Stochastic Simulation
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1 Stochastic Simulation APPM 7400 Lesson 11: Spatial Poisson Processes October 3, 2018 Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
2 Consider a spatial configuration of points in the plane: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
3 Notation: Let S be a subset of R 2. (R k ) (Assume S is normalized to have volume 1.) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
4 Notation: Let S be a subset of R 2. (R k ) (Assume S is normalized to have volume 1.) Let A be the family of all subsets of S. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
5 Notation: Let S be a subset of R 2. (R k ) (Assume S is normalized to have volume 1.) Let A be the family of all subsets of S. For A A, let A denote the size of A. (length, area, volume,...) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
6 Notation: Let S be a subset of R 2. (R k ) (Assume S is normalized to have volume 1.) Let A be the family of all subsets of S. For A A, let A denote the size of A. (length, area, volume,...) Let N(A) be the number of points in the set A. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
7 Then {N(A)} A A is a homogenous spatial Poisson process with intensity λ > 0 if: For each A A, N(A) Poisson(λ A ). For every finite collection A 1, A 2,..., A n of disjoint subsets of S, are independent. N(A 1 ), N(A 2 ),..., N(A n ) (N( ) = 0) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
8 Alternatively, a spatial Poisson process satisfies the following axioms: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
9 Alternatively, a spatial Poisson process satisfies the following axioms: i. If A 1, A 2,..., A n are disjoint regions in S, then N(A 1 ), N(A 2 ),..., N(A n ) are independent random variables and N(A 1 A 2 A n ) = N(A 1 ) + N(A 2 ) + + N(A n ) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
10 Alternatively, a spatial Poisson process satisfies the following axioms: i. If A 1, A 2,..., A n are disjoint regions in S, then N(A 1 ), N(A 2 ),..., N(A n ) are independent random variables and N(A 1 A 2 A n ) = N(A 1 ) + N(A 2 ) + + N(A n ) ii. The probability distribution of N(A) depends on the set A only through its size A. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
11 iii. There exists a λ such that P(N(A) 1) = λ A + o( A ) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
12 iii. There exists a λ such that P(N(A) 1) = λ A + o( A ) iv. There is probability zero of points overlapping: P(N(A) 2) = o( A ) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
13 If these axioms are satisfied, we have: for k = 0, 1, 2,... P(N(A) = k) = e λ A (λ A ) k k! Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
14 Consider a subset A of S: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
15 Consider a subset A of S: There are 3 points in A... how are they distributed within A? Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
16 Consider a subset A of S: There are 3 points in A... how are they distributed within A? Expect a uniform distribution... Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
17 In fact, for any B A, we have P(N(B) = 1 N(A) = 1) = B A Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
18 In fact, for any B A, we have P(N(B) = 1 N(A) = 1) = B A Proof: P(N(B) = 1 N(A) = 1) = P(N(B) = 1, N(A) = 1) P(N(A) = 1) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
19 In fact, for any B A, we have P(N(B) = 1 N(A) = 1) = B A Proof: P(N(B) = 1 N(A) = 1) = P(N(B) = 1, N(A) = 1) P(N(A) = 1) = P(N(B)=1,N(A B )=0) P(N(A)=1) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
20 In fact, for any B A, we have P(N(B) = 1 N(A) = 1) = B A Proof: P(N(B) = 1 N(A) = 1) = P(N(B) = 1, N(A) = 1) P(N(A) = 1) = P(N(B)=1,N(A B )=0) P(N(A)=1) = λ B e λ B e λ A B λ A e λ A Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
21 In fact, for any B A, we have P(N(B) = 1 N(A) = 1) = B A Proof: P(N(B) = 1 N(A) = 1) = P(N(B) = 1, N(A) = 1) P(N(A) = 1) = P(N(B)=1,N(A B )=0) P(N(A)=1) = λ B e λ B e λ A B λ A e λ A = B A Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
22 Simulating a spatial Poisson pattern over a rectangular region S = [a, b] [c, d]: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
23 Simulating a spatial Poisson pattern over a rectangular region S = [a, b] [c, d]: simulate a Poisson number of points Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
24 Simulating a spatial Poisson pattern over a rectangular region S = [a, b] [c, d]: simulate a Poisson number of points scatter that number of points uniformly over S Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
25 Simulating a spatial Poisson pattern over a rectangular region S = [a, b] [c, d]: simulate a Poisson number of points scatter that number of points uniformly over S ie: 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)) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
26 Generalization of the uniform result: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
27 Generalization of the uniform result: For any B A, we have N(B) N(A) = n bin(n, B / A ) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
28 More Generalization: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
29 More Generalization: For disjoint subsets A 1, A 2,..., A m A, P(N(A 1 ) = n 1, N(A 2 ) = n 2,..., N(A m ) = n m N(A) = n) = n! n 1!n 2! n m! for n 1 + n n m = n. ( ) A1 n1 A ( ) A2 n2 A ( ) Am nm A Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
