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General Equations
Kriging is a group of geostatistical techniques to interpolate the value $$Z(x_0)$$ of a random field $$Z(x)$$ (e.g. the elevation Z of the landscape as a function of the geographic location $$x$$) at an unobserved location $$x_0$$ from observations $$z_i=Z(x_i),\;i=1,\ldots,n$$ of the random field at nearby locations $$x_1,\ldots,x_n$$. Kriging computes the best linear unbiased estimator $$\hat{Z}(x_0)$$ of $$Z(x_0)$$ based on a stochastic model of the spatial dependence quantified either by the variogram $$\gamma(x,y)$$ or by expectation $$\mu(x)=E[Z(x)]$$ and the covariance function $$c(x,y)$$ of the random field.

The kriging estimator is given by a linear combination


 * $$\hat{Z}(x_0)=\sum_{i=1}^n w_i(x_0) Z(x_i)$$

of the observed values $$z_i=Z(x_i)$$ with weights $$w_i(x_0),\;i=1,\ldots,n$$ choosen such that the variance (also called kriging variance or kriging error):


 * $$\sigma^2_k(x_0):=var\left(\hat{Z}(x_0)-Z(x)\right)=var\left(\sum_{i=0}^n w_i(x_0)Z(x_i)\right)=\sum_{i=0}^n\sum_{j=0}^n w_i(x_0) w_j(x_0) c(x_i,x_j)$$

(with $$w_0(x_0)=-1$$) of the prediction error $$\hat{Z}(x)-Z(x)$$ is minimized subject to the unbiasedness condition:

E[\hat{Z}(x)-Z(x)]=\sum_{i=1}^n w_i(x_0)\mu(x_i) - \mu(x_0) =0 $$ Depending on the stochastic properties of the random field different types of kriging apply. For the different types of kriging the unbiasedness condition is rewritten into different linear constraints for the weights $$w_i$$.

The kriging variance must not be confused with the variance

var\left(\hat{Z}(x_0)\right)=var\left(\sum_{i=1}^n w_iZ(x_i)\right)=\sum_{i=1}^n\sum_{j=1}^n w_i w_j c(x_i,x_j) $$ of the kriging predictor $$\hat{Z}(x_0)$$ itself.

The types of kriging
Classical types of kriging are


 * Simple kriging assuming a known constant trend: $$\mu(x)=0$$.
 * Ordinary kriging assuming an unkown constant trend: $$\mu(x)=\mu$$.
 * Universal Kriging assuming a general linear trend modell $$\mu(x)=\sum_{k=0}^p \beta_k f(x)$$.
 * IRFk-Kriging assuming $$\mu(x)$$ to be polynomial in $$x$$.
 * Indicator Kriging using indicator functions instead of the process itself in order to estimate transition probabilities.
 * Multiple indicator kriging is a version of Indicatior kriging working with
 * Disjunctive Kriging is a nonlinear generalisation of kriging
 * Lognormal Kriging interpolates positive data by means of logarithms.

Simple Kriging
Simple kriging is the most simple kind of kriging. It assumes the expecation of to random field to be beforehand and relies on a covariance function. However in most real application neigther expectation nor covariance are known beforehand.

Simple Kriging Assumptions
The practical assumptions for the application of simple kriging are:
 * wide sense stationarity of the field.
 * The expectation is zero everywhere: $$\mu(x)=0$$.
 * Known covariance function $$c(x,y)=cov(Z(x),Z(y))$$

Simple Kriging Equation
The kriging weights of simple kriging have no unbiaseness condition and are given by the simple kriging equation system:
 * $$\begin{pmatrix}w_1 \\ \vdots \\ w_n \end{pmatrix}=

\begin{pmatrix}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{pmatrix}^{-1} \begin{pmatrix}c(x_1,x_0) \\ \vdots \\ c(x_n,x_0) \end{pmatrix} $$

Simple Kriging Interpolation
The interpolation by ordinary kriging is given by: $$\hat{Z}(x)=\begin{pmatrix}z_1 \\ \vdots \\ z_n \end{pmatrix}' \begin{pmatrix}c(x_1,x_1) & \cdots & c(x_1,x_n) \\ \vdots & \ddots & \vdots  \\ c(x_n,x_1) & \cdots & c(x_n,x_n) \end{pmatrix}^{-1} \begin{pmatrix}c(x_1,x_0) \\ \vdots \\ c(x_n,x_0) \\ 1\end{pmatrix} $$

Simple Kriging Error
The kriging error is given by:
 * $$var\left(\hat{Z}(x_0)-Z(x_0)\right)=\underbrace{c(x_0,x_0)}_{var(Z(x_0))}-

