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,…)
Using p=0.9, p1=0.7, and p2=0.8, explore the values of 1 and 2 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)
for n1+n2+…+nm = n.
n1
n2
nm
(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
In fact, for any B A , we have
|B| P(N(B) 1| N(A) 1) |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 BC ) 1) P(N(A 1))
|B| e e -|A| |A |e
- |B|
- |A BC|
|B| |A |
So, we know that,பைடு நூலகம்for k=0,1,…,n:
n |B| P(N(B) k | N(A) n) k |A |
k
|B| 1 - |A |
Then {N(A)}A A is a homogeneous Poisson point process with intensity 0 if: For each A A , N(A) ~ Poisson( |A|) .
For every finite collection {A1, A2, …, An} of disjoint subsets of S, N(A1), N(A2), …, N(A3) are independent.
Let A be the family of subsets of S.
For A A , 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.)
e-|A|( |A |)k P(N(A) k) k!
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…
P(N(A) 1) |A| o(|A|)
iv. There is probability zero of points overlapping:
P(N(A) 1) lim 1 |A|0 P(N(A) 1)
If these axioms are satisfied, we have:
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( pii /100) germinating seeds.
Tips on simulating this:
Keep in mind that we don’t really have to keep track of where the 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 consider a thinned Poisson number of plants of Type i with rate pii .
p

)
So, the surviving seeds continue to be distributed “at random”.
Simulation Problem:
Two types of seeds are randomly dispersed on a one-acre field according to two independent Poisson processes with intensities 1 and 2 . Type 1 and type 2 seeds will germinate with probabilities p1 and p2, 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 the one-acre field is evenly divided into 10x10 quadrats.
- U e i i 1 N 1
(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)
N(A1 U A2 U … U An) = N(A1) + N(A2) + … + N(An)
ii. The probability distribution of N(A) depends on the set A only through it’s size |A|.
iii. There exists a 0 such that
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 p . (accept probability is
n -k
ie: N(B)|N(A)=n ~ bin(n,|B|/|A|)
Generalization:
For a partition A1, A2, …, Am of A:
P(N(A1 ) n1, N(A2 ) n2 , ... , N(Am ) nm | N(A) n)
|A 1 | |A 2 | |A m | n! n1! n2 !nm ! |A | |A | | A |
Alternatively, a spatial Poisson process satisfies the following axioms: i. If A1, A2, …, An are disjoint regions, then N(A1), N(A2), …, N(An) are independent rv’s and
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.
For x>0,
FD (x) P(D x) 1 - P(D x)
1 - P(no other particles in disk centered 2 at the particle with area x )
1- e