空间泊松点过程

1、空间泊松点过程第1页，共19页。The Spatial Poisson ProcessConsider a spatial configuration of points in the plane:第2页，共19页。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,) , AA Let N(A) = the number of points in the set A

2、.(Assume S is normalized to have volume 1.)第3页，共19页。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.A AN(A)0 For each , AA .|)A|Poisson( N(A) 第4页，共19页。Alternatively, a spatial Pois

3、son process satisfies the following axioms: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)The probability distribution of N(A) depends on the set A only through its size |A|. 第5页，共19页。There exists a such that 0|)Ao(

4、| |A| 1)P(N(A)There is probability zero of points overlapping: 1 1)P(N(A)1)P(N(A) lim0|A|第6页，共19页。If these axioms are satisfied, we have: for k=0,1,2, k!|)A|(e k)P(N(A)k|A|-第7页，共19页。Consider a subset A of S: There are 3 points in A how are they distributed in A? A Expect a uniform distribution 第8页，共

5、19页。In fact, for any , we haveProof: AB |A|B 1)N(A) | 1P(N(B)1)P(N(A1)N(A) 1,P(N(B) 1)N(A) | 1P(N(B)1)P(N(A1)BN(A 1,P(N(B) C|A|-|BA|-|B|-e|A|e e |B C|A|B 第9页，共19页。So, we know that, for k=0,1,n:k-nk|A|B-1|A|B kn n)N(A) | kP(N(B)ie: N(B)|N(A)=n bin(n,|B|/|A|)第10页，共19页。Generalization:For a partition A1

6、, A2, , Am of A: n)N(A) | n)N(A , . ,n)N(A ,n)P(N(Amm2211m21nmn2n1m21|A|A| |A|A| |A|A| !n!n !nn! for n1+n2+nm = n.(Multinomial distribution)第11页，共19页。Simulating a spatial Poisson pattern with intensity over a rectangular region S=a,bxc,d. simulate a Poisson( ) number of points1N1i-ie U(perhaps by fi

7、nding 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)第12页，共19页。Consider a two-dimensional Poisson process of particles in the plane with intensity parameter .Lets determine the (random

8、) distance D between a particle and its nearest neighbor.For x0,x)P(D (x)FDx)P(D - 1 centered disk in particlesother P(no - 1 )x area withparticle the at22x-e - 1 第13页，共19页。So,for x0.2x-DDe x 2 (x)Fdxd (x)fIn 3-D we could show that:3x-De -1 (x)F343x-2DDe x 4 (x)Fdxd (x)f34第14页，共19页。Example: Spatial

9、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 random variable.The reason this tends

10、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.第15页，共19页。Suppose now that the probability that a seed germinates is p and that they are not sufficiently packed together to

11、interact at this stage.Question: What is the distribution of the number of germinated seeds?Answer: This is a thinned Poisson process.pwith rate(accept probability is )pSo, the surviving seeds continue to be distributed “at random”.第16页，共19页。Simulation Problem: Type 1 and type 2 seeds will germinate

12、 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 are randomly dispersed on a one-acre field according to two independent Poisson processes with intensities. and 21 Suppo

13、se that the one-acre field is evenly divided into 10 x10 quadrats.第17页，共19页。 Assume that the number of offshoot plants that fall into a quadrat different 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 p

14、lants.21 and 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 assumption that there is no time component to this process (and, in particular, that offshoot plants do not have further offshoots)第18页，共19页。

15、 Keep in mind that we dont really have to keep track of where the individual plants are, only the number in each quadrat. pii 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: