16 Gaussian Random Fields
In many spatial models, randomness is not attached directly to points, but to an underlying field over space.
Why?
These are known as random fields.
Random fields play a central role in Cox processes, where they are used to construct random intensity functions.
16.1 Defintion
A random field on \(\mathbb{R}^d\) is a collection of random variables:
\[ \{ Z(u) : u \in \mathbb{R}^d\} \]
That is, each location \(u\) is associated with a random variable \(Z(u)\).
Intuitively, a random field can be thought of as:
A random surface over a space.
- In 1D, a random curve
- In 2D, a random surface
- In higher dimensions, a random function
Each realization gives a deterministic function, but the function itself is random.
16.2 Mean and covaraince functions
The distribution of a random field is characterised (at second order) by:
Mean function
\[ m(u) = \mathbb{E}[Z(u)] \]
Covariance function
\[ C(u, v) = \mathrm{Cov}(Z(u),Z(v)) \]
16.3 Stationarity
A random field is second-order stationary if:
- \(m(u) = m\) (constant)
- \(C(u, v) = C(u - v)\)
16.4 Isotropy
A stationary random field is isotropic if:
\[ C(u, v) = C(\|u - v\|) \]
So covariance depends only on distance.
16.5 Gaussian random fields
A random field is Gaussian if:
Any finite collection \((Z(u_1), \dots, Z(u_n))\) has a multivariate normal distribution.
We write:
\[ Z(u) \sim \mathcal{GP}(m(u), C(u, v)) \]
16.6 Why Gaussian fields
Gaussian random fields are widely used because:
They are fully characterized by the mean and covariance
Mathematically tractable
Flexible via choice of covariance function
16.7 Common covariance functions
A common choice (used throughout these notes) is the exponential covariance:
\[ C(r) = \sigma^2 \exp(\frac{-r}{s}) \]
where:
- \(\sigma^2\) controls variance
- \(s\) controls correlation range
16.8 Correlation function
Often we write:
\[ \rho(r) = \frac{C(r)}{\sigma^2} \]
so that:
\[ C(r) = \sigma^2 \rho(r) \]
16.9 Sample paths
A realization of a random field is called a sample path.
16.10 Connection to Cox Processes
Random fields are used to define random intensity functions:
Log-Gaussian Cox process (LGCP): \[ \Lambda(u) = \exp(Z(u)) \]
Chi-square Cox process (CSCP): \[ \Lambda(u) = \mu + Z(u)^2 \]
Shot-noise Cox process: \[ \Lambda(u) = \sum_{i} k(u - x_i) \]
where \(\{x_i\}\) are random locations and \(k(\cdot)\) is a kernel function.
Gamma Cox process: \[ \Lambda(u) \sim \text{Gamma random field} \]
Permanental Cox process: \[ \Lambda(u) = \sum_{i=1}^k Z_i(u)^2 \]
where \(Z_i(u)\) are independent Gaussian random fields.
Different choices of random field and transformation lead to different clustering behaviour, even when the underlying spatial correlation structure is similar.
16.11 Summary
A random field is:
- A collection of random variables indexed by space
- Characterized by mean and covariance
- Often assumed stationary and isotropic
Gaussian random fields are particularly important, as they provide the foundation for many Cox process models.