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Kriging for Hilbert-space valued random fields: The operatorial point of view

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  • Menafoglio, Alessandra
  • Petris, Giovanni

Abstract

We develop a comprehensive framework for linear spatial prediction in Hilbert spaces. We explore the problem of Best Linear Unbiased (BLU) prediction in Hilbert spaces through an original point of view, based on a new Operatorial definition of Kriging. We ground our developments on the theory of Gaussian processes in function spaces and on the associated notion of measurable linear transformation. We prove that our new setting allows (a) to derive an explicit solution to the problem of Operatorial Ordinary Kriging, and (b) to establish the relation of our novel predictor with the key concept of conditional expectation of a Gaussian measure. Our new theory is posed as a unifying theory for Kriging, which is shown to include the Kriging predictors proposed in the literature on Functional Data through the notion of finite-dimensional approximations. Our original viewpoint to Kriging offers new relevant insights for the geostatistical analysis of either finite- or infinite-dimensional georeferenced dataset.

Suggested Citation

  • Menafoglio, Alessandra & Petris, Giovanni, 2016. "Kriging for Hilbert-space valued random fields: The operatorial point of view," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 84-94.
  • Handle: RePEc:eee:jmvana:v:146:y:2016:i:c:p:84-94
    DOI: 10.1016/j.jmva.2015.06.012
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    References listed on IDEAS

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    1. Chen, Kun & Chen, Kehui & Müller, Hans-Georg & Wang, Jane-Ling, 2011. "Stringing High-Dimensional Data for Functional Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 275-284.
    2. Pedro Delicado, 2007. "Functional k-sample problem when data are density functions," Computational Statistics, Springer, vol. 22(3), pages 391-410, September.
    3. Aneiros, Germán & Vieu, Philippe, 2014. "Variable selection in infinite-dimensional problems," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 12-20.
    4. Nerini, David & Monestiez, Pascal & Manté, Claude, 2010. "Cokriging for spatial functional data," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 409-418, February.
    5. Sangalli, Laura M. & Secchi, Piercesare & Vantini, Simone & Vitelli, Valeria, 2010. "k-mean alignment for curve clustering," Computational Statistics & Data Analysis, Elsevier, vol. 54(5), pages 1219-1233, May.
    6. Simone Vantini, 2012. "On the definition of phase and amplitude variability in functional data analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(4), pages 676-696, December.
    7. Delicado, P., 2011. "Dimensionality reduction when data are density functions," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 401-420, January.
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    Cited by:

    1. Menafoglio, Alessandra & Secchi, Piercesare, 2017. "Statistical analysis of complex and spatially dependent data: A review of Object Oriented Spatial Statistics," European Journal of Operational Research, Elsevier, vol. 258(2), pages 401-410.

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