空間泊松點過程

1、Spatial Poisson ProcessesThe Spatial Poisson ProcessConsider a spatial configuration of points in the plane: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.)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) Alternatively, a spatial Poisson process satisf

3、ies 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 set A only through its size |A|. iii. There exists a such that 0|)Ao(| |A| 1)P(N(A)iv

4、. There is probability zero of points overlapping: 1 1)P(N(A)1)P(N(A) lim0|A|If these axioms are satisfied, we have: for k=0,1,2, k!|)A|(e k)P(N(A)k|A|-Consider a subset A of S: There are 3 points in A how are they distributed in A? A Expect a uniform distribution In fact, for any , we haveProof: AB

5、 |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 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|)Generalization:For a partition A1, A2, , Am of A: n)N(A) | n)N(A , . ,n)N(A ,n)P(N(Amm2211m

6、21nmn2n1m21|A|A| |A|A| |A|A| !n!n !nn! for n1+n2+nm = n.(Multinomial distribution)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 finding the smallest number N such that) scatter that number of points

7、 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)Consider 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,x)P(D (x)FDx)P

8、(D - 1 centered disk in particlesother P(no - 1 )x area withparticle the at22x-e - 1 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)f34Example: Spatial Patterns in Statistical EcologyConsider a wide expanse of open ground of a uniform character (such

9、 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 to be true is due to the binomial approximation to the Poisson distribution which will hold if the

10、re are many seeds with an extremely small chance of falling into the quadrat.Suppose now that the probability 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

11、a thinned Poisson process.pwith rate(accept probability is )pSo, the surviving seeds continue to be distributed “at random”.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 a

12、round 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 Suppose that the one-acre field is evenly divided into 10 x10 quadrats. Assume that the number of offshoot plants that fall

13、 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 plants.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 survivi

14、ng. 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) 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

15、 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 numbers for each quadrat separately (and ignore locations) using the fact that each quadrat will contain Poisson( ) germinating seeds./100pii It would be nice if we could further modify the Pois