Kriging prediction for manifold-valued random fields

Davide Pigoli; Alessandra Menafoglio; Piercesare Secchi

doi:10.1016/j.jmva.2015.12.006

Kriging prediction for manifold-valued random fields

Davide Pigoli^*, Alessandra Menafoglio, Piercesare Secchi

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

17 Citations (Scopus)

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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.

Original language	English
Pages (from-to)	117-131
Number of pages	15
Journal	JOURNAL OF MULTIVARIATE ANALYSIS
Volume	145
Early online date	25 Dec 2015
DOIs	https://doi.org/10.1016/j.jmva.2015.12.006
Publication status	Published - 1 Mar 2016

Keywords

Non Euclidean data
Positive definite symmetric matrices
Residual kriging

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10.1016/j.jmva.2015.12.006

Pigoli et al 2015 Journal of Multivariate AnalysisAccepted author manuscript, 609 KBLicence: CC BY-NC-ND

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title = "Kriging prediction for manifold-valued random fields",

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.",

keywords = "Non Euclidean data, Positive definite symmetric matrices, Residual kriging",

author = "Davide Pigoli and Alessandra Menafoglio and Piercesare Secchi",

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AU - Menafoglio, Alessandra

AU - Secchi, Piercesare

N1 - Please note that the accepted author manuscript of this publication was deposited at time of acceptance into the repository of the University of Cambridge (url: https://www.repository.cam.ac.uk/handle/1810/253013).

PY - 2016/3/1

Y1 - 2016/3/1

N2 - 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.

AB - 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.

KW - Non Euclidean data

KW - Positive definite symmetric matrices

KW - Residual kriging

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JO - JOURNAL OF MULTIVARIATE ANALYSIS

JF - JOURNAL OF MULTIVARIATE ANALYSIS

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Kriging prediction for manifold-valued random fields

Abstract

Keywords

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