Design-based spatial interpolation with data driven selection of the smoothing parameter
Lorenzo Fattorini, Sara Franceschi, Marzia Marcheselli, Caterina Pisani, Luca Pratelli
Abstract
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 $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and the appropriate choice of the smoothing parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> is a crucial issue. In this paper, we give sufficient conditions for the design-based consistency of the inverse distance weighting interpolator when $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> 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 $$\alpha =\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , 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.