x denotes an interpolated (arbitrary) point, xi is an interpolating (known) point, is a given distance (metric operator) from the known point xi to the unknown point x, N is the total number of known points used in interpolation and is a positive real number, called the power parameter. Here weight decreases as distance increases from the interpolated points. Greater values of 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 cause the interpolated values to be dominated by points far away, since with a density of data points and neighboring points between distances to , the summed weight is approximately which diverges for and . For M dimensions, the same argument holds for . 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 tuples of interpolating points {x, u} and i tuples of interpolated points {xi, ui}, defined as: derived from the minimizing condition: 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 and is available in Netlib as algorithm 661 in the toms library. Example in 1 dimension Modified Shepard's method Another modification of Shepard's method calculates interpolated value using only nearest neighbors within R-sphere (instead of full sample). Weights are slightly modified in this case: When combined with fast spatial search structure (like kd-tree), it becomes efficient N log N interpolation method suitable for large-scale problems. See also Field (geography) Gravity model Kernel density estimation Spatial analysis Tobler's first law of geography Tobler's second law of geography References Geostatistics Multivariate interpolation