A statistical learning view of simple Kriging

• Emilia Siviero ^[2] ; Emilie Chautru ^[1] ; Stephan Clémençon ^[3]
  1. [1] Geosciences and Geoengineering Research Department

     Geosciences and Geoengineering Research Department

     Arrondissement de Fontainebleau, Francia

     
  2. [2] Institut Polytechnique de Paris
  3. [3] LTCI, Télécom Paris, Institut Polytechnique de Paris, Palaiseau, France

  Mostrar afiliaciones +
• Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 33, Nº. 1, 2024, págs. 271-296
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • In the Big Data era, with the ubiquity of geolocation sensors in particular, massive datasets exhibiting a possibly complex spatial dependence structure are becoming increasingly available. In this context, the standard probabilistic theory of statistical learning does not apply directly and guarantees of the generalization capacity of predictive rules learned from such data are left to establish. We analyze here the simple Kriging task, the flagship problem in Geostatistics, from a statistical learning perspective, i.e., by carrying out a nonparametric finite-sample predictive analysis. Given values taken by a realization of a square integrable random field , , with unknown covariance structure, at sites in S, the goal is to predict the unknown values it takes at any other location with minimum quadratic risk. The prediction rule being derived from a training spatial dataset: a single realization of X, is independent from those to be predicted, observed at locations in S. Despite the connection of this minimization problem with kernel ridge regression, establishing the generalization capacity of empirical risk minimizers is far from straightforward, due to the non-independent and identically distributed nature of the training data involved in the learning procedure. In this article, non-asymptotic bounds of order are proved for the excess risk of a plug-in predictive rule mimicking the true minimizer in the case of isotropic stationary Gaussian processes, observed at locations forming a regular grid in the learning stage. These theoretical results, as well as the role played by the technical conditions required to establish them, are illustrated by various numerical experiments, on simulated data and on real-world datasets, and hopefully pave the way for further developments in statistical learning based on spatial data.