15  Exercises

15.1 Stationarity & Isotropy

15.1.1 1. Translation invariance

Let \(X\) be a point process on \(\mathbb{R}^d\).

  1. Write down the formal definition of stationarity using counting measures.

  2. Show that if \(X\) is stationary, then for any bounded set \(B\),

\[ \mathbb{E}[N(B)] = \mathbb{E}[N(B + h)] \]

for all \(h \in \mathbb{R}^d\).

  1. Explain in words what this implies about the spatial distribution of points.

15.1.2 2. Constant intensity

Suppose \(X\) is stationary and admits an intensity function \(\lambda(u)\).

  1. Starting from the definition of intensity, show that

\[ \lambda(u) = \lambda(u + h) \]

for all \(h\).

  1. Deduce that \(\lambda(u)\) must be constant.

  2. Give an example of a process with constant intensity that is not stationary.

15.1.3 3. Second-order structure under stationarity

Let \(X\) be stationary with second-order product density \(\lambda^{(2)}(u,v)\).

  1. Show that

\[ \lambda^{(2)}(u,v) = \lambda^{(2)}(u + h, v + h) \]

for all \(h\).

  1. Let \(h = u - v\). Show that there exists a function \(\gamma\) such that

\[ \lambda^{(2)}(u,v) = \gamma(u - v). \]

  1. Explain why this reduces a function of two variables to one.

15.1.4 4. From stationarity to the PCF

Assume stationarity and that \(\lambda(u) = \lambda\).

  1. Show that the pair correlation function can be written as

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

  1. Explain why this is still a function of a vector, not a scalar.

15.1.5 5. Rotational invariance

Let \(X\) be stationary.

  1. Write down the formal definition of isotropy using rotations.

  2. Show that if \(X\) is isotropic, then for any rotation \(R\),

\[ \lambda^{(2)}(h) = \lambda^{(2)}(Rh). \]

  1. Explain what this implies about directional dependence.

15.1.6 6. Reduction to distance

Assume stationarity and isotropy.

  1. Show that there exists a function \(f\) such that

\[ \lambda^{(2)}(u,v) = f(\|u - v\|). \]

  1. Deduce that the pair correlation function can be written as

\[ g(r), \quad r = \|u - v\|. \]

  1. Explain why this reduction is crucial for practical estimation.

15.1.7 7. Interpreting anisotropy

Consider a point pattern where clustering is stronger in the horizontal direction than the vertical direction.

  1. Would this process be stationary? Justify your answer.

  2. Would this process be isotropic? Justify your answer.

  3. Sketch (qualitatively) what \(g(u,v)\) might look like in this case.

15.1.8 8. When assumptions fail

Suppose a process has intensity

\[ \lambda(u) = 2 + \sin(u_1), \]

where \(u = (u_1, u_2)\).

  1. Is the process stationary? Why or why not?

  2. Could the process be isotropic? Explain.

  3. Suggest a transformation or modelling approach to handle this situation.

15.1.9 9. Connection to practice

You are given an observed point pattern and its estimated PCF \(\hat g(r)\).

  1. What assumptions are required for \(\hat g(r)\) to be a valid summary?

  2. What would go wrong if the process were strongly anisotropic?

  3. Suggest one way to diagnose anisotropy from data.

15.1.10 10. Linking assumptions together

Complete the following chain of simplifications:

\[ g(u,v) \;\longrightarrow\; g(\,\_\_\_\_) \;\longrightarrow\; g(\,\_\_\_\_) \]

  1. Fill in the missing expressions.

  2. State the assumption used at each step.

  3. Briefly explain why this chain is central to spatial statistics.