Kriging

Kriging is a regression technique used in geostatistics to approximate or interpolate data. The theory behind interpolation by Kriging was developed by the French mathematician Georges Matheron based on the Master's thesis of Daniel Gerhardus Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex. The English verb is to krige and the most common adjective is kriging.

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).

Kriging interpolation

Figure 1. Example of one-dimensional data interpolation by Kriging, with confidence intervals. Squares indicate the location of the data. The Kriging interpolation is in red. The confidence intervals are in green.

Kriging belongs to the family of linear least squares estimation algorithms. As illustrated in Figure 1, the aim of Kriging is to estimate the value of an unknown real function $f$ at a point $x^{*}$, given the values of the function at some other points $x_{1},\ldots ,x_{n}$. A Kriging estimator is said to be linear because the predicted value ${\hat {f}}(x^{*})$ is a linear combination that may be written as

${\hat {f}}(x^{*})=\sum _{i=1}^{n}\lambda _{i}f(x_{i})$ .

The weights $\lambda _{i}$ are solutions of a system of linear equations which is obtained by assuming that $f$ is a sample-path of a random process $F(x)$, and that the error of prediction

$\varepsilon (x)=F(x)-\sum _{i=1}^{n}\lambda _{i}F(x_{i})$

is to be minimized in some sense. For instance, the so-called simple Kriging assumption is that the mean and the covariance of $F(x)$ is known and then, the Kriging predictor is the one that minimizes the variance of the prediction error.

Kriging can also 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: $N$ 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.

Applications

Applications in Geostatistics

The application of Kriging to problems in geology and mining as well as to hydrology date back a considerable time. Beginning in the mid-60's and especially in the 70's with the work of Georges Matheron, the connection between Kriging and Geostastiscs is still prevailing today.

Controversy in Mineral Exploration and Mining

The question of whether spatial dependence may be assumed or ought to be verified by applying Fisher's F-test to the variance of a set of measured values and the first variance term of the ordered set prior to interpolation by kriging is of particular relevance in mineral exploration and mining. For example, Clark’s hypothetical uranium data in Practical Geostatistics do not display a significant degree of spatial dependence but the author reports a kriged estimate for some selected coordinates within this sample space anyway. The practice of kriging lends itself to abuse, particularly when applied to a model ore distribution based on the assumption that ore concentrations display a significant degree of spatial dependency in the sample space under examination. Spatial dependence between borehole grades was assumed at Bre-X's Busang property, Hecla's Grouse Creek mine and others where grades turned out to be lower than predicted. A significant degree of spatial dependence is required to justify modelling by a Gaussian process.. Failing to pass a test for spatial dependence would indicate that a constant model cannot be distinguished from a kriging model without further information or knowledge.

Black-box modeling

Two fundamental types of methods for non-linear black-box modeling are linear prediction of random processes, or Kriging, and kernel-based regularized regression (which includes Splines, Radial Basis Functions and Support Vector Regression as special cases). Kriging-based black-box modeling was first proposed to predict the result of computer experiments .

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. Multiple Indicator Kriging (MIK) is a further development of kriging, however MIK in recent years has fallen out of favour as an interpolation technique. This is due to some inherent difficulties related to operation and model validation. Conditional Simulation is fast becoming the accepted replacement technique.

History

The theory of Kriging was developed by the French mathematician Georges Matheron based on the Master's thesis of Daniel Gerhardus Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex. The English verb is to krige and the most common adjective is kriging. The method was called 'krigeage' for the first time in Matheron's 1960 Krigeage d’un Panneau Rectangulaire par sa Périphérie. Matheron, in this Note Géostatistique No 28, derives k*, his 'estimateur' and a precursor to the kriged estimate or kriged estimator.

See also

• Sampling variogram
• variogram (also known as a semivariogram).
• Multiple Indicator Kriging

References

Historical references

1. Krige, D.G, A statistical approach to some mine valuations and allied problems at the Witwatersrand, Master's thesis of the University of Witwatersrand, 1951,
2. Matheron, G., Principles of geostatistics, Economic Geology, 58, pp 1246--1266, 1963
3. Matheron, G., "The intrinsic random functions, and their applications", Adv. Appl. Prob., 5, pp 439-468, 1973

Other references

1. Wackernagel, H., Multivariate Geostatistics, Springer, 1995

External links

• An information server about geostatistics and spatial statistics
• The Gaussian processes web site
• On-Line Library that chronicles Matheron's journey from classical statistics to the new science of geostatistics