- 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
tuple s 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 thestatistical error probability distribution s of measurement of the interpolated points.References
ee also
*
Multivariate interpolation
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