Design-based spatial interpolation with data driven selection of the smoothing parameter

Abstract

In the inverse distance weighting interpolation the interpolated, value is a weighted mean of the sampled values, with weights decreasing with the distances. The most widely adopted class of distance functions is the class of negative powers of order alpha and the appropriate choice of the smoothing parameter alpha is a crucial issue. In this paper, we give sufficient conditions for the design-based consistency of the inverse distance weighting interpolator when alpha is selected by cross-validation techniques, and a pseudo-population bootstrap approach is introduced to estimate the accuracy of the resulting interpolator. A simulation study is performed to empirically confirm the theoritical findings and to investigate the finite-sample properties of the interpolator obtained using leave-one-out cross-validation. Moreover, a comparison with the nearest neighbor interpolator, which is the limiting case for alpha =infty , is performed. Finally, the estimation of the surface of the Shannon diversity index of tree diameter at breast height in the experimental watershed of Bonis forest (Southern Italy) is described.