Stochastic Simulation APPM 7400

Lesson 11: Spatial Poisson Processes

October 3, 2018

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 1 / 24 The Spatial Poisson Process

Consider a spatial configuration of points in the plane:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 2 / 24 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.

The Spatial Poisson Process

Notation:

2 k Let S be a subset of R .(R ) (Assume S is normalized to have volume1.)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 3 / 24 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.

The Spatial Poisson Process

Notation:

2 k Let S be a subset of R .(R ) (Assume S is normalized to have volume1.)

Let A be the family of all subsets of S.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 3 / 24 Let N(A) be the number of points in the set A.

The Spatial Poisson Process

Notation:

2 k Let S be a subset of R .(R ) (Assume S is normalized to have volume1.)

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, 2018 3 / 24 The Spatial Poisson Process

Notation:

2 k Let S be a subset of R .(R ) (Assume S is normalized to have volume1.)

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, 2018 3 / 24 The Spatial Poisson Process

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 A1, A2,..., An of disjoint subsets of S,

N(A1), N(A2),..., N(An)

are independent.

(N(∅) = 0)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 4 / 24 i. If A1, A2,..., An are disjoint regions in S, then

N(A1), N(A2),..., N(An)

are independent random variables and

N(A1 ∪ A2 ∪ · · · ∪ An) = N(A1) + N(A2) + ··· + N(An)

ii. The probability distribution of N(A) depends on the set A only through its size |A|.

The Spatial Poisson Process

Alternatively, a spatial Poisson process satisfies the following axioms:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 5 / 24 ii. The probability distribution of N(A) depends on the set A only through its size |A|.

The Spatial Poisson Process

Alternatively, a spatial Poisson process satisfies the following axioms:

i. If A1, A2,..., An are disjoint regions in S, then

N(A1), N(A2),..., N(An)

are independent random variables and

N(A1 ∪ A2 ∪ · · · ∪ An) = N(A1) + N(A2) + ··· + N(An)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 5 / 24 The Spatial Poisson Process

Alternatively, a spatial Poisson process satisfies the following axioms:

i. If A1, A2,..., An are disjoint regions in S, then

N(A1), N(A2),..., N(An)

are independent random variables and

N(A1 ∪ A2 ∪ · · · ∪ An) = N(A1) + N(A2) + ··· + N(An)

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, 2018 5 / 24 iv. There is probability zero of points overlapping:

P(N(A) ≥ 2) = o(|A|)

The Spatial Poisson Process

iii. There exists a λ such that

P(N(A) ≥ 1) = λ|A| + o(|A|)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 6 / 24 The Spatial Poisson Process

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, 2018 6 / 24 The Spatial Poisson Process

If these axioms are satisfied, we have:

e−λ|A|(λ|A|)k P(N(A) = k) = k! for k = 0, 1, 2,...

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 7 / 24 There are 3 points in A... how are they distributed within A? Expect a uniform distribution...

The Spatial Poisson Process

Consider a subset A of S:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 8 / 24 Expect a uniform distribution...

The Spatial Poisson Process

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, 2018 8 / 24 The Spatial Poisson Process

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, 2018 8 / 24 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∩B0)=0) = P(N(A)=1)

−λ|B| −λ|A∩B0| = λ|B|e ·e = |B| λ|A|eλ|A| |A|

The Spatial Poisson Process

In fact, for any B ⊆ A, we have

|B| P(N(B) = 1|N(A) = 1) = |A|

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24 P(N(B)=1,N(A∩B0)=0) = P(N(A)=1)

−λ|B| −λ|A∩B0| = λ|B|e ·e = |B| λ|A|eλ|A| |A|

The Spatial Poisson Process

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)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24 −λ|B| −λ|A∩B0| = λ|B|e ·e = |B| λ|A|eλ|A| |A|

The Spatial Poisson Process

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∩B0)=0) = P(N(A)=1)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24 |B| = |A|

The Spatial Poisson Process

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∩B0)=0) = P(N(A)=1)

−λ|B| −λ|A∩B0| = λ|B|e ·e λ|A|eλ|A|

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24 The Spatial Poisson Process

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∩B0)=0) = P(N(A)=1)

