24  Pair Correlation Function of the LGCP

24.1 Motivation

From the moment calculations, the pair correlation function of a log-Gaussian Cox process is

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

where \(C(u,v)\) is the covariance function of the underlying Gaussian random field \(Z(u)\).

Under stationarity and isotropy, this simplifies to

\[ g(r) = \exp\big(C(r)\big), \qquad r = \|u-v\| \]

This gives a direct and explicit link between:

  • the covariance structure of the Gaussian field
  • the clustering behaviour of the point process

24.2 A standard covariance model

A common choice is the exponential covariance function:

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

where:

  • \(\sigma^2\) controls variability
  • \(s\) controls the spatial scale

Substituting into the LGCP PCF gives

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

24.3 Clustering strength

A natural measure of clustering is the value of the PCF at the origin:

\[ g(0) = \exp\big(C(0)\big) = \exp(\sigma^2) \]

Define the clustering strength

\[ \phi = g(0) - 1 = \exp(\sigma^2) - 1 \]

This gives a more interpretable parameter:

  • \(\phi = 0\) → no clustering (Poisson)
  • larger \(\phi\) → stronger clustering

We can invert this relationship:

\[ \sigma^2 = \log(1+\phi). \]

24.4 Reparameterised PCF

Substituting \(\sigma^2 = \log(1+\phi)\) into the PCF gives

\[ g(r) = \exp\left( \log(1+\phi)\,\exp\left(-\frac{r}{s}\right) \right) \]

This can be written more compactly as

\[ g(r) = (1+\phi)^{\exp(-r/s)} \]

24.5 Interpretation

This parameterisation makes the structure of the LGCP much clearer:

  • \(\phi\) controls how much clustering occurs
  • \(s\) controls how quickly clustering decays with distance

Key properties:

  1. At \(r=0\): \[ g(0) = 1 + \phi \]

  2. As \(r \to \infty\): \[ g(r) \to 1 \]

  3. The decay is governed by \(\exp(-r/s)\):

    • small \(s\) → rapid decay
    • large \(s\) → long-range clustering

24.6 Shape of the LGCP PCF

The LGCP PCF has a characteristic form:

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

This implies:

  • smooth decay from \(g(0)\) to \(1\)
  • strictly positive clustering (no inhibition)
  • nonlinear relationship between covariance and clustering
Note

Even moderate values of \(\sigma^2\) can produce large values of \(g(0)\) due to the exponential transformation.

24.7 Role of the mean

Recall that the intensity is

\[ \lambda = \exp\left(m + \tfrac{1}{2}\sigma^2\right). \]

So:

  • \(m\) affects the overall intensity
  • \(\sigma^2\) affects both intensity and clustering

This means that:

Important

In the LGCP, intensity and clustering are not fully independent under the \((m, \sigma^2)\) parameterisation.

The \((\lambda, \phi, s)\) parameterisation separates these roles more clearly:

  • \(\lambda\) → first-order intensity
  • \(\phi\) → clustering strength
  • \(s\) → clustering scale

24.8 Summary

For a stationary isotropic LGCP with exponential covariance:

\[ C(r) = \sigma^2 \exp(-r/s), \]

the pair correlation function can be written as

\[ g(r) = \exp(C(r)) = (1+\phi)^{\exp(-r/s)}, \]

where

\[ \phi = \exp(\sigma^2) - 1 \]

This parameterisation provides:

  • a direct interpretation of clustering strength (\(\phi\))
  • a clear notion of spatial scale (\(s\))
  • a convenient bridge between theory and empirical analysis