appm 7400 lesson 11: spatial poisson processes...the spatial poisson process consider a spatial con...
TRANSCRIPT
Stochastic SimulationAPPM 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
The Spatial Poisson Process
Notation:
Let S be a subset of R2. (Rk)(Assume S is normalized to have volume 1.)
Let A be the family of all subsets of S .
For A ā 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
Notation:
Let S be a subset of R2. (Rk)(Assume S is normalized to have volume 1.)
Let A be the family of all subsets of S .
For A ā 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
Notation:
Let S be a subset of R2. (Rk)(Assume S is normalized to have volume 1.)
Let A be the family of all subsets of S .
For A ā 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
Notation:
Let S be a subset of R2. (Rk)(Assume S is normalized to have volume 1.)
Let A be the family of all subsets of S .
For A ā 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)}AāA is a homogenous spatial Poisson process with intensityĪ» > 0 if:
For each A ā A, N(A) ā¼ Poisson(Ī»|A|).
For every finite collection A1,A2, . . . ,An of disjoint subsets of S ,
N(A1),N(A2), . . . ,N(An)
are independent.
(N(ā ) = 0)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 4 / 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 āŖ A2 āŖ Ā· Ā· Ā· āŖ An) = N(A1) + N(A2) + Ā· Ā· Ā·+ N(An)
ii. The probability distribution of N(A) depends on the set A onlythrough its size |A|.
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 āŖ A2 āŖ Ā· Ā· Ā· āŖ An) = N(A1) + N(A2) + Ā· Ā· Ā·+ N(An)
ii. The probability distribution of N(A) depends on the set A onlythrough its size |A|.
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 āŖ A2 āŖ Ā· Ā· Ā· āŖ An) = N(A1) + N(A2) + Ā· Ā· Ā·+ N(An)
ii. The probability distribution of N(A) depends on the set A onlythrough its size |A|.
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 5 / 24
The Spatial Poisson Process
iii. There exists a Ī» such that
P(N(A) ā„ 1) = Ī»|A|+ o(|A|)
iv. There is probability zero of points overlapping:
P(N(A) ā„ 2) = o(|A|)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 6 / 24
The Spatial Poisson Process
iii. There exists a Ī» such that
P(N(A) ā„ 1) = Ī»|A|+ o(|A|)
iv. There is probability zero of points overlapping:
P(N(A) ā„ 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:
P(N(A) = k) =eāĪ»|A|(Ī»|A|)k
k!
for k = 0, 1, 2, . . .
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 7 / 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
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
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
The Spatial Poisson Process
In fact, for any B ā 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(Aā©Bā²)=0)P(N(A)=1)
= Ī»|B|eāĪ»|B|Ā·eāĪ»|Aā©Bā²|
Ī»|A|eĪ»|A| = |B||A|
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24
The Spatial Poisson Process
In fact, for any B ā 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(Aā©Bā²)=0)P(N(A)=1)
= Ī»|B|eāĪ»|B|Ā·eāĪ»|Aā©Bā²|
Ī»|A|eĪ»|A| = |B||A|
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24
The Spatial Poisson Process
In fact, for any B ā 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(Aā©Bā²)=0)P(N(A)=1)
= Ī»|B|eāĪ»|B|Ā·eāĪ»|Aā©Bā²|
Ī»|A|eĪ»|A| = |B||A|
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24
The Spatial Poisson Process
In fact, for any B ā 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(Aā©Bā²)=0)P(N(A)=1)
= Ī»|B|eāĪ»|B|Ā·eāĪ»|Aā©Bā²|
Ī»|A|eĪ»|A|
= |B||A|
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24
The Spatial Poisson Process
In fact, for any B ā 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(Aā©Bā²)=0)P(N(A)=1)
= Ī»|B|eāĪ»|B|Ā·eāĪ»|Aā©Bā²|
Ī»|A|eĪ»|A| = |B||A|
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 9 / 24
The Spatial Poisson Process
Simulating a spatial Poisson pattern over a rectangular regionS = [a, b]Ć [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)ās and place it at
((b ā a)U1 + a, ((d ā c)U2 + c))
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 10 / 24
The Spatial Poisson Process
Simulating a spatial Poisson pattern over a rectangular regionS = [a, b]Ć [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)ās and place it at
((b ā a)U1 + a, ((d ā c)U2 + c))
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 10 / 24
The Spatial Poisson Process
Simulating a spatial Poisson pattern over a rectangular regionS = [a, b]Ć [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)ās and place it at
((b ā a)U1 + a, ((d ā c)U2 + c))
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 10 / 24
The Spatial Poisson Process
Simulating a spatial Poisson pattern over a rectangular regionS = [a, b]Ć [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)ās and place it at
((b ā a)U1 + a, ((d ā c)U2 + c))
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 10 / 24
The Spatial Poisson Process
Generalization of the uniform result:
For any B ā A, we have
N(B)|N(A) = n ā¼ bin(n, |B|/|A|)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 11 / 24
The Spatial Poisson Process
Generalization of the uniform result:
For any B ā A, we have
N(B)|N(A) = n ā¼ bin(n, |B|/|A|)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 11 / 24
The Spatial Poisson Process
More Generalization:
For disjoint subsets A1,A2, . . . ,Am ā A,
P(N(A1) = n1,N(A2) = n2, . . . ,N(Am) = nm|N(A) = n)
=n!
