Gaussian random fields: with and without covariances
Authors:
N. H. Bingham and Tasmin L. Symons
Journal:
Theor. Probability and Math. Statist. 106 (2022), 27-40
MSC (2020):
Primary 54C40, 14E20; Secondary 46E25, 20C20
DOI:
https://doi.org/10.1090/tpms/1163
Published electronically:
May 16, 2022
MathSciNet review:
4438442
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We begin with isotropic Gaussian random fields, and show how the Bochner–Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs). We then turn from this continuous setting to approximating discrete settings, Gaussian Markov random fields (GMRFs), and the computational advantages they bring in handling large data sets, by exploiting the sparseness properties of the relevant precision (concentration) matrices.
References
- Sofia Åberg and Krzysztof Podgórski, A class of non-Gaussian second order random fields, Extremes 14 (2011), no. 2, 187–222. MR 2794324, DOI 10.1007/s10687-010-0119-1
- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
- N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385–475 (English, with French summary). MR 143935, DOI 10.5802/aif.116
- Peiman Asadi, Anthony C. Davison, and Sebastian Engelke, Extremes on river networks, Ann. Appl. Stat. 9 (2015), no. 4, 2023–2050. MR 3456363, DOI 10.1214/15-AOAS863
- R. Askey and N. H. Bingham, Gaussian processes on compact symmetric spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 2, 127–143. MR 423000, DOI 10.1007/BF00536776
- Christian Berg and Emilio Porcu, From Schoenberg coefficients to Schoenberg functions, Constr. Approx. 45 (2017), no. 2, 217–241. MR 3619442, DOI 10.1007/s00365-016-9323-9
- N. H. Bingham, Random walk on spheres, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 169–192. MR 305485, DOI 10.1007/BF00536088
- N. H. Bingham and W. J. Krzanowski, The penetration of matrix and linear algebra into multivariate analysis and statistics. To appear, British J. Hist. Math., 2022.
- N. H. Bingham, Aleksandar Mijatović, and Tasmin L. Symons, Brownian manifolds, negative type and geo-temporal covariances, Commun. Stoch. Anal. 10 (2016), no. 4, Article 3, 421–432. MR 3654466, DOI 10.31390/cosa.10.4.03
- N. H. Bingham and A. J. Ostaszewski, Extremes and regular variation, Progress in Probability 78, 121–137. Birkhäuser, 2021, arXiv:2001.05420, 2020.
- N. H. Bingham and T. L. Symons, Probability, Statistics and Planet Earth, I: Geotemporal covariances, arXiv:1706.02972, 2017.
- N. H. Bingham and T. L. Symons, Probability, Statistics and Planet Earth, II:The Bochner-Godement theorem for symmetric spaces, arXiv:1707.05204, 2017.
- N. H. Bingham and T. L. Symons, Gaussian random fields on the sphere and sphere cross line, Stochastic Processes and their Applications (2022), https://doi.org/10.1016/j.spa.2019.08.007, arXiv:1812.02103.
- V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
- V. Borovitskiy, I. Azangulov, A. Terenin, P. Mostowsky, M. P. Diesenroth, and N. Durrande, Matérn Gaussian processes on graphs, arXiv:2010.15538v3, 2020.
- V. Borovitskiy, A. Terenin, P. Mostowsky, and M. Deisenroth, Matérn Gaussian Processes on Riemannian Manifolds, Advances in Neural Information Processing Systems (H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, eds.), vol. 33, Curran Associates, Inc., 2020, pp. 12426–12437.
- Phil Broadbridge, Alexander D. Kolesnik, Nikolai Leonenko, and Andriy Olenko, Random spherical hyperbolic diffusion, J. Stat. Phys. 177 (2019), no. 5, 889–916. MR 4031900, DOI 10.1007/s10955-019-02395-0
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
- Fan R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR 1421568
- Daniel Cooley, Douglas Nychka, and Philippe Naveau, Bayesian spatial modeling of extreme precipitation return levels, J. Amer. Statist. Assoc. 102 (2007), no. 479, 824–840. MR 2411647, DOI 10.1198/016214506000000780
- Harald Cramér and M. R. Leadbetter, Stationary and related stochastic processes, Dover Publications, Inc., Mineola, NY, 2004. Sample function properties and their applications; Reprint of the 1967 original. MR 2108670
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
- Timothy A. Davis, Sivasankaran Rajamanickam, and Wissam M. Sid-Lakhdar, A survey of direct methods for sparse linear systems, Acta Numer. 25 (2016), 383–566. MR 3509211, DOI 10.1017/S0962492916000076
- Laurens de Haan and John de Ronde, Sea and wind: multivariate extremes at work, Extremes 1 (1998), no. 1, 7–45. MR 1652944, DOI 10.1023/A:1009909800311
- A. P. Dempster, Elements of continuous multivariate analysis, Addison-Wesley, 1969.
