Multivariate isotropic random fields on spheres: nonparametric Bayesian modeling and \(L^p\) fast approximations. (English) Zbl 1471.62563
Summary: We study multivariate Gaussian random fields defined over \(d\)-dimensional spheres. First, we provide a nonparametric Bayesian framework for modeling and inference on matrix-valued covariance functions. We determine the support (under the topology of uniform convergence) of the proposed random matrices, which cover the whole class of matrix-valued geodesically isotropic covariance functions on spheres. We provide a thorough inspection of the properties of the proposed model in terms of (a) first moments, (b) posterior distributions, and (c) Lipschitz continuities. We then provide an approximation method for multivariate fields on the sphere for which measures of \(L^p\) accuracy are established. Our findings are supported through simulation studies that show the rate of convergence when truncating a spectral expansion of a multivariate random field at a finite order. To illustrate the modeling framework developed in this paper, we consider a bivariate spatial data set of two 2019 NCEP/NCAR Flux Reanalyses.
MSC:
| 62R30 | Statistics on manifolds |
| 62H12 | Estimation in multivariate analysis |
| 62G05 | Nonparametric estimation |
| 60F25 | \(L^p\)-limit theorems |
| 62P12 | Applications of statistics to environmental and related topics |
| 86A08 | Climate science and climate modeling |
Keywords:
bivariate climate data; matrix-valued covariance function; multivariate random field; nonparametric Bayes; \(L^p\) approximations; sphereReferences:
| [1] | Achim, K. (2008). Probability theory. A comprehensive course. Universitext. Springer-Verlag London, Ltd., London. · Zbl 1141.60001 |
| [2] | Alegría, A. (2020). Cross-dimple in the cross-covariance functions of bivariate isotropic random fields on spheres. Stat 9 e301. · Zbl 07851187 |
| [3] | Alegría, A., Emery, X. and Lantuéjoul, C. (2020). The turning arcs: a computationally efficient algorithm to simulate isotropic vector-valued Gaussian random fields on the \(d\)-sphere. Statistics and Computing 30 1403-1418. · Zbl 1461.62011 |
| [4] | Alegría, A., Porcu, E., Furrer, R. and Mateu, J. (2019a). Covariance functions for multivariate Gaussian fields. Stochastic Environmental Research Risk Assessment 33 1593-1608. |
| [5] | Alegría, A., Cuevas, F., Diggle, P. J. and Porcu, E. (2019b). A new class of covariance functions of random fields on spheres. Technical Report, University of Aalborg. |
| [6] | Banerjee, S. (2005). On geodetic distance computations in spatial modeling. Biometrics 61 617-625. · doi:10.1111/j.1541-0420.2005.00320.x |
| [7] | Banerjee, S., Gelfand, A. E., Finley, A. O. and Sang, H. (2008). Gaussian predictive process models for large spatial data sets. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 70 825-848. · Zbl 1533.62065 |
| [8] | Bevilacqua, M., Diggle, P. and Porcu, E. (2019). Families of covariance functions for bivariate random fields over spheres. Spatial Statistics. Under Second Review. |
| [9] | Cabella, P. and Marinucci, D. (2009). Statistical challenges in the analysis of cosmic microwave background radiation. The Annals of Applied Statistics 3 61-95. · Zbl 1160.62097 |
| [10] | Castillo, I., Kerkyacharian, G. and Picard, D. (2014). Thomas Bayes’ walk on manifolds. Probab. Theory Related Fields 158 665-710. · Zbl 1285.62028 · doi:10.1007/s00440-013-0493-0 |
| [11] | Chiles, J.-P. and Delfiner, P. (2009). Geostatistics: modeling spatial uncertainty 497. John Wiley & Sons. |
| [12] | Chopin, N., Rousseau, J. and Liseo, B. (2013). Computational aspects of Bayesian spectral density estimation. Journal of Computational and Graphical Statistics 22 533-557. |