n1!n2! Ā· Ā· Ā· nm!
(|A1||A|
)n1
Ā·(|A2||A|
)n2
Ā· Ā· Ā·(|Am||A|
)nm
for n1 + n2 + Ā· Ā· Ā·+ nm = n.
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 12 / 24
The Spatial Poisson Process
More Generalization:
For disjoint subsets A1,A2, . . . ,Am ā A,
P(N(A1) = n1,N(A2) = n2, . . . ,N(Am) = nm|N(A) = n)
=n!
n1!n2! Ā· Ā· Ā· nm!
(|A1||A|
)n1
Ā·(|A2||A|
)n2
Ā· Ā· Ā·(|Am||A|
)nm
for n1 + n2 + Ā· Ā· Ā·+ nm = n.
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 12 / 24
The Spatial Poisson Process
Consider a two-dimensional spatial Poisson process of particles in the planewith intensity parameter Ī».
Letās determine the (random) distance D between a particle and itāsnearest neighbor.
For x > 0,
FD(x) = P(D ā¤ x) = 1ā P(D > x)
= 1ā P( no other particles in disk with areaĻx2 centered at the particle )
= 1ā eāĪ»Ļx2
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 planewith intensity parameter Ī».
Letās determine the (random) distance D between a particle and itāsnearest neighbor.
For x > 0,
FD(x) = P(D ā¤ x) = 1ā P(D > x)
= 1ā P( no other particles in disk with areaĻx2 centered at the particle )
= 1ā eāĪ»Ļx2
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 planewith intensity parameter Ī».
Letās determine the (random) distance D between a particle and itāsnearest neighbor.
For x > 0,
FD(x) = P(D ā¤ x) = 1ā P(D > x)
= 1ā P( no other particles in disk with areaĻx2 centered at the particle )
= 1ā eāĪ»Ļx2
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 planewith intensity parameter Ī».
Letās determine the (random) distance D between a particle and itāsnearest neighbor.
For x > 0,
FD(x) = P(D ā¤ x) = 1ā P(D > x)
= 1ā P( no other particles in disk with areaĻx2 centered at the particle )
= 1ā eāĪ»Ļx2
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 planewith intensity parameter Ī».
Letās determine the (random) distance D between a particle and itāsnearest neighbor.
For x > 0,
FD(x) = P(D ā¤ x) = 1ā P(D > x)
= 1ā P( no other particles in disk with areaĻx2 centered at the particle )
= 1ā eāĪ»Ļx2
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 13 / 24
The Spatial Poisson Process
So,
fD(x) =d
dxFD(x) = 2Ī»ĻxeāĪ»Ļx
2
for x > 0. (Weibull distribution!)
Similarly, in 3-D:
FD(x) = 1ā eāĪ»4Ļ3x3
fD(x) =d
dxFD(x) = 4ĻĪ»x2eāĪ»
4Ļ3x3
(Weibull distribution!)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 14 / 24
The Spatial Poisson Process
So,
fD(x) =d
dxFD(x) = 2Ī»ĻxeāĪ»Ļx
2
for x > 0. (Weibull distribution!)
