Axially symmetric processes on spheres, for which the second-order dependency structure may substantially vary with shifts in latitude, are a prominent alternative to model the spatial uncertainty of natural variables located over large portions of the Earth. In this paper, we focus on Karhunen–Loève expansions of axially symmetric Gaussian processes. First, we investigate a parametric family of Karhunen–Loève coefficients that allows for versatile spatial covariance functions. The isotropy as well as the longitudinal independence can be obtained as limit cases of our proposal. Second, we introduce a strategy to render any longitudinally reversible process irreversible, which means that its covariance function could admit certain types of asymmetries along longitudes. Then, finitely truncated Karhunen–Loève expansions are used to approximate axially symmetric processes. For such approximations, bounds for the \(L^2\)-error are provided. Numerical experiments are conducted to illustrate our findings.
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Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York
Alegria A, Cuevas F, Diggle P, Porcu E (2018) A family of covariance functions for random fields on spheres. CSGB Research Reports, Department of Mathematics, Aarhus University
Alegría A, Emery X, Lantuéjoul C (2020) The turning arcs: a computationally efficient algorithm to simulate isotropic vector-valued Gaussian random fields on the \(d\)-sphere. Stat Comput. https://doi.org/10.1007/s11222-020-09952-8
Alegría A, Porcu E, Furrer R, Mateu J (2019) Covariance functions for multivariate Gaussian fields evolving temporally over planet earth. Stoch Env Res Risk Assess 33(8–9):1593–1608
Bissiri PG, Peron AP, Porcu E (2020) Strict positive definiteness under axial symmetry on the sphere. Stoch Env Res Risk Assess 34:723–732
Castruccio S (2016) Assessing the spatio-temporal structure of annual and seasonal surface temperature for CMIP5 and reanalysis. Spat Stat 18:179–193
Castruccio S, Genton MG (2014) Beyond axial symmetry: an improved class of models for global data. Stat 3(1):48–55
Clarke J, Alegría A, Porcu E (2018) Regularity properties and simulations of Gaussian random fields on the sphere cross time. Electron J Stat 12(1):399–426
Cleanthous G, Georgiadis AG, Lang A, Porcu E (2020) Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces. Stoch Process Appl 130(8):4873–4891
Creasey PE, Lang A (2018) Fast generation of isotropic Gaussian random fields on the sphere. Monte Carlo Methods Appl 24(1):1–11
Cuevas F, Allard D, Porcu E (2020) Fast and exact simulation of Gaussian random fields defined on the sphere cross time. Stat Comput 30(1):187–194
Emery X, Porcu E (2019) Simulating isotropic vector-valued Gaussian random fields on the sphere through finite harmonics approximations. Stoch Environ Res Risk Assess 33(8–9):1659–1667
Emery X, Furrer R, Porcu E (2019a) A turning bands method for simulating isotropic Gaussian random fields on the sphere. Stat Probab Lett 144:9–15
Emery X, Porcu E, Bissiri PG (2019b) A semiparametric class of axially symmetric random fields on the sphere. Stoch Environ Res Risk Assess 33(10):1863–1874
Gneiting T (2013) Strictly and non-strictly positive definite functions on spheres. Bernoulli 19(4):1327–1349
Guinness J, Fuentes M (2016) Isotropic covariance functions on spheres: some properties and modeling considerations. J Multivar Anal 143:143–152
Hansen LV, Thorarinsdottir TL, Ovcharov E, Gneiting T, Richards D (2015) Gaussian random particles with flexible Hausdorff dimension. Adv Appl Probab 47(2):307–327
Hitczenko M, Stein ML (2012) Some theory for anisotropic processes on the sphere. Stat Methodol 9(1–2):211–227
Huang C, Zhang H, Robeson SM (2012) A simplified representation of the covariance structure of axially symmetric processes on the sphere. Stat Probab Lett 82(7):1346–1351
Jeong J, Jun M, Genton MG (2017) Spherical process models for global spatial statistics. Stat Sci 32(4):501–513
Jones RH (1963) Stochastic processes on a sphere. Ann Math Stat 34(1):213–218
Jun M (2011) Non-stationary cross-covariance models for multivariate processes on a globe. Scand J Stat 38(4):726–747
Jun M, Stein ML (2008) Nonstationary covariance models for global data. Ann Appl Stat 2(4):1271–1289
Kerkyacharian G, Ogawa S, Petrushev P, Picard D (2018) Regularity of Gaussian processes on Dirichlet spaces. Constr Approx 47(2):277–320
Lang A, Schwab C (2015) Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. Ann Appl Probab 25(6):3047–3094
Lantuéjoul C, Freulon X, Renard D (2019) Spectral simulation of isotropic Gaussian random fields on a sphere. Math Geosci 51(8):999–1020
Leonenko NN, Taqqu MS, Terdik GH (2018) Estimation of the covariance function of Gaussian isotropic random fields on spheres, related Rosenblatt-type distributions and the cosmic variance problem. Electron J Stat 12(2):3114–3146
Ma C (2012) Stationary and isotropic vector random fields on spheres. Math Geosci 44(6):765–778
Marinucci D, Peccati G (2011) Random fields on the sphere: representation limit theorems and cosmological applications. Cambridge University Press, Cambridge
Peron A, Porcu E, Emery X (2018) Admissible nested covariance models over spheres cross time. Stoch Environ Res Risk Assess 32(11):3053–3066
Porcu E, Alegria A, Furrer R (2018) Modeling temporally evolving and spatially globally dependent data. Int Stat Rev 86(2):344–377
Porcu E, Castruccio S, Alegría A, Crippa P (2019) Axially symmetric models for global data: a journey between geostatistics and stochastic generators. Environmetrics 30(1):e2555
Schoenberg IJ (1942) Positive definite functions on spheres. Duke Math J 9(1):96–108
Siegel KM (1955) Bounds of the Legendre function. J Math Phys 34(1–4):43–49
Stein ML (2007) Spatial variation of total column ozone on a global scale. Ann Appl Stat 1(1):191–210
Terdik G (2015) Angular spectra for non-Gaussian isotropic fields. Braz J Probab Stat 29(4):833–865
Vanlengenberg CD, Wang W, Zhang H (2019) Data generation for axially symmetric processes on the sphere. Commun Stat Simul Comput. https://doi.org/10.1080/03610918.2019.1588309
The Associate Editor and two Referees are gratefully acknowledged for their thorough revisions that allow for a considerably improved version of the manuscript. The research work of Alfredo Alegría was partially supported by Grant FONDECYT 11190686 from the Chilean government.
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Alegría, A., Cuevas-Pacheco, F. Karhunen–Loève expansions for axially symmetric Gaussian processes: modeling strategies and \(L^2\) approximations. Stoch Environ Res Risk Assess 34, 1953–1965 (2020). https://doi.org/10.1007/s00477-020-01839-4
- Associated Legendre polynomials
- Covariance functions
- Great-circle distance
- Longitudinally independent
- Longitudinally reversible
- Spherical harmonics