18  Exercises

18.1 Random Fields & Covariance Functions

18.1.1 1. Finite-dimensional distributions

Let \(\{Z(u): u \in \mathbb{R}^d\}\) be a Gaussian random field.

  1. State precisely what it means for \(Z(u)\) to be Gaussian.

  2. Let \(u_1, \dots, u_n\) be fixed locations. Write down the joint distribution of \[ (Z(u_1), \dots, Z(u_n)). \]

  3. Show that this distribution is completely determined by the mean vector and covariance matrix.

18.1.2 2. Stationarity of a random field

Let \(Z(u)\) be a random field with mean function \(m(u)\) and covariance \(C(u,v)\).

  1. Write down the definition of second-order stationarity.

  2. Show that under stationarity, \[ \mathrm{Var}(Z(u)) = C(0) \] is constant.

  3. Give an example of a mean function \(m(u)\) that violates stationarity.

18.1.3 3. Covariance vs correlation

Let \(C(h)\) be a covariance function with variance \(\sigma^2\).

  1. Define the correlation function \(\rho(h)\).

  2. Show that \[ |\rho(h)| \le 1. \]

  3. Explain why \(\rho(h)\) is often more interpretable than \(C(h)\).

18.1.4 4. Positive definiteness

Let \(C(h)\) be a candidate covariance function.

  1. State the definition of positive definiteness.

  2. Show that if \(C(h)\) is a valid covariance function, then for any \(u_1, \dots, u_n\), the matrix \[ [C(u_i - u_j)]_{i,j} \] is non-negative definite.

  3. Consider the function \[ C(h) = \exp(\|h\|). \] Explain why this cannot be a valid covariance function.

18.1.5 5. Isotropy

Let \(Z(u)\) be a stationary random field.

  1. Define isotropy in terms of the covariance function.

  2. Show that isotropy implies \[ C(h) = C(\|h\|). \]

  3. Give an example of a covariance function that is stationary but not isotropic.

18.1.6 6. Exponential covariance structure

Consider the covariance function \[ C(r) = \sigma^2 \exp(-r/s). \]

  1. Show that \(C(0) = \sigma^2\).

  2. Compute the correlation function \(\rho(r)\).

  3. Interpret the role of the parameter \(s\).

18.1.7 7. From covariance to PCF (LGCP)

Let \(Z(u)\) be a mean-zero Gaussian random field with covariance \(C(r)\), and define \[ \Lambda(u) = \exp(Z(u)). \]

  1. Show that \[ \mathbb{E}[\Lambda(u)] = \exp\left(\frac{1}{2} C(0)\right). \]

  2. Show that \[ \mathbb{E}[\Lambda(u)\Lambda(v)] = \exp\left(C(0) + C(u-v)\right). \]

  3. Deduce that the pair correlation function is \[ g(r) = \exp(C(r)). \]

18.1.8 8. From covariance to PCF (CSCP)

Let \(Z(u)\) be a mean-zero Gaussian random field with variance \(\sigma^2\) and correlation \(\rho(r)\), and define \[ \Lambda(u) = Z(u)^2. \]

  1. Show that \[ \mathbb{E}[\Lambda(u)] = \sigma^2. \]

  2. Show that \[ \mathbb{E}[Z(u)^2 Z(v)^2] = \sigma^4 (1 + 2\rho(r)^2). \]

  3. Deduce that \[ g(r) = 1 + 2\rho(r)^2. \]

18.1.9 9. Comparing functional forms

Consider the two PCFs:

\[ g_{\text{LGCP}}(r) = \exp(C(r)), \quad g_{\text{CSCP}}(r) = 1 + 2\rho(r)^2. \]

  1. Let \(C(r) = \sigma^2 \rho(r)\). Show that for small \(\sigma^2\), \[ g_{\text{LGCP}}(r) \approx 1 + \sigma^2 \rho(r). \]

  2. Compare this approximation with the CSCP form.

  3. Explain why the two models may be difficult to distinguish based on second-order structure alone.

18.1.10 10. Effect of covariance choice

Consider two covariance functions:

  • Exponential: \(C_1(r) = \sigma^2 e^{-r/s}\)
  • Gaussian: \(C_2(r) = \sigma^2 e^{-r^2/s^2}\)

  1. Sketch (qualitatively) both functions.

  2. Describe how the decay rate differs.

  3. Explain how this would affect clustering in a Cox process driven by these fields.

18.1.11 11. Range of dependence

Let \(C(r)\) be a covariance function.

  1. Define what is meant by the range of dependence.

  2. For the exponential covariance, find \(r\) such that \[ \rho(r) = 0.05. \]

  3. Interpret this value in terms of spatial dependence.

18.1.12 12. Structural insight

Consider two Cox processes driven by the same Gaussian random field \(Z(u)\) but with different transformations:

\[ \Lambda_1(u) = \exp(Z(u)), \quad \Lambda_2(u) = Z(u)^2. \]

  1. Explain why both processes inherit the same covariance structure from \(Z(u)\).

  2. Explain why their pair correlation functions differ.

  3. Discuss why these differences may be difficult to detect in practice.