Similarly, in 3-D:
FD(x) = 1ā eāĪ»4Ļ3x3
fD(x) =d
dxFD(x) = 4ĻĪ»x2eāĪ»
4Ļ3x3
(Weibull distribution!)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 14 / 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 particularāquadratā on this surface is well modeled by a Poisson randomvariable.
The reason this tends to be true is due to the Poisson approximationto the binomial distribution which will hold if there are many seedswith 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
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 particularāquadratā on this surface is well modeled by a Poisson randomvariable.
The reason this tends to be true is due to the Poisson approximationto the binomial distribution which will hold if there are many seedswith 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
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 particularāquadratā on this surface is well modeled by a Poisson randomvariable.
The reason this tends to be true is due to the Poisson approximationto the binomial distribution which will hold if there are many seedswith 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
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 particularāquadratā on this surface is well modeled by a Poisson randomvariable.
The reason this tends to be true is due to the Poisson approximationto the binomial distribution which will hold if there are many seedswith 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 thatthey 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 pĪ».
(So, the surviving seedlings continue to be distributed āat randomā.)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 16 / 24
The Spatial Poisson Process
Simulation Problem:
Two types of seeds are randomly dispersed on a one-acre fieldaccording to two independent spatial Poisson processes withintensities Ī»1 and Ī»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 randomlyspaced around the plant where K ā¼ geom(p). (P(K = 0) = p)
Suppose that time is discretized as follows:
Time 0: seeds are dispersedTime 1: seeds germinateTime 2: offshoot plants produced
Suppose that the one-acre field is evenly divided into 10Ć 10quadrats.
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 fieldaccording to two independent spatial Poisson processes withintensities Ī»1 and Ī»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 randomlyspaced around the plant where K ā¼ geom(p). (P(K = 0) = p)
Suppose that time is discretized as follows:
Time 0: seeds are dispersedTime 1: seeds germinateTime 2: offshoot plants produced
Suppose that the one-acre field is evenly divided into 10Ć 10quadrats.
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 fieldaccording to two independent spatial Poisson processes withintensities Ī»1 and Ī»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 randomlyspaced around the plant where K ā¼ geom(p). (P(K = 0) = p)
Suppose that time is discretized as follows:
Time 0: seeds are dispersedTime 1: seeds germinateTime 2: offshoot plants produced
Suppose that the one-acre field is evenly divided into 10Ć 10quadrats.
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 fieldaccording to two independent spatial Poisson processes withintensities Ī»1 and Ī»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 randomlyspaced around the plant where K ā¼ geom(p). (P(K = 0) = p)
Suppose that time is discretized as follows:
Time 0: seeds are dispersedTime 1: seeds germinateTime 2: offshoot plants produced
Suppose that the one-acre field is evenly divided into 10Ć 10quadrats.
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 fieldaccording to two independent spatial Poisson processes withintensities Ī»1 and Ī»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 randomlyspaced around the plant where K ā¼ geom(p). (P(K = 0) = p)
Suppose that time is discretized as follows:
Time 0: seeds are dispersedTime 1: seeds germinateTime 2: offshoot plants produced
Suppose that the one-acre field is evenly divided into 10Ć 10quadrats.
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 fieldaccording to two independent spatial Poisson processes withintensities Ī»1 and Ī»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 randomlyspaced around the plant where K ā¼ geom(p). (P(K = 0) = p)
Suppose that time is discretized as follows:
Time 0: seeds are dispersedTime 1: seeds germinateTime 2: offshoot plants produced
Suppose that the one-acre field is evenly divided into 10Ć 10quadrats.
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 17 / 24
The Spatial Poisson Process
Simulation Problem:
Assume that the number of offshoot plants that fall into a quadratdifferent 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 Ī»1 andĪ»2 that will give the insect population a 95% chance of surviving.
Use the hugely simplifying assumption that there is no āreal timeācomponent of this process. (In particular, assume that offshoot plantsdo not have further offshoots.)
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 quadratdifferent 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 Ī»1 andĪ»2 that will give the insect population a 95% chance of surviving.
Use the hugely simplifying assumption that there is no āreal timeācomponent of this process. (In particular, assume that offshoot plantsdo not have further offshoots.)
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 quadratdifferent 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 Ī»1 andĪ»2 that will give the insect population a 95% chance of surviving.
