25  Exercises

These exercises are intended to consolidate the basic theory of the log-Gaussian Cox process (LGCP), with particular emphasis on moment calculations, pair correlation structure, parameter interpretation, and implementation.

Throughout, let \(X\) denote a log-Gaussian Cox process with random intensity field

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

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

Unless otherwise stated, assume all required expectations exist.

25.1 1. Conditional construction of the LGCP

Let \(X\) be an LGCP driven by a Gaussian random field \(Z(u)\).

  1. State carefully what is meant by \[ X \mid Z \sim \text{Poisson process with intensity } \exp(Z(u)). \]

  2. Explain why \(\Lambda(u)=\exp(Z(u))\) is always a valid intensity field.

  3. Why is the Gaussian random field introduced on the log-intensity scale, rather than directly on the intensity scale?

  4. Explain, in words, why an LGCP can exhibit clustering even though it is conditionally Poisson.

25.2 2. Gaussian exponential moment identity

Let \(Y \sim \mathcal{N}(\mu,\sigma^2)\).

  1. Show that the moment generating function of \(Y\) is \[ M_Y(t)=\exp\left(\mu t + \tfrac{1}{2}\sigma^2 t^2\right). \]

  2. Deduce that \[ \mathbb{E}[\exp(Y)] = \exp\left(\mu + \tfrac{1}{2}\sigma^2\right). \]

  3. More generally, if \((Y_1,Y_2)\) is jointly Gaussian, show that \[ \mathbb{E}[\exp(Y_1+Y_2)] = \exp\left( \mathbb{E}[Y_1+Y_2] + \tfrac{1}{2}\mathrm{Var}(Y_1+Y_2) \right). \]

  4. Explain why this identity is the key tool in deriving LGCP moment properties.

25.3 3. First-order intensity of an LGCP

Let \(Z(u)\) be a Gaussian random field with mean function \(m(u)\) and covariance function \(C(u,v)\).

  1. Starting from \[ \lambda(u)=\mathbb{E}[\Lambda(u)], \] derive the formula \[ \lambda(u)=\exp\left(m(u)+\tfrac{1}{2}C(u,u)\right). \]

  2. Why does the variance term \(C(u,u)\) appear in the expression for the intensity?

  3. Explain why \(\lambda(u)\neq \exp(m(u))\) in general.

  4. Interpret the formula in terms of Jensen’s inequality.

25.4 4. Second-order product density of an LGCP

Let \(u \neq v\).

  1. Starting from \[ \lambda^{(2)}(u,v)=\mathbb{E}[\Lambda(u)\Lambda(v)], \] derive the formula \[ \lambda^{(2)}(u,v) = \exp\left( m(u)+m(v)+\tfrac{1}{2}\big(C(u,u)+C(v,v)\big)+C(u,v) \right). \]

  2. In your derivation, explicitly compute: \[ \mathbb{E}[Z(u)+Z(v)] \quad \text{and} \quad \mathrm{Var}(Z(u)+Z(v)). \]

  3. Show carefully where joint Gaussianity is used.

  4. Why is this derivation much more difficult for a non-Gaussian latent field?

25.5 5. Pair correlation function of the LGCP

Using your expressions for \(\lambda(u)\) and \(\lambda^{(2)}(u,v)\), derive

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

Then answer the following.

  1. Why does the mean function \(m(u)\) cancel out of the pair correlation function?

  2. What does this tell you about the role of the covariance function in an LGCP?

  3. Explain why the pair correlation function of an LGCP is always greater than or equal to \(1\) whenever \(C(u,v)\ge 0\).

  4. Can an LGCP model inhibition at short distances? Explain.

25.6 6. Stationary and isotropic LGCPs

Suppose now that \(Z(u)\) is stationary and isotropic, with

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

  1. Show that the LGCP intensity is constant: \[ \lambda = \exp\left(m+\tfrac{1}{2}C(0)\right). \]

  2. Show that \[ g(r)=\exp(C(r)). \]

  3. Explain why stationarity and isotropy of \(Z(u)\) imply stationarity and isotropy of the induced point process, at least at first and second order.

  4. What features of the point pattern would you expect to change if isotropy were dropped?

25.7 7. Exponential covariance kernel

Suppose the Gaussian random field has covariance

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

with \(\sigma^2>0\) and \(s>0\).

