Instatistics, originally ingeostatistics,kriging orKriging (/ˈkriːɡɪŋ/), also known asGaussian process regression, is a method ofinterpolation based onGaussian process governed by priorcovariances. Under suitable assumptions of the prior, kriging gives thebest linear unbiased prediction (BLUP) at unsampled locations.^[1] Interpolating methods based on other criteria such assmoothness (e.g.,smoothing spline) may not yield the BLUP. The method is widely used in the domain ofspatial analysis andcomputer experiments. The technique is also known asWiener–Kolmogorov prediction, afterNorbert Wiener andAndrey Kolmogorov.

The theoretical basis for the method was developed by the French mathematicianGeorges Matheron in 1960, based on the master's thesis ofDanie G. Krige, the pioneering plotter of distance-weighted average gold grades at theWitwatersrand reef complex inSouth Africa. Krige sought to estimate the most likely distribution of gold based on samples from a few boreholes. The English verb isto krige, and the most common noun iskriging. The word is sometimes capitalized asKriging in the literature.

Though computationally intensive in its basic formulation, kriging can be scaled to larger problems using variousapproximation methods.

Kriging predicts the value of a function at a given point by computing a weighted average of the known values of the function in the neighborhood of the point. The method is closely related toregression analysis. Both theories derive abest linear unbiased estimator based on assumptions oncovariances, make use ofGauss–Markov theorem to prove independence of the estimate and error, and use very similar formulae. Even so, they are useful in different frameworks: kriging is made for estimation of a single realization of a random field, while regression models are based on multiple observations of a multivariate data set.

The kriging estimation may also be seen as aspline in areproducing kernel Hilbert space, with the reproducing kernel given by the covariance function.^[2] 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 withpolynomial trend surfaces is mathematically identical togeneralized least squares polynomialcurve fitting.

Kriging can also be understood as a form ofBayesian optimization.^[3] Kriging starts with apriordistribution overfunctions. This prior takes the form of a Gaussian process:${\displaystyle N}$ samples from a function will benormally distributed, where thecovariance between any two samples is the covariance function (orkernel) of the Gaussian process evaluated at the spatial location of two points. Aset 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 Gaussianlikelihood function for each of the observed values. The resultingposterior 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.

In geostatistical models, sampled data are interpreted as the result of a random process. The fact that these models incorporate uncertainty in their conceptualization doesn't mean that the phenomenon – the forest, the aquifer, the mineral deposit – has resulted from a random process, but rather it allows one to build a methodological basis for the spatial inference of quantities in unobserved locations and to quantify the uncertainty associated with the estimator.

Astochastic process is, in the context of this model, simply a way to approach the set of data collected from the samples. The first step in geostatistical modulation is to create a random process that best describes the set of observed data.

A value from location${\displaystyle x_1}$ (generic denomination of a set ofgeographic coordinates) is interpreted as a realization${\displaystyle z(x_1)}$ of therandom variable${\displaystyle Z(x_1)}$. In the space${\displaystyle A}$, where the set of samples is dispersed, there are${\displaystyle N}$ realizations of the random variables${\displaystyle Z(x_1),Z(x_2),\dots ,Z(x_N)}$, correlated between themselves.

The set of random variables constitutes a random function, of which only one realization is known – the set${\displaystyle z(x_i)}$ of observed data. With only one realization of each random variable, it's theoretically impossible to determine anystatistical parameter of the individual variables or the function. The proposed solution in the geostatistical formalism consists inassuming various degrees ofstationarity in the random function, in order to make the inference of some statistic values possible.

For instance, if one assumes, based on the homogeneity of samples in area${\displaystyle A}$ where the variable is distributed, the hypothesis that thefirst moment is stationary (i.e. all random variables have the same mean), then one is assuming that the mean can be estimated by the arithmetic mean of sampled values.

The hypothesis of stationarity related to thesecond moment is defined in the following way: the correlation between two random variables solely depends on the spatial distance between them and is independent of their location. Thus if${\displaystyle \mathbf{h}=x_2-x_1}$ and${\displaystyle h=|\mathbf{h}|}$, then:

${\displaystyle C(Z(x_1),Z(x_2))=C(Z(x_i),Z(x_i+\mathbf{h}))=C(h),}$

${\displaystyle \gamma (Z(x_1),Z(x_2))=\gamma (Z(x_i),Z(x_i+\mathbf{h}))=\gamma (h).}$

For simplicity, we define${\displaystyle C(x_i,x_j)=C(Z(x_i),Z(x_j))}$ and${\displaystyle \gamma (x_i,x_j)=\gamma (Z(x_i),Z(x_j))}$.

