Spatial Poisson Processes

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Spatial Poisson Processes
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The Spatial Poisson Process
Consider a spatial configuration of points in the
plane
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Notation

• Let S be a subset of R2. (R, R2, R3,)

• Let A be the family of subsets of S.

• Let N(A) points in the set A.

(Assume S is normalized to have volume 1.)
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Then is a homogeneous Poisson
point process with intensity if

• For every finite collection A1, A2, , An of
  disjoint subsets of S, N(A1), N(A2), , N(A3) are
  independent.

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Alternatively, a spatial Poisson process
satisfies the following axioms

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

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• There exists a such that

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If these axioms are satisfied, we have
for k0,1,2,
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Consider a subset A of S
There are 3 points in A how are they distributed
in A?
A
Expect a uniform distribution
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In fact, for any , we have
Proof
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So, we know that, for k0,1,,n
ie N(B)N(A)n bin(n,B/A)
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Generalization
For a partition A1, A2, , Am of A
(Multinomial distribution)
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Simulating a spatial Poisson pattern with
intensity over a rectangular region
Sa,bxc,d.

• 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)U1a),(d-c)U2c)
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Consider a two-dimensional Poisson process of
particles in the plane with intensity parameter
.
Lets determine the (random) distance D between a
particle and its nearest neighbor.
For xgt0,
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So,
for xgt0.
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Example Spatial Patterns in Statistical Ecology
Consider a wide expanse of open ground of a
uniform character (such as the 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
binomial approximation to the Poisson
distribution which will hold if there are many
seeds with an extremely small chance of falling
into the quadrat.
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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?
So, the surviving seeds continue to be
distributed at random.
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Simulation Problem

• 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
  Kgeom(p). (P(K0)p)

• Suppose that the one-acre field is evenly
  divided into 10x10 quadrats.

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• 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.

• 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)

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Tips on simulating this

• Keep in mind that we dont really have to keep
  track of where the individual plants are, only
  the number in each quadrat.

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How to deal with offshoot plants

• It would be nice if we could further modify the
  Poisson number of seeds for Type 1.

• We can, at least, 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.