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}) } },


: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 [* [ 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.


ee also

*Multivariate interpolation

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