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P(N(kA )e)-|A |(|A|)k
k!
for k=0,1,2,…
7
Consider a subset A of S: There are 3 points in A… how are they distributed in A?
A
Expect a uniform distribution…
8
In fact, for any B, Awe have P(N(1B | N) (A 1))|B| |A|
Proof:
P(N 1 |N (B (1 A ))P ) (N 1N (,B (1 A ))) P( N 1()A )
P(N(1 BN ,) (A BC)1) P(N(1A))
|B| e-|B| e-|ABC| | A| e-|A|
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|.
5
iii. There exists a s0uch that
4ቤተ መጻሕፍቲ ባይዱ
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
14
Example: Spatial Patterns in Statistical Ecology
|B | |A|
9
So, we know that, for k=0,1,…,n:
P(N k|(N B( )n A ) )n k ||A B || k 1 -||A B || n -k
ie: N(B)|N(A)=n ~ bin(n,|B|/|A|)
10
Generalization: For a partition A1, A2, …, Am of A: P1 ) ( n 1 , N N 2 ) n ( ( 2 , . A A , N .. m ) n ( m |A N n (A )
over a rectangular region S=[a,b]x[c,d].
simulate a Poisson( )number of points
(perhaps by finding the smallest number N such that)
N1
Ui e-
i1
scatter that number of points uniformly over S
Spatial Poisson Processes
1
The Spatial Poisson Process
Consider a spatial configuration of points in the plane:
Notation:
Let S be a subset of R2. (R, R2, R3,…) Let A be the family of subsets of S.
atthpeartw ica iltre h e x2)a
1-e-x2
13
So, for x>0.
fD (x)d dF x D (x)2x e-x2
In 3-D we could show that:
-4x3
FD(x) 1-e 3
fD (x)d dF x D (x4 ) x2e -4 3 x3
Let’s determine the (random) distance D between a particle and its nearest neighbor.
For x>0,
FD(x)P(D x)1-P(Dx)
1 -P(o nto h pe arritn id cilc s ee ksnter
(for each point, draw U1, U2, indep unif(0,1)’s and place it at ((b-a)U1+a),(d-c)U2+c)
12
Consider a two-dimensional Poisson process of
particles in the plane with intensity parameter .
For AAl,et |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.)
3
Then {N(A)A }isAa homogeneous Poisson point
process with intensity i0f:
For each AA, N(A ~P ) oiss|oA|n).(
For every finite collection {A1, A2, …, An} of disjoint subsets of S, N(A1), N(A2), …, N(A3) are independent.
n 1 !n 2 n ! n !m ! ||A A 1 || n 1 ||A A 2 || n 2 ||A A m || n m
for n1+n2+…+nm = n.
(Multinomial distribution)
11
Simulating a spatial Poisson pattern with intensity
P( N 1 ( )|A A | o )A (|) |
iv. There is probability zero of points overlapping:
limP(N(A 1) )1 |A|0 P(N(A 1))
6
If these axioms are satisfied, we have:
k!
for k=0,1,2,…
7
Consider a subset A of S: There are 3 points in A… how are they distributed in A?
A
Expect a uniform distribution…
8
In fact, for any B, Awe have P(N(1B | N) (A 1))|B| |A|
Proof:
P(N 1 |N (B (1 A ))P ) (N 1N (,B (1 A ))) P( N 1()A )
P(N(1 BN ,) (A BC)1) P(N(1A))
|B| e-|B| e-|ABC| | A| e-|A|
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|.
5
iii. There exists a s0uch that
4ቤተ መጻሕፍቲ ባይዱ
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
14
Example: Spatial Patterns in Statistical Ecology
|B | |A|
9
So, we know that, for k=0,1,…,n:
P(N k|(N B( )n A ) )n k ||A B || k 1 -||A B || n -k
ie: N(B)|N(A)=n ~ bin(n,|B|/|A|)
10
Generalization: For a partition A1, A2, …, Am of A: P1 ) ( n 1 , N N 2 ) n ( ( 2 , . A A , N .. m ) n ( m |A N n (A )
over a rectangular region S=[a,b]x[c,d].
simulate a Poisson( )number of points
(perhaps by finding the smallest number N such that)
N1
Ui e-
i1
scatter that number of points uniformly over S
Spatial Poisson Processes
1
The Spatial Poisson Process
Consider a spatial configuration of points in the plane:
Notation:
Let S be a subset of R2. (R, R2, R3,…) Let A be the family of subsets of S.
atthpeartw ica iltre h e x2)a
1-e-x2
13
So, for x>0.
fD (x)d dF x D (x)2x e-x2
In 3-D we could show that:
-4x3
FD(x) 1-e 3
fD (x)d dF x D (x4 ) x2e -4 3 x3
Let’s determine the (random) distance D between a particle and its nearest neighbor.
For x>0,
FD(x)P(D x)1-P(Dx)
1 -P(o nto h pe arritn id cilc s ee ksnter
(for each point, draw U1, U2, indep unif(0,1)’s and place it at ((b-a)U1+a),(d-c)U2+c)
12
Consider a two-dimensional Poisson process of
particles in the plane with intensity parameter .
For AAl,et |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.)
3
Then {N(A)A }isAa homogeneous Poisson point
process with intensity i0f:
For each AA, N(A ~P ) oiss|oA|n).(
For every finite collection {A1, A2, …, An} of disjoint subsets of S, N(A1), N(A2), …, N(A3) are independent.
n 1 !n 2 n ! n !m ! ||A A 1 || n 1 ||A A 2 || n 2 ||A A m || n m
for n1+n2+…+nm = n.
(Multinomial distribution)
11
Simulating a spatial Poisson pattern with intensity
P( N 1 ( )|A A | o )A (|) |
iv. There is probability zero of points overlapping:
limP(N(A 1) )1 |A|0 P(N(A 1))
6
If these axioms are satisfied, we have: