Kriging

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Kriging is a regression technique used in geostatistics. It is named after its inventor, Danie G. Krige. In the statistical community, it is more commonly known as Gaussian process regression.

Kriging can be understood as a form of Bayesian inference. Kriging starts with a prior distribution over functions. This prior takes the form of a Gaussian process: <math>N<math> samples from a function will be normally distributed, where the covariance between any two samples is the covariance function (or kernel) of the Gaussian process evaluated at the spatial location two points.

A set of values are then observed, each value associated with a spatial location. Now, a new value can be predicted at any new spatial location, by combining a the Gaussian prior with a Gaussian likelihood function for each of the observed values. The resulting posterior distribution is also a Gaussian, with a mean and covariance than can be simply computed from the observed values, their variance, and the kernel matrix derived from the prior.

From the geological point of view, Kriging uses prior knowledge about the spatial distribution of a mineral: this prior knowledge encapsulates how minerals co-occur as a function of space. Then, given a series of measurements of mineral concentrations, Kriging can predict mineral concentrations at unobserved points.

Kriging is a family of linear least squares estimation algorithms. The end result of Kriging is to obtain the conditional expectation as a best estimate for all unsampled locations in a field and consequently, a minimized error variance at each location. The conditional expectation minimizes the error variance when the optimality criterion is based on least squares residuals. The Kriging estimate is a weighted linear combination of the data. The weights that are assigned to each known datum are determined by solving the Kriging system of linear equations, where the weights are the unknown regression parameters. The optimality criterion used to arrive at the Kriging system, as mentioned above, is a minimization of the error variance in the least-squares sense.

Kriging was first developed empirically by Krige in the 1950s. Kriging was then formalized (and first named) by Georges Matheron, in the early 1960s.

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