Extrapolation
From MohidWiki
Extrapolation is often required when interpolation techniques cannot be used; often when mapping information from one dataset to a grid. Extrapolation methods vary, yielding different results both in quality and in performance. This article proposes a couple of simple extrapolation techniques that use all the information available to extrapolate. The techniques inspire from the weighted average principle.
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Weighted Average
Suppose you have a discrete dataset , with coordinates on a normed space, , and want to map the dataset to another dataset, , with a new set of coordinates . The operation is performed by constructing mapping coefficients , also called "weights", and by performing a linear combination of the elements of with the weights, , to compute each element of .
Any method of constructing the weights is valid (as long as ) and depends on the nature of the mapping. A practical use-case consists in performing an interpolation/extrapolation of the data from to and thus the weights are usually a function of the coordinates and and of the norm of the space. More complex techniques could be used, with also as a function of and its derivatives, but these will not be considered in this text.
Geometric Weighted Average
The geometric weighted average is the most intuitive approach: it calculates the geometric average of a set of values/points pairs at a given point. The weights are constructed naturally as