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Characterization theorems for pseudo cross-variograms

Part of: Stochastic processes Geophysics Geophysics (general)

Published online by Cambridge University Press:  24 April 2023

Christopher Dörr  and
Martin Schlather
Christopher Dörr*
Affiliation:
University of Mannheim
Martin Schlather*
Affiliation:
University of Mannheim
*
*Postal address: Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany.
*Postal address: Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany.
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Abstract

Pseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case.

Keywords

Multivariate geostatisticsconditionally negative definitepositive definitespace–time covariance functionsGneiting functions

MSC classification

Primary: 86A32: Geostatistics
Secondary: 60G10: Stationary processes

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1219 - 1231
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Allard, D., Emery, X., Lacaux, C. and Lantuéjoul, C. (2020). Simulating space–time random fields with nonseparable Gneiting-type covariance functions. Statist. Comput. 30, 1479–1495.CrossRefGoogle Scholar
Berg, C., Christensen, J. P. R. and Ressel, P. (1984). Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Springer, New York.CrossRefGoogle Scholar
Berg, C., Koumandos, S. and Pedersen, H. L. (2021). Nielsen’s beta function and some infinitely divisible distributions. Math. Nachr. 294, 426–449.CrossRefGoogle Scholar
Bourotte, M., Allard, D. and Porcu, E. (2016). A flexible class of non-separable cross-covariance functions for multivariate space–time data. Spat. Statist. 18, 125–146.CrossRefGoogle Scholar
Chen, W. and Genton, M. G. (2019). Parametric variogram matrices incorporating both bounded and unbounded functions. Stoch. Environ. Res. Risk Assess. 33, 1669–1679.CrossRefGoogle Scholar
Cressie, N. and Wikle, C. K. (1998). The variance-based cross-variogram: you can add apples and oranges. Math. Geol. 30, 789–799.CrossRefGoogle Scholar
Du, J. and Ma, C. (2012). Variogram matrix functions for vector random fields with second-order increments. Math. Geosci. 44, 411–425.CrossRefGoogle Scholar
Genton, M. G., Padoan, S. A. and Sang, H. (2015). Multivariate max-stable spatial processes. Biometrika 102, 215–230.CrossRefGoogle Scholar
Gesztesy, F. and Pang, M. (2016). On (conditional) positive semidefiniteness in a matrix-valued context. Available at arXiv:1602.00384.Google Scholar
Gneiting, T. (2002). Nonseparable, stationary covariance functions for space–time data. J. Amer. Statist. Assoc. 97, 590–600.CrossRefGoogle Scholar
Gneiting, T., Sasvári, Z. and Schlather, M. (2001). Analogies and correspondences between variograms and covariance functions. Adv. Appl. Prob. 33, 617–630.CrossRefGoogle Scholar
Guella, J. C. (2020). On Gaussian kernels on Hilbert spaces and kernels on hyperbolic spaces. Available at arXiv:2007.14697.Google Scholar
Ma, C. (2011). A class of variogram matrices for vector random fields in space and/or time. Math. Geosci. 43, 229–242.Google Scholar
Ma, C. (2011). Vector random fields with second-order moments or second-order increments. Stoch. Anal. Appl. 29, 197–215.CrossRefGoogle Scholar
Matheron, G. (1972). Leçon sur les fonctions aléatoire d’ordre 2. Tech. Rep. C-53, MINES Paristech – Centre de Géosciences.Google Scholar
Matheron, G. (1973). The intrinsic random functions and their applications. Adv. Appl. Prob. 5, 439–468.CrossRefGoogle Scholar
Menegatto, V. A. (2020). Positive definite functions on products of metric spaces via generalized Stieltjes functions. Proc. Amer. Math. Soc. 148, 4781–4795.CrossRefGoogle Scholar
Menegatto, V. A., Oliveira, C. P. and Porcu, E. (2020). Gneiting class, semi-metric spaces and isometric embeddings. Constr. Math. Anal. 3, 85–95.Google Scholar
Myers, D. E. (1982). Matrix formulation of co-kriging. J. Int. Assoc. Math. Geol. 14, 249–257.CrossRefGoogle Scholar
Myers, D. E. (1991). Pseudo-cross variograms, positive-definiteness, and cokriging. Math. Geol. 23, 805–816.CrossRefGoogle Scholar
Oesting, M., Schlather, M. and Friederichs, P. (2017). Statistical post-processing of forecasts for extremes using bivariate Brown–Resnick processes with an application to wind gusts. Extremes 20, 309–332.CrossRefGoogle Scholar
Papritz, A., Künsch, H. R. and Webster, R. (1993). On the pseudo cross-variogram. Math. Geol. 25, 1015–1026.CrossRefGoogle Scholar
Porcu, E., Bevilacqua, M. and Genton, M. G. (2016). Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. J. Amer. Statist. Assoc. 111, 888–898.Google Scholar
Porcu, E., Furrer, R. and Nychka, D. (2021). 30 years of space–time covariance functions. Wiley Interdiscip. Rev. Comput. Statist. 13, e1512.CrossRefGoogle Scholar
Qadir, G. and Sun, Y. (2022). Modeling and predicting spatio-temporal dynamics of ${\textbf{PM}}_{2.5}$ concentrations through time-evolving covariance models. Available at arXiv:2202.12121.Google Scholar
Schilling, R. L., Song, R. and Vondracek, Z. (2012). Bernstein Functions, 2nd rev. and ext. edn. De Gruyter, Berlin.Google Scholar
Schlather, M. (2010). Some covariance models based on normal scale mixtures. Bernoulli 16, 780–797.Google Scholar
Schoenberg, I. J. (1938). Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522–536.CrossRefGoogle Scholar
Ver Hoef, J. and Cressie, N. (1993). Multivariable spatial prediction. Math. Geol. 25, 219–240.Google Scholar
Wackernagel, H. (2003). Multivariate Geostatistics: An Introduction with Applications, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Xie, T. (1994). Positive definite matrix-valued functions and matrix variogram modeling. Doctoral thesis, The University of Arizona.Google Scholar
Zastavnyi, V. P. and Porcu, E. (2011). Characterization theorems for the Gneiting class of space–time covariances. Bernoulli 17, 456–465.Google Scholar