z Current Distribution of the PreysAccording to our assumptions, the distribution of the preys through the hunting area is already known, and represented by a two-dimensional array C . The distribution after the ith crime action represented by C (i=1…n), and we will figure out C during the simulation, the process as below:1)After the i th crime action, we can calculate the C based on the C2)According to the algorithm above, and repeat n times, we could figure out C .4.2.3.2 Points GeneratorWe develop an algorithm to search for the next possible action spaces that would welcome a crime action; we called the algorithm points-generator. The algorithm derives from Gradient descent, but we improve it so that it could be used in a discrete problem.z Gradient DescentGradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.z Gradient descent in discrete useWe replace the Gradient in the Gradient descent with the difference of concentration.Though we use a method developed from an optimization algorithm, we don’t mean to search for the local optimization result, we just use the method to simulate the moving of the criminal.Why not to search for the optimization result will be explained below. And we will obtain a series of action space that will welcome a crime action.Why not search for the optimization result?There two reasons to explain it:1)The first reason is that the total iteration number to find a local maximum result would bevery large, and a large iteration number may not be realistic because the criminal’s ability, physical strength, or his financial condition is limited and always cannot afford to go so far.2)The second reason is that even though a criminal is deeply affected by the preyconcentration, his destination may not be the optimal solution but a place that is above his standard. This is because the criminal is somewhat shortsighted and he may satisfy with the places above his standard.z Figure out the next action spaceWe replace the gradient in the Gradient descent with the difference of prey numbers between the adjacent action spaces, as follows:We still use the local area as is shown in Figure 8 (the action space on the border of the hunting area can be proposed in the similar way)11 Figure 8For that in every iterative round, the algorithm will generate a series of points, for each point/current iterative point, we do things as follows:For each action space marked from 1 to 8 around current iterative point in the figure above, we calculate the difference of prey concentration between the marked action space and the current iterative point, choose those action spaces with a corresponding difference that is larger than a certain value of h to be a part of the iterative points of the next round.In fact, the threshold h can describe the criminal’s sensibility degree to prey concentration, and if threshold h is large, indicating that the criminal is sensitive to the difference of prey concentration, otherwise the opposite.z When to Terminate the IterationAccording to the analysis above, we terminate the iteration when the number of iteration steps has reached a constant value of d, the value of d is constant as the scale of the crime varies, and in a given serial murder, it is also constant.Why it is constant as scale varies?In fact, no matter it is crime in a large scale with automobiles as its transportation or a crime in small scale with transportation just base on foot, the d is constant among all kinds of cases because we have normalized the scope of the case, thus for a large scale case one iterative step (move from current action space to another action space) represents a longer distance than in a small scale case.Evidence for a constant d in a given serial murderWe also have found that in one case, the distance between two crime-sites that happened in succession don’t vary largely, that means that to assume that the value of d is constant in a given serial murder is reasonable.We studied the case of Peter Sutcliff, and his moving track during the case is in Figure 9.12Figure 9In this case, there are 15 distances, we calculate the mean m and the standard deviation σ of them, and =0.2903, the value is minor and that means the distance don’t vary very much. On the other hand, the conclusion above can be explained as that the criminal’s physicalstrength, or his financial condition or something else would limit him.4.3 Centro-based Model (CM)4.3.1 OverviewThis is a macro-dynamic mode. We divide the hunting area into k action spaces, in which we find the possible locations of the next crime based on the time and locations of the past crime scenes. We determine the next possible crime locations by confirming the murderer’s choice of the next crime location’s direction and distance.In CM, the murderer’s choice of the