17  Covariance Functions

In the previous section, we introduced Gaussian random fields and saw that they are fully characterized by their mean and covariance functions.

In practice, the choice of covariance function plays a central role, as it determines the spatial dependence structure of the field.

17.1 Definition

A function \(C : \mathbb{R}^d \to \mathbb{R}\) is called a covariance function if there exists a random field \(Z(u)\) such that

\[ C(u,v) = \mathrm{Cov}(Z(u), Z(v)). \]

Under stationarity, this reduces to:

\[ C(u,v) = C(u - v) = C(h). \]

17.2 Positive Definiteness

Not every function is a valid covariance function.

A function \(C(h)\) is valid if and only if it is positive definite, meaning that for any points \(u_1, \dots, u_n\) and coefficients \(a_1, \dots, a_n\),

\[ \sum_{i=1}^n \sum_{j=1}^n a_i a_j C(u_i - u_j) \ge 0. \]

Note

This condition ensures that the covariance matrix of \((Z(u_1), \dots, Z(u_n))\) is always non-negative definite.

17.3 Variance and Correlation

At zero lag:

\[ C(0) = \mathrm{Var}(Z(u)) = \sigma^2. \]

We often define the correlation function:

\[ \rho(h) = \frac{C(h)}{\sigma^2}, \]

so that:

  • \(\rho(0) = 1\)
  • \(|\rho(h)| \le 1\)

17.4 Isotropy

If the field is isotropic, then:

\[ C(h) = C(\|h\|) = C(r). \]

So the covariance depends only on distance.

17.5 Common Covariance Functions

17.5.1 Exponential covariance

\[ C(r) = \sigma^2 \exp\left(-\frac{r}{s}\right) \]

  • Short-range dependence
  • Non-smooth sample paths
  • Used throughout these notes

17.5.2 Gaussian covariance

\[ C(r) = \sigma^2 \exp\left(-\frac{r^2}{s^2}\right) \]

  • Very smooth fields
  • Rapid decay of correlation

17.5.3 Matérn covariance

\[ C(r) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)} \left( \frac{r}{s} \right)^\nu K_\nu\left(\frac{r}{s}\right) \]

  • \(\nu\) controls smoothness
  • Includes exponential and Gaussian as special/limiting cases

17.6 Interpretation

The covariance function controls:

  • Strength of dependence via \(\sigma^2\)
  • Range of dependence via \(s\)
  • Smoothness of the field via functional form

17.7 Connection to Pair Correlation Functions

The covariance structure of the underlying field directly determines the second-order properties of Cox processes.

For example:

  • LGCP: \[ g(r) = \exp(C(r)) \]

  • CSCP (mean-zero, single component): \[ g(r) = 1 + 2\rho(r)^2 \]

17.8 Key Insight

Different models can produce very similar pair correlation functions when driven by the same covariance structure.

For example, if

\[ C(r) = \sigma^2 \exp(-r/s), \]

then:

  • LGCP: \[ g(r) = \exp(\sigma^2 e^{-r/s}) \]

  • CSCP: \[ g(r) = 1 + 2 e^{-2r/s} \]

These can be very similar in shape for suitable parameter choices.

17.9 Summary

  • Covariance functions determine spatial dependence in random fields
  • Valid covariance functions must be positive definite
  • Under isotropy, they depend only on distance
  • The choice of covariance function directly influences the PCF of Cox processes

Understanding covariance functions is therefore essential for modelling and interpreting spatial clustering.