- A. P. Dempster, Covariance selection, Biometrics 28 (1972), no. 1, 157–175. MR 3931974, DOI 10.2307/2528966
- J. Deny, Review of N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Mathematical Reviews 26, 1485.
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
- A. Erdélyi, Higher transcendental functions [volumes I-III], McGraw-Hill Book Company, 1953.
- V. Feller, Vvedenie v teoriyu veroyatnosteĭ i ee prilozheniya. Tom 2, 2nd ed., “Mir”, Moscow, 1984 (Russian). Translated from the second English edition and with a preface by Yu. V. Prokhorov. MR 779091
- Simon Foucart and Holger Rauhut, A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013. MR 3100033, DOI 10.1007/978-0-8176-4948-7
- I. M. Gel′fand, Spherical functions in symmetric Riemann spaces, Doklady Akad. Nauk SSSR (N.S.) 70 (1950), 5–8 (Russian). MR 0033832
- I. M. Gel′fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. MR 0435834
- Marc G. Genton and William Kleiber, Cross-covariance functions for multivariate geostatistics, Statist. Sci. 30 (2015), no. 2, 147–163. MR 3353096, DOI 10.1214/14-STS487
- R. Godement, Review of I. M. Gelfand, Spherical functions in symmetric Riemann spaces, Mathematical Reviews 11, 498b.
- Gene H. Golub and Charles F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR 1417720
- 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. 10 (2016), no. 4, 671–685. MR 3543906, DOI 10.1215/17358787-3649260
- Peter Guttorp and Tilmann Gneiting, Studies in the history of probability and statistics. XLIX. On the Matérn correlation family, Biometrika 93 (2006), no. 4, 989–995. MR 2285084, DOI 10.1093/biomet/93.4.989
- Trevor Hastie, Robert Tibshirani, and Martin Wainwright, Statistical learning with sparsity, Monographs on Statistics and Applied Probability, vol. 143, CRC Press, Boca Raton, FL, 2015. The lasso and generalizations. MR 3616141, DOI 10.1201/b18401
- Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Sigurdur Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. MR 1280714, DOI 10.1090/surv/039
- Takeyuki Hida, Hui-Hsiung Kuo, Jürgen Potthoff, and Ludwig Streit, White noise, Mathematics and its Applications, vol. 253, Kluwer Academic Publishers Group, Dordrecht, 1993. An infinite-dimensional calculus. MR 1244577, DOI 10.1007/978-94-017-3680-0
- D. Higdon, A process-convolution approach to modelling temperatures in the North Atlantic Ocean, Environmental and Ecological Statistics 5 (1998), no. 2, 173–190.
- Dave Higdon, Space and space-time modeling using process convolutions, Quantitative methods for current environmental issues, Springer, London, 2002, pp. 37–56. MR 2059819
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- Bent Jørgensen, Statistical properties of the generalized inverse Gaussian distribution, Lecture Notes in Statistics, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR 648107, DOI 10.1007/978-1-4612-5698-4
- Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169, DOI 10.1007/978-1-4757-4015-8
- Olav Kallenberg, Random measures, theory and applications, Probability Theory and Stochastic Modelling, vol. 77, Springer, Cham, 2017. MR 3642325, DOI 10.1007/978-3-319-41598-7
- Annika Lang and Christoph Schwab, Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations, Ann. Appl. Probab. 25 (2015), no. 6, 3047–3094. MR 3404631, DOI 10.1214/14-AAP1067
- Steffen L. Lauritzen, Graphical models, Oxford Statistical Science Series, vol. 17, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1419991
- Finn Lindgren, Håvard Rue, and Johan Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc. Ser. B Stat. Methodol. 73 (2011), no. 4, 423–498. With discussion and a reply by the authors. MR 2853727, DOI 10.1111/j.1467-9868.2011.00777.x
- Anatoliy Malyarenko, Invariant random fields on spaces with a group action, Probability and its Applications (New York), Springer, Heidelberg, 2013. With a foreword by Nikolai Leonenko. MR 2977490, DOI 10.1007/978-3-642-33406-1
- Domenico Marinucci and Giovanni Peccati, Random fields on the sphere, London Mathematical Society Lecture Note Series, vol. 389, Cambridge University Press, Cambridge, 2011. Representation, limit theorems and cosmological applications. MR 2840154, DOI 10.1017/CBO9780511751677
- B. Matérn, Spatial Variation, 2nd ed., Springer, 1986.