−λ|B| −λ|A∩B0| = λ|B|e ·e = |B| λ|A|eλ|A| |A|

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24 simulate a Poisson number of points

scatter that number of points uniformly over S

ie: For each point, draw U1, U2 indep. unif (0, 1)’s and place it at

((b − a)U1 + a, ((d − c)U2 + c))

The Spatial Poisson Process

Simulating a spatial Poisson pattern over a rectangular region S = [a, b] × [c, d]:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 10 / 24 scatter that number of points uniformly over S

ie: For each point, draw U1, U2 indep. unif (0, 1)’s and place it at

((b − a)U1 + a, ((d − c)U2 + c))

The Spatial Poisson Process

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, 2018 10 / 24 ie: For each point, draw U1, U2 indep. unif (0, 1)’s and place it at

((b − a)U1 + a, ((d − c)U2 + c))

The Spatial Poisson Process

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, 2018 10 / 24 The Spatial Poisson Process

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 U1, U2 indep. unif (0, 1)’s and place it at

((b − a)U1 + a, ((d − c)U2 + c))

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 10 / 24 For any B ⊆ A, we have

N(B)|N(A) = n ∼ bin(n, |B|/|A|)

The Spatial Poisson Process

Generalization of the uniform result:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 11 / 24 The Spatial Poisson Process

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, 2018 11 / 24 For disjoint subsets A1, A2,..., Am ⊆ A,

P(N(A1) = n1, N(A2) = n2,..., N(Am) = nm|N(A) = n)

n! |A |n1 |A |n2 |A |nm = 1 · 2 ··· m n1!n2! ··· nm! |A| |A| |A|

for n1 + n2 + ··· + nm = n.

The Spatial Poisson Process

More Generalization:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 12 / 24 The Spatial Poisson Process

More Generalization:

For disjoint subsets A1, A2,..., Am ⊆ A,

P(N(A1) = n1, N(A2) = n2,..., N(Am) = nm|N(A) = n)

n! |A |n1 |A |n2 |A |nm = 1 · 2 ··· m n1!n2! ··· nm! |A| |A| |A|

for n1 + n2 + ··· + nm = n.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 12 / 24 Let’s determine the (random) distance D between a particle and it’s nearest neighbor.

For x > 0,

FD (x) = P(D ≤ x) = 1 − P(D > x)

= 1 − P( no other particles in disk with area πx2 centered at the particle )

= 1 − e−λπx2

The Spatial Poisson Process

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, 2018 13 / 24 For x > 0,

FD (x) = P(D ≤ x) = 1 − P(D > x)

= 1 − P( no other particles in disk with area πx2 centered at the particle )

= 1 − e−λπx2

The Spatial Poisson Process

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, 2018 13 / 24 = 1 − P( no other particles in disk with area πx2 centered at the particle )

= 1 − e−λπx2

The Spatial Poisson Process

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,

FD (x) = P(D ≤ x) = 1 − P(D > x)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 13 / 24 = 1 − e−λπx2

The Spatial Poisson Process

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,

FD (x) = P(D ≤ x) = 1 − P(D > x)

= 1 − P( no other particles in disk with area πx2 centered at the particle )

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 13 / 24 The Spatial Poisson Process

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,

FD (x) = P(D ≤ x) = 1 − P(D > x)

= 1 − P( no other particles in disk with area πx2 centered at the particle )

= 1 − e−λπx2

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 13 / 24 Similarly, in 3-D: −λ 4π x3 FD (x) = 1 − e 3

d 2 −λ 4π x3 f (x) = F (x) = 4πλx e 3 D dx D (Weibull distribution!)

The Spatial Poisson Process

So, d 2 f (x) = F (x) = 2λπxe−λπx D dx D for x > 0.(Weibull distribution!)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 14 / 24 The Spatial Poisson Process

So, d 2 f (x) = F (x) = 2λπxe−λπx D dx D for x > 0.(Weibull distribution!)

Similarly, in 3-D: −λ 4π x3 FD (x) = 1 − e 3

d 2 −λ 4π x3 f (x) = F (x) = 4πλx e 3 D dx D (Weibull distribution!)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 14 / 24 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.

The Spatial Poisson Process

Example: Spatial Patterns in Statistical Ecology

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 15 / 24 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.

The Spatial Poisson Process

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, 2018 15 / 24 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.

