Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus
Authors:
François Bachoc, Ana Paula Peron and Emilio Porcu
Journal:
Theor. Probability and Math. Statist. 107 (2022), 3-14
MSC (2020):
Primary 62M15, 62M30; Secondary 60G12
DOI:
https://doi.org/10.1090/tpms/1176
Published electronically:
November 8, 2022
Full-text PDF
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Additional Information
Abstract:
The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings.
We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.
References
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References
- A. Alegría, F. Cuevas-Pacheco, P. Diggle, and E. Porcu, The $\mathcal {F}$-family of covariance functions: A Matérn analogue for modeling random fields on spheres, 2021. MR 4253853
- D. Allard, R. Senoussi, and E. Porcu, Anisotropy models for spatial data, Int. Assoc. Math. Geosci. Stud. Math. Geosci. 48 (2016), no. 3, 305–328. MR 3476704
- A. Arafat, E. Porcu, M. Bevilacqua, and J. Mateu, Equivalence and orthogonality of Gaussian measures on spheres, J. Multivariate Anal. 167 (2018), 306–318. MR 3830648
- F. Bachoc, F. Gamboa, J.-M. Loubes, and N. Venet, A Gaussian process regression model for distribution inputs, IEEE Transactions on Information Theory 64 (2017), no. 10, 6620–6637. MR 3860751
- F. Bachoc, E. Porcu, M. Bevilacqua, R. Furrer, and T. Faouzi, Asymptotically equivalent prediction in multivariate geostatistics, Bernoulli arXiv:2007.14684 (2020). MR 4474552
- C. Berg, J. P. R. Christensen, and P. Ressel, Harmonic analysis on semigroups: theory of positive definite and related functions, vol. 100, Springer, 1984. MR 747302
- C. Berg, A. Peron, and E. Porcu, Orthogonal expansions related to compact Gelfand pairs, Exposithiones Mathematicae 36 (2018), 259–277. MR 3907332
- —, Schoenberg’s theorem for real and complex Hilbert spheres revisited, J. Approx. Theory 228 (2018), 58–78. MR 3766896
- C. Berg and E. Porcu, From Schoenberg coefficients to Schoenberg functions, Constructive Approximation 45 (2017), no. 2, 217–241. MR 3619442
- M. Bevilacqua, T. Faouzi, R. Furrer, and E. Porcu, Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics, Ann. Statist. 47 (2019), no. 2, 828–856. MR 3909952
- H. Cramer, On the theory of stationary random processes, Ann. Math. Artif. Intell. 41 (1940), no. 1, 215–230. MR 920
- A. Estrade, A. Fariñas, and E. Porcu, Covariance functions on spheres cross time: Beyond spatial isotropy and temporal stationarity, Statist. Probab. Lett. 151 (2019), 1–7. MR 3934007
- J. C. Guella and V. A. Menegatto, Strictly positive definite kernels on a product of spheres, Aust. J. Math. Anal. Appl. 435 (2016), 286–301. MR 3423396
- —, Conditionally positive definite matrix valued kernels on Euclidean spaces, Constr. Approx. 52 (2020), no. 1, 65–92. MR 4118977
- J. C. Guella, V. A. Menegatto, and A. P. Peron, An extension of a theorem of Schoenberg to products of spheres, Banach J. Math. Anal. 435 (2015), 286–301. MR 3423396
- —, Strictly positive definite kernels on a product of spheres II, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 103, 15. MR 3566202
- —, Strictly positive definite kernels on a product of circles, Positivity 21 (2017), no. 1, 329–342. MR 3613000
- N. Leonenko and A. Malyarenko, Matérn class tensor-valued random fields and beyond, J. Stat. Phys. 168 (2017), no. 6, 1276–1301. MR 3691251
- A. Malyarenko and M. Ostoja-Starzewski, Tensor-valued random fields for continuum physics, Cambridge University Press, 2018. MR 3930601
- D. Marinucci and G. Peccati, Random Fields on the Sphere, Representation, Limit Theorems and Cosmological Applications, Cambridge, New York, 2011. MR 2840154
- G. Mastrantonio, G. Jona Lasinio, and A. Gelfand, Spatio-temporal circular models with non-separable covariance structure, Test 25 (2016), 331–350. MR 3493522
- G. Mastrantonio, G. Jona Lasinio, A. Pollice, G. Capotorti, L. Teodonio, G. Genova, and C. Blasi, A hierarchical multivariate spatio-temporal model for clustered climate data with annual cycles, Ann. Appl. Stat. 13 (2019), no. 2, 797–823. MR 3963553
- V. A. Menegatto, V. A. Oliveira, and A. P. Peron, Conditionally positive definite dot product kernels, J. Math. Anal. Appl. 321 (2006), no. 1, 223–241. MR 2236554
