Kriging

Color representation of the yield values of a field for a Kriging -Interpolation

Under Kriging (or: krigen) is meant a geostatistical method, with which one values at locations for which no sample is present, by surrounding readings interpolate or approach can. Outside of geostatistics, the method is known as Gaussian regression .

In 1951, the South African mining engineer Danie Krige tried to develop an optimal interpolation method for mining, based on the spatial dependence of measurement points. The procedure was later named after him. The French mathematician Georges Matheron (1963) developed the "theory of the regionalized variable", which forms the theoretical basis of the method developed by Danie Krige.

The main advantage over simpler methods such as inverse distance weighting is the consideration of the spatial variance , which can be determined with the help of the semivariograms . For a searched value, the weights of the measured values ​​flowing into the calculation are determined in such a way that the estimation error variance is as low as possible. The error depends on the quality of the variogram or the variogram function.

In the case of simpler interpolation methods, problems can arise if the measuring points are clustered. This is avoided in kriging by taking into account the statistical distances between the neighbors that are included in the calculation of a point and by optimizing the weighted mean. Kriging is based on efficient and unbiased estimators. If clustering occurs at one point , the weights of the points within this cluster are reduced.

Mathematical formulation

Identify the feature of interest at the location . Usually it is a random variable . If you have observed the feature at the locations with the values , then kriging is understood to be the calculation of the best linear prediction for the feature at the unobserved location . Best prediction means that the mean square error between and is minimized. ${\displaystyle Y(x)}$ ${\displaystyle x}$ ${\displaystyle Y(x)}$ ${\displaystyle x_{1},\dotsc ,x_{n}}$ ${\displaystyle y_{1},\dotsc ,y_{n}}$ ${\displaystyle {\hat {Y}}(x)}$ ${\displaystyle Y(x)}$ ${\displaystyle x}$ ${\displaystyle {\hat {Y}}(x)}$ ${\displaystyle Y(x)}$

Special cases

• Simple kriging : the expected value is constant and known, i.e. H. . ${\displaystyle EY(x)=m(x)}$ ${\displaystyle m(x)=m}$
• Ordinary Kriging : The expected value is constant, but unknown, so it has to be estimated. ${\displaystyle m(x)}$
• Universal kriging : is not constant and is modeled using a linear regression approach. The regression parameters are also estimated. ${\displaystyle m(x)}$
• Indicator kriging : For characteristics with only two values ​​(e.g. limit value exceeded - yes or no)

See also

• Inverse distance weighting
• Approximation
• Habitat modeling

literature

• Daniel G. Krige: A statistical approach to some basic mine valuation problems on the Witwatersrand. In: J. of the Chem., Metal. and Mining Soc. of South Africa. 52 (6), 1951, pp. 119-139.
• Rudolf Dutter: Mathematical Methods in Technology. Volume 2: Geostatistics. BG Teubner Verlag, Stuttgart 1985, ISBN 3-519-02614-7 .
• JP Chiles, P. Delfiner: Geostatistics: Modeling Spatial Uncertainty . Wiley, New York 1999, ISBN 0-471-08315-1 .

Web links

Commons : Kriging  - collection of images, videos and audio files