This hypothesis allows one to infer those two measures – thevariogram and thecovariogram:

${\displaystyle \gamma (h)=\frac{1}{2|N(h)|}\sum _{(i,j)\in N(h)}(Z(x_i)-Z(x_j){)}^2,}$

${\displaystyle C(h)=\frac{1}{|N(h)|}\sum _{(i,j)\in N(h)}(Z(x_i)-m(h))(Z(x_j)-m(h)),}$

where:

${\displaystyle m(h)=\frac{1}{2|N(h)|}\sum _{(i,j)\in N(h)}Z(x_i)+Z(x_j)}$;

${\displaystyle N(h)}$ denotes the set of pairs of observations${\displaystyle i,j}$ such that${\displaystyle |x_i-x_j|=h}$, and${\displaystyle |N(h)|}$ is the number of pairs in the set.

In this set,${\displaystyle (i,j)}$ and${\displaystyle (j,i)}$ denote the same element. Generally an "approximate distance"${\displaystyle h}$ is used, implemented using a certain tolerance.

Spatial inference, or estimation, of a quantity${\displaystyle Z:{\mathbb{R}}^n\to \mathbb{R}}$, at an unobserved location${\displaystyle x_0}$, is calculated from a linear combination of the observed values${\displaystyle z_i=Z(x_i)}$ and weights${\displaystyle w_i(x_0),i=1,\dots ,N}$:

The weights${\displaystyle w_i}$ are intended to summarize two extremely important procedures in a spatial inference process:

• reflect the structural "proximity" of samples to the estimation location${\displaystyle x_0}$;
• at the same time, they should have a desegregation effect, in order to avoid bias caused by eventual sampleclusters.

When calculating the weights${\displaystyle w_i}$, there are two objectives in the geostatistical formalism:unbias andminimal variance of estimation.

If the cloud of real values${\displaystyle Z(x_0)}$ is plotted against the estimated values${\displaystyle \hat{Z}(x_0)}$, the criterion for global unbias,intrinsic stationarity orwide sense stationarity of the field, implies that the mean of the estimations must be equal to mean of the real values.

The second criterion says that the mean of the squared deviations${\displaystyle (\hat{Z}(x)-Z(x))}$ must be minimal, which means that when the cloud of estimated valuesversus the cloud real values is more disperse, the estimator is more imprecise.

Depending on the stochastic properties of the random field and the various degrees of stationarity assumed, different methods for calculating the weights can be deduced, i.e. different types of kriging apply. Classical methods are:

• Ordinary kriging assumes constant unknown mean only over the search neighborhood of${\displaystyle x_0}$.
• Simple kriging assumes stationarity of thefirst moment over the entire domain with a known mean:${\displaystyle E\{Z(x)\}=E\{Z(x_0)\}=m}$, where${\displaystyle m}$ is the known mean.
• Universal kriging assumes a general polynomial trend model, such as linear trend model${\displaystyle E\{Z(x)\}=\sum _{k=0}^p{\beta }_k{f}_k(x)}$.
• IRFk-kriging assumes${\displaystyle E\{Z(x)\}}$ to be an unknownpolynomial in${\displaystyle x}$.
• Indicator kriging usesindicator functions instead of the process itself, in order to estimate transition probabilities.
  • Multiple-indicator kriging is a version of indicator kriging working with a family of indicators. Initially, MIK showed considerable promise as a new method that could more accurately estimate overall global mineral deposit concentrations or grades. However, these benefits have been outweighed by other inherent problems of practicality in modelling due to the inherently large block sizes used and also the lack of mining scale resolution. Conditional simulation is fast, becoming the accepted replacement technique in this case.^{[citation needed]}
• Disjunctive kriging is a nonlinear generalisation of kriging.
• Log-normal kriging interpolates positive data by means oflogarithms.
• Latent kriging assumes the various krigings on the latent level (second stage) of thenonlinear mixed-effects model to produce a spatial functional prediction.^[4] This technique is useful when analyzing a spatial functional data${\displaystyle \{(y_i,x_i,s_i){\}}_{i=1}^n}$, where${\displaystyle y_i=(y_{i1},y_{i2},\cdots ,y_{i{T}_i}{)}^{\mathrm{\top }}}$ is a time series data over${\displaystyle T_i}$ period,${\displaystyle x_i=(x_{i1},x_{i2},\cdots ,x_{ip}{)}^{\mathrm{\top }}}$ is a vector of${\displaystyle p}$ covariates, and${\displaystyle s_i=(s_{i1},s_{i2}{)}^{\mathrm{\top }}}$ is a spatial location (longitude, latitude) of the${\displaystyle i}$-th subject.
• Co-kriging denotes the joint kriging of data from multiple sources with a relationship between the different data sources.^[5] Co-kriging is also possible in aBayesian approach.^[6][7]
• Bayesian kriging departs from the optimization of unknown coefficients and hyperparameters, which is understood as amaximum likelihood estimate from the Bayesian perspective. Instead, the coefficients and hyperparameters are estimated from theirexpectation values. An advantage of Bayesian kriging is, that it allows to quantify the evidence for and the uncertainty of the krigingemulator.^[8] If the emulator is employed to propagate uncertainties, the quality of the kriging emulator can be assessed by comparing the emulator uncertainty to the total uncertainty (see alsoBayesian Polynomial Chaos). Bayesian kriging can also be mixed with co-kriging.^[6][7]