- François G. Meyer and Ronald R. Coifman, Brushlets: a tool for directional image analysis and image compression, Appl. Comput. Harmon. Anal. 4 (1997), no. 2, 147–187. MR 1448220, DOI 10.1006/acha.1997.0208
- Franco P. Preparata and Michael Ian Shamos, Computational geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985. An introduction. MR 805539, DOI 10.1007/978-1-4612-1098-6
- Nasir M. Rajpoot, Roland G. Wilson, François G. Meyer, and Ronald R. Coifman, Adaptive wavelet packet basis selection for zerotree image coding, IEEE Trans. Image Process. 12 (2003), no. 12, 1460–1472. MR 2067798, DOI 10.1109/TIP.2003.818115
- Marcel Riesz, Collected papers, Springer Collected Works in Mathematics, Springer, Heidelberg, 2013. Reprint of the 1988 edition [0962287]; Edited by Lars Gårding and Lars Hörmander. MR 3114980
- Alexandre Rodrigues and Peter J. Diggle, A class of convolution-based models for spatio-temporal processes with non-separable covariance structure, Scand. J. Stat. 37 (2010), no. 4, 553–567. MR 2779636, DOI 10.1111/j.1467-9469.2009.00675.x
- Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
- Håvard Rue and Leonhard Held, Gaussian Markov random fields, Monographs on Statistics and Applied Probability, vol. 104, Chapman & Hall/CRC, Boca Raton, FL, 2005. Theory and applications. MR 2130347, DOI 10.1201/9780203492024
- Håvard Rue, Sara Martino, and Nicolas Chopin, Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations, J. R. Stat. Soc. Ser. B Stat. Methodol. 71 (2009), no. 2, 319–392. MR 2649602, DOI 10.1111/j.1467-9868.2008.00700.x
- Paul Sharkey and Hugo C. Winter, A Bayesian spatial hierarchical model for extreme precipitation in Great Britain, Environmetrics 30 (2019), no. 1, e2529, 19. MR 3908107, DOI 10.1002/env.2529
- Daniel Simpson, Finn Lindgren, and Håvard Rue, In order to make spatial statistics computationally feasible, we need to forget about the covariance function, Environmetrics 23 (2012), no. 1, 65–74. MR 2873784, DOI 10.1002/env.1137
- Daniel Simpson, Finn Lindgren, and Håvard Rue, Beyond the valley of the covariance function, Statist. Sci. 30 (2015), no. 2, 164–166. MR 3353097, DOI 10.1214/15-STS515
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.
