Inverse distance weighting (IDW) is a type of deterministic method for multivariate interpolation with a known homogeneously scattered set of points. The assigned values to unknown points are calculated with a weighted average of the values available at the known points. This method can also be used to create spatial weights matrices in spatial autocorrelation analyses (e.g. Moran's I). The name given to this type of method was motivated by the weighted average applied, since it resorts to the inverse of the distance to each known point ("amount of proximity") when assigning weights.
Definition of the problem The expected result is a discrete assignment of the unknown function u {\displaystyle u} in a study region:
u ( x ) : x → R , x ∈ D ⊂ R n , {\displaystyle u(x):x\to \mathbb {R} ,\quad x\in \mathbf {D} \subset \mathbb {R} ^{n},}
where D {\displaystyle \mathbf {D} } is the study region. The set of N {\displaystyle N} known data points can be described as a list of tuples:
[ ( x 1 , u 1 ) , ( x 2 , u 2 ) , . . . , ( x N , u N ) ] . {\displaystyle [(x_{1},u_{1}),(x_{2},u_{2}),...,(x_{N},u_{N})].}
The function is to be "smooth" (continuous and once differentiable), to be exact ( u ( x i ) = u i {\displaystyle u(x_{i})=u_{i}} ) and to meet the user's intuitive expectations about the phenomenon under investigation. Furthermore, the function should be suitable for a computer application at a reasonable cost (nowadays, a basic implementation will probably make use of parallel resources).
Shepard's method
Historical reference At the Harvard Laboratory for Computer Graphics and Spatial Analysis, beginning in 1964, a varied collection of scientists converged to rethink, among other things, what are now called geographic information systems. The motive force behind the Laboratory, Howard Fisher, conceived an improved computer mapping program that he called SYMAP, which, from the start, Fisher wanted to improve on the interpolation. He showed Harvard College freshmen his work on SYMAP, and many of them participated in Laboratory events. One freshman, Donald Shepard, decided to overhaul the interpolation in SYMAP, resulting in his famous article from 1968. Shepard's algorithm was also influenced by the theoretical approach of William Warntz and others at the Lab who worked with spatial analysis. He conducted a number of experiments with the exponent of distance, deciding on something closer to the gravity model (exponent of -2). Shepard implemented not just basic inverse distance weighting, but also allowed barriers (permeable and absolute) to interpolation. Other research centers were working on interpolation at this time, particularly University of Kansas and their SURFACE II program. Still, the features of SYMAP were state-of-the-art, even though programmed by an undergraduate.
Basic form
Given a set of sample points { x i , u i | for x i ∈ R n , u i ∈ R } i = 1 N {\displaystyle \{\mathbf {x} _{i},u_{i}|{\text{for }}\mathbf {x} _{i}\in \mathbb {R} ^{n},u_{i}\in \mathbb {R} \}_{i=1}^{N}} , the IDW interpolation function u ( x ) : R n → R {\displaystyle u(\mathbf {x} ):\mathbb {R} ^{n}\to \mathbb {R} } is defined as:
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