Gaussian random fields: properties
GRFs can be:
- strictly stationary: if \(Z(\cdot)\) is invariant to shifts: constant distribution
- weakly stationary (2nd order stationarity):
- process has constant mean \(E[Z(s)]\)
- covariances depend only on differences between locations: \(C(h) = Cov(Z(s), Z(s + h))\)
- if ‘differences between locations’ only involve distance, not direction, then the process is isotropic (opposite: anisotropic).
- intrinsically stationary:
- weakly stationary
- the variogram \(Var[Z(s_i) - Z(s_j)]\) follows from the distance \(s_i - s_j\).