Inverse distance weighting

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The inverse distance weighting (more rarely also -weighting ) is a non- statistical interpolation method of geostatistics and is used for the simple interpolation of the spatial dependency of georeferenced data. The basic assumption here is that the similarity of an unknown value to the known measured value decreases with the distance from it, i.e. the further apart the data are, the more dissimilar. In inverse distance weighting, this relationship is expressed in that the measured value is multiplied by a weight that is proportional to the inverse of the distance between the estimated point and the measurement location.

Procedure

A finite number n of measurements with the measurement locations and the measurement values ​​is assumed, with the index i standing for the natural numbers from 1 to n. The value sought at the point was not measured and must therefore be estimated. The measured values are weighted with . The estimator for this unknown value is then calculated according to: ${\displaystyle x_{i}}$ ${\displaystyle z(x_{i})}$ ${\displaystyle z(x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{i}}$ ${\displaystyle \textstyle {\frac {1}{Abstand(x_{0},x_{i})}}=\lambda (x_{0},x_{i})}$ ${\displaystyle z^{*}(x_{0})}$

${\displaystyle z^{*}(x_{0})={\frac {1}{\sum _{l=1}^{n}\lambda _{l}(x_{0},x_{l})}}\sum _{i=1}^{n}\lambda _{i}(x_{0},x_{i})\cdot z(x_{i})={\frac {1}{\sum _{l=1}^{n}\lambda _{l}(x_{0},x_{l})}}(\lambda _{1}(x_{0},x_{1})\cdot z(x_{1})+\lambda _{2}(x_{0},x_{2})\cdot z(x_{2})+...+\lambda _{n}(x_{0},x_{n})\cdot z(x_{n}))}$

Are there

${\displaystyle \lambda _{i}(x_{0},x_{i})={\frac {1}{distance(x_{0},x_{i})}}}$

the weights or values ​​of the weight function for , and is the normalization factor. ${\displaystyle z(x_{i})}$ ${\displaystyle {\frac {1}{\sum _{l=1}^{n}\lambda _{l}(x_{0},x_{l})}}}$

The requirement here is that the estimation function should be identical to the measured values ​​at the measuring points themselves. The above equation can be used with the normalized values ​​of the weights for the measured values ${\displaystyle \textstyle z^{*}(x_{i})=z(x_{i})}$

${\displaystyle \lambda _{i}^{*}(x_{0})={\frac {\lambda _{i}(x_{0})}{\sum _{l=1}^{n}\lambda _{l}(x_{0})}}}$

also write:

${\displaystyle z^{*}(x_{0})=\sum _{i=1}^{n}\lambda _{i}^{*}(x_{0})\cdot z(x_{i})=\lambda _{1}(x_{0})\cdot z(x_{1})+\lambda _{2}(x_{0})\cdot z(x_{2})+...+\lambda _{n}(x_{0})\cdot z(x_{n})}$

Sometimes the decrease in weights with the distance is also increased by a power with the exponent in order to be able to better take into account the given physical situation. This power must be determined and should approximate the data situation as closely as possible. Often, however, an exponent of is simply estimated, which means that the interpolation is similar to gravity , for example (weighting decreases with , i.e. quadratically). This results in the following equation for the estimated value: ${\displaystyle m}$ ${\displaystyle m=2}$ ${\displaystyle \textstyle {\frac {1}{r^{2}}}}$

${\displaystyle z^{*}(x_{0})={\frac {\sum _{i=1}^{n}{\frac {z(x_{i})}{d^{m}(x_{0},x_{i})}}}{\sum _{i=1}^{n}{\frac {1}{d^{m}(x_{0},x_{i})}}}}}$

meaning

The inverse distance weighting does not follow the increase in dissimilarity with the distance present in the data, but defines this as a prerequisite in the form of a freely selectable exponent within the weighting function. The choice of the measured values ​​included in the weighting can also be used, whereby the range of the data must be taken into account. Since the method only allows the distance between the measured data to flow into it, it does not take into account all the information contained in the data set that can in principle be used for geostatistical purposes, which is why the estimator does not have the smallest possible estimation error and, depending on the properties of the data set and the choice of the estimation function, it is more or less above the Minimum value is. This is especially true when the measuring locations are distributed in clusters, i.e. concentrate at certain points and thin out at others. On the other hand, there is the simplicity of the method, which is why it is used, for example, in advance of the kriging method or in the case of lower demands on the quality of the interpolation.