Use the hugely simplifying assumption that there is no āreal timeācomponent of this process. (In particular, assume that offshoot plantsdo not have further offshoots.)
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 quadratdifferent 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 Ī»1 andĪ»2 that will give the insect population a 95% chance of surviving.
Use the hugely simplifying assumption that there is no āreal timeācomponent of this process. (In particular, assume that offshoot plantsdo not have further offshoots.)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 18 / 24
The Spatial Poisson Process
Tips on Simulating This:
Keep in mind that we donāt really have to keep track of whereindividual plants are, only the number in each quadrat.
Note that we donāt have to consider germination of the plants as asecond step after the arrival of the seedsā instead onsider a thinnedspatial Poisson number of plants of type i with intensity piĪ»i .
Rather than drawing uniformly distributed locations for the seeds, wecan simulate the number for each quadrat separately (and ignorelocations) using the fact that each quadrat contains aPoisson(piĪ»i/100) number of germinating seeds.
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 donāt really have to keep track of whereindividual plants are, only the number in each quadrat.
Note that we donāt have to consider germination of the plants as asecond step after the arrival of the seedsā instead onsider a thinnedspatial Poisson number of plants of type i with intensity piĪ»i .
Rather than drawing uniformly distributed locations for the seeds, wecan simulate the number for each quadrat separately (and ignorelocations) using the fact that each quadrat contains aPoisson(piĪ»i/100) number of germinating seeds.
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 donāt really have to keep track of whereindividual plants are, only the number in each quadrat.
Note that we donāt have to consider germination of the plants as asecond step after the arrival of the seedsā instead onsider a thinnedspatial Poisson number of plants of type i with intensity piĪ»i .
Rather than drawing uniformly distributed locations for the seeds, wecan simulate the number for each quadrat separately (and ignorelocations) using the fact that each quadrat contains aPoisson(piĪ»i/100) number of germinating seeds.
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 donāt really have to keep track of whereindividual plants are, only the number in each quadrat.
Note that we donāt have to consider germination of the plants as asecond step after the arrival of the seedsā instead onsider a thinnedspatial Poisson number of plants of type i with intensity piĪ»i .
Rather than drawing uniformly distributed locations for the seeds, wecan simulate the number for each quadrat separately (and ignorelocations) using the fact that each quadrat contains aPoisson(piĪ»i/100) number of germinating seeds.
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 19 / 24
The Spatial Poisson Process
How to deal with offshoot plants...
It would be nice if we could further modify the Poisson number ofseeds for Type 1 plants.
We canāt. :(
We can, however, simplify the generation of offshoot plants, dealingwith all plants in a particular quadrat together by adding a negativebinomial number of plants to each quadrat.
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 ofseeds for Type 1 plants.
We canāt. :(
We can, however, simplify the generation of offshoot plants, dealingwith all plants in a particular quadrat together by adding a negativebinomial number of plants to each quadrat.
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 ofseeds for Type 1 plants.
We canāt. :(
We can, however, simplify the generation of offshoot plants, dealingwith all plants in a particular quadrat together by adding a negativebinomial number of plants to each quadrat.
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 ofseeds for Type 1 plants.
We canāt. :(
We can, however, simplify the generation of offshoot plants, dealingwith all plants in a particular quadrat together by adding a negativebinomial number of plants to each quadrat.
