Poisson Flashcards
(26 cards)
What is a spatial point process?
A spatial point process X is a random subset of R^d which is locally finite.
What is ρ?
It is the intensity function for the Poisson point process. It is defined from a subset of R^d into [0, ∞).
What is μ?
It is the intensity measure for the Poisson point process. It is defined as an integral over B of ρ(u)du, where B is any subset of S, which is a subset of R^d.
We require that μ(B) < ∞ for B ∈ B_0. Furthermore N(B) ∼ po(μ(B)).
Binomial point process
Let S ⊆ Rd . Suppose f is a probability density function (pdf) on S.
X is a binomial point process on S with n points if X = {X_1, . . . , X_n} where X_1, . . . , X_n are i.i.d. points with X_i ∼ f.
It is called ”binomial” because N(B) is binomially distributed b(n, p) with p = P(X_1 ∈ B) = ∫_B f (x) dx.
What is the Poisson point process?
When conditioned on the Poisson distributed amount of points, it is a binomial point process with pdf f (u) = ρ(u)/μ(B) for u ∈ B.
What is the unit rate Poisson point process?
Po(S, 1), i.e. ρ=1.
What are void probabilities and what are they used for?
The void probability v (B) = P(N(B) = 0) is the probability that there are no points in B ∈ B_0.
The distribution of X is uniquely determined by the void
probabilities.
What are the void probabilities for the Poisson?
The void probabilities for the Poisson process with intensity
measure μ are
v(B) = exp(−μ(B)).
What is a homogeneous Poisson point process?
ρ is constant
What is a inhomogeneous Poisson point process?
ρ is not constant
What is a isotropic point process?
X on R^d is isotropic if its distribution is invariant under rotations:
X ∼ RX = {Ru | u ∈ X}
for all rotations R around the origin.
What is a stationary point process?
X on R^d is stationary if its distribution is invariant under translations:
X ∼ X + s = {u + s | u ∈ X}, ∀s ∈ R^d .
What is the Poisson expansion?
Let X be a Poisson process, A ∈ Nlf , and B ∈ B with μ(B) < ∞.
P(X_B ∈ A) = e^{−μ(B)} ∑ _{n=0}^∞ 1/n! ∫ _B … ∫ _ B 1{{x_1, . . . , x _n} ∈ A} ∏ _{i=1}^n ρ(x _i)dx_1 … dx _n.
What is the standard proof?
The standard proof shows a statement for all non-negative measurable h in three steps:
1. True for 1{· ∈ A} for all measurable A.
2. If it is true for non-negative measurable functions f and g , then it is true for af + bg for a, b ≥ 0.
3. If it is true for an increasing sequence of measurable functions
0 ≤ f1 ≤ f2 ≤ · · · , then it is true for f = lim_{n→∞} f_n.
Note: Typically, only 1. needs to be verified. 2. usually follows from linearity, 3. follows from monotone convergence.
What is the unique property of the Poisson point process?
Independent scattering.
Suppose X is a Poisson process on S and B1, B2, . . . are disjoint subsets of S. Then X_B1 , X_B2 , . . . are independent.
What is N_lf?
The collection of locally finite subsets of R^d.
What are the properties of the Poisson point process?
The Poisson process po(S, ρ) exists.
The po(S, ρ)-distribution is completely determined by N(B).
The restriction of a Poisson process is again a Poisson process.
The union of independent Poisson processes in countably many disjoint domains is again a Poisson process.
Unique: Independent scattering, if B1, B2, . . . are disjoint subsets of S, then X_B1 , X_B2 , . . . are independent Poisson processes.
What can you do to a Poisson process so it is still a Poisson process?
- Independent thinning
- Superposition
- Displacements (move points with a probability)
What is the Slivnyak-Mecke theorem?
Suppose X ∼ Poisson(S, ρ). Let h : S × N_{lf} → [0, ∞) be
measurable. Then,
E ∑_{u∈X} h(u, X \ {u}) = ∫_S E[ h(u, X)ρ(u) du].
What is the extended Slivnyak-Mecke theorem?
Suppose X ∼ Poisson(S, ρ). Let h : S^k × N_{lf} → [0, ∞) be
measurable.E∑_{u_1,...,u_k ∈X}^≠ h(u _1, . . . , u _k , X \ {u _1, . . . , u _k }) = ∫_{S^k} E h(u _1, . . . , u _k, X) ∏_{i=1}^k ρ(u _i) du _1 ... du _k .
What is a superposition?
A disjoint union U_{i=1}^∞ X_i of point processes X_1, X_2, . . .
What is independent thinning?
Let X be a point process on S and p : S → [0, 1]. Given a realization of X, we remove points independently of each other, where u ∈ X is kept with probability p(u).
What are marked Poisson point processes?
Poisson point process but with more information, like the diameter of a tree.
Marked point process with finite mark space,
e.g. M = {1, . . . , k}. If the points are times of earthquakes, the
marks could be both position and magnitude. A marked point processes on S may be viewed as a point process on S × M.
What is absolutely continuous?
Let μ_1, μ_2 be measures on the same space. Then μ_2 is absolutely continuous with respect to μ_1 (written μ_2 ≪ μ_1) if μ_1(F ) = 0 implies μ_2(F ) = 0 for all measurable F .
