12  Exercises

These exercises cover:

12.1 A. Core Concepts

  1. Define the second-order product density \(\lambda^{(2)}(u,v)\) and explain its interpretation in terms of joint occurrence of points.

  2. Explain why the condition \(u \neq v\) is required in the definition of \(\lambda^{(2)}(u,v)\).

  3. Show that for a process with independent points, \[ \lambda^{(2)}(u,v) = \lambda(u)\lambda(v). \]

  4. Explain, in words, what additional information \(\lambda^{(2)}(u,v)\) provides beyond the intensity function.

  5. Define the pair correlation function (PCF) and explain why it is a normalized quantity.

  6. Show that for a Poisson process, \(g(u,v) = 1\).

  7. Interpret the cases:

    • \(g(u,v) > 1\)
    • \(g(u,v) < 1\) in terms of spatial interaction.

  8. Explain why the PCF is generally easier to interpret than \(\lambda^{(2)}(u,v)\).

  9. Explain what is meant by writing the PCF as \(g(r)\), and what assumptions this implicitly encodes.

  10. Define the \(K\)-function and explain its interpretation in terms of expected neighbours.

  11. Show that for a Poisson process, \(K(r) = \pi r^2\).

  12. Explain the key conceptual difference between:

    • the PCF \(g(r)\)
    • the \(K\)-function \(K(r)\)

12.2 B. Interpretation & Structure

  1. Two point processes have identical intensity functions. Explain how their PCFs could still differ.

  2. Suppose \(g(r)\) is large near \(r=0\) and decays to 1. What does this imply about clustering and its scale?

  3. Suppose \(g(r)<1\) for small \(r\) but approaches 1. What type of interaction does this indicate?

  4. Explain how the PCF encodes both strength and range of interaction.

  5. Why does the PCF provide more detailed information about interaction than the \(K\)-function?

  6. Explain why the \(K\)-function can obscure local features of interaction.

  7. Give an example (conceptual or constructed) where two different PCFs could produce very similar \(K\)-functions.

  8. Explain why cumulative summaries tend to be smoother than local summaries.

  9. Why is the PCF typically noisier to estimate than the \(K\)-function?

  10. Explain why estimates near \(r=0\) are often unstable.

  11. Explain why estimates at large \(r\) are unreliable.

  12. Why are moderate distances typically the most informative region for second-order analysis?

12.3 C. PCF–K Relationship

  1. Show that if \(g(r) = 1\) for all \(r\), then \(K(r) = \pi r^2\).

  2. Given \[ g(r) = 1 + A e^{-br}, \] write down an expression for \(K(r)\).

  3. Explain how the parameters in \(g(r) = 1 + \phi e^{-r/s}\) relate to clustering strength and scale.

  4. Construct two PCFs:

    • one with a sharp peak near \(r=0\)
    • one with a broader peak

    Predict qualitatively how their \(K\)-functions will differ.

  5. Explain why differentiating an estimated \(K(r)\) is not a stable way to estimate \(g(r)\) in practice.

  6. Explain why estimating a density-type function (PCF) is inherently harder than estimating a cumulative function (\(K\)).

12.4 D. Estimation Theory

  1. Explain why all second-order estimators are based on pairwise distances.

  2. Write down the standard estimator of the \(K\)-function and explain the role of each component.

  3. Write down a kernel-based estimator of the PCF and explain:

    • the kernel
    • the bandwidth
    • the \(2\pi r\) factor

  4. Explain why edge effects arise in second-order estimation.

  5. Describe two different approaches to handling edge effects.

  6. Explain how bandwidth affects bias and variance in PCF estimation.

  7. Why can different bandwidth choices lead to different scientific conclusions?

  8. Explain why intensity estimation is required for second-order estimation.

  9. What goes wrong if a homogeneous estimator is applied to strongly inhomogeneous data?

  10. Explain the purpose of inhomogeneous second-order estimators.

12.5 E. Interpretation of Estimates

  1. An estimated PCF shows a strong spike near \(r=0\) and quickly decays. What does this suggest?

  2. An estimated PCF is approximately flat at 1. What does this suggest?

  3. An estimated PCF dips below 1 near the origin and rises above 1 later. Give a possible interpretation.

  4. An estimated \(K(r)\) lies above \(\pi r^2\) for small \(r\). What does this indicate?

  5. Explain how it is possible for two processes to have nearly identical \(K\)-functions but different PCFs.

  6. Explain why visual inspection of a point pattern may disagree with second-order summaries.

  7. Why should small fluctuations in estimated PCFs be interpreted cautiously?

  8. Explain how first-order inhomogeneity can be mistaken for clustering.

12.6 F. Implementation (spatstat)

  1. Simulate a homogeneous Poisson process using rpoispp(). Plot the result.

  2. Compute and plot its \(K\)-function and compare it to \(\pi r^2\).

  3. Compute and plot its PCF and compare it to \(g(r)=1\).

  4. Repeat the above for several realizations. Describe the variability.

  5. Simulate a clustered process (e.g. Thomas process) and compare its PCF to a Poisson process.

  6. Compare the \(K\)-functions of a clustered process and a Poisson process. Which shows clustering more clearly?

  7. Simulate patterns with different intensities. Compare their PCFs. What remains invariant?

12.7 G. Edge Effects & Bandwidth

  1. Estimate the \(K\)-function of a point pattern using two different edge corrections. Compare the results.

  2. Estimate the PCF using multiple bandwidths and overlay the curves. Describe the effect of bandwidth.

  3. Explain why edge correction differences become more pronounced at larger distances.

  4. Compare estimates from a small window and a large window. Which is more stable and why?

12.8 H. Inhomogeneity

  1. Simulate an inhomogeneous Poisson process and compare:
    • homogeneous PCF / K estimates
    • inhomogeneous versions
    Explain the differences and why they arise.