- V. A. Menegatto and A. P. Peron, Conditionally positive definite kernels on Euclidean domains, J. Math. Anal. Appl. 294 (2004), no. 1, 345–359. MR 2059891
- M. Morimoto, Analytic functionals on the sphere, Translations of Mathematical Monographs, vol. 178, American Mathematical Society, Providence, RI, 1998. MR 1641900
- F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST handbook of mathematical functions hardback and CD-ROM, Cambridge University Press, 2010. MR 2655349
- E. Porcu, A. Alegría, and R. Furrer, Modeling spatially global and temporally evolving data, International Statistical Review 86 (2018), 344–377. MR 3852415
- E. Porcu, M. Bevilacqua, and M. G. Genton, Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere, J. Amer. Statist. Assoc. 111 (2016), no. 514, 888–898. MR 3538713
- E. Porcu, S. Castruccio, A. Alegria, and P. Crippa, Axially symmetric models for global data: A journey between geostatistics and stochastic generators, Environmetrics 30 (2019), no. 1, e2555. MR 3908104
- E. Porcu, R. Furrer, and D. Nychka, 30 years of space–time covariance functions, Wiley Interdiscip. Rev. Comput. Stat. (2020), e1512. MR 4218945
- E. Porcu, R. Senoussi, E. Mendoza, and M. Bevilacqua, Reduction problems and deformation approaches to nonstationary covariance functions over spheres, Electron. J. Stat. 14 (2020), no. 1, 890–916. MR 4062758
- E. Porcu and P. A. White, Random Fields on the Hypertorus. Part I: their Covariance Modelling and Applications, Submitted (2020). MR 4376811
- W. Rudin, Principles of mathematical analysis, vol. 3, McGraw-Hill New York, 1964. MR 0166310
- M. Schlather, Some covariance models based on normal scale mixtures, Bernoulli 16 (2010), no. 3, 780–797. MR 2730648
- I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J. 9 (1942), no. 1, 96–108. MR 5922
- R. Senoussi and E. Porcu, Nonstationary space–time covariance functions induced by dynamical systems, Scand. J. Stat. 49 (2022), 211–235. MR 4391052
- S. Shirota and A. Gelfand, Space and circular time log Gaussian Cox processes with application to crime event data, Ann. Appl. Stat. 11 (2017), no. 2, 481–503. MR 3693535
- M.L. Stein, Interpolation of spatial data: Some theory for kriging, Springer, New York, 1999. MR 1697409
- G. Szegő, Orthogonal Polynomials, Colloquium Publications, vol. XXIII, American Mathematical Society, 1939. MR 0000077
- G. Terdik, Angular spectra for non-Gaussian isotropic fields, Braz. J. Probab. Stat. 29 (2015), no. 4, 833–865. MR 3397396
- P. A. White and E.Porcu, Nonseparable covariance models on circles cross time: A study of Mexico City ozone, Environmetrics (2019), e2558. MR 3999535
- P. A. White and E. Porcu, Towards a complete picture of stationary covariance functions on spheres cross time, Electron. J. Stat. 13 (2019), 2566–2594. MR 3988087
- M. I. Yadrenko, Spectral theory of random fields, Optimization Software, 1983. MR 697386
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Additional Information
François Bachoc
Affiliation:
Department of Mathematics, Université Paul Sabatier, Toulouse, France.
Email:
francois.bachoc@math.univ-toulouse.fr
Ana Paula Peron
Affiliation:
Department of Mathematics, ICMC, University of São Paulo, São Carlos, Brazil.
Email:
apperon@icmc.usp.br
Emilio Porcu
Affiliation:
Department of Mathematics, Khalifa University, The United Arab Emirates, $\&$ School of Computer Science and Statistics, Trinity College Dublin.
Email:
emilio.porcu@ku.ac.ae
Keywords:
Matrix valued covariance functions,
multivariate random fields,
torus,
matrix spectral density
Received by editor(s):
June 9, 2021
Accepted for publication:
November 15, 2021
Published electronically:
November 8, 2022
Additional Notes:
A. P. Peron was partially supported by FAPESP # 2021/04269-0.
E. Porcu acknowledges this publication is based upon work supported by the Khalifa University of Science and Technology under Award No. FSU-2021-016
Dedicated:
This paper is dedicated to Professor M. Yadrenko, who has largely inspired our research since the times of our PhD studies.
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv