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 20 / 24
The Spatial Poisson Process
Specifics of my simulation:
created two 10Ć 10 arrays A1 and A2
filled the arrays by drawing 100 values from the Poisson(p1Ī»1/100)distribution and 100 values from the Poisson(p2Ī»2/100) 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 binomialdistribution with r = A1[i , j ] and p = 0.9
let A = A1 + A2 and determined the proportion of entries in A with35 or more plants
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24
The Spatial Poisson Process
Specifics of my simulation:
created two 10Ć 10 arrays A1 and A2
filled the arrays by drawing 100 values from the Poisson(p1Ī»1/100)distribution and 100 values from the Poisson(p2Ī»2/100) 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 binomialdistribution with r = A1[i , j ] and p = 0.9
let A = A1 + A2 and determined the proportion of entries in A with35 or more plants
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24
The Spatial Poisson Process
Specifics of my simulation:
created two 10Ć 10 arrays A1 and A2
filled the arrays by drawing 100 values from the Poisson(p1Ī»1/100)distribution and 100 values from the Poisson(p2Ī»2/100) 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 binomialdistribution with r = A1[i , j ] and p = 0.9
let A = A1 + A2 and determined the proportion of entries in A with35 or more plants
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24
The Spatial Poisson Process
Specifics of my simulation:
created two 10Ć 10 arrays A1 and A2
filled the arrays by drawing 100 values from the Poisson(p1Ī»1/100)distribution and 100 values from the Poisson(p2Ī»2/100) 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 binomialdistribution with r = A1[i , j ] and p = 0.9
let A = A1 + A2 and determined the proportion of entries in A with35 or more plants
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24
The Spatial Poisson Process
Specifics of my simulation:
created two 10Ć 10 arrays A1 and A2
filled the arrays by drawing 100 values from the Poisson(p1Ī»1/100)distribution and 100 values from the Poisson(p2Ī»2/100) 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 binomialdistribution with r = A1[i , j ] and p = 0.9
let A = A1 + A2 and determined the proportion of entries in A with35 or more plants
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 21 / 24
The Spatial Poisson Process
Specifics of my simulation:
Repeat... Want to see a proportion of 0.75 or greater inapproximately 95% of simulations
Explore by changing the intensities (the Ī»ās)
Depending on the efficiency of your simulation, this could be timeconsuming, so you might think about choosing Ī»ās in the right ball park.
Ignoring offshoot plants, we know to expect
p1Ī»1/100 + p2Ī»2/100
in each quadrat.
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 of 0.75 or greater inapproximately 95% of simulations
Explore by changing the intensities (the Ī»ās)
Depending on the efficiency of your simulation, this could be timeconsuming, so you might think about choosing Ī»ās in the right ball park.
Ignoring offshoot plants, we know to expect
p1Ī»1/100 + p2Ī»2/100
in each quadrat.
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 of 0.75 or greater inapproximately 95% of simulations
Explore by changing the intensities (the Ī»ās)
Depending on the efficiency of your simulation, this could be timeconsuming, so you might think about choosing Ī»ās in the right ball park.
Ignoring offshoot plants, we know to expect
p1Ī»1/100 + p2Ī»2/100
in each quadrat.
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 of 0.75 or greater inapproximately 95% of simulations
Explore by changing the intensities (the Ī»ās)
Depending on the efficiency of your simulation, this could be timeconsuming, so you might think about choosing Ī»ās in the right ball park.
Ignoring offshoot plants, we know to expect
p1Ī»1/100 + p2Ī»2/100
in each quadrat.
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 22 / 24
The Spatial Poisson Process
I started with Ī»1 = Ī»2 = 2500. (Then p1Ī»1/100 + p2Ī»2/100 = 37.5.)
Results for some simulations:
Sim Prop. of Quadrats Containing Support?at Least 35 Plants
1 0.72 No2 0.75 Yes3 0.82 Yes4 0.83 Yes5 0.81 Yes6 0.79 Yes7 0.71 No
(Continuing on for 10, 000 simuations, I found that the popoulation couldbe supported using these Ī»ās roughly 87% of the time.)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 23 / 24
The Spatial Poisson Process
I started with Ī»1 = Ī»2 = 2500. (Then p1Ī»1/100 + p2Ī»2/100 = 37.5.)Results for some simulations:
Sim Prop. of Quadrats Containing Support?at Least 35 Plants
1 0.72 No2 0.75 Yes3 0.82 Yes4 0.83 Yes5 0.81 Yes6 0.79 Yes7 0.71 No
(Continuing on for 10, 000 simuations, I found that the popoulation couldbe supported using these Ī»ās roughly 87% of the time.)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 23 / 24
The Spatial Poisson Process
I started with Ī»1 = Ī»2 = 2500. (Then p1Ī»1/100 + p2Ī»2/100 = 37.5.)Results for some simulations:
Sim Prop. of Quadrats Containing Support?at Least 35 Plants
1 0.72 No2 0.75 Yes3 0.82 Yes4 0.83 Yes5 0.81 Yes6 0.79 Yes7 0.71 No
(Continuing on for 10, 000 simuations, I found that the popoulation couldbe supported using these Ī»ās roughly 87% of the time.)
Lesson 11: Spatial Poisson Processes Stochastic Simulation October 3, 2018 23 / 24