Extrapolation

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.

Weighted Average

Suppose you have a discrete dataset $I$, with coordinates $X$ on a normed space, $d\left(.\,,\,.\right)$, and want to map the dataset to another dataset, $O$, with a new set of coordinates $Y$. The operation is performed by constructing mapping coefficients $w$, also called "weights", and by performing a linear combination of the elements of $I$ with the weights, $w$, to compute each element of $O$.

$O_k = \frac{ \sum_i w_{ki} \, I_i }{\sum_i w_{ki}};$

Any method of constructing the weights is valid (as long as $\sum_i w_{ki} \neq 0$ ) and depends on the nature of the mapping. A practical use-case consists in performing an interpolation/extrapolation of the data from $I$ to $O$ and thus the weights $w$ are usually a function of the coordinates $X$ and $Y$ and of the norm of the space. More complex techniques could be used, with $w$ also as a function of $I$ 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

$w_{ki} = \frac{1}{d\left(X_i,\;Y_k\right)}$

See also

• ConvertToHdf5