direction of the next possible crime location is based on the time and locations of the past crime scenes. The direction is a stochastic variable, following a certain distribution. The distribution is influenced by the time and locations of the past crime scenes. We design the probability density function of the murderer’s next direction choice on the basis of his past crime scenes’ time and locations. And then we generate a crime location’s direction through a simulation on the basis of the distribution.We got a buffered distance-decay function [2], and construct a probability density function of the ND of the murderer’s next crime location in accordance with the curve’s variation law. Then we run a simulation to get the murderer’s next crime location’s distance.4.3.2 Model Definition4.3.2.1 Random Direction Algorithm1. Calculate the weights f t of the murderer’s past crime scene at the time of t . We opt for thefunction defined as follows:f t a lg b t 1 ,a 0, 0(4.3.2.1-1) Wheret is the time of the murderer’s past crime scenes. The crime happened earlier as the valueof t getting larger. a and b are determined by concrete serial crime case.13Figure 10Figure 10 shows the function curve of this function.We can see that f 0 0, which means the murdererwon’t choose the same direction as that of his latest crimelocation. This may be on account of the large possibility ofbeing arrested at the very direction. We can also find thatf t tends to get larger as t is increasing. This is becausethat as the crime happened earlier the impact of itsdirection on the murderer’s next direction choice is gettingsmaller and smaller. 2. Using the weights f t direction choice as follows:g θ w f t 1 cos θ θ,θ π,π(4.3.2.1-2) Whereθ is the argument that can uniquely determine an action space;θ is the argument of the murderer’s past crime location at the time of t ; and w ∑.Figure 11 shows the definition of the argument.14 Figure 113. Calculate the function of possibility distribution as follows:G θ g θ d θθ π,θ π,π(4.3.2.1-3) 4. Divide the range of θ into h equal non-overlapping intervals. Accordingly, we divide the rangeof G θ into h non-overlapping intervals that are not always of equal width.5. We run a simulation to get a random sampling on an interval 0,1 . If this sampling fall into thei th interval of G θ , we calculate the average of the two endpoints of the i th interval of θ. And it is the direction of the murderer’s next crime location.4.3.2.2 Random Distance AlgorithmBuffered distance-decay function:It is a dynamic process of a murderer’s target selection. Crimes occur in those areas where suitable targets overlap the offender’s awareness space. Offenders then search outward from these areas, the search behavior following some form of distance-decay function. There is usually a “buffer zone,” however, centered the criminal’s residence. Within this zone, targets are viewed as less desirable because of the perceived level of risk associated with operating too close to home. For the offender, this area represents an optimized balance between the maximization of opportunity and the minimization of risk. Figure 12 shows an example, derived from a serial rape case, of a typical buffered distance-decay function .Algorithm:1. Confirm the type of the serial crime, and determine the gradient of a in Equation 4.1.1;2. Calculate the maximum distance from residence to crime site y on the basis of the maximumdistance between the past crime sites x, using the Equation 4.1.1;3. Calculate the ND of the actual distance d by using the equation ND; 4. Construct a probability density function d ND of the ND of the murderer’s next crime locationin accordance with the buffered distance-decay function curve’s variation law. We study Figure 12, and design the probability density function in form of the formula below: d ND q 1 q 2 10 ND q 3 2,0 ND 0.2k x b,0.2 1 (4.3.2.2-1) Figure 13 shows the probability density function curve. We can find that it preserves the approximate characters of Figure 12, and we can use this probability density function to simulate the ND of the murderer’s next crime location.15 Figure 1216Figure 135. Calculate the function of possibility distribution as follows:D ND D ND d ND D,ND 0,1 (4.3.2.2-2)6. Divide the range of ND into h equal non-overlapping intervals. Accordingly, we divide the rangeof D ND into h non-overlapping intervals that are not always of equal width;7. We run a simulation to get a random sampling on an interval 0,1 . If this sampling fall into thei th interval of D ND , we calculate the average of the two endpoints of the i th interval of ND ;8. Calculate the result of ND y , and it is the distance of the murderer’s next crime location.。
2010年高教杯数学建模竞赛B题全国一等奖
5.2 世博会对就业的贡献率模型 根据经济学的就业弹性思想,以及劳动生产理论,以就业人数的增加量与 GDP 增长的比值,测算世博会投资拉动的 GDP 对就业人数的影响。为方便计算,我们令
