30  Exercises

30.1 Likelihood

  1. Consider a homogeneous Poisson process with intensity \(\lambda\) on a window \(W\).

    1. Starting from the Poisson distribution of \(N(W)\), derive the likelihood \[ L(\lambda; X) = e^{-\lambda |W|} \lambda^n. \]

    2. Show that the MLE is \[ \hat{\lambda} = \frac{n}{|W|}. \]

Answer

  1. \(N(W) \sim \text{Poisson}(\lambda |W|)\), so
    \[ \mathbb{P}(N(W)=n) = \frac{(\lambda |W|)^n}{n!} e^{-\lambda |W|}. \] Ignoring constants not depending on \(\lambda\) gives
    \[ L(\lambda) \propto e^{-\lambda |W|} \lambda^n. \]

  2. Log-likelihood: \[ \ell(\lambda) = n \log \lambda - \lambda |W|. \] Differentiate: \[ \frac{d\ell}{d\lambda} = \frac{n}{\lambda} - |W| = 0 \Rightarrow \hat{\lambda} = \frac{n}{|W|}. \]

  1. Starting from the partition argument in the notes, explain precisely where the term \[ \int_W \lambda(u)\,du \] comes from.

Then explain why this term corresponds to the expected number of points.

Answer

It arises as the limit of the sum over small cells: \[ \sum_j \lambda(u_j)|B_j| \to \int_W \lambda(u)\,du. \]

For a Poisson process, \[ \mathbb{E}[N(W)] = \int_W \lambda(u)\,du, \] so this term represents the expected total number of points.

  1. Let \(\Lambda(u)\) be a random intensity field.

Show that the Cox process likelihood can be written as \[ L(\theta;X) = \mathbb{E}_\Lambda\left[L(\theta;X \mid \Lambda)\right]. \]

Explain why this is analogous to a mixture model.

Answer

By definition: \[ L(\theta;X) = \int L(\theta;X \mid \Lambda)\, dP(\Lambda). \]

This is identical in structure to mixture models: \[ f(x) = \int f(x \mid z)\, dP(z). \]

Here, \(\Lambda\) plays the role of a latent variable.

30.2 Minimum Contrast

  1. Show that the minimum contrast estimator based on the PCF,

\[ C(\theta) = \int w(r)\,[\hat{g}(r) - g(r;\theta)]^2 dr, \]

is a functional analogue of ordinary least squares.

What plays the role of: - response variable? - predictor? - residual?

Answer

  • response: \(\hat{g}(r)\)
  • predictor: \(g(r;\theta)\)
  • residual: \(\hat{g}(r) - g(r;\theta)\)

So the contrast is an \(L^2\) loss over functions.

  1. Suppose \[ g(r;\theta) = 1 + A e^{-2r/s}. \]

  1. Show that \[ \log(g(r)-1) = \log A - \frac{2}{s} r. \]

  2. Explain how this leads to a linear regression estimator.

  3. Discuss when this transformation may be unstable.

Answer

  1. Direct log: \[ \log(g(r)-1) = \log A - \frac{2}{s}r. \]

  2. Regress \(y = \log(\hat{g}(r)-1)\) on \(r\).

  3. When \(\hat{g}(r) \approx 1\), the log is unstable.

  1. Explain why the choice of fitting range \([r_{\min}, r_{\max}]\) is critical.

Your answer should discuss:

  • behaviour near \(r=0\)
  • behaviour for large \(r\)
  • bias–variance trade-off
  1. Suppose two models have identical PCFs \(g(r)\) but different higher-order structure.

Show that minimum contrast based on the PCF cannot distinguish between them.

30.3 Composite Likelihood

  1. Starting from \[ \lambda^{(2)}(u,v) = \lambda(u)\lambda(v) g(u,v), \]

derive the pairwise composite likelihood

\[ L_{\text{pair}}(\theta) \propto \prod_{i<j} \lambda^{(2)}(x_i,x_j). \]

Then express it in terms of \(\lambda\) and \(g\).

  1. In the stationary case, show that the pairwise log-likelihood reduces to

\[ \ell(\theta) = \sum_{i<j} \log g(r_{ij}) + \text{constant}. \]

Explain why this makes the estimator depend only on the PCF.

  1. Compare minimum contrast and composite likelihood:
  1. What data do they use?
  2. What structure do they rely on?
  3. Why might they give similar estimates?

30.4 Pseudolikelihood

  1. Show that for a Poisson process,

\[ \lambda(u \mid X) = \lambda(u). \]

Explain why this implies pseudolikelihood equals the true likelihood.

  1. For a Cox process, show that

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

Explain why this is difficult to compute.

  1. Compare Cox and Gibbs processes in terms of conditional intensity:
  • which has tractable \(\lambda(u \mid X)\)?
  • which is easier for pseudolikelihood?

30.5 Simulation and Applied Exercises

  1. Poisson likelihood simulation

Simulate a homogeneous Poisson process with known \(\lambda\).

  1. Estimate \(\lambda\) using the likelihood.
  2. Repeat over many simulations.
  3. Verify empirically that the estimator is unbiased.
  1. Minimum contrast fitting

Simulate a clustered point process (e.g. LGCP or Thomas).

  1. Estimate \(\hat{g}(r)\)
  2. Fit parameters using minimum contrast
  3. Plot \(\hat{g}(r)\) and fitted \(g(r)\)

Investigate sensitivity to:

  • fitting range
  • weighting
  1. Composite likelihood vs minimum contrast

Using the same simulated dataset:

  1. Fit parameters using minimum contrast
  2. Fit parameters using pairwise likelihood (if implemented)
  3. Compare estimates and fitted PCFs
  1. Identifiability experiment (important)

Simulate two processes with similar PCFs:

  • one LGCP
  • one CSCP
  1. Compute empirical \(\hat{g}(r)\)
  2. Fit both models using minimum contrast
  3. Compare fitted curves

Discuss:

  • how similar the fits are
  • what this implies about identifiability
  1. Effect of sample size

Repeat Exercise 17 for increasing window sizes.

Explain:

  • how estimation improves
  • whether model distinguishability improves

30.6 Challenging / Theory Extensions

  1. Suppose \(g(r) = 1 + \epsilon h(r)\) for small \(\epsilon\).

Show that, to first order,

\[ \log g(r) \approx \epsilon h(r). \]

Explain why this suggests different models may look similar under weak clustering.

  1. Consider two models:
  • Model A: \(g_A(r) = \exp(C(r))\)
  • Model B: \(g_B(r) = 1 + C(r)\)

Show that for small \(C(r)\),

\[ g_A(r) \approx g_B(r). \]

Explain why this is relevant for LGCP vs CSCP.