19  Introduction to Cox Processes

19.1 Definition

A Cox process (or doubly stochastic Poisson process) is a point process defined as follows:

Let \(\Lambda(u)\) be a non-negative random field on a spatial domain \(W \subset \mathbb{R}^d\).

Conditional on \(\Lambda\), the point process \(X\) is a Poisson process with intensity function \(\Lambda(u)\):

\[ X \mid \Lambda \sim \text{Poisson process with intensity } \Lambda(u) \]

Equivalently, for any Borel set \(A \subset W\):

\[ N(A) \mid \Lambda \sim \text{Poisson}\left(\int_A \Lambda(u)\,du\right), \]

and counts in disjoint sets are conditionally independent given \(\Lambda\).

19.2 Intuition

A Cox process introduces two layers of randomness:

  1. A random intensity surface \(\Lambda(u)\) is generated
  2. A Poisson process is sampled conditional on this surface
Note

A Cox process can be interpreted as a Poisson process in a random environment.

This contrasts with a standard Poisson process, where the intensity \(\lambda(u)\) is deterministic.

Important

I feel like this is EXTREMELY important.

A Cox process allows clustering, but that clustering is asssumed to be due to latent environment variables, NOT because of attraction between points themselves.

This is why we have a Poisson process when conditioning on \(\Lambda\).

A question I don’t know the answer to is: how do we model processes where there is attraction between the points themselves (see Gibbs processes?).

19.3 Conditional vs Unconditional Perspectives

19.3.1 Conditional (given \(\Lambda\))

  • \(X\) is a Poisson process
  • All Poisson properties hold:
    • Independent increments
    • No interaction between points

19.3.2 Unconditional (marginal)

  • \(X\) is not Poisson in general
  • Points exhibit dependence
  • Clustering arises through variation in \(\Lambda(u)\)
Note

All dependence in a Cox process is induced by the shared random intensity field.

(I guess this is just reiterating my callout above).

19.4 Connection to the Poisson Process

The Poisson process is a special case of a Cox process where:

\[ \Lambda(u) = \lambda(u) \quad \text{(deterministic)} \]

Thus, Cox processes generalise Poisson processes by allowing random intensity functions.

19.5 Examples

19.5.1 Log-Gaussian Cox Process (LGCP)

Let \(Z(u)\) be a Gaussian random field. Define:

\[ \Lambda(u) = \exp(Z(u)) \]

  • Ensures \(\Lambda(u) > 0\)
  • Widely used in applications
  • Leads to strong clustering

19.5.2 Chi-Square Cox Process (CSCP)

Let \(Z(u)\) be a Gaussian random field. Define:

\[ \Lambda(u) = \mu + Z(u)^2, \quad \mu \ge 0. \]

  • Quadratic transformation of a GRF
  • Produces interpretable clustering structure
  • Central to these notes

19.6 Interpretation via Random Fields

Cox processes provide a direct link between:

  • Random fields (continuous stochastic structure)
  • Point processes (discrete event patterns)

The random field \(\Lambda(u)\) acts as a latent driver of spatial variation.

Note

Regions where \(\Lambda(u)\) is large produce higher point density, leading to clustering.

19.7 Consequences

  • First- and second-order properties of \(X\) depend on the moments of \(\Lambda(u)\)
  • The pair correlation function is determined by the covariance structure of \(\Lambda(u)\)
  • Cox processes are a natural framework for modelling heterogeneity and clustering

19.8 Summary

  • A Cox process is a Poisson process with random intensity
  • Conditional on \(\Lambda\), the process is Poisson
  • Marginally, it exhibits dependence and clustering
  • Key examples (LGCP, CSCP) arise from transformations of Gaussian random fields
  • All higher-order structure is driven by the random intensity field