20 Moment Properties
20.1 Motivation
The defining feature of a Cox process is that its intensity function is itself random.
If \(X\) is a Cox process driven by a random intensity field \(\Lambda(u)\), then
\[ X \mid \Lambda \sim \text{Poisson process with intensity } \Lambda(u). \]
So, conditional on \(\Lambda\), the process is Poisson.
But marginally, the process is no longer Poisson: its first- and second-order structure now depends on the moments of the random field \(\Lambda(u)\).
Not exactly confident in the meaning of marginal here.
This gives a very clean principle:
For a Cox process, the moments of the point process are determined by the moments of the random intensity field.
In this section, we derive the intensity, second-order product density, and pair correlation function of a Cox process directly from this idea.
20.2 Setup
Let \(X\) be a Cox process on a spatial domain \(W \subset \mathbb{R}^d\), driven by a non-negative random field \(\Lambda(u)\).
Thus,
\[ X \mid \Lambda \sim \text{Poisson process with intensity } \Lambda(u). \]
We write:
- \(\lambda(u)\) for the first-order intensity of \(X\)
- \(\lambda^{(2)}(u,v)\) for the second-order product density
- \(g(u,v)\) for the pair correlation function
Our goal is to express each of these in terms of the moments of \(\Lambda\).
20.3 First-order intensity
Recall that the intensity function is defined so that, for a small region \(du\) around \(u\),
\[ \mathbb{E}[N(du)] \approx \lambda(u)\ |du| \]
For a Cox process, we use conditioning:
\[ \mathbb{E}[N(du)] = \mathbb{E}\big[\mathbb{E}[N(du)\mid \Lambda]\big] \]
Conditional on \(\Lambda\), the process is Poisson with intensity \(\Lambda(u)\), so
\[ \mathbb{E}[N(du)\mid \Lambda] \approx \Lambda(u)\ |du| \]
Taking expectations gives
\[ \mathbb{E}[N(du)] \approx \mathbb{E}[\Lambda(u)]\,|du| \]
Therefore,
\[ \lambda(u) = \mathbb{E}[\Lambda(u)] \]
20.3.1 Interpretation
The intensity of a Cox process is just the mean of the random intensity field.
A Cox process may be highly clustered, but its first-order intensity only records the average level of the latent field. It does not capture the variability of \(\Lambda(u)\).
20.4 Mean measure
Integrating the intensity over a region \(B \subset W\) gives
\[ \mathbb{E}[N(B)] = \int_B \lambda(u)\ du = \int_B \mathbb{E}[\Lambda(u)]\ du \]
Equivalently,
\[ \mathbb{E}[N(B)] = \mathbb{E}\left[\int_B \Lambda(u)\ du\right] \]
So the expected number of points in a region is the expected total latent intensity over that region.
Need to flesh this out a little, too condensed as is.
20.5 Second-order product density
We now derive the second-order product density.
Recall that for disjoint small regions \(du\) around \(u\) and \(dv\) around \(v\),
\[ \mathbb{E}[N(du)N(dv)] \approx \lambda^{(2)}(u,v)\ |du|\ |dv| \]
Again, condition on \(\Lambda\):
\[ \mathbb{E}[N(du)N(dv)] = \mathbb{E}\big[\mathbb{E}[N(du)N(dv)\mid \Lambda]\big] \]
Conditional on \(\Lambda\), the process is Poisson, so counts in disjoint regions are independent. Hence
\[ \mathbb{E}[N(du)N(dv)\mid \Lambda] = \mathbb{E}[N(du)\mid \Lambda]\ \mathbb{E}[N(dv)\mid \Lambda] \]
Using the local Poisson approximation,
\[ \mathbb{E}[N(du)\mid \Lambda] \approx \Lambda(u)\ |du|, \qquad \mathbb{E}[N(dv)\mid \Lambda] \approx \Lambda(v)\ |dv| \]
Therefore,
\[ \mathbb{E}[N(du)N(dv)\mid \Lambda] \approx \Lambda(u)\Lambda(v)\ |du|\ |dv| \]
Taking expectations gives
\[ \mathbb{E}[N(du)N(dv)] \approx \mathbb{E}[\Lambda(u)\Lambda(v)]\ |du|\ |dv| \]
Hence
\[ \lambda^{(2)}(u,v) = \mathbb{E}[\Lambda(u)\Lambda(v)], \qquad u \neq v \]
20.5.1 Interpretation
The second-order product density is the second moment of the random intensity field.
This is the key mechanism by which Cox processes generate clustering.
If \(\Lambda(u)\) and \(\Lambda(v)\) tend to be simultaneously large, then points near \(u\) and \(v\) tend to occur together more often than under independence.
20.6 Covariance form
Using the identity
\[ \mathbb{E}[\Lambda(u)\Lambda(v)] = \mathrm{Cov}(\Lambda(u),\Lambda(v)) + \mathbb{E}[\Lambda(u)]\mathbb{E}[\Lambda(v)], \]
we obtain
\[ \lambda^{(2)}(u,v) = \lambda(u)\lambda(v) + \mathrm{Cov}(\Lambda(u),\Lambda(v)) \]
This decomposition is extremely useful.
It shows that the second-order structure of a Cox process is equal to:
- the Poisson baseline \(\lambda(u)\lambda(v)\)
- plus an extra covariance term induced by the latent field
A Cox process differs from a Poisson process at second order precisely through the covariance structure of its random intensity field.
