Abstract
Interpolation of spatial data is a very general mathematical problem with many applications, such as surface reconstruction, the numerical solution of partial differential equations, learning theory, and the prediction of environmental variables, to name a few. One important statistical approach to this problem is known as Kriging, the geostatistical term for optimal linear prediction of spatial processes. It is identical to a method called kernel interpolation used in numerical analysis for the same problem, but derived under completely different modelling assumptions. Despite their similarity, these two approaches have so far been developed largely independently within the two different mathematical communities.Synthesizing results from both statistical and numerical analysis literature with many new results, this monograph presents and contrasts the two modelling paradigms, yielding an understanding of the different notions of optimality and the different concepts to quantify the interpolation error. New results are presented which allow for a comprehensive characterization of the sample path regularity of second-order random fields (the common model in geostatistics), showing that the typical modelling assumptions in numerical analysis are also made implicitly in the statistical model. Finally we explore both theoretically and in simulation studies in how far methods for identifying the covariance parameters of a random field and for selecting a good interpolation kernel can be used in the respective other framework.This PhD thesis is entirely self-contained providing a concise introduction to both probability theory and to the theory of reproducing kernel Hilbert spaces. It is addressed to researchers, lectures and students with a background and interest in either statistics or numerical analysis and may serve as a lecture note as well as a reference manual for questions concerning models and methods for spatial interpolation.
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