  1. Show that the LGCP PCF is \[ g(r)=\exp\left(\sigma^2\exp(-r/s)\right). \]

  2. Show that \[ g(0)=\exp(\sigma^2). \]

  3. Show that \[ \lim_{r\to\infty} g(r)=1. \]

  4. Explain the roles of \(\sigma^2\) and \(s\) in shaping the PCF.

  5. Sketch the qualitative shape of \(g(r)\) for:

    • small \(\sigma^2\), small \(s\)
    • small \(\sigma^2\), large \(s\)
    • large \(\sigma^2\), small \(s\)
    • large \(\sigma^2\), large \(s\)

25.8 8. Gaussian covariance kernel

Suppose instead that

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

  1. Derive the corresponding LGCP pair correlation function.

  2. Compute \(g(0)\).

  3. Show that \(g(r)\to 1\) as \(r\to\infty\).

  4. Compare qualitatively with the exponential covariance case:

    • which decays faster near the origin?
    • which produces a smoother PCF near \(r=0\)?

  5. Why might the choice of covariance kernel matter for fitted clustering behaviour?

25.9 9. Matérn-type covariance kernel

Suppose that the covariance of the Gaussian field is taken from a Matérn family,

\[ C(r)=\sigma^2 M_{\nu,s}(r), \]

where \(M_{\nu,s}(r)\) is a valid correlation function with smoothness parameter \(\nu\) and scale parameter \(s\).

  1. Write down the corresponding LGCP PCF.

  2. Show that \(g(0)=\exp(\sigma^2)\) regardless of the value of \(\nu\).

  3. Explain what changing \(\nu\) is expected to do to the shape of the PCF.

  4. Why is the Matérn class often attractive in spatial modelling?

  5. What practical difficulties arise when trying to estimate both \(\nu\) and \(s\) from point-pattern data?

25.10 10. Reparameterisation by clustering strength

For the exponential covariance model,

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

define

\[ \phi = g(0)-1. \]

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

  2. Invert this relationship to obtain \(\sigma^2\) in terms of \(\phi\).

  3. Show that the PCF can be written as \[ g(r) = (1+\phi)^{\exp(-r/s)}. \]

  4. Explain why \((\phi,s)\) is often more interpretable than \((\sigma^2,s)\).

  5. What does the case \(\phi=0\) correspond to?

25.11 11. Incorporating the intensity parameter

For a stationary LGCP with exponential covariance kernel, recall that

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

  1. Solve for \(m\) in terms of \(\lambda\) and \(\sigma^2\).

  2. Then substitute \(\sigma^2=\log(1+\phi)\) to express \(m\) in terms of \((\lambda,\phi)\).

  3. Show that \[ m = \log \lambda - \tfrac{1}{2}\log(1+\phi). \]

  4. Explain why the \((\lambda,\phi,s)\) parameterisation separates first- and second-order interpretation more clearly than \((m,\sigma^2,s)\).

  5. Why is this useful for simulation studies?

25.12 12. Small-covariance approximation

Suppose \(C(r)\) is small.

  1. Use the Taylor expansion of the exponential function to show that \[ g(r)=\exp(C(r)) \approx 1 + C(r) \] when \(C(r)\) is close to zero.

  2. Write down the next non-zero term in the expansion.

  3. Explain why this approximation is most accurate under weak clustering.

  4. What does this tell you about the relation between covariance decay and PCF decay in the weak-clustering regime?

  5. Why might this approximation be useful when comparing different point-process models?

25.13 13. Covariance of the intensity field

For an LGCP, the intensity field is

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

  1. Show that \[ \mathbb{E}[\Lambda(u)] = \lambda(u). \]

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

  3. Deduce the general Cox-process identity \[ g(u,v) = 1+\frac{\mathrm{Cov}(\Lambda(u),\Lambda(v))} {\lambda(u)\lambda(v)}. \]

  4. Interpret the covariance formula in words.

  5. Why is the covariance of the intensity field more directly relevant here than the covariance of the observed counts themselves?

25.14 14. Local behaviour near the origin

Consider the exponential-kernel LGCP:

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

  1. Compute \(g(0)\).

  2. Show that \(g(r)\) is decreasing in \(r\).

  3. Compute the derivative \(g'(r)\).

  4. Evaluate \(g'(0)\).

  5. Explain how the derivative near the origin reflects the clustering scale.

25.15 15. Large-distance behaviour

Let \(g(r)=\exp(C(r))\) for a stationary isotropic LGCP with \(C(r)\to 0\) as \(r\to\infty\).

