Editing Inverse distance weighting (section)

=== Basic form ===
[[Image:Shepard interpolation 2.png|thumb|640px|center|Shepard's interpolation for different power parameters ''p'', from scattered points on the surface <math>z=\exp(-x^2-y^2)</math>]]

Given a set of sample points <math>\{ \mathbf{x}_i, u_i | \text{for } \mathbf{x}_i \in \mathbb{R}^n,  u_i \in \mathbb{R}\}_{i=1}^N</math>, the IDW interpolation function <math>u(\mathbf{x}): \mathbb{R}^n \to \mathbb{R}</math> is defined as:

:<math>u(\mathbf{x}) = \begin{cases}
 \dfrac{\sum_{i = 1}^{N}{ w_i(\mathbf{x}) u_i } }{ \sum_{i = 1}^{N}{ w_i(\mathbf{x}) } }, & \text{if } d(\mathbf{x},\mathbf{x}_i) \neq 0 \text{ for all } i, \\
 u_i, & \text{if } d(\mathbf{x},\mathbf{x}_i) = 0 \text{ for some } i,
\end{cases} </math>

where

:<math>w_i(\mathbf{x}) =  \frac{1}{d(\mathbf{x},\mathbf{x}_i)^p}</math>

is a simple IDW weighting function, as defined by Shepard,<ref name=shepardArticle/> '''x''' denotes an interpolated (arbitrary) point, '''x'''<sub>''i''</sub> is an interpolating (known) point, <math>d</math> is a given distance ([[Metric (mathematics)|metric]] operator) from the known point '''x'''<sub>''i''</sub> to the unknown point '''x''', ''N'' is the total number of known points used in interpolation and <math>p</math> is a positive real number, called the power parameter.

Here weight decreases as distance increases from the interpolated points. Greater values of <math>p</math> assign greater influence to values closest to the interpolated point, with the result turning into a mosaic of tiles (a [[Voronoi diagram]]) with nearly constant interpolated value for large values of ''p''. For two dimensions, power parameters <math>p \leq 2</math> cause the interpolated values to be dominated by points far away, since with a density <math>\rho</math> of data points and neighboring points between distances <math>r_0</math> to <math>R</math>, the summed weight is approximately
:<math>\sum_j w_j \approx \int_{r_0}^R \frac{2\pi r\rho \,dr}{r^p} = 2\pi\rho\int_{r_0}^R r^{1-p} \,dr,</math>
which diverges for <math>R\rightarrow\infty</math> and <math>p\leq2</math>. For ''M'' dimensions, the same argument holds for <math>p\leq M</math>. For the choice of value for ''p'', one can consider the degree of smoothing desired in the interpolation, the density and distribution of samples being interpolated, and the maximum distance over which an individual sample is allowed to influence the surrounding ones.

''Shepard's method'' is a consequence of minimization of a functional related to a measure of deviations between [[tuple]]s of interpolating points {'''x''', ''u''} and ''i'' tuples of interpolated points {'''x'''<sub>''i''</sub>, ''u<sub>i</sub>''}, defined as:
:<math>\phi(\mathbf{x}, u) = \left( \sum_{i = 0}^{N}{\frac{(u-u_i)^2}{d(\mathbf{x},\mathbf{x}_i)^p}} \right)^{\frac{1}{p}} ,</math>
derived from the minimizing condition:
:<math>\frac{\partial \phi(\mathbf{x}, u)}{\partial u} = 0.</math>

The method can easily be extended to other dimensional spaces and it is in fact a generalization of Lagrange approximation into a multidimensional spaces. A modified version of the algorithm designed for trivariate interpolation was developed by Robert J. Renka <ref>[https://computerscience.engineering.unt.edu/people/faculty/robert-renka Robert Renka, Professor Emeritus], [[University of North Texas]]</ref> and is available in [[Netlib]] as [https://people.sc.fsu.edu/~jburkardt/f77_src/toms661/toms661.f algorithm 661] in the [[ACM Transactions on Mathematical Software|TOMS Library]].