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In
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, originally in
geostatistics Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including pe ...
, kriging or Kriging, also known as Gaussian process regression, is a method of
interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has ...
based on
Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. ...
governed by prior
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
s. Under suitable assumptions of the prior, kriging gives the best linear unbiased prediction (BLUP) at unsampled locations. Interpolating methods based on other criteria such as
smoothness In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
(e.g.,
smoothing spline Smoothing splines are function estimates, \hat f(x), obtained from a set of noisy observations y_i of the target f(x_i), in order to balance a measure of goodness of fit of \hat f(x_i) to y_i with a derivative based measure of the smoothness of ...
) may not yield the BLUP. The method is widely used in the domain of
spatial analysis Spatial analysis or spatial statistics includes any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early deve ...
and
computer experiment A computer experiment or simulation experiment is an experiment used to study a computer simulation, also referred to as an in silico system. This area includes computational physics, computational chemistry, computational biology and other similar ...
s. The technique is also known as Wiener–Kolmogorov prediction, after
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher ...
and
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
. The theoretical basis for the method was developed by the French mathematician Georges Matheron in 1960, based on the master's thesis of Danie G. Krige, the pioneering plotter of distance-weighted average gold grades at the
Witwatersrand The Witwatersrand () (locally the Rand or, less commonly, the Reef) is a , north-facing scarp in South Africa. It consists of a hard, erosion-resistant quartzite metamorphic rock, over which several north-flowing rivers form waterfalls, which ...
reef complex in
South Africa South Africa, officially the Republic of South Africa (RSA), is the southernmost country in Africa. It is bounded to the south by of coastline that stretch along the South Atlantic and Indian Oceans; to the north by the neighbouring coun ...
. Krige sought to estimate the most likely distribution of gold based on samples from a few boreholes. The English verb is ''to krige'', and the most common noun is ''kriging''; both are often pronounced with a hard "g", following an Anglicized pronunciation of the name "Krige". The word is sometimes capitalized as ''Kriging'' in the literature. Though computationally intensive in its basic formulation, kriging can be scaled to larger problems using various approximation methods.


Main principles


Related terms and techniques

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 to
regression analysis In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
. Both theories derive a best linear unbiased estimator based on assumptions on
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
s, make use of
Gauss–Markov theorem In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the ...
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 a spline in a reproducing kernel Hilbert space, with the reproducing kernel given by the covariance function. 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 In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinar ...
polynomial curve fitting. Kriging can also be understood as a form of Bayesian optimization. Kriging starts with a
prior Prior (or prioress) is an ecclesiastical title for a superior in some religious orders. The word is derived from the Latin for "earlier" or "first". Its earlier generic usage referred to any monastic superior. In abbeys, a prior would be low ...
distribution Distribution may refer to: Mathematics * Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a vari ...
over functions. This prior takes the form of a Gaussian process: N samples from a function will be normally distributed, where the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
between any two samples is the covariance function (or kernel) of the Gaussian process evaluated at the spatial location of two points. A
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
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 The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood functi ...
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.


Geostatistical estimator

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. A
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
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 x_1 (generic denomination of a set of
geographic coordinates The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...
) is interpreted as a realization z(x_1) of the
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
Z(x_1). In the space A, where the set of samples is dispersed, there are N realizations of the random variables Z(x_1), Z(x_2), \ldots, Z(x_N), correlated between themselves. The set of random variables constitutes a random function, of which only one realization is known – the set z(x_i) of observed data. With only one realization of each random variable, it's theoretically impossible to determine any
statistical parameter In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population ...
of the individual variables or the function. The proposed solution in the geostatistical formalism consists in ''assuming'' various degrees of ''stationarity'' 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 A where the variable is distributed, the hypothesis that the first 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 the second 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 \mathbf = x_2 - x_1 and , \mathbf, = h, then: : C\big(Z(x_1), Z(x_2)\big) = C\big(Z(x_i), Z(x_i + \mathbf)\big) = C(h), : \gamma\big(Z(x_1), Z(x_2)\big) = \gamma\big(Z(x_i), Z(x_i + \mathbf)\big) = \gamma(h). For simplicity, we define C(x_i, x_j) = C\big(Z(x_i), Z(x_j)\big) and \gamma(x_i, x_j) = \gamma\big(Z(x_i), Z(x_j)\big). This hypothesis allows one to infer those two measures – the
variogram In spatial statistics the theoretical variogram 2\gamma(\mathbf_1,\mathbf_2) is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z(\mathbf). The semivariogram \gamma(\mathbf_1,\mathbf_2) is ha ...
and the covariogram: : \gamma(h) = \frac \sum_ \big(Z(x_i) - Z(x_j)\big)^2, : C(h) = \frac \sum_ \big(Z(x_i) - m(h)\big)\big(Z(x_j) - m(h)\big), where: : m(h) = \frac \sum_ Z(x_i) + Z(x_j); : N(h) denotes the set of pairs of observations i,\;j such that , x_i - x_j, = h, and , N(h), is the number of pairs in the set. In this set, (i,\;j) and (j,\;i) denote the same element. Generally an "approximate distance" h is used, implemented using a certain tolerance.


