Difference between revisions of "Extrapolation"
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==Squared Weighted Average== | ==Squared Weighted Average== | ||
| + | The same as above, except that the squared norm is used instead of the norm. | ||
| + | |||
| + | <math> w_{ki} = \frac{1}{d^2\left(X_i,\;Y_k\right)} </math> | ||
==Gaussian Weighted Average== | ==Gaussian Weighted Average== | ||
Revision as of 10:51, 10 February 2012
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.
Contents
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
Squared Weighted Average
The same as above, except that the squared norm is used instead of the norm.