The Spatial Poisson Process

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, 2018 15 / 24 The Spatial Poisson Process

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, 2018 15 / 24 The Spatial Poisson Process

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, 2018 16 / 24 Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ1 and λ2.

Type1 and Type2 seeds will germinate with probabilities p1 and p2, respectively. Type1 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 10 × 10 quadrats.

The Spatial Poisson Process

Simulation Problem:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 17 / 24 Type1 and Type2 seeds will germinate with probabilities p1 and p2, respectively. Type1 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 10 × 10 quadrats.

The Spatial Poisson Process

Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ1 and λ2.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 17 / 24 Type1 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 10 × 10 quadrats.

The Spatial Poisson Process

Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ1 and λ2.

Type1 and Type2 seeds will germinate with probabilities p1 and p2, respectively.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 17 / 24 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 10 × 10 quadrats.

The Spatial Poisson Process

Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ1 and λ2.

Type1 and Type2 seeds will germinate with probabilities p1 and p2, respectively. Type1 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, 2018 17 / 24 Suppose that the one-acre field is evenly divided into 10 × 10 quadrats.

The Spatial Poisson Process

Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ1 and λ2.

Type1 and Type2 seeds will germinate with probabilities p1 and p2, respectively. Type1 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, 2018 17 / 24 The Spatial Poisson Process

Simulation Problem: Two types of seeds are randomly dispersed on a one-acre field according to two independent spatial Poisson processes with intensities λ1 and λ2.

Type1 and Type2 seeds will germinate with probabilities p1 and p2, respectively. Type1 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 10 × 10 quadrats.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 17 / 24 A particular insect population can only be supported if at least 75% of the quadrats contain at least 35 plants.

Using p = 0.9, p1 = 0.7, and p2 = 0.8, explore the values of λ1 and λ2 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.)

The Spatial Poisson Process

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, 2018 18 / 24 Using p = 0.9, p1 = 0.7, and p2 = 0.8, explore the values of λ1 and λ2 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.)

The Spatial Poisson Process

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, 2018 18 / 24 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.)

The Spatial Poisson Process

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, p1 = 0.7, and p2 = 0.8, explore the values of λ1 and λ2 that will give the insect population a 95% chance of surviving.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 18 / 24 The Spatial Poisson Process

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, p1 = 0.7, and p2 = 0.8, explore the values of λ1 and λ2 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, 2018 18 / 24 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 pi λi .

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(pi λi /100) number of germinating seeds.

The Spatial Poisson Process

Tips on Simulating This:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 19 / 24 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 pi λi .

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(pi λi /100) number of germinating seeds.

The Spatial Poisson Process

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, 2018 19 / 24 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(pi λi /100) number of germinating seeds.

The Spatial Poisson Process

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 pi λi .

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 19 / 24 The Spatial Poisson Process

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 pi λi .

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(pi λi /100) number of germinating seeds.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 19 / 24 It would be nice if we could further modify the Poisson number of seeds for Type1 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.

The Spatial Poisson Process

How to deal with offshoot plants...

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 20 / 24 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.

The Spatial Poisson Process

How to deal with offshoot plants...

It would be nice if we could further modify the Poisson number of seeds for Type1 plants.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 20 / 24 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.

The Spatial Poisson Process

How to deal with offshoot plants...

It would be nice if we could further modify the Poisson number of seeds for Type1 plants.

We can’t. :(

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 20 / 24 The Spatial Poisson Process

How to deal with offshoot plants...

It would be nice if we could further modify the Poisson number of seeds for Type1 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, 2018 20 / 24 created two 10 × 10 arrays A1 and A2

filled the arrays by drawing 100 values from the Poisson(p1λ1/100) distribution and 100 values from the Poisson(p2λ2/100) distribution

went through the first array and replaced A1[i, j] with A1[i, j] + Ki,j where the Ki,j are drawn independently from the negative binomial distribution with r = A1[i, j] and p = 0.9

let A = A1 + A2 and determined the proportion of entries in A with 35 or more plants

The Spatial Poisson Process

Specifics of my simulation:

