Geostatistics

Geostatistics - characterisation of spatial variability, creation of maps from point observations through spatial interpolation, and quantification of the accuracy of interpolated maps

Autocorrelation function for a time series

For a time series \(Z(t_1),...,Z(t_n)\) with \(E[Z(t_i)] = 0\) and \(var(Z(t_i)) = \sigma^2, i = 1,...,n\) the covariance function is a function of the distance in time \(t_j - t_i\) \[cov(Z(t_i),Z(t_j)) = E[Z(t_i)Z(t_j)] = C(t_j-t_i).\]

The autocorrelation function is \[R(t_j-t_i) = \frac{cov(Z(t_i),Z(t_j))}{\sqrt{var(Z(t_i))var(Z(t_j))}} = \frac{C(t_j- t_i)}{C(0)}.\]

But time has direction, space does not…

Autocorrelation function in space

Covariance function: \(C(s,h) = cov(Z(s),Z(s+h)) = E[(Z(s) - \mu(s))(Z(s+h) - \mu(s+h))]\)

Autocorrelation function (correlogram): \[R(s,h) = \frac{C(s,h)}{\sqrt{var(Z(s)) var(Z(s+h))}}\]

Edge effects are more important than in time series since potentially many points located at the domain boundary.