4  Exercises

These exercises are designed to reinforce understanding of point processes and the Poisson process. They should be attempted without notes first, then refined.

4.1 A. Core Definitions

  1. What is a counting measure?
    • Give both a formal definition and an intuitive interpretation.

Formal definition:
A counting measure \(N(\cdot)\) assigns to each set \(B \subset W\) the number of points in \(B\):

\[ N(B) = \sum_{i=1}^n \mathbf{1}\{x_i \in B\}. \]

Intuition:
A counting measure simply counts how many points fall inside any region \(B\).

  1. What is a point process?

      1. Definition via counting measure
      1. Definition via random set / configuration

  2. What does it mean for a point process to be simple?

  3. What is a Poisson process?

      1. Definition via counts
      1. Definition via conditional construction

  4. What is the difference between:

    • a homogeneous Poisson process (HPPP)
    • an inhomogeneous Poisson process (IPPP)?

4.2 B. Interpretation & Intuition

  1. What does the intensity parameter ( ) represent in a Poisson process?

  2. Why is the Poisson process called complete spatial randomness (CSR)?

  3. What does independent increments mean, and why is it important?

  4. Why can’t a density function model a spatial point pattern?

  5. Explain the difference between:

  • randomness in counts
  • randomness in locations
  1. Give an example of:
    1. clustered data
    1. regular (inhibited) data that would violate a Poisson process assumption.

4.3 C. Basic Derivations

  1. Show that for a homogeneous Poisson process: [ [N(B)] = |B|]

  2. Show that: [ (N(B)) = |B|]

  3. Show that for disjoint sets ( B_1, B_2 ): [ (N(B_1), N(B_2)) = 0]

  4. For an inhomogeneous Poisson process, show that: [ [N(B)] = _B (u),du]

4.4 D. Conditional Structure

  1. Show that for a HPPP, conditional on ( N = n ), the points are i.i.d. uniform on ( W ).

  2. Write down the joint density: [ (N=n, x_1,,x_n)] for a homogeneous Poisson process.

  3. Explain why the Poisson process can be viewed as:

  • random number of points
  • i.i.d. locations given the count

4.5 E. Construction & Simulation

  1. Describe how to simulate a homogeneous Poisson process using the count-first method.

  2. Describe how to simulate an inhomogeneous Poisson process using thinning.

  3. Explain why thinning works.

  4. Explain why the Poisson process avoids the variable dimension problem.

4.6 F. R Coding Exercises

  1. Write R code to simulate a homogeneous Poisson process on a rectangular window.

  2. Write R code to simulate an inhomogeneous Poisson process using thinning.

  3. Modify your code to:

  • simulate multiple realisations
  • compute the average number of points
  1. Plot:
  • a single realisation
  • several realisations side-by-side

  1. Empirically verify that: [ [N(W)] |W|] by simulation.

  2. Simulate an IPPP with intensity: [ (u) = 100 (-|u|)] and plot the resulting point pattern.

4.7 G. Conceptual Understanding

  1. Compare:
  • Poisson process vs general point process
  1. Compare:
  • Poisson process vs random field

  1. Explain how a Cox process generalises a Poisson process (no formulas required).

  2. What assumption of the Poisson process is violated if:

  • points appear in clusters?
  • points repel each other?

4.8 H. Stretch Exercises

  1. Show that independent increments imply no interaction between points.

  2. Explain why the Poisson process has no second-order structure.

  3. Explain what kind of statistical summaries would detect deviations from a Poisson process.

4.9 I. Oral / Conceptual Practice

  1. Explain the Poisson process to a non-statistician in under 2 minutes.

  2. Describe what a Poisson point pattern “looks like”.

  3. Why is the Poisson process used as a baseline in spatial statistics?