30 Consider a two-dimensional spatial Poisson process of particles in the plane with intensity parameter λ. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
31 Consider a two-dimensional spatial Poisson process of particles in the plane with intensity parameter λ. Let s determine the (random) distance D between a particle and it s nearest neighbor. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
32 Consider a two-dimensional spatial Poisson process of particles in the plane with intensity parameter λ. Let s determine the (random) distance D between a particle and it s nearest neighbor. For x > 0, F D (x) = P(D x) = 1 P(D > x) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
33 Consider a two-dimensional spatial Poisson process of particles in the plane with intensity parameter λ. Let s determine the (random) distance D between a particle and it s nearest neighbor. For x > 0, F D (x) = P(D x) = 1 P(D > x) = 1 P( no other particles in disk with area πx 2 centered at the particle ) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
34 Consider a two-dimensional spatial Poisson process of particles in the plane with intensity parameter λ. Let s determine the (random) distance D between a particle and it s nearest neighbor. For x > 0, F D (x) = P(D x) = 1 P(D > x) = 1 P( no other particles in disk with area πx 2 centered at the particle ) = 1 e λπx2 Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
35 So, f D (x) = d dx F D(x) = 2λπxe λπx2 for x > 0. (Weibull distribution!) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
36 So, f D (x) = d dx F D(x) = 2λπxe λπx2 for x > 0. (Weibull distribution!) Similarly, in 3-D: F D (x) = 1 e λ 4π 3 x3 f D (x) = d dx F D(x) = 4πλx 2 e λ 4π 3 x3 (Weibull distribution!) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
37 Example: Spatial Patterns in Statistical Ecology Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
38 Example: Spatial Patterns in Statistical Ecology Consider a wide expanse of open ground of a uniform character. (example: muddy bed of a recently drained lake) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
39 Example: Spatial Patterns in Statistical Ecology Consider a wide expanse of open ground of a uniform character. (example: 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. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
40 Example: Spatial Patterns in Statistical Ecology Consider a wide expanse of open ground of a uniform character. (example: 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 Poisson approximation to the binomial distribution which will hold if there are many seeds with an extremely small chance of falling into the quadrat. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
41 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 spatial Poisson process with intensity pλ. (So, the surviving seedlings continue to be distributed at random.) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
42 Simulation Problem: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
43 Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ 1 and λ 2. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
44 Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ 1 and λ 2. Type 1 and Type 2 seeds will germinate with probabilities p 1 and p 2, respectively. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
45 Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ 1 and λ 2. 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 K geom(p). (P(K = 0) = p) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
46 Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ 1 and λ 2. 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 K geom(p). (P(K = 0) = p) Suppose that time is discretized as follows: Time 0: seeds are dispersed Time 1: seeds germinate Time 2: offshoot plants produced Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
47 Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ 1 and λ 2. 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 K geom(p). (P(K = 0) = p) Suppose that time is discretized as follows: Time 0: seeds are dispersed Time 1: seeds germinate Time 2: offshoot plants produced Suppose that the one-acre field is evenly divided into quadrats. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
48 Simulation Problem: Assume that the number of offshoot plants that fall into a quadrat different from their parent plans is negligible. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
49 Simulation Problem: Assume that the number of offshoot plants that fall into a quadrat different from their parent plans is negligible. A particular insect population can only be supported if at least 75% of the quadrats contain at least 35 plants. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
50 Simulation Problem: Assume that the number of offshoot plants that fall into a quadrat different from their parent plans 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 λ 1 and λ 2 that will give the insect population a 95% chance of surviving. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
51 Simulation Problem: Assume that the number of offshoot plants that fall into a quadrat different from their parent plans 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 λ 1 and λ 2 that will give the insect population a 95% chance of surviving. Use the hugely simplifying assumption that there is no real time component of this process. (In particular, assume that offshoot plants do not have further offshoots.) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
