Cox point process

In previous posts I have stressed the importance of the Poisson point process, but it can be unsuitable for certain mathematical models.  We can generalize this point process by first considering a non-negative random measure, called a driving or directing measure. Then a Poisson point process, which is independent of the random driving measure, is generated by using the random measure as its intensity or mean measure. This doubly stochastic construction gives what is called a Cox point process.

Typically, we don’t observe the driving measure, so it’s impossible to distinguish a Cox point process from a Poisson point if there’s only one realization available. The properties of a Cox point process are derived by conditioning on the random driving measure, and then using the properties of the Poisson point process.

Cox point processes are also known as doubly stochastic Poisson point processes. Guttorp and Thorarinsdottir argue that it should be called the Quenouille point process, as Maurice Quenouille introduced an example of it before Sir David Cox, but I opt for the more common name.

In this post I’ll cover a couple examples of Cox point processes, but first I’ll give a more precise mathematical definition.

Definition

We consider a point process defined on some underlying mathematical space \(\mathbb{S}\), which is sometimes called the carrier space or state space.  For applications, the underlying space is often the real line \(\mathbb{R}\), the plane \(\mathbb{R}^2\), or some other familiar mathematical space like a square lattice.

For the first definition, we use the concept of a random measure.

Let \(M\) be a non-negative random measure on \(\mathbb{S} \). Then a point process \(\Phi\) defined on some underlying space \(\mathbb{S}\) is a Cox point process driven by the intensity measure \(M\) if the conditional distribution of \(\Phi\) is a Poisson point process with intensity function \(M\).

Alternatively, we can give a slightly less general definition of a Cox  point process by using a random intensity function.

Let \(Z=\{Z(x):x\in\mathbb{S} \}\) be a non-negative random field such that with probability one, \(x\rightarrow Z(x)\) is a locally integrable function. Then a point process \(\Phi\) defined on some underlying space \(\mathbb{S}\) is a Cox point process driven by \(Z\) if the conditional distribution of \(\Phi\) is a Poisson point process with intensity function \(Z\).

The random driving measure \(M\) is then simply the integral
$$
M(B)=\int_B Z(x)\, dx , \quad B\subseteq S.
$$

Over-dispersion

We will soon see that the random driving measures take different forms, giving different Cox point processes. But there is a general observation that can be made for all Cox point processes. For any region \(B \subseteq S\), it can be shown that the number of points \(\Phi (B)\) adheres to the inequality
$$
\mathbb{Var} [\Phi (B)] \geq \mathbb{E} [\Phi (B)],
$$

where \(\mathbb{Var} [\Phi (B)] \) is the variance of the random variable \(\Phi (B)\).  As a comparison, for a Poisson point process \(\Phi’\), the variance of \(\Phi’ (B)\) is simply \(\mathbb{Var} [\Phi’ (B)] =\mathbb{E} [\Phi’ (B)]\).  Due to its greater variance, the Cox point process is said to be over-dispersed compared to the Poisson point process.

Special cases

There is an virtually unlimited number of ways to define a random driving measure, where each one yields a different a Cox point process. But in general mathematicians are and practitioners are restricted by examining only tractable and interesting Cox point processes. I will give some common examples of such Cox point processes, but I stress that the Cox point process family is very large.

Mixed Poisson point process

For the random driving measure \(M\), an obvious example is the product form \(M= Y \mu \),  where \(Y\) is some independent non-negative random variable and \(\mu\) is the Lebesgue measure on \(\mathbb{S}\), meaning \(Y\) is the only source of randomness. This driving measure gives the mixed Poisson point process.

Log-Gaussian Cox point process

Instead of a random variable, we can use a non-negative random field to define a random driving measure.  We then have the product form \(M= Y \mu \), where \(Y\) is now some independent non-negative random field. (A random field is a collection of random variables indexed by some set, which in this case is the underlying space \(\mathbb{S}\).)

Arguably the most tractable and used random field is the Gaussian random field. This random field, like Gaussian or normal random variables, takes negative and positive values, but if we define the random field such that its logarithm is a Gaussian field \(Z\), then we obtain the non-negative random driving measure \(M=\mu e^Z \), which gives the log-Gaussian Cox point process.

This point process has found applications in spatial statistics.

Cox-Poisson line-point process

To construct this Cox point process, we first consider a Poisson line process, which I discussed previously.  Given a Poisson line process, we then place an independent one-dimensional Poisson point process on each line, then we obtain an example of a Cox point process, which we could call a Cox line-point process orCox-Poisson line-point process, but I am not sure of the name.

Researchers have recently used this point process to study wireless communication networks in cities, where the streets correspond to Poisson lines. For example, see these two preprints:

  1. Continuum percolation for Cox point processes
  2. Poisson Cox Point Processes for Vehicular Networks

Shot-noise Cox point process

We construct the next Cox point process by first considering a Poisson point process on the space \(\mathbb{S}\) to create a shot noise term, which we then use as the driving measure of the Cox point process.

More specifically, we first introduce a kernel function \(k(\cdot,\cdot)\) on \(\mathbb{S}\), where \(k(x,\cdot)\) is a probability density function for all points \(x\in \mathbb{S}\). We then consider a Poisson point process \(\Phi’\) on \(\mathbb{S}\times (0,\infty)\), which we assume has a locally integrable intensity function \(\mu \). (We can interpret  the point process \(\Phi’\) as a spatially-dependent marked Poisson point process,  where the unmarked Poisson point process is defined on \(\mathbb{S}\), and each point \(X\) of this unmarked point process has a mark  \(T \in (0,\infty)\) with probability density \(\mu(X,t)\).) The resulting shot noise

$$
Z(x)= \sum_{(Y,T)\in \Phi’} T \, k(Y,x)\,.
$$

gives the random field, which we then use as the random intensity function to drive the shot-noise Cox point process.

In previous posts, I have detailed how to simulate non-Poisson point processes such as the Matérn and Thomas cluster point processes. These are, more specifically, examples of a Neyman-Scott point process, which is a special case of a shot noise Cox point process. All these point processes find applications in spatial statistics.

Simulation

Unfortunately, there is no universal way to simulate all Cox point processes — and even if there were one, it would not be the most optimal way for every Cox point process. The simulation method depends on how the Cox point process is constructed, which usually means how its directing or driving measure is defined.

In previous posts I have presented ways (with code) to simulate the following Cox point processes:

In addition to the Matérn and Thomas point processes, there are ways to simulate to more general shot noise Cox point processes, which I will cover in another post.

Further reading

For general Cox point processes, I suggest the following: Chapter 6 in the monograph Poisson processes by Kingman; Chapter 5 in Statistical Inference and Simulation for Spatial Point Processes by Møller and Waagepetersen; and Section 5.2 in Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke. For a much more mathematical treatment, see Chapter 13 in Lectures on the Poisson Process by Last and Penrose. Grandell wrote two detailed monographs titled Mixed Poisson Process and Doubly Stochastic Poisson Processes.

Motivated by applications in spatial statistics, Jesper Møller has (co)-written papers on specific Cox point processes. For example:

  • 2001, Møller, Syversveen, and Waagepetersen, Log Gaussian Cox Processes;
  • 2003, Møller, Shot noise Cox Processes;
  • 2005, Møller and Torrisi,Generalised shot noise Cox processes.

I also suggest the survey article:

  • 2003, Møller and Waagepetersen, Modern statistics for spatial point processes.