Inverse distance weighting


Inverse distance weighting

Inverse distance weighting (IDW) is a method for multivariate interpolation, a process of assigning values to unknown points by using values from usually scattered set of known points.

A general form of finding an interpolated value "u" for a given point x using IDW is an interpolating function:

:u(mathbf{x}) = frac{ sum_{k = 0}^{N}{ w_k(mathbf{x}) u_k } }{ sum_{k = 0}^{N}{ w_k(mathbf{x}) } },

where:

:w_k(mathbf{x}) = frac{1}{d(mathbf{x},mathbf{x}_k)^p},

is a simple IDW weighting function, as defined by Shepard [cite conference |last=Shepard |first=Donald |year=1968 |title=A two-dimensional interpolation function for irregularly-spaced data |booktitle=Proceedings of the 1968 ACM National Conference |pages = 517–524 |doi=10.1145/800186.810616 ] , x denotes an interpolated (arbitrary) point, x"k" is an interpolating (known) point, d is a given distance (metric operator) from the known point x"k" to the unknown point x, "N" is the total number of known points used in interpolation and p is a positive real number, called the power parameter. Here weight decreases as distance increases from the interpolated points. Greater values of p assign greater influence to values closest to the interpolated point. For 0 < "p" < 1 "u"(x) has sharp peaks over the interpolated points xk, while for "p" > 1 the peaks are smooth. The most common value of p is 2.

The "Shepard's method" is a consequence of minimization of a functional related to a measure of deviations between tuples of interpolating points {x, "u"} and "k" tuples of interpolated points {x"k", "uk"}, defined as:

:phi(mathbf{x}, u) = left( sum_{k = 0}^{N}{frac{(u-u_k)^2}{d(mathbf{x},mathbf{x}_k)^p ight)^{frac{1}{p ,

derived from the minimizing condition:

:frac{part phi(mathbf{x}, u)}{part u} = 0.

The method can easily be extended to higher dimensional space and it is in fact a generalization of Lagrangeapproximation into a multidimensional spaces. A modified version of the algorithm designed for trivariate interpolation was developed by Robert J. Renka and is available in Netlib as algorithm 661 in the toms library.

Liszka's method

A modification of the Shepard's method was proposed by Liszka [cite journal | last = Liszka | first = T. | year = 1984 | title = An interpolation method for an irregular net of nodes | journal = International Journal for Numerical Methods in Engineering | volume = 20 | issue = 9 | pages = 1599–1612 | doi = 10.1002/nme.1620200905 ] in applications to experimental mechanics, who proposed to use:

:w_k(mathbf{x}) = frac{1}{(d(mathbf{x},mathbf{x}_k)^2+ varepsilon^2)^frac{1}{2,as a weighting function, where "ε" is chosen in dependence of the statistical error of measurement of the interpolated points.

Probability metric

Yet another modification of the Shepard's method was proposed by Łukaszyk [* [http://www.springerlink.com/content/y4fbdb0m0r12701p/ A new concept of probability metric and its applications in approximation of scattered data sets] ] also in applications to experimental mechanics. The proposed weighting function had the form:

:w_k(mathbf{x}) = frac{1}{(D_{**}(mathbf{x}, mathbf{x}_k) )^frac{1}{2,where D_{**}(mathbf{x}, mathbf{x}_k) is a probability metric chosen also with regard to the statistical error probability distributions of measurement of the interpolated points.

References

ee also

*Multivariate interpolation


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Pondération inverse à la distance — La Pondération Inverse à la Distance (PID) est une méthode d interpolation spatiale, un processus permettant d assigner une valeur à un espace non connu à partir d un semis de points connus. Une forme courante pour trouver une valeur interpolée u …   Wikipédia en Français

  • Glossaire du data mining — Exploration de données Articles principaux Exploration de données Fouille de données spatiales Fouille du web Fouille de flots de données Fouille de textes …   Wikipédia en Français

  • List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia

  • Multivariate interpolation — In numerical analysis, multivariate interpolation or spatial interpolation is interpolation on functions of more than one variable. The function to be interpolated is known at given points and the interpolation problem consist of yielding values… …   Wikipedia

  • Fouille de données spatiales — Exploration de données Articles principaux Exploration de données Fouille de données spatiales Fouille du web Fouille de flots de données Fouille de textes …   Wikipédia en Français

  • Rockworks — First developed in 1985 by RockWare Inc, RockWorks is used by the mining, petroleum, and environmental industry for subsurface visualization, borehole database management as well as the creation of grids, solid models, calculating volumetric ana …   Wikipedia

  • List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

  • Spatial analysis — In statistics, spatial analysis or spatial statistics includes any of the formal techniques which study entities using their topological, geometric, or geographic properties. The phrase properly refers to a variety of techniques, many still in… …   Wikipedia

  • Regionalisierung (Geostatistik) — Unter Regionalisierung versteht man in der Geostatistik die Übertragung von Punktdaten auf die Fläche. Dieser Vorgang spielt natürlich nicht nur in diesem Bereich eine Rolle, der sich schwerpunktmäßig damit befasst. Regionalisierung ist vielmehr… …   Deutsch Wikipedia

  • Geographic information system — GIS redirects here. For other uses, see GIS (disambiguation). A geographic information system, geographical information science, or geospatial information studies is a system designed to capture, store, manipulate, analyze, manage, and present… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.