- Michael L. Stein, Interpolation of spatial data, Springer Series in Statistics, Springer-Verlag, New York, 1999. Some theory for Kriging. MR 1697409, DOI 10.1007/978-1-4612-1494-6
- Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1959. Revised ed. MR 0106295
- Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463, DOI 10.1515/9781400886104
- Gerrit van Dijk, Introduction to harmonic analysis and generalized Gelfand pairs, De Gruyter Studies in Mathematics, vol. 36, Walter de Gruyter & Co., Berlin, 2009. MR 2640609, DOI 10.1515/9783110220209
- John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI 10.1007/BFb0074920
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
- Nanny Wermuth, Analogies between multiplicative models in contingency tables and covariance selection, Biometrics 32 (1976), no. 1, 95–108 (English, with French summary). MR 403088, DOI 10.2307/2529341
- P. Whittle, On stationary processes in the plane, Biometrika 41 (1954), 434–449. MR 67450, DOI 10.1093/biomet/41.3-4.434
- P. Whittle, Stochastic processes in several dimensions, Bull. Inst. Internat. Statist. 40 (1963), 974–994 (English, with French summary). MR 173287
- Peter Whittle, Systems in stochastic equilibrium, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1986. MR 850012
- David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. MR 1155402, DOI 10.1017/CBO9780511813658
- David Williams, Weighing the odds, Cambridge University Press, Cambridge, 2001. A course in probability and statistics. MR 1854128, DOI 10.1017/CBO9781139164795
- Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740
- Joseph A. Wolf, Harmonic analysis on commutative spaces, Mathematical Surveys and Monographs, vol. 142, American Mathematical Society, Providence, RI, 2007. MR 2328043, DOI 10.1090/surv/142
References
- S. Åberg and K. Podgórski, A class of non-Gaussian second order random fields, Extremes 14 (2011), no. 2, 187–222. MR 2794324
- G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
- N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385–475. MR 143935
- P. Asadi, A. C. Davison, and S. Engelke, Extremes on river networks, Ann. Appl. Stat. 9 (2015), no. 4, 2023–2050. MR 3456363
- R. Askey and N. H. Bingham, Gaussian processes on compact symmetric spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 2, 127–143. MR 423000
- C. Berg and E. Porcu, From Schoenberg coefficients to Schoenberg functions, Constr. Approx. 45 (2017), no. 2, 217–241. MR 3619442
- N. H. Bingham, Random walk on spheres, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 169–192. MR 305485
- N. H. Bingham and W. J. Krzanowski, The penetration of matrix and linear algebra into multivariate analysis and statistics. To appear, British J. Hist. Math., 2022.
- N. H. Bingham, A. Mijatović, and T. L. Symons, Brownian manifolds, negative type and geo-temporal covariances, Commun. Stoch. Anal. 10 (2016), no. 4, Article 3, 421–432. MR 3654466
- N. H. Bingham and A. J. Ostaszewski, Extremes and regular variation, Progress in Probability 78, 121–137. Birkhäuser, 2021, arXiv:2001.05420, 2020.
- N. H. Bingham and T. L. Symons, Probability, Statistics and Planet Earth, I: Geotemporal covariances, arXiv:1706.02972, 2017.
- N. H. Bingham and T. L. Symons, Probability, Statistics and Planet Earth, II:The Bochner-Godement theorem for symmetric spaces, arXiv:1707.05204, 2017.
- N. H. Bingham and T. L. Symons, Gaussian random fields on the sphere and sphere cross line, Stochastic Processes and their Applications (2022), https://doi.org/10.1016/j.spa.2019.08.007, arXiv:1812.02103.
- V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655
- V. Borovitskiy, I. Azangulov, A. Terenin, P. Mostowsky, M. P. Diesenroth, and N. Durrande, Matérn Gaussian processes on graphs, arXiv:2010.15538v3, 2020.
- V. Borovitskiy, A. Terenin, P. Mostowsky, and M. Deisenroth, Matérn Gaussian Processes on Riemannian Manifolds, Advances in Neural Information Processing Systems (H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, eds.), vol. 33, Curran Associates, Inc., 2020, pp. 12426–12437.
- P. Broadbridge, A. D. Kolesnik, N. Leonenko, and A. Olenko, Random spherical hyperbolic diffusion, J. Stat. Phys. 177 (2019), no. 5, 889–916. MR 4031900
- I. Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. MR 768584
- I. Chavel, Riemannian geometry, second ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. MR 2229062
- F. R. K. Chung, Spectral graph theory, no. 92, American Mathematical Society, 1997. MR 1421568
- D. Cooley, D. Nychka, and P. Naveau, Bayesian spatial modeling of extreme precipitation return levels, J. Amer. Statist. Assoc. 102 (2007), no. 479, 824–840. MR 2411647
- H. Cramér and M. R. Leadbetter, Stationary and related stochastic processes, Dover Publications, Inc., Mineola, NY, 2004. MR 2108670
- I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
- T. A. Davis, S. Rajamanickam, and W. M. Sid-Lakhdar, A survey of direct methods for sparse linear systems, Acta Numer. 25 (2016), 383–566. MR 3509211
- L. de Haan and J. de Ronde, Sea and wind: multivariate extremes at work, Extremes 1 (1998), no. 1, 7–45. MR 1652944
- A. P. Dempster, Elements of continuous multivariate analysis, Addison-Wesley, 1969.
