21  Exercises

These exercises are intended to consolidate the general theory of Cox processes before moving to specific models such as the log-Gaussian Cox process.

Throughout, let \(X\) denote a Cox process driven by a non-negative random intensity field \(\Lambda(u)\) on a window \(W \subset \mathbb{R}^d\), so that

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

Unless stated otherwise, assume all required expectations exist.

21.1 1. Conditional vs unconditional structure

Let \(X\) be a Cox process driven by \(\Lambda(u)\).

  1. Explain carefully what is meant by the statement \[ X \mid \Lambda \sim \text{Poisson process with intensity } \Lambda(u). \]

  2. Show that conditional on \(\Lambda\), counts in disjoint sets are independent.

  3. Explain why this does not imply that counts in disjoint sets are independent unconditionally.

  4. Give an intuitive explanation for why a Cox process may appear clustered even though it is conditionally Poisson.

21.2 2. Mean measure of a Cox process

Let \(B \subset W\) be a Borel set.

  1. Show that \[ \mathbb{E}[N(B)\mid \Lambda] = \int_B \Lambda(u)\,du. \]

  2. Deduce that \[ \mathbb{E}[N(B)] = \mathbb{E}\left[\int_B \Lambda(u)\,du\right]. \]

  3. Under suitable regularity conditions, justify the interchange of expectation and integration, and conclude that \[ \mathbb{E}[N(B)] = \int_B \mathbb{E}[\Lambda(u)]\,du. \]

  4. Interpret this result in words.

21.3 3. Deriving the intensity function

Using only the conditional Poisson definition of a Cox process, derive from first principles that

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

Your derivation should begin with a small region \(du\) around \(u\) and should make clear where the approximation enters.

Note

The goal here is not just to state the formula, but to derive it in the same style used for ordinary point-process intensities.

21.4 4. Deriving the second-order product density

Let \(u \neq v\).

  1. Starting from disjoint infinitesimal regions \(du\) and \(dv\), derive \[ \lambda^{(2)}(u,v) = \mathbb{E}[\Lambda(u)\Lambda(v)]. \]

  2. Identify exactly where conditional independence is used.

  3. Show that this implies \[ \lambda^{(2)}(u,v) = \lambda(u)\lambda(v) + \mathrm{Cov}(\Lambda(u),\Lambda(v)). \]

  4. Explain why this decomposition is important for understanding clustering in Cox processes.

21.5 5. Pair correlation function of a Cox process

Starting from the definition

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

show that for a Cox process,

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

Then answer the following:

  1. What does the case \(g(u,v)=1\) correspond to?
  2. Why do Cox processes typically satisfy \(g(u,v)\ge 1\) for nearby locations?
  3. Can a Cox process ever have \(g(u,v)<1\)? Discuss carefully.

21.6 6. Poisson process as a special case

Suppose \(\Lambda(u)=\lambda(u)\) is deterministic.

  1. Show that the Cox process reduces to an inhomogeneous Poisson process.
  2. Verify directly that \[ \lambda(u)=\lambda(u), \qquad \lambda^{(2)}(u,v)=\lambda(u)\lambda(v), \qquad g(u,v)=1. \]
  3. Explain why this confirms that Cox processes are a genuine generalisation of Poisson processes.

21.7 7. Stationary and isotropic Cox processes

Suppose \(\Lambda(u)\) is stationary, so that \(\mathbb{E}[\Lambda(u)]=\lambda\) is constant and \(\mathrm{Cov}(\Lambda(u),\Lambda(v))\) depends only on \(u-v\).

  1. Show that the Cox process \(X\) is stationary at first and second order.

  2. If \(\Lambda(u)\) is also isotropic, show that \[ g(r) = 1 + \frac{\mathrm{Cov}(\Lambda(u),\Lambda(v))}{\lambda^2}, \qquad r=\|u-v\|. \]

  3. Explain why, in this case, the pair correlation function is essentially a rescaled covariance function of the latent intensity field.

