Kriging is a group of geostatistical techniques to interpolate the value of a random field (e.g., the elevation, z, of the landscape as a function of the geographic location) at an unobserved location from observations of its value at nearby locations.
The theory behind interpolation and extrapolation 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 in South Africa. The English verb is to krige and the most common noun is kriging; both are often pronounced with a hard "g", following the pronunciation of the name "Krige".
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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-valued function, , at a point, , given the values of the function at some other points, . A kriging estimator is said to be linear because the predicted value is a linear combination that may be written as
The weights are solutions of a system of linear equations which is obtained by assuming that is a sample-path of a random process , and that the error of prediction
is to be minimized in some sense. For instance, the so-called simple kriging assumption is that the mean and the covariance of is known and then, the kriging predictor is the one that minimizes the variance of the prediction error.
Although kriging was developed originally for applications in geostatistics, it is a general method of statistical interpolation that can be applied within any discipline to sampled data from random fields that satisfy the appropriate mathematical assumptions.
To date kriging has been used in a variety of disciplines, including the following:
and many others.
Kriging interpolates the value of a random field (e.g. the elevation of the landscape as a function of the geographic location ) at an unobserved location from observations of the random field at nearby locations . Kriging computes the best linear unbiased estimator of based on a stochastic model of the spatial dependence quantified either by the variogram or by expectation and the covariance function of the random field.
The kriging estimator is given by a linear combination
of the observed values with weights chosen such that the variance (also called kriging variance or kriging error):
is minimized subject to the unbiasedness condition:
The kriging variance must not be confused with the variance
of the kriging predictor itself.
Depending on the stochastic properties of the random field different types of kriging apply. The type of kriging determines the linear constraint on the weights implied by the unbiasedness condition; i.e. the linear constraint, and hence the method for calculating the weights, depends upon the type of kriging.
Classical methods of kriging are
Simple kriging is mathematically the simplest, but the least general. It assumes the expectation of the random field to be known, and relies on a covariance function. However, in most applications neither the expectation nor the covariance are known beforehand.
The practical assumptions for the application of simple kriging are:
The kriging weights of simple kriging have no unbiasedness condition and are given by the simple kriging equation system:
This is analogous to a linear regression of on the other .
The interpolation by simple kriging is given by:
The kriging error is given by:
which leads to the generalised least squares version of the Gauss-Markov theorem (Chiles & Delfiner 1999, p. 159):
Ordinary kriging is the most commonly used type of kriging. It assumes a constant but unknown mean.
The typical assumptions for the practical application of ordinary kriging are:
The mathematical condition for applicability of ordinary kriging are:
The kriging weights of ordinary kriging fulfill the unbiasedness condition
and are given by the ordinary kriging equation system:
the additional parameter is a Lagrange multiplier used in the minimization of the kriging error to honor the unbiasedness condition.
The interpolation by ordinary kriging is given by:
The kriging error is given by:
(Cressie 1993, Chiles&Delfiner 1999, Wackernagel 1995)
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.
Kriging is mathematically closely related to regression analysis. Both theories derive a best linear unbiased estimator, based on assumptions on covariances, make use of Gauss-Markov theorem to prove independence of the estimate and error, and make use of very similar formulae. They are nevertheless useful in different frameworks: kriging is made for interpolation of a single realisation of a random field, while regression models are based on multiple observations of a multivariate dataset.
In the statistical community the same technique is also known as Gaussian process regression, Kolmogorov Wiener prediction, or best linear unbiased prediction.
The kriging interpolation may also be seen as a spline in a reproducing kernel Hilbert space, with reproducing kernel given by the covariance function.[12] The difference with the classical kriging approach is provided by the interpretation: while the spline is motivated by a minimum norm interpolation based on a Hilbert space structure, kriging is motivated by an expected squared prediction error based on a stochastic model.
Kriging with polynomial trend surfaces is mathematically identical to generalized least squares polynomial curve fitting.
Kriging can also be understood as a form of Bayesian inference.[13] Kriging starts with a prior distribution over functions. This prior takes the form of a Gaussian process: 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 is 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 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.
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. In classical statistics, Matheron's k* is the length-weighted average grade of each of his panneaux in his set. What Matheron failed to derive was var(k*), the variance of his estimateur. On the contrary, he computed the length-weighted average grade of each panneau but did not compute the variance of its central value. In time, he replaced length-weighted average grades for three-dimensional sample spaces such as Matheronian blocks of ore with more abundant distance-weighted average grades for zero-dimensional sample spaces such as Matheronian points.