If \(\Lambda(u)\) is deterministic, then \(\mathrm{Cov}(\Lambda(u),\Lambda(v)) = 0\), and we recover the Poisson case.
20.7 Pair correlation function
Recall that the pair correlation function is defined by
\[ g(u,v) = \frac{\lambda^{(2)}(u,v)}{\lambda(u)\lambda(v)}, \qquad u \neq v. \]
Substituting the Cox process formulas gives
\[ g(u,v) = \frac{\mathbb{E}[\Lambda(u)\Lambda(v)]} {\mathbb{E}[\Lambda(u)]\,\mathbb{E}[\Lambda(v)]}. \]
Using the covariance form above, we can also write
\[ g(u,v) = 1 + \frac{\mathrm{Cov}(\Lambda(u),\Lambda(v))} {\mathbb{E}[\Lambda(u)]\,\mathbb{E}[\Lambda(v)]}. \]
20.7.1 Consequences
Since covariance can be positive, zero, or negative, this formula describes how dependence in the latent field affects dependence in the point process.
For most Cox process models used in practice, nearby values of \(\Lambda(u)\) and \(\Lambda(v)\) are positively correlated, so
\[ g(u,v) > 1 \]
for small distances.
This corresponds to clustering.
Moreover, if \(\Lambda(u)\) and \(\Lambda(v)\) are uncorrelated, then
\[ g(u,v) = 1 \]
So the PCF of a Cox process is directly determined by the covariance structure of the random intensity field.
20.8 Stationary Cox processes
Suppose now that the Cox process is stationary.
Then:
- \(\lambda(u) = \lambda\) is constant
- \(\lambda^{(2)}(u,v)\) depends only on \(u-v\)
- \(g(u,v)\) depends only on \(u-v\)
If the process is also isotropic, then these depend only on the distance
\[ r = \|u-v\| \]
In that case, we write
\[ g(r) = \frac{\mathbb{E}[\Lambda(u)\Lambda(v)]}{\lambda^2} = 1 + \frac{\mathrm{Cov}(\Lambda(u),\Lambda(v))}{\lambda^2}, \qquad r=\|u-v\|. \]
Thus, under stationarity and isotropy, the PCF is just a rescaled version of the covariance function of the latent intensity field.
This is one of the main reasons Cox processes are so convenient: once the moment structure of \(\Lambda\) is known, the PCF follows immediately.
20.9 Why Cox processes cluster
We can now make the source of clustering explicit.
Conditional on \(\Lambda\), the process is Poisson, so there is no interaction between points.
But marginally, two nearby locations may both lie in a region where the latent intensity is large.
This creates positive dependence in the counts, and hence clustering.
So for a Cox process:
- points do not attract each other directly
- clustering arises because points respond to the same random environment
This is exactly what the formula
\[ g(u,v) = 1 + \frac{\mathrm{Cov}(\Lambda(u),\Lambda(v))} {\mathbb{E}[\Lambda(u)]\,\mathbb{E}[\Lambda(v)]} \]
is telling us.
20.10 Example: Log-Gaussian Cox process
Let
\[ \Lambda(u) = \exp(Z(u)), \]
where \(Z(u)\) is a Gaussian random field.
Then the intensity and PCF can be derived from Gaussian moment formulas.
In particular, if \(Z\) is stationary with covariance function \(C(r)\), then
\[ g(r) = \exp(C(r)). \]
So for an LGCP, the PCF is an exponential transformation of the covariance function of the underlying Gaussian random field.
20.11 Example: Chi-square Cox process
Let
\[ \Lambda(u) = \mu + Z(u)^2, \qquad \mu \ge 0, \]
where \(Z(u)\) is a Gaussian random field.
Then:
- the intensity depends on \(\mathbb{E}[Z(u)^2]\)
- the second-order product density depends on \(\mathbb{E}[Z(u)^2 Z(v)^2]\)
- the PCF depends on fourth moments of the Gaussian field
For a mean-zero stationary Gaussian random field with variance \(\sigma^2\) and correlation function \(\rho(r)\), this leads to a PCF of the form
\[ g(r) = 1 + \frac{2\sigma^4}{(\mu+\sigma^2)^2}\rho(r)^2. \]
So unlike the LGCP case, the PCF now depends on a quadratic function of the correlation structure.
20.12 Summary
For a Cox process driven by a random intensity field \(\Lambda(u)\):
\[ \lambda(u) = \mathbb{E}[\Lambda(u)] \]
\[ \lambda^{(2)}(u,v) = \mathbb{E}[\Lambda(u)\Lambda(v)] \]
\[ g(u,v) = \frac{\mathbb{E}[\Lambda(u)\Lambda(v)]} {\mathbb{E}[\Lambda(u)]\mathbb{E}[\Lambda(v)]} = 1 + \frac{\mathrm{Cov}(\Lambda(u),\Lambda(v))} {\mathbb{E}[\Lambda(u)]\mathbb{E}[\Lambda(v)]} \]
These identities are fundamental.
They show that:
- first-order structure is determined by the mean of \(\Lambda\)
- second-order structure is determined by the second moments of \(\Lambda\)
- clustering in a Cox process is induced by covariation in the latent intensity field
This makes Cox processes a natural bridge between random fields and clustered point processes.