| [13] | Combes, V., Hormázabal, S. and Di Lorenzo, E. (2017). Interannual variability of the subsurface eddy field in the Southeast Pacific. Journal of Geophysical Research-Oceans 120 2769-2783. |
| [14] | Cressie, N. and Johannesson, G. (2008). Fixed rank kriging for very large spatial data sets. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 70 209-226. · Zbl 05563351 |
| [15] | Da Prato, G. and Zabczyk, J. (1992). Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge. · Zbl 0761.60052 |
| [16] | Dai, F. and Xu, Y. (2013). Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, New York. · Zbl 1275.42001 · doi:10.1007/978-1-4614-6660-4 |
| [17] | Dashti, M. and Stuart, A. (2016). The Bayesian approach to inverse problems. In Handbook of uncertainty quantification (R. Ghanem, D. Higdon and H. Owhadi, eds.) 1-118. |
| [18] | Dette, H., Konstantinou, M., Schorning, K. and Gösmann, J. (2019). Optimal designs for regression with spherical data. Electronic Journal of Statistics 13 361-390. · Zbl 1407.62285 |
| [19] | Di Lorenzo, E., Combes, V., Keister, J., Strub, P., Andrew, T., Peter, F., Marck, O., Furtado, J. ., Bracco, A., Bograd, S., Peterson, W., Schwing, F., Taguchi, B., Hormázabal, S. and Parada, C. (2014). Synthesis of Pacific Ocean Climate and ecosystem dynamics. Oceanography 26 68-81. |
| [20] | Du, J., Ma, C. and Li, Y. (2013). Isotropic variogram matrix functions on spheres. Math. Geosci. 45 341-357. · Zbl 1321.86021 · doi:10.1007/s11004-013-9441-x |
| [21] | Duan, J. A., Guindani, M. and Gelfand, A. E. (2007). Generalized spatial Dirichlet process models. Biometrika 94 809-825. · Zbl 1156.62064 |
| [22] | Durrer, R. (2020). The cosmic microwave background. Cambridge University Press. |
| [23] | Edwards, M., Castruccio, S. and Hammerling, D. (2019). A multivariate Global Spatio-Temporal Stochastic Generator for Climate Ensembles. Journal of Agricultural, Biological and Environmental Sciences. · Zbl 1426.62339 |
| [24] | Emery, X. and Lantuéjoul, C. (2006). Tbsim: A computer program for conditional simulation of three-dimensional gaussian random fields via the turning bands method. Computers & Geosciences 32 1615-1628. |
| [25] | Emery, X. and Porcu, E. (2019). Simulating isotropic vector-valued Gaussian random fields on the sphere through finite harmonics approximations. Stochastic Environmental Research and Risk Assessment 33 1659-1667. |
| [26] | Fixsen, D. (2009). The temperature of the cosmic microwave background. The Astrophysical Journal 707 916. |
| [27] | Forster, P., Ramaswamy, V., Artaxo, P., Berntsen, T., Betts, R., Fahey, D. W., Haywood, J., Lean, J., Lowe, D. C. and Myhre, G. (2007). Changes in atmospheric constituents and in radiative forcing. Chapter 2. In Climate Change 2007. The Physical Science Basis. |
| [28] | Gelfand, A. E., Kottas, A. and MacEachern, S. N. (2005). Bayesian nonparametric spatial modeling with Dirichlet process mixing. Journal of the American Statistical Association 100 1021-1035. · Zbl 1117.62342 |
| [29] | Gneiting, T. (2013). Strictly and non-strictly positive definite functions on spheres. Bernoulli 19 1327-1349. · Zbl 1283.62200 · doi:10.3150/12-BEJSP06 |
| [30] | Gorski, K. M., Hivon, E., Banday, A. J., Wandelt, B. D., Hansen, F. K., Reinecke, M. and Bartelmann, M. (2005). HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere. The Astrophysical Journal 622 759. |
| [31] | Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82 711-732. · Zbl 0861.62023 |
| [32] | Guella, J. C. and Menegatto, V. A. (2018). Schoenberg’s theorem for positive definite functions on products: a unifying framework. J. Fourier Anal Appl. To appear. · Zbl 1420.42005 |
| [33] | Guinness, J. and Fuentes, M. (2016). Isotropic covariance functions on spheres: some properties and modeling considerations. Journal of Multivariate Analysis 143 143-152. · Zbl 1328.62543 |
