Dr. SaMeH S. Ahmed

Associate Prof. of Environmental Engineering

Kriging

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Kriging is 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 ta group of geostatistical techniques to interpolate the value of a random field (e.g., the elevation, z, of the landscape as ahe 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 iskriging; both are often pronounced with a hard "g", following

the pronunciation of the name "Krige".

Simple krging

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.

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Simple kriging assumptions

The practical assumptions for the application of simple kriging are:

[edit]
Simple kriging equation

The kriging weights of simple kriging have no unbiasedness condition and are given by the simple kriging equation system:

egin{pmatrix}w_1  vdots  w_n end{pmatrix}=
egin{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}
egin{pmatrix}c(x_1,x_0)  vdots  c(x_n,x_0) end{pmatrix}

This is analogous to a linear regression of Z(x_0) on the other z_1 , ldots, z_n.

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Simple kriging interpolation

The interpolation by simple kriging is given by:

hat{Z}(x_0)=egin{pmatrix}z_1  vdots  z_n  end{pmatrix}'
egin{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}
egin{pmatrix}c(x_1,x_0)  vdots  c(x_n,x_0)end{pmatrix}

[edit]
Simple kriging error

The kriging error is given by:

mathrm{Var}left(hat{Z}(x_0)-Z(x_0)
ight)=underbrace{c(x_0,x_0)}_{mathrm{Var}(Z(x_0))}-
underbrace{egin{pmatrix}c(x_1,x_0)  vdots  c(x_n,x_0)end{pmatrix}'
egin{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}
egin{pmatrix}c(x_1,x_0)  vdots  c(x_n,x_0) end{pmatrix}}_{mathrm{Var}(hat{Z}(x_0))}

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

mathrm{Var}(Z(x_0))=mathrm{Var}(hat{Z}(x_0))+mathrm{Var}left(hat{Z}(x_0)-Z(x_0)
ight).

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Ordinary kriging

Ordinary kriging is the most commonly used type of kriging. It assumes a constant but unknown mean.

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Typical ordinary kriging assumptions

The typical assumptions for the practical application of ordinary kriging are:

The mathematical condition for applicability of ordinary kriging are:

  • The mean E[Z(x)]=mu is unknown but constant
  • The variogram gamma(x,y)=E[(Z(x)-Z(y))^2] of Z(x) is known.

[edit]
Ordinary kriging equation

The kriging weights of ordinary kriging fulfill the unbiasedness condition

sum_{i=1}^n lambda_i = 1

and are given by the ordinary kriging equation system:

egin{pmatrix}lambda_1  vdots  lambda_n  mu end{pmatrix}=
egin{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}
egin{pmatrix}gamma(x_1,x^*)  vdots  gamma(x_n,x^*)  1end{pmatrix}

the additional parameter mu is a Lagrange multiplier used in the minimization of the kriging error sigma_k^2(x) to honor the unbiasedness condition.

[edit]
Ordinary kriging interpolation

The interpolation by ordinary kriging is given by:

hat{Z}(x^*)=egin{pmatrix}lambda_1  vdots  lambda_n  end{pmatrix}'
egin{pmatrix}Z(x_1)  vdots  Z(x_n) end{pmatrix}

[edit]
Ordinary kriging error

The kriging error is given by:

varleft(hat{Z}(x^*)-Z(x^*)
ight)=
egin{pmatrix}lambda_1  vdots  lambda_n  mu end{pmatrix}'
egin{pmatrix}gamma(x_1,x^*)  vdots  gamma(x_n,x^*)  1end{pmatrix}

[edit]
Properties

(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) (assuming no measurement error is incurred)
  • 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_k^2 as a measure of precision. However this measure relies on the correctness of the variogram.

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Related terms and techniques

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Terms

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.

[edit]
Related methods

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 forinterpolation 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: Nsamples 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.


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