  4. Why is this observation so useful when constructing parametric Cox process models?

21.8 8. A shifted exponential random-field Cox process

Suppose the random intensity field is

\[ \Lambda(u) = \mu + Y(u), \qquad \mu > 0, \]

where \(Y(u)\) is a non-negative random field with mean \(m(u)\) and covariance function \(C_Y(u,v)\).

  1. Derive \(\lambda(u)\).

  2. Derive \(\lambda^{(2)}(u,v)\).

  3. Derive \(g(u,v)\).

  4. Suppose now that \(Y(u)\) is stationary with mean \(m\) and covariance \(C_Y(r)\). Write \(g(r)\) explicitly.

  5. What happens to \(g(r)\) as \(\mu \to \infty\)? Interpret this result.

21.9 9. A binary-environment Cox process

Let \(Z(u)\) be a random field taking values in \(\{0,1\}\), and define

\[ \Lambda(u)=\lambda_0 + \alpha Z(u), \qquad \lambda_0>0,\ \alpha>0. \]

This may be interpreted as a model with random β€œon/off” environmental hotspots.

  1. Derive \(\lambda(u)\) in terms of \(\mathbb{E}[Z(u)]\).

  2. Derive \(\lambda^{(2)}(u,v)\) in terms of \(\mathbb{E}[Z(u)Z(v)]\).

  3. Derive the pair correlation function \(g(u,v)\).

  4. Show that if \(Z(u)\) and \(Z(v)\) are positively correlated, then the Cox process is clustered.

  5. Compare this model conceptually with a Cox process driven by a Gaussian random field. What kinds of latent environments might each be better suited to represent?

21.10 10. Quadratic transformation of a Gaussian random field

Let \(Z(u)\) be a mean-zero Gaussian random field with variance \(\sigma^2\) and correlation function \(\rho(u,v)\), and define

\[ \Lambda(u)=\mu + Z(u)^2, \qquad \mu \ge 0. \]

  1. Show that \[ \lambda(u)=\mu+\sigma^2. \]

  2. Using Gaussian moment identities, show that \[ \mathbb{E}[Z(u)^2 Z(v)^2] = \sigma^4 + 2\,\mathrm{Cov}(Z(u),Z(v))^2. \]

  3. Deduce that \[ \lambda^{(2)}(u,v) = (\mu+\sigma^2)^2 + 2\,\mathrm{Cov}(Z(u),Z(v))^2. \]

  4. If \(Z\) is stationary and isotropic with \[ \mathrm{Cov}(Z(u),Z(v))=\sigma^2\rho(r), \] show that \[ g(r) = 1 + \frac{2\sigma^4}{(\mu+\sigma^2)^2}\rho(r)^2. \]

  5. Explain why squaring the Gaussian field produces a qualitatively different PCF form from a model whose PCF is linear in \(\rho(r)\).

21.11 11. A higher-order transformed Cox process

Let \(Z(u)\) be a mean-zero Gaussian random field, and consider the intensity

\[ \Lambda(u)=\mu + Z(u)^4, \qquad \mu \ge 0. \]

This is a valid Cox process whenever \(\Lambda(u)\ge 0\) almost surely.

  1. Derive \(\lambda(u)\).

  2. Write down an expression for \(\lambda^{(2)}(u,v)\) in terms of the joint moments of \((Z(u),Z(v))\).

  3. Show that the resulting pair correlation function depends on moments up to order eight.

  4. Explain why transformed Gaussian-field Cox processes quickly become algebraically complicated as the transformation becomes more nonlinear.

  5. Discuss whether increasing the nonlinearity of the transformation is likely to make the process easier or harder to distinguish from other Cox models using only second-order structure.

Note

You do not need to fully simplify the eighth-order Gaussian moments unless you want an additional challenge.

21.12 12. Clustering without interaction

A recurring theme in Cox processes is that clustering is induced by the latent intensity field rather than by direct attraction between points.