从业增量 增量
( 表示年份)。
5
其中 年份 2003 2004 2005 2006 2007 2008 2009
4
参数置信区间 [-348.28142 1616.31812] [0.65802 0.74749]
表 2 上海市人均消费与收入
年份(年) 2003 2004 2005 2006 2007 2008 2009
人均消费(元) 11040.00 12631.00 13773.00 14762.00 17255.00 19398.00 20992.00
表示 t 年的从业人数, GDP(亿元) 6251 8072.8 9125 10297 11658 13698 14900 1821.8 1052.2 1172 1361 2040 1202 GDP 增量
表示 t 年的 GDP 值。 从业人数 (万人) 813 837 863 886 909 1053 1064 23.82 26.45 22.19 23.57 144.16 11.18 0.01308 0.02513 0.01893 0.01732 0.07067 0.00930 从业增量
(5.3) 其中, 表示人均消费变动。 根据简单消费函数可建立如下模型: (5.4) 其中, 为人均消费, 为人均收入。 表二给出 2003 到 2009 年上海市人均消费与人均收入。利用 MATLAB 软件的 regress 命令求解它们之间的线性回归方程如下: (5.5)
表 1 回归参数表 参数 参数估计值 634.02 0.703
2010年数学建模B题—上海世博会
上海世博会影响力的定量评估摘要本文是一个对上海世博会影响力的定量评估问题,首先我们收集了与世博会有关的数据,如国内来沪旅游人数,国外来沪旅游人数等。
并用灰色预测对相应的数据进行了预处理,然后我们从横向(本届世博对上海的影响)和纵向(本届世博和历届世博的影响比较)两个角度对世博影响力进行了研究,最后还应用了多目标优化模型求出在不同投资增长系数下上海世博对当地旅游经济最大影响力系数。
第一步,我们横向考虑世博会对本地旅游业的影响力,并将该影响分为对旅游经济的影响和对旅游文化的影响两方面。
首先应用本底趋势线模型得出相应数据的本底值,再分别建立对旅游经济和旅游文化的影响力系数模型,然后利用本底值和统计值得出相底值增加了579.39亿元的旅游收入。
而世博对旅游文化的影响力系数为1.29。
第二步,我们纵向考虑上海世博会与历届世博会相比的影响力。
根据收集的历届世博会相关的规模数据,将世博会影响力等级从低到高分为1-5等,从而建立了世博会综合影响力的模糊评价模型。
对历届世博会的影响力做出综合评价并得出了相应的综合影第三步,我们从环保,旅游收入以及后世博效应三个角度对上海世博的影响重新进行了思考。
综合权衡这三个方面因素,我们建立了一个多目标优化的模型。
得出了在不同投资增长系数下的一个合理的旅游经济影响力系数和世博年最优的旅游者的人数。
当投资增长系数为0.4时,其对旅游经济的影响力系数为1.297,则该年最大的旅客人数为13415.54万人。
而我们根据预测值得出2010年总旅客人数为12695万人,说明预测的旅客人数未超过最大人数限制。
最后,我们根据所求得的影响力系数,对上海世博会写了一篇影响力评估报告。
关键词:本底趋势线模型模糊评价模型多目标优化旅游文化影响力系数1.问题重述1.1问题背景中国2010年上海世界博览会(Expo 2010),是第41届世界博览会。
于2010年5月1日至10月31日期间,在中国上海市举行。
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SummaryFaced with serial crimes,we usually estimate the possible location of next crime by narrowing search area.We build three models to determine the geographical profile of a suspected serial criminal based on the locations of the existing crimes.Model One assumes that the crime site only depends on the average distance between the anchor point and the crime site.To ground this model in reality,we incorporate the geographic features G,the decay function D and a normalization factor N.Then we can get the geographical profile by calculating the probability density.Model Two is Based on the assumption that the choice of crime site depends on ten factors which is specifically described in Table5in this paper.By using analytic hierarchy process (AHP)to generate the geographical profile.Take into account these two geographical profiles and the two most likely future crime sites.By using mathematical dynamic programming method,we further estimate the possible location of next crime to narrow the search area.To demonstrate how our model works,we apply it to Peter's case and make a prediction about some uncertainties which will affect the sensitivity of the program.Both Model One and Model Two have their own strengths and weaknesses.The former is quite rigorous while it lacks considerations of practical factors.The latter takes these into account while it is too subjective in application. Combined these two models with further analysis and actual conditions,our last method has both good precision and operability.We show that this strategy is not optimal but can be improved by finding out more links between Model One and Model Two to get a more comprehensive result with smaller deviation.Key words:geographic profiling,the probability density,anchor point, expected utilityE xecutive S ummaryNowadays,in a serial crime,the spatial distribution of crime sites is arousing the more and more attention of the criminologists and the geographers.The Serial criminals,in a spirit of defiance,have endangered public security,gravely infringed on the citizens'personal safety,lives and property,and are abhorred by the people across the country.Since the offenders usually have no stable residence,it is very difficult for the police to find out and arrest them.Therefore,in order to help the police solve the crime as soon as possible to maintain social security and stability,a more sophisticated technique is in urgent need to be developed to determine the “geographical profile”of a suspected serial criminal based on the locations of the crimes.This paper presents three methods,especially a new mathematical method combining the advantages of the two previous models,which can generate a useful prediction for law enforcement officers about possible locations of the next crime.Based on Bayesian statistical methods,Model One makes explicit connections between assumptions on offender behavior and the components of the mathematical model.It also takes into account local geographic features that either influence the selection of a crime site or influence the selection of an offender’s anchor point. What’s more,with rigorous inference formulas,this model has both precision and operability.Model Two uses