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A fuzzy interval analysis approach to kriging with ill-known variogram and data


Geostatistics is a branch of statistics dealing with spatial phenomena. Kriging consists in estimating or predicting a spatial phenomenon at non-sampled locations from an estimated random function. It is assumed that, under some well-chosen simplifying hypotheses of stationarity, the probabilistic model, i.e. the random function describing spatial variability dependencies, can be completely assessed from the dataset. However, in the usual kriging approach, the choice of the random function is mostly made at a glance by the experts (i.e. geostatisticians), via the selection of a variogram from a thorough descriptive analysis of the dataset. Although information necessary to properly select a unique random function model seems to be partially lacking, geostatistics, in general, and the kriging methodology, in particular, does not account for the incompleteness of the information that seems to pervade the procedure. The paper proposes an approach to handle epistemic uncertainty appearing in the kriging methodology. On the one hand, the collected data may be tainted with errors that can be modelled by intervals or fuzzy intervals. On the other hand, the choice of parameter values for the theoretical variogram, an essential step, contains some degrees of freedom that are seldom acknowledged. In this paper, we propose to account for epistemic uncertainty pervading the variogram parameters, and possibly the dataset, and lay bare its impact on the kriging results, improving on previous attempts by Bardossy and colleagues in the late 1980s.

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  1. See page 24 of Chilès and Delfiner (1999). It is interesting to cite what they say in more details: “…no statistical test can disprove stationarity in general. We choose to consider \(z(x)\) as a realization of \(Z(x)\) over \(D.\) It does not mean that this decision is arbitrary - in practice, it is suggested by the homogeneity of the data - but simply that it cannot be refuted… ergodicity is also not an objective property”.


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This work is supported by the French Research National Agency (ANR) through the \({\text{CO}}_2\) program (project CRISCO2 ANR-06-CO2-003). The data were kindly provided to us by Jean-Paul Chilès and originates from the French institute IRSN (Institut de Radioprotection et de Sûreté nucléaire). The authors wish to thank Jean-Paul Chilès and Nicolas Desassis for their comments on a first draft of this paper and their support during the project.

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Loquin, K., Dubois, D. A fuzzy interval analysis approach to kriging with ill-known variogram and data. Soft Comput 16, 769–784 (2012).

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