Kriging
Kriging is Gaussian process interpolation developed in geostatistics, predicting a spatial field with a best linear unbiased estimator.
Origins and equivalence
Kriging arose in geostatistics to estimate mineral concentrations from scattered boreholes and is named after the mining engineer D. G. Krige. Mathematically it is the same object as Gaussian process regression: a best linear unbiased predictor of a spatial field, with a variance that quantifies the estimate's reliability. The vocabulary differs, but the equations coincide.
The variogram
Geostatistics describes spatial correlation through the variogram, which measures how the expected squared difference between two field values grows with the distance between them. The variogram is fit to the data and then translated into the covariance function used for prediction. Its range, sill, and nugget correspond to a GP's length scale, signal variance, and noise variance.
Kinds of kriging
- Simple kriging: assumes a known constant mean
- Ordinary kriging: estimates an unknown constant mean from the data
- Universal kriging: allows a spatially varying trend, such as a polynomial
- Co-kriging: uses correlated secondary variables to improve estimates
Why the variance matters
Kriging returns not only an estimate at each unsampled location but also a kriging variance that is zero at observations and rises with distance from them. This map of uncertainty is used to decide where new samples would be most informative, the foundation of sampling design and, in modern surrogate optimization, of acquisition functions that choose the next expensive evaluation.
Modern use
Beyond geoscience, kriging is a workhorse for building surrogate models of expensive simulations. An engineer runs a computer experiment at a limited set of design points, fits a kriging surrogate, and uses it to explore the design space cheaply while the variance flags regions where the surrogate should not yet be trusted.