Kriging
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Kriging is a regression technique used in geostatistics to approximate or interpolate data. The theory of Kriging was developed from the seminal work of its inventor, Danie G. Krige, who found the infinite set of distance-weighted averages but did not know that each distance-weighted average has its own variance. Neither did the French mathematician Georges Matheron know in the early sixties that both the kriging variance and the kriging covariance of a subset of some infinite set of kriged estimate conflict with the requirement of functional independence and concept of degrees of freedom. In the statistical community, it is also known as Gaussian process regression. Kriging is also a reproducing kernel method (like splines and support vector machines).
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What is kriging?
Figure: example of one-dimensional data interpolation by Kriging, with confidence intervals
Kriging can be understood as linear prediction or a form of Bayesian inference.<ref>Template:Cite book</ref> 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 of 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 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 that 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.
Controversy
Kriging is a commonly applied technique to model distribution of ore that assumes, a priori, that the ore concentration has a spatial dependency that is modelled by a Gaussian process.<ref>Template:Cite book</ref> However, some practitioners question the assumption that spatial dependence follows a stochastic process, and that the stochastic process can be correctly estimated from an empirical variogram.<ref>Template:Cite journal</ref> Other practitioners recommend using statistical tests to test the assumption of spatial dependency.<ref>Template:Cite book</ref><ref>Template:Cite book</ref><ref>Template:Cite book</ref>. For example, in the figure above, the function fits the graph perfectly, but the primary data set may not have a statistically significant degree of spatial dependence. Failure of a statistical test would indicate that a constant model cannot be distinguished from a kriging model without further information or knowledge.
Related terms and techniques
A series of related terms were also named after Krige, including kriged estimate, kriged estimator, kriging variance, kriging covariance, zero kriging variance, unity kriging covariance, kriging matrix, kriging method, kriging model, kriging plan, kriging process, kriging system, block kriging, co-kriging, disjunctive kriging, linear kriging, ordinary kriging, point kriging, random kriging, regular grid kriging, simple kriging and universal kriging.
See also: Variogram (also known as a semivariogram).
References
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More historical references
- Armstrong, M and Champigny, N, 1988, A Study on Kriging Small Blocks, CIM Bulletin, Vol 82, No 923
- Clark, I, 1979, Practical Geostatistics, Applied Science Publishers, London
- David, M, 1977, Geostatistical Ore Reserve Estimation, Elsevier Scientific Publishing Company, Amsterdam
- Hald, A, 1952, Statistical Theory with Engineering Applications, John Wiley & Sons, New York
- Lipschutz, S, 1968, Theory and Problems of Probability, McCraw-Hill Book Company, New York
- Volk, W, 1980, Applied Statistics for Engineers, Krieger Publishing Company, Huntington, New York
- Youden, W J, 1951, Statistical Methods for Chemists: John Wiley & Sons, New York