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Quasi-arithmetic means of covariance functions with potential applications to space-time data

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  • Porcu, Emilio
  • Mateu, Jorge
  • Christakos, George

Abstract

The theory of quasi-arithmetic means represents a powerful tool in the study of covariance functions across space-time. In the present study we use quasi-arithmetic functionals to make inferences about the permissibility of averages of functions that are not, in general, permissible covariance functions. This is the case, e.g., of the geometric and harmonic averages, for which we obtain permissibility criteria. Also, some important inequalities involving covariance functions and preference relations as well as algebraic properties can be derived by means of the proposed approach. In particular, quasi-arithmetic covariances allow for ordering and preference relations, for a Jensen-type inequality and for a minimal and maximal element of their class. The general results shown in this paper are then applied to the study of spatial and spatio-temporal random fields. In particular, we discuss the representation and smoothness properties of a weakly stationary random field with a quasi-arithmetic covariance function. Also, we show that the generator of the quasi-arithmetic means can be used as a link function in order to build a space-time nonseparable structure starting from the spatial and temporal margins, a procedure that is technically sound for those working with copulas. Several examples of new families of stationary covariances obtainable with this procedure are shown. Finally, we use quasi-arithmetic functionals to generalise existing results concerning the construction of nonstationary spatial covariances, and discuss the applicability and limits of this generalisation.

Suggested Citation

  • Porcu, Emilio & Mateu, Jorge & Christakos, George, 2009. "Quasi-arithmetic means of covariance functions with potential applications to space-time data," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1830-1844, September.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:8:p:1830-1844
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    References listed on IDEAS

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    1. Porcu, E. & Mateu, J. & Zini, A. & Pini, R., 2007. "Modelling spatio-temporal data: A new variogram and covariance structure proposal," Statistics & Probability Letters, Elsevier, vol. 77(1), pages 83-89, January.
    2. Marc Hallin & Thomas S. Ferguson & Christian Genest, 2000. "Kendall's tau for serial dependence," ULB Institutional Repository 2013/2093, ULB -- Universite Libre de Bruxelles.
    3. Ma, Chunsheng, 2003. "Spatio-temporal stationary covariance models," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 97-107, July.
    4. Banerjee, Sudipto & Gelfand, Alan E., 2006. "Bayesian Wombling: Curvilinear Gradient Assessment Under Spatial Process Models," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1487-1501, December.
    5. Banerjee S. & Gelfand A.E. & Sirmans C.F., 2003. "Directional Rates of Change Under Spatial Process Models," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 946-954, January.
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    Cited by:

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    2. Fernández-Avilés, G & Montero, JM & Mateu, J, 2011. "Mathematical Genesis of the Spatio-Temporal Covariance Functions," MPRA Paper 35874, University Library of Munich, Germany.
    3. Hristo S. Sendov & Ričardas Zitikis, 2014. "The Shape of the Borwein–Affleck–Girgensohn Function Generated by Completely Monotone and Bernstein Functions," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 67-89, January.
    4. Schlather, Martin & Malinowski, Alexander & Menck, Peter J. & Oesting, Marco & Strokorb, Kirstin, 2015. "Analysis, Simulation and Prediction of Multivariate Random Fields with Package RandomFields," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 63(i08).
    5. Jun, Mikyoung, 2014. "Matérn-based nonstationary cross-covariance models for global processes," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 134-146.
    6. Mehdi Omidi & Mohsen Mohammadzadeh, 2016. "A new method to build spatio-temporal covariance functions: analysis of ozone data," Statistical Papers, Springer, vol. 57(3), pages 689-703, September.
    7. 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.
    8. Kleiber, William & Nychka, Douglas, 2012. "Nonstationary modeling for multivariate spatial processes," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 76-91.
    9. Christopher J. Geoga & Mihai Anitescu & Michael L. Stein, 2021. "Flexible nonstationary spatiotemporal modeling of high‐frequency monitoring data," Environmetrics, John Wiley & Sons, Ltd., vol. 32(5), August.
    10. Montero, José-María, 2018. "Geostatistics: Unde venis et quo vadis? /Geoestadística:¿De dónde vienes y a dónde vas?," Estudios de Economia Aplicada, Estudios de Economia Aplicada, vol. 36, pages 81-106, Enero.
    11. José-María Montero & Gema Fernández-Avilés & Tiziana Laureti, 2021. "A Local Spatial STIRPAT Model for Outdoor NO x Concentrations in the Community of Madrid, Spain," Mathematics, MDPI, vol. 9(6), pages 1-33, March.
    12. Nishiyama, Tomohiro, 2018. "Generalized Bregman and Jensen divergences which include some f-divergences," OSF Preprints ybmdx, Center for Open Science.
    13. 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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