| [34] | Haario, H., Saksman, E. and Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7 223-242. · Zbl 0989.65004 |
| [35] | Hannan, E. J. (2009). Multiple Time Series. Wiley Series in Probability and Statistics. Wiley. |
| [36] | Higdon, D. (1998). A process-convolution approach to modelling temperatures in the North Atlantic Ocean. Environmental and Ecological Statistics 5 173-190. |
| [37] | Holbrook, A., Lan, S., Vandenberg-Rodes, A. and Shahbaba, B. (2018). Geodesic Lagrangian Monte Carlo over the space of positive definite matrices: with application to Bayesian spectral density estimation. Journal of Statistical Computation and Simulation 88 982-1002. · Zbl 07192588 |
| [38] | Jona-Lasinio, G., Gelfand, A. and Jona-Lasinio, M. (2012). Spatial analysis of wave direction data using wrapped Gaussian processes. The Annals of Applied Statistics 6 1478-1498. · Zbl 1257.62094 |
| [39] | Jones, R. H. (1963). Stochastic processes on a sphere. Ann. Math. Statist. 34 213-218. · Zbl 0202.46702 |
| [40] | Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., Iredell, M., Saha, S., White, G. and Woollen, J. (1996). The NCEP/NCAR 40-year reanalysis project. Bulletin of the American Meteorological Society 77 437-472. |
| [41] | Katzfuss, M. and Guinness, J. (2021). A general framework for Vecchia approximations of Gaussian processes. Statist. Sci. 36 124-141. · Zbl 07368223 · doi:10.1214/19-STS755 |
| [42] | Kerkyacharian, G., Ogawa, S., Petrushev, P. and Picard, D. (2018). Regularity of Gaussian processes on Dirichlet spaces. Constr. Approx. 47 277-320. · Zbl 1390.58022 · doi:10.1007/s00365-018-9416-8 |
| [43] | Lang, A. a. and Schwab, C. (2015). Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations. Ann. Appl. Probab. 25 3047-3094. · Zbl 1328.60126 · doi:10.1214/14-AAP1067 |
| [44] | Leonenko, N. and Malyarenko, A. (2017). Matérn class tensor-valued random fields and beyond. Journal of Statistical Physics 168 1276-1301. · Zbl 1373.60088 |
| [45] | Lu, T., Leonenko, N. and Ma, C. (2020). Series representations of isotropic vector random fields on balls. Statist. Probab. Lett. 156 108583, 7. · Zbl 1460.60045 · doi:10.1016/j.spl.2019.108583 |
| [46] | Lu, T. and Ma, C. (2020). Isotropic covariance matrix functions on compact two-point homogeneous spaces. J. Theoret. Probab. 33 1630-1656. · Zbl 1464.60051 · doi:10.1007/s10959-019-00920-1 |
| [47] | Ma, C. (2012). Stationary and isotropic vector random fields on spheres. Math. Geosci. 44 765-778. · Zbl 1321.62117 · doi:10.1007/s11004-012-9411-8 |
| [48] | Ma, C. (2015). Isotropic covariance matrix functions on all spheres. Math. Geosci. 47 699-717. · Zbl 1323.62092 · doi:10.1007/s11004-014-9566-6 |
| [49] | Ma, C. (2016). Stochastic representations of isotropic vector random fields on spheres. Stoch. Anal. Appl. 34 389-403. · Zbl 1341.60043 · doi:10.1080/07362994.2015.1136562 |
| [50] | Marinucci, D. and Peccati, G. (2011). Random fields on the sphere: representation, limit theorems and cosmological applications 389. Cambridge University Press. · Zbl 1260.60004 |
| [51] | Mastrantonio, G., Lasinio, G. J., Pollice, A., Capotorti, G., Teodonio, L., Genova, G. and Blasi, C. (2019). A hierarchical multivariate spatio-temporal model for clustered climate data with annual cycles. Annals of Applied Statistics 13 797-823. · Zbl 1423.62159 |
| [52] | Müeller, P., Quintana, F. A. and Page, G. (2018). Nonparametric Bayesian inference in applications. Statistical Methods & Applications 27 175-206. · Zbl 1396.62067 |
| [53] | Nychka, D., Bandyopadhyay, S., Hammerling, D., Lindgren, F. and Sain, S. (2015). A multi-resolution Gaussian process model for the analysis of large spatial datasets. Journal of Computational and Graphical Statistics 24 579-599. |