Linear estimation

Spatial inference, or estimation, of a quantity Z \colon \mathbb^n \to \mathbb, at an unobserved location x_0, is calculated from a linear combination of the observed values z_i = Z(x_i) and weights w_i(x_0),\; i = 1, \ldots, N: : \hat(x_0) = \begin w_1 & w_2 & \cdots & w_N \end \begin z_1 \\ z_2 \\ \vdots \\ z_N \end = \sum_^N w_i(x_0) Z(x_i). The weights 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 x_0; * at the same time, they should have a desegregation effect, in order to avoid bias caused by eventual sample ''clusters''. When calculating the weights w_i, there are two objectives in the geostatistical formalism: ''unbias'' and ''minimal variance of estimation''. If the cloud of real values Z(x_0) is plotted against the estimated values \hat(x_0), the criterion for global unbias, ''intrinsic stationarity'' or wide 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 \big(\hat(x) - Z(x)\big) must be minimal, which means that when the cloud of estimated values ''versus'' the cloud real values is more disperse, the estimator is more imprecise.


Methods

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 x_0. * ''Simple kriging'' assumes stationarity of the first moment over the entire domain with a known mean: E\ = E\ = m, where m is the known mean. * '' Universal kriging'' assumes a general polynomial trend model, such as linear trend model \textstyle E\ = \sum_^p \beta_k f_k(x). * ''IRFk-kriging'' assumes E\ to be an unknown
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
in x. * ''Indicator kriging'' uses
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
s 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. * ''Disjunctive kriging'' is a nonlinear generalisation of kriging. * ''
Log-normal In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed, ...
kriging'' interpolates positive data by means of
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s. * ''Latent kriging'' assumes the various krigings on the latent level (second stage) of the
nonlinear mixed-effects model Nonlinear mixed-effects models constitute a class of statistical models generalizing mixed model, linear mixed-effects models. Like linear mixed-effects models, they are particularly useful in settings where there are multiple measurements within t ...
to produce a spatial functional prediction. This technique is useful when analyzing a spatial functional data \_^n, where y_i = (y_ ,y_, \cdots, y_)^\top is a time series data over T_i period, x_i = (x_, x_, \cdots, x_)^\top is a vector of p covariates, and s_i = (s_, s_)^\top is a spatial location (longitude, latitude) of the i-th subject. *''Co-kriging'' denotes the joint kriging of data from multiple sources with a relationship between the different data sources. Co-kriging is also possible in a
Bayesian Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...
approach. *''Bayesian kriging'' departs from the optimization of unknown coefficients and hyperparameters, which is understood as a maximum likelihood estimate from the Bayesian perspective. Instead, the coefficients and hyperparameters are estimated from their
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
s. An advantage of Bayesian kriging is, that it allows to quantify the evidence for and the uncertainty of the kriging
emulator In computing, an emulator is hardware or software that enables one computer system (called the ''host'') to behave like another computer system (called the ''guest''). An emulator typically enables the host system to run software or use pe ...
. 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 also Bayesian Polynomial Chaos). Bayesian kriging can also be mixed with co-kriging.