The unknown value${\displaystyle Z(x_0)}$ is interpreted as a random variable located in${\displaystyle x_0}$, as well as the values of neighbors samples${\displaystyle Z(x_i),i=1,\dots ,N}$. The estimator${\displaystyle \hat{Z}(x_0)}$ is also interpreted as a random variable located in${\displaystyle x_0}$, a result of the linear combination of variables.

Kriging seeks to minimize the mean square value of the following error in estimating${\displaystyle Z(x_0)}$, subject to lack of bias:

The two quality criteria referred to previously can now be expressed in terms of the mean and variance of the new random variable${\displaystyle ϵ(x_0)}$:

Lack of bias

Since the random function is stationary,${\displaystyle E[Z(x_i)]=E[Z(x_0)]=m}$, the weights must sum to 1 in order to ensure that the model is unbiased. This can be seen as follows:

${\displaystyle E[ϵ(x_0)]=0\iff \sum _{i=1}^N{w}_i(x_0)\times E[Z(x_i)]-E[Z(x_0)]=0}$

${\displaystyle \iff m\sum _{i=1}^N{w}_i(x_0)-m=0\iff \sum _{i=1}^N{w}_i(x_0)=1\iff {\mathbf{1}}^T\cdot W=1.}$

Minimum variance

Two estimators can have${\displaystyle E[ϵ(x_0)]=0}$, but the dispersion around their mean determines the difference between the quality of estimators. To find an estimator with minimum variance, we need to minimize${\displaystyle E[ϵ(x_0{)}^2]}$.

Seecovariance matrix for a detailed explanation.

where the literals${\displaystyle \left\{{\mathrm{Var}}_{x_i},{\mathrm{Var}}_{x_0},{\mathrm{Cov}}_{x_i{x}_0}\right\}}$ stand for

Once defined the covariance model orvariogram,${\displaystyle C(\mathbf{h})}$ or${\displaystyle \gamma (\mathbf{h})}$, valid in all field of analysis of${\displaystyle Z(x)}$, then we can write an expression for the estimation variance of any estimator in function of the covariance between the samples and the covariances between the samples and the point to estimate:

${\displaystyle \{\begin{array}{l}\mathrm{Var}(ϵ(x_0))=W^T\cdot {\mathrm{Var}}_{x_i}\cdot W-{\mathrm{Cov}}_{x_i{x}_0}^T\cdot W-W^T\cdot {\mathrm{Cov}}_{x_i{x}_0}+{\mathrm{Var}}_{x_0},\\ \mathrm{Var}(ϵ(x_0))=\mathrm{Cov}(0)+\sum _i\sum _j{w}_i{w}_j\mathrm{Cov}(x_i,x_j)-2\sum _i{w}_{i}C(x_i,x_0).\end{array}}$

Some conclusions can be asserted from this expression. The variance of estimation:

• is not quantifiable to any linear estimator, once the stationarity of the mean and of the spatial covariances, or variograms, are assumed;
• grows when the covariance between the samples and the point to estimate decreases. This means that, when the samples are farther away from${\displaystyle x_0}$, the estimation becomes worse;
• grows with the a priori variance${\displaystyle C(0)}$ of the variable${\displaystyle Z(x)}$; when the variable is less disperse, the variance is lower in any point of the area${\displaystyle A}$;
• does not depend on the values of the samples, which means that the same spatial configuration (with the same geometrical relations between samples and the point to estimate) always reproduces the same estimation variance in any part of the area${\displaystyle A}$; this way, the variance does not measure the uncertainty of estimation produced by the local variable.