\underbrace{\begin{pmatrix}c(x_1,x_0) \\ \vdots \\ c(x_n,x_0)\end{pmatrix}' \begin{pmatrix} 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{pmatrix}^{-1} \begin{pmatrix}c(x_1,_0) \\ \vdots \\ c(x_n,x_0) \end{pmatrix}}_{var(\hat{Z}(x))} $$ which leads to the generalised least squares version of the Gauss-Markov theorem (Chiles&Delfiner 1999, p. 159):
 * $$var(Z(x_0))=var(\hat{Z}(x_0))+var\left(\hat{Z}(x_0)-Z(x_0)\right)$$

Ordinary Kriging
Ordinary kriging is to most commonly used type of kriging. It assumes a constant but unkown mean.

Typical Ordinary Kriging Assumptions
The typical practical assumptions for the application of ordinary kriging are: The mathematical condition for applicability of ordinary kriging are:
 * Intrinsic Stationarity or wide sense stationarity of the field.
 * Enough observations to estimate the variogram.
 * The mean $$E[Z(x)]=\mu$$ is unkown but constant
 * The variogram $$\gamma(x,y)=E[(Z(x)-Z(y))^2]$$ of $$Z(x)$$ is known.

Ordinary Kriging Equation
The kriging weights of ordinary kriging solve the unbiasedness condition


 * $$\sum_{i=1}^n w_i = 1$$

and are given by the ordinary kriging equation system:
 * $$\begin{pmatrix}w_1 \\ \vdots \\ w_n \\ \lambda \end{pmatrix}=

\begin{pmatrix}\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{pmatrix}^{-1} \begin{pmatrix}\gamma(x_1,x^*) \\ \vdots \\ \gamma(x_n,x^*) \\ 1\end{pmatrix} $$ the additional parameter $$\lambda$$ is a Lagrange multiplier used in the minisation of the kriging error $$\sigma_k^2(x)$$ to honor the unbiasedness condition.

Ordinary Kriging Interpolation
The interpolation by ordinary kriging is given by: $$\hat{Z}(x)=\begin{pmatrix}z_1 \\ \vdots \\ z_n \\ 0 \end{pmatrix}' \begin{pmatrix}\gamma(x_1,x_1) & \cdots & \gamma(x_1,x_n) &1 \\ \vdots & \ddots & \vdots & \vdots \\ \gamma(x_n,x_1) & \cdots & \gamma(x_n,x_n) & 1 \\ 1 &\cdots& 1 & 0 \end{pmatrix}^{-1} \begin{pmatrix}\gamma(x_1,x^*) \\ \vdots \\ \gamma(x_n,x^*) \\ 1\end{pmatrix} $$

Ordinary Kriging Error
The kriging error is given by: $$var\left(\hat{Z}(x)-Z(x)\right)= \begin{pmatrix}\gamma(x_1,x^*) \\ \vdots \\ \gamma(x_n,x^*) \\ 1\end{pmatrix}' \begin{pmatrix} \gamma(x_1,x_1) & \cdots & \gamma(x_1,x_n) &1 \\ \vdots & \ddots & \vdots & \vdots \\ \gamma(x_n,x_1) & \cdots & \gamma(x_n,x_n) & 1 \\ 1 &\cdots& 1 & 0 \end{pmatrix}^{-1} \begin{pmatrix}\gamma(x_1,x^*) \\ \vdots \\ \gamma(x_n,x^*) \\ 1\end{pmatrix} $$

Properties of Kriging
(Cressie 1993, Chiles&Delfiner 1999, Wackernagel 1995)
 * The kriging estimation is unbiased: $$E[\hat{Z}(x_i)]=E[Z(x_i)]$$
 * The kriging estimation honors the actually observed value: $$\hat{Z}(x_i)=Z(x_i)$$
 * The kriging estimation $$\hat{Z}(x)$$ 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 $$\sigma_f^2$$ as a measure of precision. However this measure relies on the correctness of the variogram.

Books on kriging

 * Cressie, N (1993) Statistics for spatial data, Wiley, New York
 * Journel, A.G. and C.J. Huijbregts (1978) Mining Geostatistics, Academic Press London
 * Goovaerts, P. (1997) Geostatistics for Natural Resources Evaluation, Oxford University Press, New York
 * Wackernagel, H. (1995) Multivariate Geostatistics - An Introduction with Applications., Springer Berlin
 * Chiles, J.-P. and P. Delfiner (1999) Geostatistics, Modeling Spatial uncertainty, Wiley Series in Probability and statistics.