  1. Explain carefully why points in a Cox process do not interact directly.

  2. Show how the formula \[ g(u,v) = 1 + \frac{\mathrm{Cov}(\Lambda(u),\Lambda(v))} {\mathbb{E}[\Lambda(u)]\mathbb{E}[\Lambda(v)]} \] formalises this idea.

  3. Give an example of a real-world situation where a Cox process would be a natural modelling choice.

  4. Give an example of a situation where a Cox process may be inappropriate because direct interaction between points is scientifically important.

  5. Briefly describe what class of point-process models is typically used in the latter case.

21.13 13. Simulation algorithm for a Cox process

Suppose you want to simulate a Cox process on a bounded window \(W\).

  1. Describe a generic simulation algorithm based on the conditional Poisson representation.

  2. What must be simulated first?

  3. Once a realisation of \(\Lambda(u)\) is available on \(W\), how can a point pattern be generated?

  4. If \(\Lambda(u)\) is only available on a fine grid, how might you approximately simulate the process?

  5. What numerical issues arise if the field is highly variable or strongly peaked?

21.14 14. Implementation exercise: empirical verification of the first-order moment identity

Using R and spatstat, design a simulation study to verify empirically that for a Cox process,

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

Your task should include:

  1. choosing a specific Cox model,
  2. simulating many realisations,
  3. estimating the average number of points per unit area,
  4. comparing this empirical estimate with the theoretical value.

Your write-up should state:

  • the window used,
  • the model parameters,
  • how many simulations were run,
  • how the empirical intensity estimate was computed,
  • what discrepancies you would expect for finite sample sizes.

21.15 15. Implementation exercise: empirical estimation of the PCF

Choose a stationary Cox process model and investigate how well the empirical pair correlation function recovers the theoretical one.

  1. Simulate a point pattern from the model.

  2. Estimate the empirical PCF using a nonparametric estimator.

  3. Plot the empirical and theoretical PCFs together.

  4. Repeat this for at least two different window sizes.

  5. Discuss:

    • the effect of sample size,
    • instability near \(r=0\),
    • behaviour at large \(r\),
    • the impact of bandwidth choice.

21.16 16. Comparing two Cox models through second-order structure

Consider two different Cox process models with the same first-order intensity \(\lambda\).

  1. Explain why they may still produce very similar empirical point patterns.

  2. Show that if their pair correlation functions are very similar, then they will be difficult to distinguish using only second-order methods.

  3. Why is this especially relevant when fitting models by minimum contrast to an estimated PCF?

  4. Suggest at least two possible ways one might try to distinguish such models beyond second-order structure.

21.17 17. From latent-field moments to model construction

Suppose you would like to construct a Cox process with a prescribed first-order intensity \(\lambda\) and a desired pair correlation function \(g(r)\).

  1. Using the general Cox process formulas, derive the condition that the latent field must satisfy.

  2. Explain why not every arbitrary function \(g(r)\) can necessarily arise from a valid Cox process.

  3. What properties must the covariance function of the latent intensity field satisfy?

  4. Why does this make random-field theory important for Cox process construction?

21.18 18. Theory-style challenge: covariance and clustering

Let \(X\) be a Cox process and let \(A,B \subset W\) be disjoint Borel sets.

  1. Show that \[ \mathrm{Cov}(N(A),N(B)) = \mathrm{Cov}\left(\int_A \Lambda(u)\,du,\ \int_B \Lambda(v)\,dv\right). \]

  2. Deduce that disjoint counts in a Cox process are generally positively correlated when the latent field has positive spatial dependence.

  3. Explain why this is impossible for an ordinary Poisson process.

  4. Interpret this result as a large-scale analogue of the pair correlation function formula.

21.19 Summary questions

These are intended as short synthesis questions rather than full derivations.

  1. In one sentence, what distinguishes a Cox process from a Poisson process?

  2. In one sentence, where does clustering come from in a Cox process?

  3. In one sentence, why is the pair correlation function of a Cox process determined by the moments of \(\Lambda(u)\)?

  4. In one sentence, why can two different Cox process models be difficult to distinguish using only second-order statistics?