analytic hierarchy process(AHP).It takes full account of a variety of factors relevant to the crime sites,assisting the police to take measures adapted to local conditions so as to improve our work.The last method is composed from Model One and Model Two.Taken the expected utility and other practical factors into consideration,it further estimate the geographic profile generated by the previous two models.To conclude,we suggest the policemen put this mathematical method about geographical profiling into practice for it will be of significant assistance to law enforcement.The technical details are as follows:First of all,enhance the consciousness of the public social security and its improvement in the potential criminal area and inform the local people that a series of crimes have occurred recently,reminding them to keep vigilant and not to go into remote areas alone.Second,the police departments should focus their activities,geographically prioritize suspects,and concentrate saturation or directed patrolling efforts in those zones where the criminal predator is most likely to be active.Third,after arresting the criminal,the police need to make experiential analysis on all these kinds of serial crimes to prevent a similar case.What’s more,the police departments set up an enhanced intelligence exchange network,especially in the area which is the next crime site according to our prediction.Finally,search the suspect in the predicted criminal location or the likelyresidence of the offender.Generally speaking,our method has good maneuverability and practicability. However,there are various uncertain factors in the situation which cannot be predicted such as the weather condition and the traffic conditions.Additionally,the determination and analysis of weight-coefficients on the various factors is very subjective.As a result,the actual site scene of next crime may be outside of our geographical profiling.Therefore,the police deployments must be based on the analysis of local actual condition instead of applying our model blindly.Furthermore,our prerequisite that the offender has the only one stable anchor point differs from the actual conditions.Since the offender is very likely to change his residence,the police had better search the suspect according to the latest information.Maintaining social stability and ensuring the safety of residents will bring well-being and peace to all mankind.Every one of us should make effort to make our society become better.Hope that this paper will be of some help to prevent the criminal behaviors.1.IntroductionClues derived from the locations connected to violent repeat criminal offenders, such as serial murderers,arsonists,and rapists,can be of significant assistance to law enforcement.Such information helps police departments to focus their activities, geographically prioritize suspects,and to concentrate directed patrolling efforts in those zones where the criminal offender is most likely to be active.By examining spatial data connected to a series of crime sites,this methodological model generates a probability map that indicates the area most likely to be the locations of the next crime.This paper presents two mathematical models to illustrate how geographical analysis of serial crime conducted within a geographic information system can assist crime investigation.Techniques are illustrated for determining the possible residence of offenders and for predicting the location of the next crime based on the time and locations of the existing crimes.First,we present a mathematical survey of some of the algorithms that have been used to solve the geographic profiling problem.The geographic profiling problem is the problem of constructing an estimate for the location of the anchor point of a serial offender from the locations of the offender’s crime sites.The approach that we develop will make use of at these two different schemes to generate a geographical profile.What’s more,we develop a third technique to combine the results of the two previous schemes and generate a useful prediction for law enforcement officers.The prediction provides some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes.Our method will also provide some kind of estimate about how reliable the estimate will be in a given situation,including appropriate warnings.The executive summary will provide a broad overview of the potential issues.It will also provide an overview of our approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool.The purpose is to apply geographical analysis to serial crime investigations to predict the location of future targets and determine offender residence.1.1.11Restatement of the ProblemIn order to indicate the origin of geographical profiling problems,the following background is worth