22  Introduction to Log-Gaussian Cox Processes

22.1 Definition

A log-Gaussian Cox process (LGCP) is a Cox process whose random intensity field is obtained by exponentiating a Gaussian random field.

Let \(Z(u)\) be a Gaussian random field on \(W \subset \mathbb{R}^d\). Define

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

Then the point process \(X\) defined by

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

is called a log-Gaussian Cox process.

22.2 Why the exponential?

The exponential transformation serves two key purposes:

  1. Positivity

    Since intensities must satisfy \(\Lambda(u) \ge 0\), exponentiation ensures validity: \[ \Lambda(u) = \exp(Z(u)) > 0 \]

  2. Flexible variability

    A Gaussian random field \(Z(u)\) can take any real value, so exponentiation allows:

    • small values of \(Z(u)\) → very low intensity
    • large values of \(Z(u)\) → very high intensity
Note

The exponential transformation turns additive Gaussian structure into multiplicative variation in the intensity field.

22.3 Conditional vs unconditional structure

Conditional on \(Z\)

Given the Gaussian random field \(Z(u)\):

  • \(X\) is a Poisson process with intensity \(\exp(Z(u))\)
  • points are conditionally independent
  • no interaction between points

Unconditional (marginal)

Marginally:

  • \(X\) is not Poisson
  • clustering arises through variation in \(Z(u)\)
  • dependence is entirely induced by the latent field
Important

In an LGCP, points do not interact directly, they cluster because they respond to the same Gaussian random field.

22.4 Gaussian random field specification

To fully specify an LGCP, we must define the Gaussian random field \(Z(u)\).

Typically, we assume:

  • Mean function: \[ \mathbb{E}[Z(u)] = m(u) \]

  • Covariance function: \[ \mathrm{Cov}(Z(u), Z(v)) = C(u,v) \]

22.5 Stationary LGCP

A common and important special case is when \(Z(u)\) is stationary:

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

Then:

  • \(\Lambda(u)\) is also stationary in distribution
  • the LGCP has constant first-order intensity
  • second-order structure depends only on \(u-v\)

22.6 Isotropic LGCP

If, in addition, the covariance depends only on distance,

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

then the process is isotropic.

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 spatial scale

22.7 Interpretation of parameters

Although the LGCP is defined through \(Z(u)\), interpretation is typically done through the induced intensity \(\Lambda(u)\).

  • \(m(u)\) (or \(m\) under stationarity):
    • controls the baseline level of the log-intensity
  • \(C(u,v)\):
    • controls the spatial dependence in the log-intensity
    • determines the clustering structure of the point process
Note

Unlike simpler models, LGCP parameters do not directly correspond to intensity or clustering strength — these emerge after exponentiation.

This is one reason why alternative parameterisations (e.g. in terms of \(\lambda\) and \(g(0)\)) are often useful in practice.

22.8 Log-intensity interpretation

Taking logarithms gives

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

So an LGCP can be viewed as:

A model where the log-intensity is a Gaussian random field.

This interpretation is widely used in applications, especially in spatial statistics and machine learning.

22.9 Consequences (preview)

Using the general Cox process moment formulas, we will show in the next sections that:

  • the intensity is \[ \lambda(u) = \mathbb{E}[\exp(Z(u))] \]

  • the pair correlation function takes the form \[ g(u,v) = \exp(C(u,v)) \]

under stationarity

Important

For an LGCP, the pair correlation function is an exponential transformation of the covariance function of the underlying Gaussian random field.

This is a key structural property, and will be central to later comparisons with other Cox processes.

22.10 Interpretation as a latent Gaussian model

An LGCP can be viewed as a hierarchical model:

  1. Draw a Gaussian random field \(Z(u)\)
  2. Transform to obtain \(\Lambda(u) = \exp(Z(u))\)
  3. Generate a Poisson process with intensity \(\Lambda(u)\)

This makes LGCPs a natural extension of:

  • Gaussian random fields (continuous structure)
  • Poisson processes (discrete events)

22.11 Advantages of LGCP

LGCPs are widely used because they:

  • guarantee positivity of the intensity
  • allow flexible spatial dependence
  • are mathematically tractable (moments can be derived explicitly)
  • connect naturally to Gaussian process modelling

22.12 Limitations

However, LGCPs also have important limitations:

  • parameters are not directly interpretable in terms of clustering
  • likelihood-based inference is computationally challenging
  • different LGCP parameter settings can produce very similar second-order structure
Note

In particular, LGCPs can be difficult to distinguish from other Cox processes using only second-order statistics.

22.13 Summary

  • An LGCP is a Cox process with intensity \[ \Lambda(u) = \exp(Z(u)) \] where \(Z(u)\) is a Gaussian random field

  • Conditional on \(Z\), the process is Poisson

  • Marginally, clustering arises from spatial dependence in \(Z(u)\)

  • The covariance structure of \(Z(u)\) determines the clustering behaviour of the point process

  • LGCPs provide a flexible and widely used framework for modelling spatial clustering