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Generalized random fields on Riemannian manifolds : theory and practice

Abstract : Geostatistics is the branch of statistics attached to model spatial phenomena through probabilistic models. In particular, the spatial phenomenon is described by a (generally Gaussian) random field, and the observed data are considered as resulting from a particular realization of this random field. To facilitate the modeling and the subsequent geostatistical operations applied to the data, the random field is usually assumed to be stationary, thus meaning that the spatial structure of the data replicates across the domain of study. However, when dealing with complex spatial datasets, this assumption becomes ill-adapted. Indeed, how can the notion of stationarity be defined (and applied) when the data lie on non-Euclidean domains (such as spheres or other smooth surfaces)? Also, what about the case where the data clearly display a spatial structure that varies across the domain? Besides, using more complex models (when it is possible) generally comes at the price of a drastic increase in operational costs (computational and storage-wise), rendering them impossible to apply to large datasets. In this work, we propose a solution to both problems, which relies on the definition of generalized random fields on Riemannian manifolds. On one hand, working with generalized random fields allows to naturally extend ongoing work that is done to leverage a characterization of random fields used in Geostatistics as solutions of stochastic partial differential equations. On the other hand, working on Riemannian manifolds allows to define such fields on both (only) locally Euclidean domains and on locally deformed spaces (thus yielding a framework to account for non-stationary cases). The discretization of these generalized random fields is undertaken using a finite element approach, and we provide an explicit formula for a large class of fields comprising those generally used in applications. Finally, to solve the scalability problem,we propose algorithms inspired from graph signal processing to tackle the simulation, the estimation and the inference of these fields using matrix-free approaches.
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Mike Pereira. Generalized random fields on Riemannian manifolds : theory and practice. Signal and Image processing. Université Paris sciences et lettres, 2019. English. ⟨NNT : 2019PSLEM055⟩. ⟨tel-02499376⟩

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