| [54] | Porcu, E., Alegría, A. and Furrer, R. (2018). Modeling spatially global and temporally evolving data. International Statistical Review 86 344-377. · Zbl 07763598 |
| [55] | Porcu, E., Bevilacqua, M. and Genton, M. G. (2016). Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. Journal of the American Statistical Association 111 888-898. · doi:10.1080/01621459.2015.1072541 |
| [56] | Porcu, E., Bissiri, P. G., Tagle, F. and Quintana, F. (2019). Nonparametric Bayesian Modeling and Estimation of Spatial Covariance Functions for Global Data. Technical Report. |
| [57] | Reich, B. J. and Fuentes, M. (2012). Nonparametric Bayesian models for a spatial covariance. Statistical Methodology 9 265-274. Special Issue on Astrostatistics + Special Issue on Spatial Statistics. · Zbl 1248.62170 · doi:10.1016/j.stamet.2011.01.007 |
| [58] | Reinsel, G., Tiao, G., Wang, M., Lewis, R. and Nychka, D. (1981). Statistical analysis of stratospheric ozone data for the detection of trends. Atmospheric Environment (1967) 15 1569-1577. |
| [59] | Roberts, G. O., Gelman, A. and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. The Annals of Applied Probability 7 110-120. · Zbl 0876.60015 |
| [60] | Roberts, G. O. and Rosenthal, J. S. (2009). Examples of adaptive MCMC. Journal of Computational and Graphical Statistics 18 349-367. |
| [61] | Sahoo, I., Guinness, J. and Reich, B. J. (2019). A test for isotropy on a sphere using spherical harmonic functions. Statistica Sinica 29 1253-1276. · Zbl 1421.62160 |
| [62] | Schervish, M. J. (1995). Theory of statistics. Springer Series in Statistics. Springer-Verlag, New York. · Zbl 0834.62002 |
| [63] | Schmidt, A. M. and O’Hagan, A. (2003). Bayesian inference for non-stationary spatial covariance structure via spatial deformations. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65 743-758. · Zbl 1063.62034 |
| [64] | Schoenberg, I. J. (1942). Positive definite functions on spheres. Duke Math. J. 9 96-108. · Zbl 0063.06808 · doi:10.1215/S0012-7094-42-00908-6 |
| [65] | Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica 639-650. · Zbl 0823.62007 |
| [66] | Shirota, S. and Gelfand, A. E. (2017). Space and circular time log Gaussian Cox processes with application to crime event data. The Annals of Applied Statistics 11 481-503. · Zbl 1416.62689 |
| [67] | Stein, M. L. (2007). Spatial variation of total column ozone on a global scale. The Annals of Applied Statistics 1 191-210. · Zbl 1129.62115 |
| [68] | Stephens, M. (2000). Bayesian analysis of mixture models with an unknown number of components – an alternative to reversible jump methods. Annals of Statistics 40-74. · Zbl 1106.62316 |
| [69] | Stuart, A. M. (2010). Inverse problems: a Bayesian perspective. Acta Numerica 19 451-559. · Zbl 1242.65142 |
| [70] | Terdik, G. (2015). Angular spectra for non-Gaussian isotropic fields. Brazilian Journal of Probability and Statistics 29 833-865. · Zbl 1328.83219 |
| [71] | Wang, F. and Gelfand, A. E. (2014). Modeling space and space-time directional data using projected Gaussian processes. Journal of the American Statistical Association 109 1565-1580. · Zbl 1368.62304 |
| [72] | White, P. A. and Porcu, E. (2019). Nonseparable covariance models on circles cross time: A study of Mexico City ozone. Environmetrics 30 e2558. |
| [73] | Wild, M. (2009). Global dimming and brightening: A review. Journal of Geophysical Research: Atmospheres 114. |
| [74] | Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results. Springer, New York. · Zbl 0685.62077 |
| [75] | Zheng, Y., Zhu, J. and Roy, A. (2010). Nonparametric Bayesian inference for the spectral density function of a random field. Biometrika 97 238-245. · Zbl 1182.62070 |
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