52 Tips on Simulating This: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
53 Tips on Simulating This: Keep in mind that we don t really have to keep track of where individual plants are, only the number in each quadrat. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
54 Tips on Simulating This: Keep in mind that we don t really have to keep track of where individual plants are, only the number in each quadrat. Note that we don t have to consider germination of the plants as a second step after the arrival of the seeds instead onsider a thinned spatial Poisson number of plants of type i with intensity p i λ i. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
55 Tips on Simulating This: Keep in mind that we don t really have to keep track of where individual plants are, only the number in each quadrat. Note that we don t have to consider germination of the plants as a second step after the arrival of the seeds instead onsider a thinned spatial Poisson number of plants of type i with intensity p i λ i. Rather than drawing uniformly distributed locations for the seeds, we can simulate the number for each quadrat separately (and ignore locations) using the fact that each quadrat contains a Poisson(p i λ i /100) number of germinating seeds. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
56 How to deal with offshoot plants... Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
57 How to deal with offshoot plants... It would be nice if we could further modify the Poisson number of seeds for Type 1 plants. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
58 How to deal with offshoot plants... It would be nice if we could further modify the Poisson number of seeds for Type 1 plants. We can t. :( Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
59 How to deal with offshoot plants... It would be nice if we could further modify the Poisson number of seeds for Type 1 plants. We can t. :( We can, however, 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. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
60 Specifics of my simulation: Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
61 Specifics of my simulation: created two arrays A 1 and A 2 Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
62 Specifics of my simulation: created two arrays A 1 and A 2 filled the arrays by drawing 100 values from the Poisson(p 1 λ 1 /100) distribution and 100 values from the Poisson(p 2 λ 2 /100) distribution Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
63 Specifics of my simulation: created two arrays A 1 and A 2 filled the arrays by drawing 100 values from the Poisson(p 1 λ 1 /100) distribution and 100 values from the Poisson(p 2 λ 2 /100) distribution went through the first array and replaced A 1 [i, j] with A 1 [i, j] + K i,j where the K i,j are drawn independently from the negative binomial distribution with r = A 1 [i, j] and p = 0.9 Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
64 Specifics of my simulation: created two arrays A 1 and A 2 filled the arrays by drawing 100 values from the Poisson(p 1 λ 1 /100) distribution and 100 values from the Poisson(p 2 λ 2 /100) distribution went through the first array and replaced A 1 [i, j] with A 1 [i, j] + K i,j where the K i,j are drawn independently from the negative binomial distribution with r = A 1 [i, j] and p = 0.9 let A = A 1 + A 2 and determined the proportion of entries in A with 35 or more plants Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
65 Specifics of my simulation: Repeat... Want to see a proportion of 0.75 or greater in approximately 95% of simulations Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
66 Specifics of my simulation: Repeat... Want to see a proportion of 0.75 or greater in approximately 95% of simulations Explore by changing the intensities (the λ s) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
67 Specifics of my simulation: Repeat... Want to see a proportion of 0.75 or greater in approximately 95% of simulations Explore by changing the intensities (the λ s) Depending on the efficiency of your simulation, this could be time consuming, so you might think about choosing λ s in the right ball park. Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
68 Specifics of my simulation: Repeat... Want to see a proportion of 0.75 or greater in approximately 95% of simulations Explore by changing the intensities (the λ s) Depending on the efficiency of your simulation, this could be time consuming, so you might think about choosing λ s in the right ball park. Ignoring offshoot plants, we know to expect in each quadrat. p 1 λ 1 /100 + p 2 λ 2 /100 Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
69 I started with λ 1 = λ 2 = (Then p 1 λ 1 /100 + p 2 λ 2 /100 = 37.5.) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
70 I started with λ 1 = λ 2 = (Then p 1 λ 1 /100 + p 2 λ 2 /100 = 37.5.) Results for some simulations: Sim Prop. of Quadrats Containing Support? at Least 35 Plants No Yes Yes Yes Yes Yes No Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
71 I started with λ 1 = λ 2 = (Then p 1 λ 1 /100 + p 2 λ 2 /100 = 37.5.) Results for some simulations: Sim Prop. of Quadrats Containing Support? at Least 35 Plants No Yes Yes Yes Yes Yes No (Continuing on for 10, 000 simuations, I found that the popoulation could be supported using these λ s roughly 87% of the time.) Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, / 24
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