System of equations

${\displaystyle W=\underset{{\mathbf{1}}^T\cdot W=1}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\left(W^T\cdot {\mathrm{Var}}_{x_i}\cdot W-{\mathrm{Cov}}_{x_i{x}_0}^T\cdot W-W^T\cdot {\mathrm{Cov}}_{x_i{x}_0}+{\mathrm{Var}}_{x_0}\right).}$

Solving this optimization problem (seeLagrange multipliers) results in thekriging system:

The additional parameter${\displaystyle \mu }$ is aLagrange multiplier used in the minimization of the kriging error${\displaystyle {\sigma }_k^2(x)}$ to honor the unbiasedness condition.

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Simple kriging is mathematically the simplest, but the least general.^[9] It assumes theexpectation of therandom field is known and relies on acovariance function. However, in most applications neither the expectation nor the covariance are known beforehand.

The practical assumptions for the application ofsimple kriging are:

• Wide-sense stationarity of the field (variance stationary).
• The expectation is zero everywhere:${\displaystyle \mu (x)=0}$.
• Knowncovariance function${\displaystyle c(x,y)=\mathrm{Cov}(Z(x),Z(y))}$.

The covariance function is a crucial design choice, since it stipulates the properties of the Gaussian process and thereby the behaviour of the model. The covariance function encodes information about, for instance, smoothness and periodicity, which is reflected in the estimate produced. A very common covariance function is the squared exponential, which heavily favours smooth function estimates.^[10] For this reason, it can produce poor estimates in many real-world applications, especially when the true underlying function contains discontinuities and rapid changes.

System of equations

Thekriging weights ofsimple kriging have no unbiasedness condition and are given by thesimple kriging equation system:

This is analogous to a linear regression of${\displaystyle Z(x_0)}$ on the other${\displaystyle z_1,\dots ,z_n}$.

Estimation

The interpolation by simple kriging is given by

The kriging error is given by

which leads to the generalised least-squares version of theGauss–Markov theorem (Chiles & Delfiner 1999, p. 159):

${\displaystyle \mathrm{Var}(Z(x_0))=\mathrm{Var}(\hat{Z}(x_0))+\mathrm{Var}(\hat{Z}(x_0)-Z(x_0)).}$

See alsoBayesian Polynomial Chaos

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• The kriging estimation is unbiased:${\displaystyle E[\hat{Z}(x_i)]=E[Z(x_i)]}$.
• The kriging estimation honors the actually observed value:${\displaystyle \hat{Z}(x_i)=Z(x_i)}$ (assuming no measurement error is incurred).
• The kriging estimation${\displaystyle \hat{Z}(x)}$ is thebest linear unbiased estimator of${\displaystyle Z(x)}$ if the assumptions hold. However (e.g. Cressie 1993):^[11]
  • As with any method, if the assumptions do not hold, kriging might be bad.
  • There might be better nonlinear and/or biased methods.
  • No properties are guaranteed when the wrong variogram is used. However, typically still a "good" interpolation is achieved.
  • Best is not necessarily good: e.g. in case of no spatial dependence the kriging interpolation is only as good as the arithmetic mean.
• Kriging provides${\displaystyle {\sigma }_k^2}$ as a measure of precision. However, this measure relies on the correctness of the variogram.

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Although kriging was developed originally for applications in geostatistics, it is a general method of statistical interpolation and can be applied within any discipline to sampled data from random fields that satisfy the appropriate mathematical assumptions. It can be used where spatially related data has been collected (in 2-D or 3-D) and estimates of "fill-in" data are desired in the locations (spatial gaps) between the actual measurements.

