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Strictly positive definite multivariate covariance functions on spheres

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  • Guella, Jean Carlo
  • Menegatto, Valdir Antonio
  • Porcu, Emilio

Abstract

We study the strict positive definiteness of matrix-valued covariance functions associated to multivariate random fields defined over d-dimensional spheres of the (d+1)-dimensional Euclidean space. Characterization of strict positive definiteness is crucial to both estimation and cokriging prediction in classical geostatistical routines. We provide characterization theorems for high dimensional spheres as well as for the Hilbert sphere. We offer a necessary condition for positive definiteness on the circle. Finally, we discuss a parametric example which might turn to be useful for geostatistical applications.

Suggested Citation

  • Guella, Jean Carlo & Menegatto, Valdir Antonio & Porcu, Emilio, 2018. "Strictly positive definite multivariate covariance functions on spheres," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 150-159.
    • Handle: RePEc:eee:jmvana:v:166:y:2018:i:c:p:150-159
    DOI: 10.1016/j.jmva.2018.03.001
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    References listed on IDEAS

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    1. Alonso-Malaver, C.E. & Porcu, E. & Giraldo, R., 2015. "Multivariate and multiradial Schoenberg measures with their dimension walks," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 251-265.
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    3. P. Gregori & E. Porcu & J. Mateu & Z. Sasvári, 2008. "On potentially negative space time covariances obtained as sum of products of marginal ones," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(4), pages 865-882, December.
    4. Stefano Castruccio & Joseph Guinness, 2017. "An evolutionary spectrum approach to incorporate large-scale geographical descriptors on global processes," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(2), pages 329-344, February.
    5. Bonfim, Rafaela N. & Menegatto, Valdir A., 2016. "Strict positive definiteness of multivariate covariance functions on compact two-point homogeneous spaces," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 237-248.
    6. Emilio Porcu & Moreno Bevilacqua & Marc G. Genton, 2016. "Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(514), pages 888-898, April.
    7. Sudipto Banerjee, 2005. "On Geodetic Distance Computations in Spatial Modeling," Biometrics, The International Biometric Society, vol. 61(2), pages 617-625, June.
    8. Porcu, Emilio & Zastavnyi, Viktor, 2011. "Characterization theorems for some classes of covariance functions associated to vector valued random fields," Journal of Multivariate Analysis, Elsevier, vol. 102(9), pages 1293-1301, October.
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