- A. P. Dempster, Covariance selection, Biometrics 28 (1972), no. 1, 157–175. MR 3931974
- J. Deny, Review of N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Mathematical Reviews 26, 1485.
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR 0058896
- A. Erdélyi, Higher transcendental functions [volumes I-III], McGraw-Hill Book Company, 1953.
- W. Feller, An introduction to probability theory and its applications, volume II, 2nd ed., Wiley, 1971. MR 779091
- S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013. MR 3100033
- I. M. Gelfand, Spherical functions in symmetric Riemann spaces, Doklady Akac. Nauk SSSR 50 (1950), 5–8, Article in Russian. MR 0033832
- I. M. Gel’fand and N. Ya. Vilenkin, Generalized functions. Vol. 4: Applications of harmonic analysis, Academic Press, New York-London, 1964. MR 0435834
- M. G. Genton and W. Kleiber, Cross-covariance functions for multivariate geostatistics, Statist. Sci. 30 (2015), no. 2, 147–163. MR 3353096
- R. Godement, Review of I. M. Gelfand, Spherical functions in symmetric Riemann spaces, Mathematical Reviews 11, 498b.
- G. H. Golub and C. F. Van Loan, Matrix computations, third ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR 1417720
- 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. 10 (2016), no. 4, 671–685. MR 3543906
- P. Guttorp and T. Gneiting, Studies in the history of probability and statistics. XLIX. On the Matérn correlation family, Biometrika 93 (2006), no. 4, 989–995. MR 2285084
- T. Hastie, R. Tibshirani, and M. Wainwright, Statistical learning with sparsity, Monographs on Statistics and Applied Probability, vol. 143, CRC Press, Boca Raton, FL, 2015, The lasso and generalizations. MR 3616141
- S. Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
- S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc., New York-London, 1978. MR 514561
- S. Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, Academic Press, 1978. MR 514561
- S. Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, no. 39, American Mathematical Society, 1994. MR 1280714
- T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White noise, Mathematics and its Applications, vol. 253, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1244577
- D. Higdon, A process-convolution approach to modelling temperatures in the North Atlantic Ocean, Environmental and Ecological Statistics 5 (1998), no. 2, 173–190.
- D. Higdon, Space and space-time modeling using process convolutions, Quantitative methods for current environmental issues (C. W. Anderson, V. Barnett, P. C. Chatwin, and A. H. El-Shaarwi, eds.), Springer, 2002, pp. 37–56. MR 2059819
- R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183
- B. Jørgensen, Statistical properties of the generalized inverse Gaussian distribution, Lecture Notes in Statistics, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR 648107
- O. Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169
- O. Kallenberg, Random measures, theory and applications, Probability Theory and Stochastic Modelling, vol. 77, Springer, Cham, 2017. MR 3642325
- A. Lang and C. Schwab, Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations, Ann. Appl. Probab. 25 (2015), no. 6, 3047–3094. MR 3404631
- S. L. Lauritzen, Graphical models, Oxford Statistical Science Series, vol. 17, The Clarendon Press, Oxford University Press, New York, 1996. MR 1419991
- F. Lindgren, H. Rue, and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc. Ser. B Stat. Methodol. 73 (2011), no. 4, 423–498. MR 2853727
- A. Malyarenko, Invariant random fields on spaces with a group action, Probability and its Applications (New York), Springer, Heidelberg, 2013. MR 2977490
- D. Marinucci and G. Peccati, Random fields on the sphere, London Mathematical Society Lecture Note Series, vol. 389, Cambridge University Press, Cambridge, 2011. MR 2840154
- B. Matérn, Spatial Variation, 2nd ed., Springer, 1986.