Ordinary kriging

The unknown value Z(x_0) is interpreted as a random variable located in x_0, as well as the values of neighbors samples Z(x_i),\ i = 1, \ldots, N. The estimator \hat(x_0) is also interpreted as a random variable located in x_0, a result of the linear combination of variables. In order to deduce the kriging system for the assumptions of the model, the following error committed while estimating Z(x) in x_0 is declared: : \epsilon(x_0) = \hat(x_0) - Z(x_0) = \begin W^T & -1 \end \cdot \begin Z(x_1) & \cdots & Z(x_N) & Z(x_0) \end^T = \sum^N_ w_i(x_0) \times Z(x_i) - Z(x_0). The two quality criteria referred to previously can now be expressed in terms of the mean and variance of the new random variable \epsilon(x_0): ; Lack of bias Since the random function is stationary, E
(x_i) X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
= E (x_0)= m, the following constraint is observed: : E epsilon(x_0)= 0 \Leftrightarrow \sum^N_ w_i(x_0) \times E
(x_i) X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
- E (x_0)= 0 \Leftrightarrow : \Leftrightarrow m \sum^N_ w_i(x_0) - m = 0 \Leftrightarrow \sum^N_ w_i(x_0) = 1 \Leftrightarrow \mathbf^T \cdot W = 1. In order to ensure that the model is unbiased, the weights must sum to one. ; Minimum variance Two estimators can have E epsilon(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 E epsilon(x_0)^2/math>. : \begin \operatorname(\epsilon(x_0)) &= \operatorname\left(\begin W^T & -1 \end \cdot \begin Z(x_1) & \cdots & Z(x_N) & Z(x_0) \end^T\right) \\ &= \begin W^T & -1 \end \cdot \operatorname\left(\begin Z(x_1) & \cdots & Z(x_N) & Z(x_0) \end^T\right) \cdot \begin W \\ -1 \end. \end See
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
for a detailed explanation. : \operatorname(\epsilon(x_0)) = \begin W^T & -1 \end \cdot \begin \operatorname_ & \operatorname_ \\ \operatorname_^T & \operatorname_ \end \cdot \begin W \\ -1 \end, where the literals \left\ stand for : \left\. Once defined the covariance model or
variogram In spatial statistics the theoretical variogram 2\gamma(\mathbf_1,\mathbf_2) is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z(\mathbf). The semivariogram \gamma(\mathbf_1,\mathbf_2) is ha ...
, C(\mathbf) or \gamma(\mathbf), valid in all field of analysis of 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: : \begin \operatorname\big(\epsilon(x_0)\big) = W^T \cdot \operatorname_ \cdot W - \operatorname_^T \cdot W - W^T \cdot \operatorname_ + \operatorname_, \\ \operatorname\big(\epsilon(x_0)\big) = \operatorname(0) + \sum_i \sum_j w_i w_j \operatorname(x_i,x_j) - 2 \sum_iw_i C(x_i,x_0). \end 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 x_0, the estimation becomes worse; * grows with the a priori variance C(0) of the variable Z(x); when the variable is less disperse, the variance is lower in any point of the area 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 A; this way, the variance does not measure the uncertainty of estimation produced by the local variable. ; System of equations : W = \underset\left( W^T \cdot \operatorname_ \cdot W - \operatorname_^T \cdot W - W^T \cdot \operatorname_ + \operatorname_ \right). Solving this optimization problem (see
Lagrange multipliers In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
) results in the ''kriging system'': : \begin\hat\\\mu\end = \begin \operatorname_& \mathbf\\ \mathbf^T& 0 \end^\cdot \begin \operatorname_\\ 1\end = \begin \gamma(x_1,x_1) & \cdots & \gamma(x_1,x_n) &1 \\ \vdots & \ddots & \vdots & \vdots \\ \gamma(x_n,x_1) & \cdots & \gamma(x_n,x_n) & 1 \\ 1 &\cdots& 1 & 0 \end^ \begin\gamma(x_1,x^*) \\ \vdots \\ \gamma(x_n,x^*) \\ 1\end. The additional parameter \mu is a
Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
used in the minimization of the kriging error \sigma_k^2(x) to honor the unbiasedness condition.


Simple kriging

Simple kriging is mathematically the simplest, but the least general. It assumes the expectation of the
random field In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other ...
is known and relies on a
covariance function In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a doma ...
. However, in most applications neither the expectation nor the covariance are known beforehand. The practical assumptions for the application of ''simple kriging'' are: * Wide-sense stationarity of the field (variance stationary). * The expectation is zero everywhere: \mu(x) = 0. * Known
covariance function In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a doma ...
c(x, y) = \operatorname\big(Z(x), Z(y)\big). 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. 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 The ''kriging weights'' of ''simple kriging'' have no unbiasedness condition and are given by the ''simple kriging equation system'': : \begin w_1 \\ \vdots \\ w_n \end = \begin c(x_1, x_1) & \cdots & c(x_1, x_n) \\ \vdots & \ddots & \vdots \\ c(x_n, x_1) & \cdots & c(x_n, x_n) \end^ \begin c(x_1,x_0) \\ \vdots \\ c(x_n,x_0) \end. This is analogous to a linear regression of Z(x_0) on the other z_1, \ldots, z_n. ; Estimation The interpolation by simple kriging is given by : \hat(x_0) = \begin z_1 \\ \vdots \\ z_n \end' \begin c(x_1, x_1) & \cdots & c(x_1, x_n) \\ \vdots & \ddots & \vdots \\ c(x_n, x_1) & \cdots & c(x_n, x_n) \end^ \begin c(x_1,x_0) \\ \vdots \\ c(x_n,x_0)\end. The kriging error is given by : \operatorname\big(\hat(x_0) - Z(x_0)\big) = \underbrace_ - \underbrace_, which leads to the generalised least-squares version of the
Gauss–Markov theorem In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the ...
(Chiles & Delfiner 1999, p. 159): : \operatorname\big(Z(x_0)\big) = \operatorname\big(\hat(x_0)\big) + \operatorname\big(\hat(x_0) - Z(x_0)\big).