mentioning.In1981Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks.One of the methods used to narrow the search for Mr.Sutcliffe was to find a“center of mass”of the locations of the attacks. In the end,the suspect happened to live in the same town predicted by this technique. Since that time,a number of more sophisticated techniques have been developed to determine the“geographical profile”of a suspected serial criminal based on the locations of the crimes.Our team has been asked by a local police agency to develop a method to aid intheir investigations of serial criminals.The approach that we develop will make use of at least two different schemes to generate a geographical profile.We also develop a technique to combine the results of the different schemes and generate a useful prediction for law enforcement officers.The prediction will provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes.Our method will also provide some kind of estimate about how reliable the estimate will be in a given situation,including appropriate warnings.1.1.22Survey of Previous Research Existing MethodsTo understand how we might proceed let us begin by adopting some common notation:●A point will have two components .x 12(,,)x x x =●These can be latitude and longitude●These can be the distances from a pair of reference axes ●The series consists of n crimes at the locations .12,,,n x x x ⋯●The offender’s anchor point will be denoted by .z ●Distance between the points and will be .x y (,)d x y Existing algorithms begin by first making a choice of distance metric ;they d then select a decay function and construct a hit score function by computingf ()S y 11()((,))((,))((,))ni n i S y f d x y f d x y f d x y ===++∑⋯Regions with a high hit score are considered to be more likely to contain the offender’s anchor point than regions with a low hit score.In practice,the hit score is not evaluated everywhere,but simply on some rectangular array of ()S y points for and giving us the array of 12(,)jk j k y y y ={}1,2,,j J ∈⋯{}1,2,,k K ∈⋯values [4].()jk jk S S y =Rossmo’s method,as described in (Rossmo,2000,Chapter10)chooses the Manhattan distance function for d and the decay functionif () if (2)h g h g k d B d f d kB d B B d −⎧<⎪⎪=⎨⎪≥⎪−⎩Other StudiesIn the study of Crime Analyst’s task,Bryan Hill considers that the use of the “probability grid method”(PGM)can narrow the search as it pertains to tactical crime analysis in the ArcView Geographic Information Systems (GIS)environment.The main point of his theory will be that any current statistical method of predicting the next hit location in a crime series is operationally ineffectual when the suspect covers a large geographic area.When the analyst combines several statistical methods and intuitive,logical thought processes into a combined “grid”score,the analysis product can be made more operationally effective.This new grid surface allows the analyst to make a better prediction of the next hit in a crime series and is useful in isolating specific target locations for law enforcement deployment efforts.This easy to apply “PGM method”allows the analyst to use sound statistical methods,as well as their experience and knowledge of a crime series to narrow the focus and potential hit area.In addition,when the crime series has sufficient suspect information,journey to crime analysis using the CrimeStat software can be used to provide investigators with a list of probable offenders from law enforcement records available to the analyst [4].2.Model Overview1、Model OneIn Model One,we assume that the crime sites depend on the distance between the anchor point and the crime site.Generate a hot zone model according to probability formula,then combine the geographic features G with the decay function D and add a normalization factor N into the model .At last,we can get the “geographical profile”.2、Model TwoIn Model Two,we take account of other factors relevant to crime site location,we have studied various literature to summary two aspects denoted by U(Utility from crimes)and P(the probability of success)specifically including the following ten factors:the responding speed of the police,public security situation,resistances’diathesis,density of registered inhabitants,the advantageous position,the number of offenses,the distance from the anchor point of the offender,the time required for committing the crime,number of target persons,offender’s mental satisfaction from the crime.Then,we use Analytic Hierarchy Process (AHP)to get weighted factors.Finally,according to the formula we can draw E of some areas.A wide range of criminal areacan then be divided into several small areas.Then,we give the proper scores for each small area according to the actual condition.When the area is small enough,we can get the geographical profile.Sites