Difference between revisions of "Extrapolation"
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==Geometric Weighted Average== | ==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 | + | 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 idea is that closer points should have a bigger weight than farther points in the weighted average. In this case, the weight is inversely proportional to the distance: |
<math> w_{ki} = \frac{1}{d\left(X_i,\;Y_k\right)} </math> | <math> w_{ki} = \frac{1}{d\left(X_i,\;Y_k\right)} </math> | ||
==Squared Weighted Average== | ==Squared Weighted Average== | ||
| − | The same as above, except that the squared norm is used instead of the norm. | + | The same as above, except that the squared norm is used instead of the norm. This gives an even bigger relative weight to the points that are closer, proportionally to the inverse of the squared distance. |
<math> w_{ki} = \frac{1}{d^2\left(X_i,\;Y_k\right)} </math> | <math> w_{ki} = \frac{1}{d^2\left(X_i,\;Y_k\right)} </math> | ||
Revision as of 10:55, 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 idea is that closer points should have a bigger weight than farther points in the weighted average. In this case, the weight is inversely proportional to the distance:
Squared Weighted Average
The same as above, except that the squared norm is used instead of the norm. This gives an even bigger relative weight to the points that are closer, proportionally to the inverse of the squared distance.