Bayesian kriging

''See also Bayesian Polynomial Chaos''


Properties

* The kriging estimation is unbiased: E hat(x_i)= E
(x_i) X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
/math>. * The kriging estimation honors the actually observed value: \hat(x_i) = Z(x_i) (assuming no measurement error is incurred). * The kriging estimation \hat(x) is the best linear unbiased estimator of Z(x) if the assumptions hold. However (e.g. Cressie 1993): ** 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 \sigma_k^2 as a measure of precision. However, this measure relies on the correctness of the variogram.


Applications

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 Environmental science is an interdisciplinary academic field that integrates physics, biology, and geography (including ecology, chemistry, plant science, zoology, mineralogy, oceanography, limnology, soil science, geology and physical geog ...
*
Hydrogeology Hydrogeology (''hydro-'' meaning water, and ''-geology'' meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aq ...
Chiles, J.-P. and P. Delfiner (1999) ''Geostatistics, Modeling Spatial Uncertainty'', Wiley Series in Probability and statistics. *
Mining Mining is the extraction of valuable minerals or other geological materials from the Earth, usually from an ore body, lode, vein, seam, reef, or placer deposit. The exploitation of these deposits for raw material is based on the econom ...
*
Natural resource Natural resources are resources that are drawn from nature and used with few modifications. This includes the sources of valued characteristics such as commercial and industrial use, aesthetic value, scientific interest and cultural value. ...
sGoovaerts (1997) ''Geostatistics for natural resource evaluation'', OUP. *
Remote sensing Remote sensing is the acquisition of information about an object or phenomenon without making physical contact with the object, in contrast to in situ or on-site observation. The term is applied especially to acquiring information about Ear ...
*
Real estate appraisal Real estate appraisal, property valuation or land valuation is the process of developing an opinion of value for real property (usually market value). Real estate transactions often require appraisals because they occur infrequently and every pr ...
* Integrated circuit analysis and optimization * Modelling of microwave devices *
Astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
* Prediction of oil production curve of shale oil wells


Design and analysis of computer experiments

Another very important and rapidly growing field of application, in
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, is the interpolation of data coming out as response variables of deterministic computer simulations, e.g.
finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
(FEM) simulations. In this case, kriging is used as a metamodeling tool, i.e. a black-box model built over a designed set of
computer experiment A computer experiment or simulation experiment is an experiment used to study a computer simulation, also referred to as an in silico system. This area includes computational physics, computational chemistry, computational biology and other similar ...
s. In many practical engineering problems, such as the design of a
metal forming Forming, metal forming, is the metalworking process of fashioning metal parts and objects through mechanical deformation; the workpiece is reshaped without adding or removing material, and its mass remains unchanged. Forming operates on the mater ...
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-called
surrogate model A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design ...
, implemented inside
optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
routines.


See also

* Bayes linear statistics *
Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. ...
* Multivariate interpolation *
Nonparametric regression Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship ...
* Radial basis function interpolation * Space mapping *
Spatial dependence Spatial analysis or spatial statistics includes any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early deve ...
*
Variogram In spatial statistics the theoretical variogram 2\gamma(\mathbf_1,\mathbf_2) is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z(\mathbf). The semivariogram \gamma(\mathbf_1,\mathbf_2) is ha ...
* Gradient-enhanced kriging (GEK) *
Surrogate model A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design ...
*
Information field theory Information field theory (IFT) is a Bayesian statistical field theory relating to signal reconstruction, cosmography, and other related areas. IFT summarizes the information available on a physical field using Bayesian probabilities. It uses comput ...


References


Further reading


Historical references

# # Agterberg, F. P., ''Geomathematics, Mathematical Background and Geo-Science Applications'', Elsevier Scientific Publishing Company, Amsterdam, 1974. # Cressie, N. A. C., ''The origins of kriging, Mathematical Geology'', v. 22, pp. 239–252, 1990. # 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. # Link, R. F. and Koch, G. S., ''Experimental Designs and Trend-Surface Analsysis, Geostatistics'', A colloquium, Plenum Press, New York, 1970. # Matheron, G., "Principles of geostatistics", ''Economic Geology'', 58, pp. 1246–1266, 1963. # Matheron, G., "The intrinsic random functions, and their applications", ''Adv. Appl. Prob.'', 5, pp. 439–468, 1973. # Merriam, D. F. (editor), ''Geostatistics'', a colloquium, Plenum Press, New York, 1970.


Books

* 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, . * 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. * . 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 {{Authority control Curve fitting Geostatistics Interpolation Multivariate interpolation