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1、Spatial Poisson ProcessesThe Spatial Poisson ProcessConsider a spatial configuration of points in the plane:严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Notation:Let S be a subset of R2.(R,R2,R3,)Let A be the family of subsets of S.For let|A|denote the size of A.(length,area,volume,)Let N
2、(A)=the number of points in the set A.(Assume S is normalized to have volume 1.)严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Then is a homogeneous Poisson point process with intensity if:For every finite collection A1,A2,An of disjoint subsets of S,N(A1),N(A2),N(A3)are independent.For eac
3、h 严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Alternatively,a spatial Poisson process satisfies the following axioms:i.If A1,A2,An are disjoint regions,then N(A1),N(A2),N(An)are independent rvs and N(A1 U A2 U U An)=N(A1)+N(A2)+N(An)ii.The probability distribution of N(A)depends on the s
4、et A only through its size|A|.严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。iii.There exists a such that iv.There is probability zero of points overlapping:严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。If these axioms are satisfied,we have:for k=0,1,2,严格执行突发事件上报制度、校外活动报批制度等相关规章制度
5、。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Consider a subset A of S:There are 3 points in A how are they distributed in A?A Expect a uniform distribution 严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。In fact,for any ,we haveProof:严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。So,we know that,for
6、 k=0,1,n:ie:N(B)|N(A)=n bin(n,|B|/|A|)严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Generalization:For a partition A1,A2,Am of A:for n1+n2+nm=n.(Multinomial distribution)严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Simulating a spatial Poisson pattern with intensity over a recta
7、ngular region S=a,bxc,d.simulate a Poisson()number of points(perhaps by finding the smallest number N such that)scatter that number of points uniformly over S(for each point,draw U1,U2,indep unif(0,1)s and place it at(b-a)U1+a),(d-c)U2+c)严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Consid
8、er a two-dimensional Poisson process of particles in the plane with intensity parameter .Lets determine the(random)distance D between a particle and its nearest neighbor.For x0,严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。So,for x0.In 3-D we could show that:严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到
9、及时发现、制止、汇报并处理各类违纪行为或突发事件。Example:Spatial Patterns in Statistical EcologyConsider a wide expanse of open ground of a uniform character(such as the muddy bed of a recently drained lake).The number of wind-dispersed seeds occurring in any particular“quadrat”on this surface is well modeled by a Poisson
10、random variable.The reason this tends to be true is due to the binomial approximation to the Poisson distribution which will hold if there are many seeds with an extremely small chance of falling into the quadrat.严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Suppose now that the probabilit
11、y that a seed germinates is p and that they are not sufficiently packed together to interact at this stage.Question:What is the distribution of the number of germinated seeds?Answer:This is a thinned Poisson processwith rate(accept probability is )So,the surviving seeds continue to be distributed“at
12、 random”.严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Simulation Problem:Type 1 and type 2 seeds will germinate with probabilities p1 and p2,respectively.Type 1 plants will produce K offshoot plants on runners randomly spaced around the plant where Kgeom(p).(P(K=0)=p)Two types of seeds ar
13、e randomly dispersed on a one-acre field according to two independent Poisson processes with intensities Suppose that the one-acre field is evenly divided into 10 x10 quadrats.严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Assume that the number of offshoot plants that fall into a quadrat d
14、ifferent from their parent plants is negligible.A particular insect population can only be supported if at least 75%of the quadrats contain at least 35 plants.Using p=0.9,p1=0.7,and p2=0.8,explore the values of that will give the insect population a 95%chance of surviving.Use the hugely simplifying
15、assumption that there is no time component to this process(and,in particular,that offshoot plants do not have further offshoots)严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。Keep in mind that we dont really have to keep track of where the individual plants are,only the number in each quadr
16、at.Note that we dont have to consider germination of the plants as a second step after the arrival of the seeds instead consider a thinned Poisson number of plants of Type i with rate Tips on simulating this:Rather than drawing uniformly distributed locations for the seeds,we can simulate the number
17、s for each quadrat separately(and ignore locations)using the fact that each quadrat will contain Poisson()germinating seeds.严格执行突发事件上报制度、校外活动报批制度等相关规章制度。做到及时发现、制止、汇报并处理各类违纪行为或突发事件。It would be nice if we could further modify the Poisson number of seeds for Type 1.We can,at least,simplify the generation of offshoot plants,dealing with all plants in a particular quadrat together by adding a negative-binomial number of plants to each quadrat.How to deal with offshoot plants