23  Moment Properties of the LGCP

23.1 Motivation

For a Cox process, all first- and second-order structure is determined by the moments of the random intensity field:

\[ \lambda(u) = \mathbb{E}[\Lambda(u)], \qquad \lambda^{(2)}(u,v) = \mathbb{E}[\Lambda(u)\Lambda(v)] \]

For a log-Gaussian Cox process (LGCP), the intensity is

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

where \(Z(u)\) is a Gaussian random field.

So the problem reduces to computing expectations of the form:

  • \(\mathbb{E}[\exp(Z(u))]\)
  • \(\mathbb{E}[\exp(Z(u)+Z(v))]\)

These can be derived using standard Gaussian moment identities.

23.2 Setup

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

  • mean function: \[ \mathbb{E}[Z(u)] = m(u) \]
  • covariance function: \[ \mathrm{Cov}(Z(u),Z(v)) = C(u,v) \]

Define the LGCP intensity:

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

23.3 A key Gaussian identity

We will repeatedly use the following result:

Important

If \(X \sim \mathcal{N}(\mu, \sigma^2)\), then \[ \mathbb{E}[\exp(X)] = \exp\left(\mu + \tfrac{1}{2}\sigma^2\right). \]

More generally, if \((X,Y)\) is jointly Gaussian, then

\[ \mathbb{E}[\exp(X+Y)] = \exp\left( \mathbb{E}[X+Y] + \tfrac{1}{2}\mathrm{Var}(X+Y) \right) \]

23.4 First-order intensity

23.4.1 Proposition

For an LGCP,

\[ \lambda(u) = \exp\left( m(u) + \tfrac{1}{2}C(u,u) \right) \]

23.4.2 Proof

Since \(\Lambda(u) = \exp(Z(u))\), we have

\[ \lambda(u) = \mathbb{E}[\exp(Z(u))]. \]

Now \(Z(u) \sim \mathcal{N}(m(u), C(u,u))\), so applying the Gaussian identity gives

\[ \lambda(u) = \exp\left( m(u) + \tfrac{1}{2}C(u,u) \right) \]

23.4.3 Interpretation

The intensity is not \(\exp(m(u))\), but is inflated by the variance term:

\[ \lambda(u) = \exp(m(u)) \times \exp\left(\tfrac{1}{2}C(u,u)\right) \]

Note

Variability in the latent Gaussian field increases the expected intensity.

23.5 Second-order product density

23.5.1 Proposition

For an LGCP,

\[ \lambda^{(2)}(u,v) = \exp\left( m(u) + m(v) + \tfrac{1}{2}\big(C(u,u) + C(v,v)\big) + C(u,v) \right) \]

23.5.2 Proof

We compute:

\[ \lambda^{(2)}(u,v) = \mathbb{E}[\exp(Z(u))\exp(Z(v))] = \mathbb{E}[\exp(Z(u)+Z(v))] \]

Since \((Z(u), Z(v))\) is jointly Gaussian, the sum \(Z(u)+Z(v)\) is Gaussian with:

  • mean: \[ m(u) + m(v) \]
  • variance: \[ \mathrm{Var}(Z(u)+Z(v)) = C(u,u) + C(v,v) + 2C(u,v) \]

Applying the Gaussian identity gives

\[ \lambda^{(2)}(u,v) = \exp\left( m(u)+m(v) + \tfrac{1}{2}\big(C(u,u)+C(v,v)+2C(u,v)\big) \right) \]

Rearranging yields the result.

23.6 Pair correlation function

23.6.1 Proposition

For an LGCP,

\[ g(u,v) = \exp\big(C(u,v)\big) \]

23.6.2 Proof

Recall that

\[ g(u,v) = \frac{\lambda^{(2)}(u,v)}{\lambda(u)\lambda(v)} \]

Substitute the expressions derived above.

The numerator is

\[ \exp\left( m(u)+m(v) + \tfrac{1}{2}(C(u,u)+C(v,v)) + C(u,v) \right), \]

and the denominator is

\[ \exp\left( m(u)+\tfrac{1}{2}C(u,u) \right) \exp\left( m(v)+\tfrac{1}{2}C(v,v) \right) \]

Cancelling terms gives

\[ g(u,v) = \exp(C(u,v)) \]

23.7 Stationary LGCP

Suppose \(Z(u)\) is stationary with:

\[ \mathbb{E}[Z(u)] = m, \qquad \mathrm{Cov}(Z(u),Z(v)) = C(u-v) \]

Then:

  • intensity: \[ \lambda = \exp\left(m + \tfrac{1}{2}C(0)\right) \]

  • pair correlation function: \[ g(u,v) = \exp(C(u-v)) \]

23.8 Isotropic LGCP

If \(C(u,v)=C(r)\) with \(r=\|u-v\|\), then

\[ g(r) = \exp(C(r)) \]

For example, with exponential covariance:

\[ C(r) = \sigma^2 \exp\left(-\frac{r}{s}\right), \]

we obtain

\[ g(r) = \exp\left( \sigma^2 \exp\left(-\frac{r}{s}\right) \right) \]

23.9 Key consequences

23.9.1 Clustering is always positive

Since \(C(u,v) \ge 0\) for nearby points in typical covariance models,

\[ g(u,v) \ge 1. \]

So LGCPs always exhibit clustering at short distances.

23.9.2 Nonlinear amplification

The PCF is an exponential transformation:

\[ g(u,v) = \exp(C(u,v)) \]

So even moderate covariance can produce strong clustering.

23.9.3 Separation of roles

  • \(m(u)\) controls overall intensity level
  • \(C(u,v)\) controls clustering

However, they are not fully independent:

Note

The variance \(C(u,u)\) affects the intensity \(\lambda(u)\), so intensity and clustering are implicitly linked.

23.9.4 Connection to general Cox theory

Comparing with the general Cox formula:

\[ g(u,v) = 1 + \frac{\mathrm{Cov}(\Lambda(u),\Lambda(v))} {\mathbb{E}[\Lambda(u)]\mathbb{E}[\Lambda(v)]}, \]

we see that for LGCP,

\[ \mathrm{Cov}(\Lambda(u),\Lambda(v)) = \lambda(u)\lambda(v)\big(\exp(C(u,v)) - 1\big) \]

So the exponential structure emerges naturally from Gaussian moments.

23.10 Interpretation

An LGCP transforms additive Gaussian dependence into multiplicative clustering.

  • Gaussian field:
    • linear covariance structure
  • LGCP:
    • exponential clustering structure

This explains why LGCPs can produce very strong clustering behaviour.

23.11 Summary

For an LGCP with \(\Lambda(u)=\exp(Z(u))\):

\[ \lambda(u) = \exp\left(m(u) + \tfrac{1}{2}C(u,u)\right) \]

\[ \lambda^{(2)}(u,v) = \exp\left( m(u)+m(v) + \tfrac{1}{2}(C(u,u)+C(v,v)) + C(u,v) \right) \]

\[ g(u,v) = \exp(C(u,v)) \]

These results show that:

  • first-order structure depends on both mean and variance of \(Z(u)\)
  • second-order structure depends directly on the covariance of \(Z(u)\)
  • clustering is an exponential transformation of Gaussian dependence

This makes the LGCP one of the most tractable and widely used Cox process models.