- F. G. Meyer and R. R. Coifman, Brushlets: a tool for directional image analysis and image compression, Appl. Comput. Harmon. Anal. 4 (1997), no. 2, 147–187. MR 1448220
- F. P. Preparata and M. I. Shamos, Computational geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985. MR 805539
- N. M. Rajpoot, R. G. Wilson, F. G. Meyer, and R. R. Coifman, Adaptive wavelet packet basis selection for zerotree image coding, IEEE Trans. Image Process. 12 (2003), no. 12, 1460–1472. MR 2067798
- M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Szeged Sect. Math. 9 (1937-1938), 1–42, Reprinted in Collected Papers, L. Gårding and L. Hörmander, editors, Springer, 1988, pp. 482–526. MR 3114980
- A. Rodrigues and P. J. Diggle, A class of convolution-based models for spatio-temporal processes with non-separable covariance structure, Scand. J. Stat. 37 (2010), no. 4, 553–567. MR 2779636
- W. Rudin, Functional analysis, 2nd ed., McGraw-Hill international series in pure and applied mathematics, McGraw-Hill, 1991. MR 1157815
- H. Rue and L. Held, Gaussian Markov random fields, Monographs on Statistics and Applied Probability, vol. 104, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2130347
- H. Rue, S. Martino, and N. Chopin, Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations, J. R. Stat. Soc. Ser. B Stat. Methodol. 71 (2009), no. 2, 319–392. MR 2649602
- P. Sharkey and H. C. Winter, A Bayesian spatial hierarchical model for extreme precipitation in Great Britain, Environmetrics 30 (2019), no. 1, e2529, 19. MR 3908107
- D. Simpson, F. Lindgren, and H. Rue, In order to make spatial statistics computationally feasible, we need to forget about the covariance function, Environmetrics 23 (2012), no. 1, 65–74. MR 2873784
- D. Simpson, F. Lindgren, and H. Rue, Beyond the valley of the covariance function, Statist. Sci. 30 (2015), no. 2, 164–166. MR 3353097
- E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 1232192
- E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.
- M. L. Stein, Interpolation of spatial data, Springer Series in Statistics, Springer-Verlag, New York, 1999, Some theory for Kriging. MR 1697409
- G. Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1959, Revised ed. MR 0106295
- M. E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463
- G. van Dijk, Introduction to harmonic analysis and generalized Gelfand pairs, De Gruyter Studies in Mathematics, vol. 36, Walter de Gruyter & Co., Berlin, 2009. MR 2640609
- J. B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; Macmillan Company, New York, 1944. MR 1349110
- N. Wermuth, Analogies between multiplicative models in contingency tables and covariance selection, Biometrics 32 (1976), no. 1, 95–108. MR 403088
- P. Whittle, On stationary processes in the plane, Biometrika 41 (1954), 434–449. MR 67450
- P. Whittle, Stochastic processes in several dimensions, Bull. Inst. Internat. Statist. 40 (1963), 974–994. MR 173287
- P. Whittle, Systems in stochastic equilibrium, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1986. MR 850012
- D. Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. MR 1155402
- D. Williams, Weighing the odds, Cambridge University Press, Cambridge, 2001, A course in probability and statistics. MR 1854128
- J. A. Wolf, Spaces of Constant Curvature, American Mathematical Society, 1967. MR 0217740
- J. A. Wolf, Harmonic analysis on commutative spaces, Mathematical Surveys and Monographs, no. 142, American Mathematical Society, 2007. MR 2328043
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2020):
54C40,
14E20,
46E25,
20C20
Retrieve articles in all journals
with MSC (2020):
54C40,
14E20,
46E25,
20C20
Additional Information
N. H. Bingham
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
Email:
n.bingham@imperial.ac.uk
Tasmin L. Symons
Affiliation:
Telethon Kids Institute, 15 Hospital Avenue, Perth, WA 6009, Australia
Email:
tasmin.symons@telethonkids.org.au
Keywords:
Gaussian random field,
covariance function,
Bessel potential,
stochastic partial differential equation,
Matérn process,
Gaussian Markov random field,
precision matrix,
sparseness,
numerical linear algebra
Received by editor(s):
August 4, 2021
Accepted for publication:
October 18, 2021
Published electronically:
May 16, 2022
Dedicated:
To the memory of Mikhailo Iosifovich Yadrenko, on his 90th birthday
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv
