Year 2020,
Volume: 3 Issue: 2, 85 - 95, 01.06.2020
Valdir Menegatto
,
Claudemir Oliveira
,
Emilio Porcu
References
- N. I. Akhiezer: Lectures on Integral transforms. Translated from Russian by H. H. McFaden. Translations of
Mathematical Monographs, 70. American Mathematical Society, Providence, RI, 1988.
- T. Apanasovich, M. Genton: Cross-covariance functions for multivariate random fields based on latent dimensions.
Biometrika 97 (2010), 15-30.
- A. Belton, D. Guillot, A. Khare, and M. Putinar: A Panorama of Positivity I: Dimension Free. In: Aleman A.,
Hedenmalm H., Khavinson D., Putinar M. (eds) Analysis of Operators on Function Spaces. Trends in Mathematics.
Birkhäuser, Cham, 2019.
- C. Berg, E. Porcu: From Schoenberg coefficients to Schoenberg functions. Constr. Approx. 45 (2017), 217-241.
- D. J. Daley, E. Porcu: Dimension walks and Schoenberg spectral measures. Proc. Amer. Math. Soc. 141 (2013), 1813-
1824.
- T. Fonseca, M. Steel: A general class of nonseparable space-time covariance models. Environmetrics 22 (2011), 224-242.
- T. Gneiting: Nonseparable, stationary covariance functions for space-time data. J. Amer. Statist. Assoc. 97 (2002), 590-
600.
- T. Gneiting, M. Genton and P. Guttorp: Geostatistical space-time models, stationarity, separability and full symmetry.
Finkenstaedt, B., Held, L. and Isham, V. (eds.), Statistics of Spatio-Temporal Systems, Chapman & Hall/CRC
Press, pp. 151-175, 2007.
- I. S. Gradshteyn, I. Ryzhik: Table of integrals, series, and products. Fourth edition prepared by Ju. V. Geronimus
and M. Ju. Ceitlin. Translated from Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey Academic
Press, New York-London, 1965.
- J. C. Guella, V. A. Menegatto: Schoenberg’s theorem for positive definite functions on products: a unifying framework.
J. Fourier Anal. Appl. 25 (2019), 1424-1446.
- R. Horn, C. Johnson: Topics in matrix analysis. Corrected reprint of the 1991 original. Cambridge University Press,
Cambridge, 1994.
- D. Karp, E. Prilepkina: Generalized Stieltjes functions and their exact order. J. Class. Anal. 1 (2012), 143-152.
- V. A. Menegatto: Positive definite functions on products of metric spaces via generalized Stieltjes functions, Proc. Amer.
Math. Soc (2020), to appear.
- V. A. Menegatto: Strictly positive definite kernels on the Hilbert sphere. Appl. Anal. 55 (1994), 91-101.
- E. Porcu, M. Bevilacqua and M. Genton: Spatio-temporal covariance and cross-covariance functions of the great circle
distance on a sphere. J. Amer. Stat. Assoc. 97 (2016), 590-600.
- E. Porcu, P. Gregori and J. Mateu: Nonseparable stationary anisotropic space-time covariance functions. Stoch. Environ.
Res. Risk Assess. 21 (2006), 113-122.
- E. Porcu, J. Mateu: Mixture-based modeling for space-time data. Environmetrics 18 (2007), 285-302.
- E. Porcu, J. Mateu and G. Christakos: Quasi-arithmetic means of covariance functions with potential applications to
space-time data. J. Multivariate Anal. 100 (2009), 1830-1844.
- A. Poularikas: The handbook of formulas and tables for signal processing. CRC Press, Boca Ratón, 1999.
- R. L. Schilling, R. Song and Z. Vondracek: Bernstein functions. Theory and applications. Second edition. De Gruyter
Studies in Mathematics, 37. Walter de Gruyter & Co., Berlin, 2012.
- M. Schlather: Some covariance models based on normal scale mixtures. Bernoulli 16 (2010), 780-797.
- I. J. Schoenberg: Metric spaces and completely monotone functions. Ann. of Math. 39 (1938), 811-841.
- I. J. Schoenberg: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44 (1938), 522-536.
- I. J. Schoenberg: Positive definite functions on spheres. Duke Math. J. 9 (1942), 96-108.
- K. Triméche: Generalized harmonic analysis and wavelet packets. Gordon and Breach Science Publishers, 2001.
- D. Widder: The Laplace Transform. Princeton University Press, Princeton, 1966.
- P. White, E. Porcu: Towards a complete picture of covariance functions on spheres cross time. Electron. J. Stat. 13 (2019),
2566-2594.
- J. H. Wells, L. R. Williams: Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete,
Band 84. Springer-Verlag, New York-Heidelberg, 1975.
- H. Wendland: Scattered data approximation. Cambridge Monographs on Applied and Computational Mathematics
Volume 17, Cambridge University Press, 2001.
- V. Zastavnyi, E. Porcu: Characterization theorems for the Gneiting class of space-time covariances. Bernoulli 17 (2011),
456-465.
Gneiting Class, Semi-Metric Spaces and Isometric Embeddings
Year 2020,
Volume: 3 Issue: 2, 85 - 95, 01.06.2020
Valdir Menegatto
,
Claudemir Oliveira
,
Emilio Porcu
Abstract
This paper revisits the Gneiting class of positive definite kernels originally proposed as a class of covariance functions for space-time processes.\ Under the framework of quasi-metric spaces and isometric embeddings, the paper proposes a general and unifying framework that encompasses results provided by earlier literature.\ Our results allow to study the positive definiteness of the Gneiting class over products of either Euclidean spaces or high dimensional spheres and quasi-metric spaces.\ In turn, Gneiting's theorem is proved here by a direct construction, eluding Fourier inversion (the so-called Gneiting's lemma) and convergence arguments that are required by Gneiting to preserve an integrability assumption.
Supporting Institution
None
References
- N. I. Akhiezer: Lectures on Integral transforms. Translated from Russian by H. H. McFaden. Translations of
Mathematical Monographs, 70. American Mathematical Society, Providence, RI, 1988.
- T. Apanasovich, M. Genton: Cross-covariance functions for multivariate random fields based on latent dimensions.
Biometrika 97 (2010), 15-30.
- A. Belton, D. Guillot, A. Khare, and M. Putinar: A Panorama of Positivity I: Dimension Free. In: Aleman A.,
Hedenmalm H., Khavinson D., Putinar M. (eds) Analysis of Operators on Function Spaces. Trends in Mathematics.
Birkhäuser, Cham, 2019.
- C. Berg, E. Porcu: From Schoenberg coefficients to Schoenberg functions. Constr. Approx. 45 (2017), 217-241.
- D. J. Daley, E. Porcu: Dimension walks and Schoenberg spectral measures. Proc. Amer. Math. Soc. 141 (2013), 1813-
1824.
- T. Fonseca, M. Steel: A general class of nonseparable space-time covariance models. Environmetrics 22 (2011), 224-242.
- T. Gneiting: Nonseparable, stationary covariance functions for space-time data. J. Amer. Statist. Assoc. 97 (2002), 590-
600.
- T. Gneiting, M. Genton and P. Guttorp: Geostatistical space-time models, stationarity, separability and full symmetry.
Finkenstaedt, B., Held, L. and Isham, V. (eds.), Statistics of Spatio-Temporal Systems, Chapman & Hall/CRC
Press, pp. 151-175, 2007.
- I. S. Gradshteyn, I. Ryzhik: Table of integrals, series, and products. Fourth edition prepared by Ju. V. Geronimus
and M. Ju. Ceitlin. Translated from Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey Academic
Press, New York-London, 1965.
- J. C. Guella, V. A. Menegatto: Schoenberg’s theorem for positive definite functions on products: a unifying framework.
J. Fourier Anal. Appl. 25 (2019), 1424-1446.
- R. Horn, C. Johnson: Topics in matrix analysis. Corrected reprint of the 1991 original. Cambridge University Press,
Cambridge, 1994.
- D. Karp, E. Prilepkina: Generalized Stieltjes functions and their exact order. J. Class. Anal. 1 (2012), 143-152.
- V. A. Menegatto: Positive definite functions on products of metric spaces via generalized Stieltjes functions, Proc. Amer.
Math. Soc (2020), to appear.
- V. A. Menegatto: Strictly positive definite kernels on the Hilbert sphere. Appl. Anal. 55 (1994), 91-101.
- E. Porcu, M. Bevilacqua and M. Genton: Spatio-temporal covariance and cross-covariance functions of the great circle
distance on a sphere. J. Amer. Stat. Assoc. 97 (2016), 590-600.
- E. Porcu, P. Gregori and J. Mateu: Nonseparable stationary anisotropic space-time covariance functions. Stoch. Environ.
Res. Risk Assess. 21 (2006), 113-122.
- E. Porcu, J. Mateu: Mixture-based modeling for space-time data. Environmetrics 18 (2007), 285-302.
- E. Porcu, J. Mateu and G. Christakos: Quasi-arithmetic means of covariance functions with potential applications to
space-time data. J. Multivariate Anal. 100 (2009), 1830-1844.
- A. Poularikas: The handbook of formulas and tables for signal processing. CRC Press, Boca Ratón, 1999.
- R. L. Schilling, R. Song and Z. Vondracek: Bernstein functions. Theory and applications. Second edition. De Gruyter
Studies in Mathematics, 37. Walter de Gruyter & Co., Berlin, 2012.
- M. Schlather: Some covariance models based on normal scale mixtures. Bernoulli 16 (2010), 780-797.
- I. J. Schoenberg: Metric spaces and completely monotone functions. Ann. of Math. 39 (1938), 811-841.
- I. J. Schoenberg: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44 (1938), 522-536.
- I. J. Schoenberg: Positive definite functions on spheres. Duke Math. J. 9 (1942), 96-108.
- K. Triméche: Generalized harmonic analysis and wavelet packets. Gordon and Breach Science Publishers, 2001.
- D. Widder: The Laplace Transform. Princeton University Press, Princeton, 1966.
- P. White, E. Porcu: Towards a complete picture of covariance functions on spheres cross time. Electron. J. Stat. 13 (2019),
2566-2594.
- J. H. Wells, L. R. Williams: Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete,
Band 84. Springer-Verlag, New York-Heidelberg, 1975.
- H. Wendland: Scattered data approximation. Cambridge Monographs on Applied and Computational Mathematics
Volume 17, Cambridge University Press, 2001.
- V. Zastavnyi, E. Porcu: Characterization theorems for the Gneiting class of space-time covariances. Bernoulli 17 (2011),
456-465.