To date kriging has been used in a variety of disciplines, including the following:

• Environmental science^[12]
• Hydrogeology^[13][14][15]
• Mining^[16][17]
• Natural resources^[18][19]
• Remote sensing^[20]
• Real estate appraisal^[21]
• Integrated circuit analysis and optimization^[22]
• Modelling of microwave devices^[23]
• Astronomy^[24][25][26]
• Prediction of oil production curve of shale oil wells^[27]

Another very important and rapidly growing field of application, inengineering, is the interpolation of data coming out as response variables of deterministic computer simulations,^[28] e.g.finite element method (FEM) simulations. In this case, kriging is used as ametamodeling tool, i.e. a black-box model built over a designed set ofcomputer experiments. In many practical engineering problems, such as the design of ametal forming process, a single FEM simulation might be several hours or even a few days long. It is therefore more efficient to design and run a limited number of computer simulations, and then use a kriging interpolator to rapidly predict the response in any other design point. Kriging is therefore used very often as a so-calledsurrogate model, implemented insideoptimization routines.^[29] Kriging-based surrogate models may also be used in the case of mixed integer inputs.^[30]

• Bayes linear statistics
• Gaussian process
• Multivariate interpolation
• Nonparametric regression
• Radial basis function interpolation
• Space mapping
• Spatial dependence
• Variogram
• Gradient-enhanced kriging (GEK)
• Surrogate model
• Information field theory
• Inverse distance weighting

Wikimedia Commons has media related toKriging.

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1. Chilès, Jean-Paul; Desassis, Nicolas (2018). "Fifty Years of Kriging".Handbook of Mathematical Geosciences. Cham: Springer International Publishing. pp. 589–612.doi:10.1007/978-3-319-78999-6_29.ISBN 978-3-319-78998-9.S2CID 125362741.
2. Agterberg, F. P.,Geomathematics, Mathematical Background and Geo-Science Applications, Elsevier Scientific Publishing Company, Amsterdam, 1974.
3. Cressie, N. A. C.,The origins of kriging, Mathematical Geology, v. 22, pp. 239–252, 1990.
4. 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.
5. Link, R. F. and Koch, G. S.,Experimental Designs and Trend-Surface Analsysis, Geostatistics, A colloquium, Plenum Press, New York, 1970.
6. Matheron, G., "Principles of geostatistics",Economic Geology, 58, pp. 1246–1266, 1963.
7. Matheron, G., "The intrinsic random functions, and their applications",Adv. Appl. Prob., 5, pp. 439–468, 1973.
8. Merriam, D. F. (editor),Geostatistics, a colloquium, Plenum Press, New York, 1970.
9. Mockus, J., "On Bayesian methods for seeking the extremum." Proceedings of the IFIP Technical Conference. 1974.

• Abramowitz, M., and Stegun, I. (1972), Handbook of Mathematical Functions, Dover Publications, New York.
• Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall/CRC Press, Taylor and Francis Group.
• Chiles, J.-P. and P. Delfiner (1999)Geostatistics, Modeling Spatial uncertainty, Wiley Series in Probability and statistics.
• Clark, I., and Harper, W. V., (2000)Practical Geostatistics 2000, Ecosse North America, USA.
• Cressie, N. (1993)Statistics for spatial data, Wiley, New York.
• David, M. (1988)Handbook of Applied Advanced Geostatistical Ore Reserve Estimation, Elsevier Scientific Publishing
• Deutsch, C. V., and Journel, A. G. (1992), GSLIB – Geostatistical Software Library and User's Guide, Oxford University Press, New York, 338 pp.
• Goovaerts, P. (1997)Geostatistics for Natural Resources Evaluation, Oxford University Press, New York,ISBN 0-19-511538-4.
• Isaaks, E. H., and Srivastava, R. M. (1989), An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 pp.
• Journel, A. G. and C. J. Huijbregts (1978)Mining Geostatistics, Academic Press London.
• Journel, A. G. (1989), Fundamentals of Geostatistics in Five Lessons, American Geophysical Union, Washington D.C.
• Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007),"Section 3.7.4. Interpolation by Kriging",Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press,ISBN 978-0-521-88068-8. Also,"Section 15.9. Gaussian Process Regression".
• Stein, M. L. (1999),Statistical Interpolation of Spatial Data: Some Theory for Kriging, Springer, New York.
• Wackernagel, H. (1995)Multivariate Geostatistics - An Introduction with Applications, Springer Berlin

• Curve fitting
• Geostatistics
• Interpolation
• Multivariate interpolation

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