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Kriging prediction for manifold-valued random fields

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  • Pigoli, Davide
  • Menafoglio, Alessandra
  • Secchi, Piercesare

Abstract

The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications, such as shape analysis, diffusion tensor imaging and the analysis of covariance matrices. In many cases, data are spatially distributed but it is not trivial to take into account spatial dependence in the analysis because of the non linear geometry of the manifold. This work proposes a solution to the problem of spatial prediction for manifold valued data, with a particular focus on the case of positive definite symmetric matrices. Under the hypothesis that the dispersion of the observations on the manifold is not too large, data can be projected on a suitably chosen tangent space, where an additive model can be used to describe the relationship between response variable and covariates. Thus, we generalize classical kriging prediction, dealing with the spatial dependence in this tangent space, where well established Euclidean methods can be used. The proposed kriging prediction is applied to the matrix field of covariances between temperature and precipitation in Quebec, Canada.

Suggested Citation

  • Pigoli, Davide & Menafoglio, Alessandra & Secchi, Piercesare, 2016. "Kriging prediction for manifold-valued random fields," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 117-131.
  • Handle: RePEc:eee:jmvana:v:145:y:2016:i:c:p:117-131
    DOI: 10.1016/j.jmva.2015.12.006
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    References listed on IDEAS

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    2. Davide Pigoli & John A. D. Aston & Ian L. Dryden & Piercesare Secchi, 2014. "Distances and inference for covariance operators," Biometrika, Biometrika Trust, vol. 101(2), pages 409-422.
    3. Nerini, David & Monestiez, Pascal & Manté, Claude, 2010. "Cokriging for spatial functional data," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 409-418, February.
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    5. Osborne, Daniel & Patrangenaru, Vic & Ellingson, Leif & Groisser, David & Schwartzman, Armin, 2013. "Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 163-175.
    6. Ying Yuan & Hongtu Zhu & Weili Lin & J. S. Marron, 2012. "Local polynomial regression for symmetric positive definite matrices," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(4), pages 697-719, September.
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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.
    2. Antonio Balzanella & Antonio Irpino, 2020. "Spatial prediction and spatial dependence monitoring on georeferenced data streams," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(1), pages 101-128, March.

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