with higher scores have high probability to be crime areas.3、The two models can be synthesized as one methodFor their different emphasis (The theory of Model One is quite rigorous,but it ignores the factors such as the geographic features and criminal motivation while these factors is important for selecting criminal crime sites.On the other hand,Model Two takes these into account,while it is too subjective in practical application which may easily cause the deviation),the geographical profiles we get from two models must be different.By using mathematical dynamic programming method to establish the model,this method explains the common rules of the offender’s choice on the crime sites.Therefore,when we get several geographical profiles from the previous two models,we can use this method to predict the next offence site.To sum up ,we can generate the geographical profile in the investigations of serial criminals,providing some kind of estimate or guidance about possible locations of the next crime and narrow the search.3.Model OneSymbols:Hypothesis :1、We assume that our offender chooses potential locations to commit crimesrandomly according to some unknown probability density function .P 2、We assume that depends upon z and a.P 3、We suppose that that the values of the anchor point z and average offensedistance are unknown,but the form of the distribution is known.a P 4、The offender has only one residence which is unchanged.symbolsmeaning (x)P probability density function xthe crime sites zthe location of the offender’s anchor point aThe average distance that the offender is willing to travel to offend.Dthe effect of distance decay Gthe geographic features N a normalization factor3.1A Mathematical ApproachIn order to looking for an appropriate model,we start with the simplest possible situation.Since we know nothing about the offender,we assume that our offender chooses potential locations to commit crimes randomly according to some unknown probability density function .For any geographic region ,the probability that our P R offender will choose a crime location in can be found by adding up the values of R in ,giving us the probabilityP R 21(x)()()R P d x d x ∫∫Upon what sorts of variables should the probability density depend?()P x The fundamental assumption of geographic profiling is that the choice of an offender’s target locations is influenced by the location of the offender’s anchor point z.Therefore,we first assume that depends upon z.(Provided that the offender has P a single anchor point and that it is stable during the crime series.)A second important factor is the distance the offender is willing to travel to commit a crime from their anchor point.(Different offender’s have different levels of mobility-an offender will need to travel farther to commit some types of crimes than others )Let a denote the average distance that the offender is willing to travel to offend.So varies between a offenders and crime types [4].Let us suppose that that the values of the anchor point z and average offense distance are unknown,but the form of the distribution is known.Then the a P problem can be stated mathematically as:Given a sample (the crime sites)from the distribution with 1,,n x x ⋯(,)P x z a parameters z and to determine the best way to estimate the parameter z (the anchor a point).One approach of this mathematical problem is the theory of maximum likelihood.First,construct the likelihood function [4]:11(y,)(,)(,)(,)ni n i L a P x y a P x y a P x y a ===∏⋯In order to get the best choice of z,make the likelihood as large as possible by maximizing the log-likelihood:11(y,)(,)n (,)n (,)ni n i a lnP x y a l P x y a l P x y a λ===++∑⋯This approach is rigorous;however,it is unsuitable as simple point estimates for the offender’s anchor point are not operationally useful.Therefore,we continue our analysis by using Bayes Theorem [4].Bayes Theorem then implies:(1)(z,)(,)(z,)()P a x z a P a x p x π=Here is the prior distribution.(,)z a πIt represents our knowledge of the probability density for the anchor point z and the average offense distance before we incorporate information about the crime.a If we assume that the choice of anchor point is independent of the average offense distance,we can write:(2)(,)()()z a H z a ππ=is the prior distribution of anchor points,and is the prior distribution ()H z ()a πof average offense distances.We will assume that the offender’s choices of crime sites are mutually independent,so that (3)11(,,,)()()n n P x x z a P x z a P x z a =⋯⋯,,Suppose that an unknown offender has committed crimes at ,and that 1,,n x x ⋯The offender has a unique stable anchor point z.The offender chooses targets to offend according to the probability density where is the average distance the offender is willing to travel.