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Interpolation of spatial data – A stochastic or a deterministic problem?

Published online by Cambridge University Press:  07 February 2013

M. SCHEUERER
Affiliation:
Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany email: michael.scheuerer@uni-heidelberg.de
R. SCHABACK
Affiliation:
Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestr, 16-18, D-37083 Göttingen, Germany email: schaback@math.uni-goettingen.de
M. SCHLATHER
Affiliation:
Institut für Mathematik, Universität Mannheim, A5, 6, D-68131 Mannheim, Germany email: schlather@math.uni-mannheim.de

Abstract

Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this paper we present the two approaches and discuss their modelling assumptions, notions of optimality and different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection and discuss the scope of these methods with a data example from the computer experiments literature.

Type
A Survey in Mathematics for Industry
Copyright
Copyright © Cambridge University Press 2013 

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