# 关于泊松点过程的生成方法-Generating Homogeneous Poisson Processes - PDF.pdf

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(2)Sett←t-是log(a) (3) Deliver t. 4)Go to Step (Standard packages use more elaborate techniques that avoid the use of the "log function in Step (2).)Another straight forward method to generate such realizations arises from a well known prop- erty of a homogeneous Poisson process. Conditioned on the number of events N(to)=n that occur in the fixed interval [0, tol, the event times T1,T2,.,In of a homogeneous Poisson process are distributed as order statistics from a uniform distribution supported on [0, tol [7.(See"Generating Nonhomogeneous Poisson Processes"for a more general result and a proof. )This property gives rise to an algorithn to generate events from a homogeneous Poisson process confined to the fixed interval 0, to. We list this as Algorithm 2 Algorithm 2 (1)Generate T Poisson(At) (2) If n=0 then exit. Otherwise independently generate u1 0(0, 1),L2 U(0, 1),..,up U(0, 1), and sort (in increasing orde er)to obtain u(1),u(2),+,(n) (3)Sett←to()-i=1,2,…,n (4)Deliver ti, i=1, 2,.,n In Step (1), the Poisson random variate n ean be generated using any number of available methods, Step most notably table-based methods 3, 5, 9, 15] and cdf-inverse adaptations 6. See 4] for a complete treatment. The sorting operation in Step(2) is O(nlog n) and so the efficieney of Algorithm 2 depends on the magnitudes of to and A Generation on the plane A counting process IN(),t 2 0 is said to constitute a two-dimensional homogeneous Poisson process on CCk2 with rate入≥0if (i) the number of events in a region R s C is Poisson distributed with parameter A(R) JRAdv-AA(R), where A(R) is the volume of the region R; (ii) the number of events occurring in any finite set, of nonoverlapping regions are mutually inde- pendent For a more general definition of a poisson process on a plane, and corresponding generation methods on a digital computer, see Generating Nonhomogeneous Poisson Processes. Generation from a two-dimensional homogeneous Poisson process with rate A on a circle of radius r>0 turns out to be easy based on the following result by Lewis and Shedler [ 8.(See also Rubinstein and Kroese[13, pp. 70J-) Theorem 1. Lewis and Shedler, 1979/ Suppose(R1, 01),(R2, 02),.,(RN, ON) are the polar coordi nates nfN>0 events from a homogeneous Poisson pmeess on the cirele C=1(7,m): r2+y2<r21 Then, conditional on N=n, the ordered event radi R(), R(2),., R(a) are order statistics from the density f(z)=2e/r2,2 E 0, r]. Furthermore, 01, 02, -., 0n are i.i.d untlorir on 0 2al and independent of R1, R2,..., Rn The resulting algorithm for generating events from a homogenous Poisson process on a circle of radius r thus involves first generating the number of events N-n from a Poisson distribution with parameter Tr A, then generating the radii R (1),R(2),--, Rin) as order statistics from the density f(2), and then finally generating the angles 01, 02,..1 0n uniformly(and independently) from (0, 2xl Algorithrn 3 (0) Generate 1 Poisson(Tr A). If 7-0 then exit. Otherwise generate a1 N U(0, 1),u2 U(0, 1).,an [(0, 1)independently (1)Set1←rya;2←T√2, (2)Sort R1, R2, -. Rn in ascending order to obtain R(), R(2) (3)Generate un+I n U(0, 1), 1n+ NU(0, 1),, u2n U(0, 1) independently (4)Set61+2vn+1,b2←2mn+21……,an+27t2n (5) Deliver(R(u,1),(B(20b2),…,(B(n),Bn) (In the above algorithm, Step(2) is not required if there is no notion of ordering that is implicit in the abeissa and ordinate. Ross [12 gives a method to extend Algorithm 3 to the context of more irregular regions. For instance, suppose we want to generate events from a honogeneous Poisson process in the region bounded by the first quadrant, the positive valued funetion f(r), and the line r-I, i.e., the region of interest C=I(3):120,y20,y<f(),r<T). Ross [12 shows that the order statistics Xq), X(2)., X() of the x-coordinates of the events are such that JX o /(ar)dr,Xo=0,i 2, .., are i. i d. exponential with mean 1/A. The corresponding ith y-coordinate Yi is uniformly distributed between 0 and f(Xo). This gives rise to an algorithm that is straightforward, at least in principle Algorithm (o) Initialize j-1, 1=0, U=0, Io=0 (1)Generate ui n U(0, 1)independentl (2)Setv←-(1/)ln(1-v) (3) Seu←u+j (4)If ws f f(r)dr then set n+n+1 and go to Step(1) (5)If n0 then exit. Otherwise solve for I(), i-1, 2,. n satisfying Mr(o f(r)dr=wi (6)Generate un+I n [(0, 1), 1 +2 [(0, 1),.,u2n U(0, 1)independently (7)Setv←tn+if(r;) (8) Deliver (ro, yi),i-1, 2,...,. The efficiency of Algorithm 4 depends critically on the inversion in Step(5). Theorem 1 can be generalixed even further to give a result that is the multidi Imensional ala alogue of the order statis tics property discussed in SecTion 1. Specifically, consider a two-dimensional homogeneous Poisson process defined on an arbitrary region C. Then it is seen somewhat easily that the loeation (X,Y of an event on the plane, conditional on the number of events N-n, is uniformly distributed over the region C [10, pp. 359. This then gives rise to a very simple algorithm that is a analogous to Algorithm 2, and generalizes Algorithm 3 Algorithrn 5 (1)Generate n n Poisson(AA(C)) (2)If n=0 then exit. Otherwise independently generate n points(fi, yi), i=1, 2, -. n that are uniformly distributed in C. (3) Deliver (Ii, yi),i-1,2,-.,n. If the region C is irregular, Step 2 may not be straightforward. In such a case, a two-dimensional homogeneous process can be generated on any convenient region C(e. g, a rectangle) that covers C, and the events falling within C can be delivered as a realization of the required process

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