Nonparametric Bayesian modelling of longitudinally integrated covariance functions on spheres. (English) Zbl 07584291
Summary: Taking into account axial symmetry in the covariance function of a Gaussian random field is essential when the purpose is modelling data defined over a large portion of the sphere representing our planet. Axially symmetric covariance functions admit a convoluted spectral representation that makes modelling and inference difficult. This motivates the interest in devising alternative strategies to attain axial symmetry, an appealing option being longitudinal integration of isotropic random fields on the sphere. This paper provides a comprehensive theoretical framework to model longitudinal integration on spheres through a nonparametric Bayesian approach. Longitudinally integrated covariances are treated as random objects, where the randomness is implied by the randomised spectrum associated with the covariance function. After investigating the topological support induced by our construction, we give the posterior distribution a thorough inspection. A Bayesian nonparametric model for the analysis of data defined on the sphere is described and implemented, its performance investigated by means of the analysis of both simulated and real data sets.
Keywords:
axial symmetry; Bayesian nonparametrics; covariance functions; global processes; longitudinal integration; data on spheresReferences:
| [1] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, vol. 55 (1964), Courier Corporation · Zbl 0171.38503 |
| [2] | Castruccio, S.; Genton, M. G., Beyond axial symmetry: an improved class of models for global data, Stat, 3, 1, 48-55 (2014) · Zbl 07847426 |
| [3] | Chopin, N.; Rousseau, J.; Liseo, B., Computational aspects of Bayesian spectral density estimation, J. Comput. Graph. Stat., 22, 3, 533-557 (2013) |
| [4] | Daley, D. J.; Porcu, E., Dimension walks and Schoenberg spectral measures, Proc. Am. Math. Soc., 141, 1813-1824 (2013) · Zbl 1343.62085 |
| [5] | Duan, J. A.; Guindani, M.; Gelfand, A. E., Generalized spatial Dirichlet process models, Biometrika, 94, 4, 809-825 (2007) · Zbl 1156.62064 |
| [6] | Emery, X.; Alegría, A., A spectral algorithm to simulate nonstationary random fields on spheres and multifractal star-shaped random sets, Stoch. Environ. Res. Risk Assess., 34, 12, 2301-2311 (2020) |
| [7] | Emery, X.; Porcu, E.; Bissiri, P. G., A semiparametric class of axially symmetric random fields on the sphere, Stoch. Environ. Res. Risk Assess., 33, 10, 1863-1874 (2019) |
| [8] | Geisser, S.; Eddy, W. F., A predictive approach to model selection, J. Am. Stat. Assoc., 74, 365, 153-160 (1979) · Zbl 0401.62036 |
| [9] | Gelfand, A.; Kottas, A.; MacEachern, S. N., Bayesian nonparametric spatial modeling with Dirichlet process mixing, J. Am. Stat. Assoc., 100, 471, 1021-1035 (2005) · Zbl 1117.62342 |
| [10] | Hitczenko, M.; Stein, M. L., Some theory for anisotropic processes on the sphere, Stat. Methodol., 9, 211-227 (2012) · Zbl 1248.60040 |
| [11] | Holbrook, A.; Lan, S.; Vandenberg-Rodes, A.; Shahbaba, B., Geodesic Lagrangian Monte Carlo over the space of positive definite matrices: with application to Bayesian spectral density estimation, J. Stat. Comput. Simul., 88, 5, 982-1002 (2018) · Zbl 07192588 |
| [12] | Ishwaran, H.; James, L. F., Gibbs sampling methods for stick-breaking priors, J. Am. Stat. Assoc., 96, 453, 161-173 (2001) · Zbl 1014.62006 |
| [13] | Jones, R. H., Stochastic processes on a sphere, Ann. Math. Stat., 34, 213-218 (1963) · Zbl 0202.46702 |
| [14] | Jun, M.; Stein, M. L., An approach to producing space-time covariance functions on spheres, Technometrics, 49, 468-479 (2007) |
| [15] | Jun, M.; Stein, M. L., Nonstationary covariance models for global data, Ann. Appl. Stat., 2, 4, 1271-1289 (2008) · Zbl 1168.62381 |
| [16] | Kalnay, E.; Kanamitsu, M.; Kistler, R.; Collins, W.; Deaven, D.; Gandin, L.; Iredell, M.; Saha, S.; White, G.; Woollen, J.; Zhu, Y.; Chelliah, M.; Ebisuzaki, W.; Higgins, W.; Janowiak, J.; Mo, K.; Ropelewski, C.; Wang, J.; Leetmaa, A.; Reynolds, R.; Jenne, R.; Joseph, D., The NCEP/NCAR 40-year reanalysis project, Bull. Am. Meteorol. Soc., 77, 3, 437-472 (1996) |
| [17] | Marinucci, D.; Peccati, G., Random Fields on the Sphere, Representation, Limit Theorems and Cosmological Applications (2011), Cambridge University Press: Cambridge University Press Cambridge, New York · Zbl 1260.60004 |
| [18] | Müller, P.; Quintana, F. A.; Page, G., Nonparametric Bayesian inference in applications, Stat. Methods Appl., 27, 2, 175-206 (2018) · Zbl 1396.62067 |
| [19] | Pitman, J.; Yor, M., The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Ann. Probab., 25, 2, 855-900 (April 1997) · Zbl 0880.60076 |
| [20] | Porcu, E.; Alegría, A.; Furrer, R., Modeling temporally evolving and spatially globally dependent data, Int. Stat. Rev., 86, 2, 344-377 (2018) · Zbl 07763598 |
| [21] | Porcu, E.; Castruccio, S.; Alegria, A.; Crippa, P., Axially symmetric models for global data: a journey between geostatistics and stochastic generators, Environmetrics, 30, 1, Article e2555 pp. (2019) |
| [22] | Porcu, E.; Bissiri, P. G.; Tagle, F.; Soza, R.; Quintana, F. A., Nonparametric Bayesian modeling and estimation of spatial correlation functions for global data, Bayesian Anal., 16, 3, 845-873 (2020) · Zbl 07807544 |
| [23] | Porcu, E.; Furrer, R.; Nychka, D., 30 years of space-time covariance functions, Wiley Interdiscip. Rev.: Comput. Stat., Article e1512 pp. (2020) · Zbl 07910735 |
| [24] | Reich, B. J.; Fuentes, M., Nonparametric Bayesian models for a spatial covariance, Stat. Methodol., 9, 1, 265-274 (2012) · Zbl 1248.62170 |
| [25] | Schervish, M. J., Theory of Statistics, Springer Series in Statistics (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0834.62002 |
| [26] | Schmidt, A. M.; O’Hagan, A., Bayesian inference for non-stationary spatial covariance structure via spatial deformations, J. R. Stat. Soc., Ser. B, Stat. Methodol., 65, 3, 743-758 (2003) · Zbl 1063.62034 |
| [27] | Schoenberg, I. J., Positive definite functions on spheres, Duke Math. J., 9, 1, 96-108 (1942) · Zbl 0063.06808 |
| [28] | Stein, M. L., Spatial variation of total column ozone on a global scale, Ann. Appl. Stat., 1, 1, 191-210 (2007) · Zbl 1129.62115 |
| [29] | Winch, D.; Ivers, D.; Turner, J.; Stening, R., Geomagnetism and Schmidt quasi-normalization, Geophys. J. Int., 160, 2, 487-504 (2005) |
| [30] | Zheng, Y.; Zhu, J.; Roy, A., Nonparametric Bayesian inference for the spectral density function of a random field, Biometrika, 97, 8, 238-245 (2010) · Zbl 1182.62070 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
