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, 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 smooth peaks over the interpolated points x_k, while as p > 1 the peaks become sharp. The choice of value for p is therefore a function of 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.

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, u_k}, defined as:

$\phi (\mathbf {x} ,u)=\left(\sum _{k=0}^{N}{\frac {(u-u_{k})^{2}}{d(\mathbf {x} ,\mathbf {x} _{k})^{p}}}\right)^{\frac {1}{p}},$

derived from the minimizing condition:

${\frac {\partial \phi (\mathbf {x} ,u)}{\partial u}}=0.$

The method can easily be extended to higher dimensional space 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 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 [This reference should be checked! In the mentioned paper that formula simply does not appear, and the paper talks about a method based on the Taylor expansion. Moreover, at the last row of page 1609 of the concerned article, it is written "No explicit form of the function is available"] 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.

Lukaszyk-Karmowski metric

Yet another modification of the Shepard's method was proposed by Łukaszyk 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 the Lukaszyk-Karmowski metric chosen also with regard to the statistical error probability distributions of measurement of the interpolated points.

See also

• Multivariate interpolation