《空间泊松点过程》PPT课件.ppt

1、Spatial Poisson Processes,The Spatial Poisson Process,Consider 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,),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.,For each,Alternatively, a spatial Poisson process satisfies the following axioms:,The pr

3、obability distribution of N(A) depends on the set A only through its size |A|.,There exists a such that,There is probability zero of points overlapping:,If these axioms are satisfied, we have:,for k=0,1,2,Consider a subset A of S:,There are 3 points in A how are they distributed in A?,A,Expect a uni

4、form distribution,In fact, for any , we have,Proof:,So, we know that, for k=0,1,n:,ie: N(B)|N(A)=n bin(n,|B|/|A|),Generalization:,For a partition A1, A2, , Am of A:,for n1+n2+nm = n.,(Multinomial distribution),Simulating a spatial Poisson pattern with intensity over a rectangular region S=a,bxc,d.,s

5、catter 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),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

6、 neighbor.,For x0,So,for x0.,In 3-D we could show that:,Example: Spatial Patterns in Statistical Ecology,Consider 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

7、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 there are many seeds with an extremely small chance of falling into the quadrat.,Suppose now that the probability that a seed ger

8、minates 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?,(accept probability is ),So, the surviving seeds continue to be distributed “at random”.,Simulation Problem:,Type 1 and type 2 seeds will g

9、erminate 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,Suppo

10、se that the one-acre field is evenly divided into 10 x10 quadrats.,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 plants.,Usi

11、ng 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),Keep in mind that we dont

12、 really have to keep track of where the individual plants are, only the number in each quadrat.,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

13、:,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.,It would be nice if we could further modify the Poisson number of seeds for Type 1.,We can, at