Cox processes Flashcards
(31 cards)
Describe the motivation for introducing Cox processes
Suppose we want to model a point pattern of trees by an inhomogeneous Poisson process. The intensity will depend on soil quality. But what if we don’t know the soil type? Or if there are other covariates that we are unaware of?
Solution: Consider a Poisson process with a random intensity, i.e. Cox process.
What is a random field?
A random field is a random function Z : S → R such that each
Z(u), u ∈ S, is a random variable.
Define a Cox point process
Let Z be a non-negative measurable random field which is locally integrable with probability 1. A Cox process with random intensity function Z is a point process X such that given Z = z, X is a Poisson process with intensity z.
Define the intensity, product density and the pair correlation function for a Cox process
ρ(u) = E Z(u),
ρ^(2)(u, v ) = E [Z(u) Z(v)],
g (u, v ) = E [Z(u) Z(v)] / E Z(u) E Z(v).
What is Cox processes typically used for?
Modelling of clustering
What functional summary statistics can be performed with Cox processes?
Often for specific Cox process models, g, the pair correlation, is expressible on closed form. Hence K and L can be calculated (at least numerically).
In general, F , G , J are hard to obtain.
How can a Cox process be simulated?
Two-step procedure:
Simulate a realization z of Z .
Given the result Z = z, simulate X as an inhomogeneous Poisson process with intensity z.
What are the void probabilities of a Cox process?
P(X ∩ B = ∅) = E [P(X ∩ B = ∅|Z )] = E [exp( - ∫_B Z(u) du) ]
What is a Neyman-Scott process?
A Neyman-Scott process is a Poisson cluster process X such that
▶ C is a homogeneous Poisson process with intensity κ.
▶ Given C, the clusters X_c , c ∈ C, are independent.
▶ X_c is an inhomogeneous Poisson process with intensity function
u → αk(u − c),
where α > 0 and k is a kernel (pdf).
What is a Matérn cluster process?
Neyman-Scott process with uniform density kernel on b(0, r):
k(u) = 1{∥u∥ ≤ r } / (ω_d r^d)
What is a Thomas process?
Neyman-Scott process with d-dimensional gauss pdf kernel with mean 0 and covariance matrix ω^2I:k(u) = 1 / (2πω^2)^{d/2} exp(−∥u∥^2 / 2ω^2)
What is the pair correlation function of a Neyman-Scott process?
g (u, v ) = g (u − v ) = 1 + 1 / κ ∫_{R^d} k(w) k(u − v + w) dw.
How can a Neyman-Scott process be simulated?
Use extended window W_{ext} ⊇ W, as we otherwise miss clusters with center outside the window. Should be chosen large enough, relative to κ and ω.
Simulate centre process C ∩ W_{ext} as a homogeneous Poisson process with intensity κ on W_{ext}.
For each c ∈ C ∩ W_{ext} , simulate its cluster X_c:
▶ Simulate the number N of points in the cluster, N ∼ po(α).
▶ Simulate n i.i.d. points with density k(· − c).
Return ∪_{c∈C ∩ W_{ext}} X_c ∩ W as a simulation X ∩ W .
What is shot-noise Cox processes?
It is the collection of 3 generalizations of the Neymann-Scott process.
- More flexible cluster sizes: For each center c ∈ C, could replace α by independent random variables γ > 0.
Model the process of centers and expected cluster size by a Poisson process Φ = {(c, γ) ∈ Rd × (0, ∞) | c ∈ C}. - Allow kernel to depend on c by using a kernel k(c, ·) which depends on c.
- Allow inhomogeneous center process.
What is the pair correlation function for the shot-noise Cox processes?
For X a SNCP andβ(u, v ) = ∫ ∫ γ^2 k(c, u) k(c, v) ζ(c, γ) dc dγ
a locally integrable function on R^d × R^d ,
g (u, v ) = 1 + β(u, v ) / ρ(u)ρ(v )
In particular, g ≥ 1.
What is minimum contrast estimation?
It is a method for fitting a model from a family parametrized by θ based on an observed point pattern x.
Compute a non-parametric functional summary statistic ˆT (r ) from the data and let T_θ(r) be its theoretical value under the model parametrized by θ.
Estimate θ such that T_θ(r ) is as close as possible to ˆT (r) for a range of r -values.
Find the minimum contrast estimate ˆθ which minimises the contrast d(ˆT , T_θ) = ∫_{a_1}^{a^2} [ˆT(r) − T_θ(r) ]^2 dr
for some chosen 0 ≤ a1 < a2.
What is the pair correlation function for a Thomas process?
g_θ(r) = g_{(κ,ω2)} (r) = 1 + exp(−r^2 / (4ω^2) ) / (4πω^2 κ)
What is a Gaussian random field?
Let S ⊆ R^d be a region. A random field Y : S → R is a Gaussian random field, if the vector (Y (u_1), . . . , Y (u_n))^T follows a multivariate normal distribution for any n ≥ 1 and u_1, . . . , u_n ∈ S, i.e.
∑_{i=1}^n a_i Y_i,
for a_i ∈ R.
What is a log-Gaussian random field?
If Y is a Gaussian random field then Z = exp(Y) is a log-Gaussian random field
What is a log-Gaussian Cox process?
An LGCP is a Cox process whose random intensity function is a log-Gaussian random field.
How can a GRF be split into two GRF’s?
Y = m + Y_0 where
m is the mean function (the ”systematic part”),
Y_0 is a mean zero GRF with covariance function c (the ”random part”).
We may use a constant mean if we know nothing of the underlying distribution, but also a non-constant mean if we have some useful information.
What is the covariance function of a GRF?
A function c : S × S → R that is symmetric and positive definite.
What is the power exponential covariance function?
c(u, v ) = σ^2 exp( −∥u − v∥^δ / α ) ,
for σ, α > 0 and 0 ≤ δ ≤ 2.
δ = 1:
The exponential covariance function. Has ”jagged” realizations.
δ = 2:
The Gaussian covariance function. Has very smooth realizations.
What is necessary for a complex covariance function of a GRF?
A function c : S × S → C is a covariance function if and only if it is hermitian and positive semi-definite.
