Optimal Spatial Prediction with Kriging

[latexpage]Suppose we are modeling a spatial process (for instance, the amount of rainfall around the world, the distribution of natural resources, or the population density of an endangered species). We’ve measured the latent function $Z$ at some locations ${\bf s}_1, \ldots, {\bf s}_N$, and we’d like to predict the function’s value at some new location ${\bf s}_0$. Kriging is a technique for extrapolating our measurements to arbitrary locations. For an in-depth discussion, see Cressie and Wikle (2011). Here I derive Kriging in a simplified case.

I will assume that $Z$ is an intrinsically stationary process. In other words, there exists some semivariogram $\gamma({\bf h})$ such that

\[ \text{var}[Z({\bf s}+{\bf h}) – Z({\bf s})] = 2\gamma({\bf h}) . \]

Furthermore, I will assume that the process is isotropic, (i.e. that $\gamma({\bf h})$ is a function only of $||h||$). As Andy described here, the existence of a covariance function implies intrinsic stationarity. In addition, I will assume that the process has a constant mean, $\mathbb E[Z({\bf s})] = \mu$. We would like to estimate $Z({\bf s})$ with a linear combination of our current observations. Our estimator will be Continue reading “Optimal Spatial Prediction with Kriging”