  1. Show that \[ g(r)\to 1 \qquad \text{as } r\to\infty. \]

  2. Use a first-order expansion to show that, for large \(r\), \[ g(r)-1 \approx C(r). \]

  3. Interpret this in terms of long-range clustering.

  4. Why does every reasonable stationary LGCP eventually look approximately Poisson at sufficiently large distance?

  5. Does this mean the process is Poisson? Explain carefully.

25.16 16. Theory-style question: identifiability from second order

Suppose two stationary isotropic LGCPs have the same pair correlation function \(g(r)\) for all \(r\).

  1. Show that they must have the same covariance function \(C(r)\).

  2. Do they necessarily have the same intensity \(\lambda\)? Explain.

  3. Do they necessarily have the same mean parameter \(m\)? Explain.

  4. Explain what this tells you about what can and cannot be identified from the PCF alone.

  5. Why is it important to keep first- and second-order structure conceptually separate?

25.17 17. Simulation algorithm for an LGCP

Describe a practical simulation algorithm for an LGCP on a bounded window \(W\).

Your answer should address the following.

  1. How would you simulate the Gaussian random field \(Z(u)\)?

  2. Once \(Z(u)\) has been simulated on a grid, how would you obtain an approximation to \(\Lambda(u)\)?

  3. Given \(\Lambda(u)\) on a grid, how would you simulate the point process approximately?

  4. What sources of approximation error arise in this procedure?

  5. Why can highly variable LGCPs be computationally challenging to simulate accurately?

25.18 18. Implementation exercise: empirical verification of the LGCP intensity

Using R and spatstat, design a simulation study to verify empirically that a stationary LGCP has intensity

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

Your exercise should include:

  1. choosing values of \(m\), \(\sigma^2\), and \(s\),
  2. simulating many realisations,
  3. estimating the empirical average number of points per unit area,
  4. comparing this with the theoretical value,
  5. discussing finite-sample discrepancy.

You should also comment on how the realised number of points varies as \(\sigma^2\) increases.

25.19 19. Implementation exercise: empirical PCF vs theoretical PCF

Choose a stationary isotropic LGCP with a specified covariance kernel.

  1. Simulate one realisation on a bounded window.

  2. Estimate the empirical pair correlation function using a nonparametric estimator.

  3. Plot the empirical PCF against the theoretical curve.

  4. Repeat for at least two different values of the window size.

  5. Discuss:

    • instability near \(r=0\),
    • variability at large \(r\),
    • the effect of bandwidth,
    • the effect of stronger versus weaker clustering.

25.20 20. Implementation exercise: comparing covariance kernels

Choose two different covariance kernels for the Gaussian random field, for example:

  • exponential,
  • Gaussian,
  • Matérn.

Try to match them so that they have approximately the same:

  • intensity \(\lambda\),
  • clustering strength \(\phi=g(0)-1\),
  • clustering scale \(s\).

Then:

  1. write down the theoretical PCF for each model,
  2. plot the two PCFs on the same axes,
  3. describe where the main differences occur,
  4. simulate point patterns from both models,
  5. comment on how easy the two models are to distinguish visually and through empirical PCF estimation.

25.21 21. Implementation exercise: parameter sensitivity

Fix \(\lambda\) and vary one parameter at a time.

  1. Hold \(\lambda\) and \(s\) fixed, and vary \(\phi\).
  2. Hold \(\lambda\) and \(\phi\) fixed, and vary \(s\).
  3. For each case, plot the theoretical PCF.
  4. Simulate representative point patterns.
  5. Summarise what visual and second-order features are most sensitive to each parameter.

25.22 22. Theory and implementation bridge

Suppose you estimate the empirical PCF of a point pattern and it lies above \(1\) for short distances, decaying smoothly toward \(1\).

  1. Explain why an LGCP may be a plausible model.

  2. Explain why this observation alone is not enough to conclude that the data arise from an LGCP.

  3. What additional modelling assumptions are required?

  4. What features of the empirical PCF would be especially informative about \(\phi\) and \(s\)?

  5. What features of the point pattern itself are not fully captured by the PCF?

25.23 Summary questions

These are short synthesis questions.

  1. In one sentence, what is an LGCP?

  2. In one sentence, why does an LGCP cluster?

  3. In one sentence, what determines the LGCP pair correlation function?

  4. In one sentence, what is the role of \(\phi\)?

  5. In one sentence, what is the role of \(s\)?