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24 filled the arrays by drawing 100 values from the Poisson(p1λ1/100) distribution and 100 values from the Poisson(p2λ2/100) distribution

went through the first array and replaced A1[i, j] with A1[i, j] + Ki,j where the Ki,j are drawn independently from the negative binomial distribution with r = A1[i, j] and p = 0.9

let A = A1 + A2 and determined the proportion of entries in A with 35 or more plants

The Spatial Poisson Process

Specifics of my simulation:

created two 10 × 10 arrays A1 and A2

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24 went through the first array and replaced A1[i, j] with A1[i, j] + Ki,j where the Ki,j are drawn independently from the negative binomial distribution with r = A1[i, j] and p = 0.9

let A = A1 + A2 and determined the proportion of entries in A with 35 or more plants

The Spatial Poisson Process

Specifics of my simulation:

created two 10 × 10 arrays A1 and A2

filled the arrays by drawing 100 values from the Poisson(p1λ1/100) distribution and 100 values from the Poisson(p2λ2/100) distribution

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24 let A = A1 + A2 and determined the proportion of entries in A with 35 or more plants

The Spatial Poisson Process

Specifics of my simulation:

created two 10 × 10 arrays A1 and A2

filled the arrays by drawing 100 values from the Poisson(p1λ1/100) distribution and 100 values from the Poisson(p2λ2/100) distribution

went through the first array and replaced A1[i, j] with A1[i, j] + Ki,j where the Ki,j are drawn independently from the negative binomial distribution with r = A1[i, j] and p = 0.9

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24 The Spatial Poisson Process

Specifics of my simulation:

created two 10 × 10 arrays A1 and A2

filled the arrays by drawing 100 values from the Poisson(p1λ1/100) distribution and 100 values from the Poisson(p2λ2/100) distribution

went through the first array and replaced A1[i, j] with A1[i, j] + Ki,j where the Ki,j are drawn independently from the negative binomial distribution with r = A1[i, j] and p = 0.9

let A = A1 + A2 and determined the proportion of entries in A with 35 or more plants

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24 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

p1λ1/100 + p2λ2/100

in each quadrat.

The Spatial Poisson Process

Specifics of my simulation:

Repeat... Want to see a proportion of0 .75 or greater in approximately 95% of simulations

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 22 / 24 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

p1λ1/100 + p2λ2/100

in each quadrat.

The Spatial Poisson Process

Specifics of my simulation:

Repeat... Want to see a proportion of0 .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, 2018 22 / 24 Ignoring offshoot plants, we know to expect

p1λ1/100 + p2λ2/100

in each quadrat.

The Spatial Poisson Process

Specifics of my simulation:

Repeat... Want to see a proportion of0 .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, 2018 22 / 24 The Spatial Poisson Process

Specifics of my simulation:

Repeat... Want to see a proportion of0 .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

p1λ1/100 + p2λ2/100

in each quadrat.

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 22 / 24 Results for some simulations:

Sim Prop. of Quadrats Containing Support? at Least 35 Plants 1 0.72 No 2 0.75 Yes 3 0.82 Yes 4 0.83 Yes 5 0.81 Yes 6 0.79 Yes 7 0.71 No

(Continuing on for 10, 000 simuations, I found that the popoulation could be supported using these λ’s roughly 87% of the time.)

The Spatial Poisson Process

I started with λ1 = λ2 = 2500. (Then p1λ1/100 + p2λ2/100 = 37.5.)

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 23 / 24 (Continuing on for 10, 000 simuations, I found that the popoulation could be supported using these λ’s roughly 87% of the time.)

The Spatial Poisson Process

I started with λ1 = λ2 = 2500. (Then p1λ1/100 + p2λ2/100 = 37.5.) Results for some simulations:

Sim Prop. of Quadrats Containing Support? at Least 35 Plants 1 0.72 No 2 0.75 Yes 3 0.82 Yes 4 0.83 Yes 5 0.81 Yes 6 0.79 Yes 7 0.71 No

Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 23 / 24 The Spatial Poisson Process

I started with λ1 = λ2 = 2500. (Then p1λ1/100 + p2λ2/100 = 37.5.) Results for some simulations:

Sim Prop. of Quadrats Containing Support? at Least 35 Plants 1 0.72 No 2 0.75 Yes 3 0.82 Yes 4 0.83 Yes 5 0.81 Yes 6 0.79 Yes 7 0.71 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, 2018 23 / 24