(,)P x z a a The target locations in the series are chosen independently.The prior distribution of anchor points is ,the prior distribution of the ()H z average offense distance is and these are independent of one another.()a πThen the probability density that the offender has anchor point at the location z satisfies (4)11(,, ),,()()n n P z x x P x z a P x z a H z a da π∝∫⋯⋯()()3.2Simple Models for Offender BehaviorWhat’s more,we need to be able to construct reasonable choices for our model of offender behavior,if our fundamental mathematical result is to have any practical or investigative value.One simple model is to assume that the offender chooses a target location based only on the Euclidean distance from the offender’s anchor point to the offense location and that this distribution is normal.In this case we obtain(5)2221P(,a ) =exp ()44x z x z a a π⎛⎞−−⎜⎟⎝⎠If we make the prior assumptions that the average offense distance and the anchor point are unchanged,and the offender commits n crimes at the crime site locations ,then1,,n x x ⋯212211P(,,) =(exp ()44n n n i i z x x x z a a π=⎛⎞−−⎜⎟⎝⎠∑⋯We see that the anchor point probability distribution is just a product of normal distributions;the maximum likelihood estimate for the anchor point is simply the mean center of the crime site locations.We also mention that in this model of offender behavior,this is also the mode of the posterior anchor point probability distribution [4].Another reasonable model for offender behavior is to assume that the offender still chooses a target location based only on the Euclidean distance from the offense location to the offender’s anchor point,but now the distribution is a negative exponential so that (6)222P(z,a ) =exp x x z a a π⎛⎞−−⎜⎟⎝⎠Once again,because of our prior assumptions that the average offense distance and the anchor point are unchanged,and the offender commits n crimes at the crime site locations ,then we have1,,n x x ⋯12122P(,, ) =()exp n n n i i z x x x z a a π=⎛⎞−−⎜⎟⎝⎠∑⋯We see that this is just a product of negative exponentials centered at each crime site.Further,the corresponding maximum likelihood estimate for the offender’s anchor point is simply the center of minimum distance for the crime series locations [4].This preceding analysis was predicated on the prior assumptions that the average offense distance and it is known in advance.Similarly,the existing methods mentioned in this paper all rely on decay functions f with one or more parameters which also need to be determined in advance.Unlike those methods,our method does not require that we make a choice for the parameter in advance.3.3Realistic Models for O ff ffenderender Behavior What would a more realistic model for offender behavior look like?Consider a model in the form :(7)(,) ((,),)()()P x z a D d x z a G x N z =▲D models the effect of distance decay using the distance metric d(x,z)1、We can specify a normal decay,so that2221(, ) exp 44D d a d a a π⎛⎞=−⎜⎟⎝⎠2、We can specify a negative exponential decay,so that222(, ) exp D d a d a a π⎛⎞=−⎜⎟⎝⎠Any choice can be made for the distance metric (Euclidean,Manhattan,et.al)▲G models the geographic features that influence crime site selection High values for G(x)indicate that x is a likely target for typical offenders;Low values for G(x)indicate that x is a less likely target▲N is a normalization factor,required to ensure that P is a probability distribution121()((,),)(y)()()N z D d y z a G d y d y =∫∫N is completely determined by the choices for D and G.G models the geographic features that influence crime site selection,with high values indicating the location was more likely to be targeted by an offender.Then,how can we calculate G?Use available geographic and demographic data and the correlations between crime rates and these variables that have already been published to construct an appropriate choice for G(x).Different crime types have different etiologies;in particular their relationship to the local geographic and demographic backcloth depends strongly on the particular type of crime.This would limit the method to only those crimes where this relationship has been well studied.Some crimes can only occur at certain,well-known locations,which are known to law enforcement For example,gas station robberies,ATM robberies,bank robberies;liquor store robberies .This does not apply to all crime types-e.g.street robberies,vehicle thefts.We can assume that historical crime patterns are good predictors of the likelihood that a particular location will be the site of a crime.Suppose that historical crimes have occurred at the locations c 1,c 2...c n .Choose a kernel density functionK y λ()is the bandwidth of the kernel λdensity function.Figure 1Calculate(8)i 1(x)(x-c )Ni G K λ==∑The bandwidth can be e.g.the mean nearest neighbor distance.