Kriging and splines: Theoretical approach to linking spatial prediction methods

Abstract

Spatial Statistics refers to a class of models and methods for spatial data that aim at providing quantitative descriptions of natural variables distributed in space or space and time (see Chiles and Delfiner [2], Cressie [3]). Examples for such variables are ore grades collected in a mineral field,density of trees of a certain species in a forest or CD (critical dimension) measurements in semiconductor productions. A typical problem in spatial statistics is to predict values of measurements at places where they were not observed, or if measured with error, to estimate a smooth spatial surface from the data. (Estimation of a regionalized variable.) A family of techniques, stochastic and non stochastic ones were developed in geostatistics for that interpolation problem. The general approach is to consider a class of unbiased estimators, usually linear in the observations and to find the one with minimum uncertainty, as measured by the error variance. A group of techniques, known loosely as kriging, is a popular method among different interpolation techniques developed in geostatistics by Krige [10], Matheron [11] and Journel and Huijbregts [8]. An interesting comparison of ten classes of interpolation techniques with characteristics can be found in Burrough and McDonnell [1] and in a lot of papers published recently a comparison of several interpolation techniques was made. The goal of kriging like that of nonparametric regression is that the understanding of spatial estimation is enriched by the interpretation as smoothing estimates. On the other hand random process models are also valuable in setting uncertainty estimates for function estimates, specially in low noise situations. There are close connections between different mathematical subjects such as kriging, radial basis functions (RBF) interpolations, spline interpolations, reproducing hilbert space kernels (rhsk), PDE, Markov Random Fields (MRF) etc. A short discussion of these links is given in Horiwitz et al. [7], see Fig. 7. Splines link different fields of mathematics and statistics and are used in statistics for spatial modelling (see more in Wahba [15]). © 2009 Springer Berlin Heidelberg.