空间泊松点过程

Click to edit Master title style,*,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Spatial Poisson Processes,The Spatial Poisson Process,Consider a spatial configuration of points in the plane:,Notation:,Let S be a subset of,R,2,.(,R,R,2,R,3,),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.,(Assume S is normalized to have volume 1.),Then is a homogeneous Poisson point process with intensity if:,For every finite collection A,1,A,2,A,n,of disjoint subsets of S,N(A,1,),N(A,2,),N(A,3,)are independent.,For each,Alternatively,a spatial Poisson process satisfies the following axioms:,If A,1,A,2,A,n,are disjoint regions,then N(A,1,),N(A,2,),N(A,n,)are independent rvs and,N(A,1,U A,2,U U A,n,)=N(A,1,)+N(A,2,)+N(A,n,),The probability 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 uniform 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 A,1,A,2,A,m,of A:,for n,1,+n,2,+n,m,=n.,(Multinomial distribution),Simulating a spatial Poisson pattern with intensity over a rectangular 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 U,1,U,2,indep unif(0,1)s and place it at(b-a)U,1,+a),(d-c)U,2,+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,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 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 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 process,with rate,(accept probability is ),So,the surviving seeds continue to be distributed“at random”.,Simulation Problem:,Type 1 and type 2 seeds will germinate with probabilities p,1,and p,2,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,Suppose 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.,Using p=0.9,p,1,=0.7,and p,2,=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 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:,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 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,