空间泊松点过程.ppt

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. (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: If A1, A2, …, An are disjoint regions, then N(A1), N(A2), …, N(An) are independent rv’s 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 it’s 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 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,b]x[c,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) Consider a two-dimensional Poisson process of particles in the plane with intensity parameter . Let’s determine the (random) distance D between a particle and its nearest neighbor. For x0, So, for x0. In 3-D we could show that: *