λWe have assumed:Each offender has a unique,well-defined anchor point that is stable throughout the crime seriesThe function H(z)represents our prior knowledge of the distribution of anchor points before we incorporate information about the crime series.Suppose that anchor points are residences-can we estimate H(z)?●Population density information is available from the U.S.Census at the block level,sorted by age,sex,and race/ethnic group.1、We can use available demographic information about the offender2、Set1()(blocks N ii H z p K z q===−∑3、Here block i has population pi,center q i ,and area A i .●Distribution of residences of past offenders can be used.Calculate using the same techniques used to calculate G(x).()H z 3.4Future O ff ffenseense Prediction Given a series of crimes at the locations committed by a single serial 1,,n x x ⋯offender,estimate the probability density ,that X next will be the 1(,,)next n P x x x ⋯location of the next offense [4].The Bayesian approach to this problem is to calculatethe posterior predictive distribution:1211(,,)(,)(,,,)()()()next n next n P x x x P x z a P z a x x d z d z d a =∫∫∫⋯⋯We can use the method above to simplify,and so obtain the expression:1211(,,)(,)(,,,)()()()()()next n next n P x x x P x z a P z a x x H z a d z d z d a π∝∫∫∫⋯⋯This approach makes the same independence assumptions about offender behavior as our fundamental result.4.Model TwoSymbols:Assumption :The selection for crime sites depends on ten factors that demonstrated in Table 5in the paper.4.1Analysis of modelsThe selection for crime sites depends on various factors instead of only one factor that the distance from the anchor point to the crime site.Therefore,Combine a large number of previous studies and related papers with our own thinking,we have summed up 10key factors and a grading system about how to make the decisions on where the criminal locations will be (see Table 5),which are listed as follows:the responding speed of the police,public security situation,resistances’diathesis,density of registered inhabitants,the advantageous position,the number of offenses,the distance from the anchor point of the offender,the time required for committing the crime,number of target persons,offender’s mental satisfaction from the crime.Classify these 10index factors into two categories P and U ,based on the actual situation.Let the former six factors belong to P and the latter four belong to U .Next,we use Analytic hierarchy process (AHP)to get weighted factors (Table 4).Then we give the proper scores for each small area according to the actual condition.(Table 5).(There are some data which is difficult for us to achieve but they may be easily achieved by the local police .)Finally,we can work out the total score of E:16141614P P U U P P U U r r r r E ww w w ×++×+×++×=⋯⋯4.2Use AHP to get the weighted factorsThen we can get a comparison matrix as follows:The probability of success (P )is as important as the expected utility (U ),so E PU =,E UP =11symbols Meaning w Weight ETotal score r B1,…r B10Each score m B1,…m B10Various factorsP The probability of success UThe utility from crimes1111E ⎡⎤=⎢⎥⎣⎦and the judgment matrix of U :Ui 1752111173711315517512U ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦Pi and the judgment matrix of P :137546116423311111176275111121546511761342111551633P ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦According to all levels of comparison matrix,let’s obtain its maximum of eigenvalue,coincidence indicator and the weight corresponding to all factors by using Matlab.The results are shown in the table below:Table 1The coincidence indicator can be computed by the formulacomparison matrix Maximum eigenvalue max λcoincidence indicator IC the weight of evaluation indexes wE=2maxλ0[0.5000;0.5000]U=4.1341maxλ0.0447[0.4971;0.0495;0.1016;0.3519]P=6.5633max λ0.1127[0.4319;0.2105;0.0298;0.0448;0.1798;0.1033]=I C max1nn λ−−The results are shown in the form below.Find the average random coincidence indicator .The average random coincidence indicator (exponent number is within I R 15)is as follows:Table 2Based on=,we can work out coincidence ratios:RC I ICRTable 3、、are all less than 0.1within the scope of consistency,which showsRwCRU C RP C that the index factors we set meet the requirements,that is,they have some credibility within the error range.Combination weight for all levels.Calculate the weight of the index factors Ui ,Pi .For example,=0.49710.5000=0.248551U w×In the same way,we can work out (i =1,2,…6),and(i =1,2,…4).Pi w UiwSee Table 4:Table 4exponent number12345678IR 000.520.89 1.12 1.26 1.36 1.41exponent number9101112131415R1.461.491.521.54 1.561.581.59RwCRUCRPC00.05020.08941P w 2P w 3P w 4P w 5P w6P w0.215950.105250.01490.02240.08990.051651U w2U w3U w4U w0.024750.05080.17595。